{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\n\u22a2 \u2200 (b : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) b) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) b =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) b) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) b\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i).Y) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i).Y) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i\n[PROOFSTEP]\nerw [\u2190 (E.\u03c0.app k).naturality, \u2190 (E.\u03c0.app k).naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n (((Functor.const K).obj E.pt).obj k).map i.g\u2081.op \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z) =\n Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n (((Functor.const K).obj E.pt).obj k).map i.g\u2082.op \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n E.pt.map i.g\u2081.op \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z) =\n Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n E.pt.map i.g\u2082.op \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z)\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b E.pt.map i.g\u2081.op) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z) =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b E.pt.map i.g\u2082.op) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Z)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nk : K\ni : (GrothendieckTopology.Cover.index W (F.obj k).val).R\n\u22a2 Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b E.pt.map i.g\u2081.op =\n Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b E.pt.map i.g\u2082.op\n[PROOFSTEP]\napply S.condition\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\n\u22a2 \u2200 \u2983X_1 Y : K\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const K).obj S.pt).map f \u226b\n (fun k =>\n IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))\n Y =\n (fun k =>\n IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))\n X_1 \u226b\n (F \u22d9 sheafToPresheaf J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f\n[PROOFSTEP]\nintro i j f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\n\u22a2 ((Functor.const K).obj S.pt).map f \u226b\n (fun k =>\n IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))\n j =\n (fun k =>\n IsLimit.lift (Presheaf.isLimitOfIsSheaf J (F.obj k).val W (_ : Presheaf.IsSheaf J (F.obj k).val))\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i)))\n i \u226b\n (F \u22d9 sheafToPresheaf J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f\n[PROOFSTEP]\ndsimp [Presheaf.isLimitOfIsSheaf]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\n\u22a2 \ud835\udfd9 S.pt \u226b\n Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj j).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) I) =\n Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj i).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n NatTrans.app (F.map f).val (op X)\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\n\u22a2 Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj j).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) I) =\n Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj i).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n NatTrans.app (F.map f).val (op X)\n[PROOFSTEP]\napply Presheaf.IsSheaf.hom_ext (F.obj j).2 W\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\n\u22a2 \u2200 (I : GrothendieckTopology.Cover.Arrow W),\n Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj j).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n (F.obj j).val.map I.f.op =\n (Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj i).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n NatTrans.app (F.map f).val (op X)) \u226b\n (F.obj j).val.map I.f.op\n[PROOFSTEP]\nintro ii\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\nii : GrothendieckTopology.Cover.Arrow W\n\u22a2 Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj j).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) I) \u226b\n (F.obj j).val.map ii.f.op =\n (Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj i).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val)\n i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) I) \u226b\n NatTrans.app (F.map f).val (op X)) \u226b\n (F.obj j).val.map ii.f.op\n[PROOFSTEP]\nrw [Presheaf.IsSheaf.amalgamate_map, Category.assoc, \u2190 (F.map f).val.naturality, \u2190 Category.assoc,\n Presheaf.IsSheaf.amalgamate_map]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\nii : GrothendieckTopology.Cover.Arrow W\n\u22a2 Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj j).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj j).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) i =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj j).val) i).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj j).val) i))\n ii =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj i).val) S.pt\n (fun i_1 => Multifork.\u03b9 S i_1 \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op i_1.Y))\n (_ :\n \u2200 (i_1 : (GrothendieckTopology.Cover.index W (F.obj i).val).R),\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) i_1 =\n (Multifork.\u03b9 S (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj i).val) i_1).Y)) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj i).val) i_1))\n ii \u226b\n NatTrans.app (F.map f).val (op ii.Y)\n[PROOFSTEP]\ndsimp [Multifork.of\u03b9]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\nii : GrothendieckTopology.Cover.Arrow W\n\u22a2 Multifork.\u03b9\n { pt := S.pt,\n \u03c0 :=\n NatTrans.mk fun x =>\n match x with\n | WalkingMulticospan.left a => Multifork.\u03b9 S a \u226b NatTrans.app (NatTrans.app E.\u03c0 j) (op a.Y)\n | WalkingMulticospan.right b =>\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) b) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 j)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) b).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) b }\n ii =\n Multifork.\u03b9\n { pt := S.pt,\n \u03c0 :=\n NatTrans.mk fun x =>\n match x with\n | WalkingMulticospan.left a => Multifork.\u03b9 S a \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op a.Y)\n | WalkingMulticospan.right b =>\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) b) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj i).val) b).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj i).val) b }\n ii \u226b\n NatTrans.app (F.map f).val (op ii.Y)\n[PROOFSTEP]\nerw [Category.assoc, \u2190 E.w f]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\nK : Type z\ninst\u271d : SmallCategory K\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni j : K\nf : i \u27f6 j\nii : GrothendieckTopology.Cover.Arrow W\n\u22a2 Multifork.\u03b9\n { pt := S.pt,\n \u03c0 :=\n NatTrans.mk fun x =>\n match x with\n | WalkingMulticospan.left a =>\n Multifork.\u03b9 S a \u226b NatTrans.app (NatTrans.app E.\u03c0 i \u226b (F \u22d9 sheafToPresheaf J D).map f) (op a.Y)\n | WalkingMulticospan.right b =>\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) b) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 i \u226b (F \u22d9 sheafToPresheaf J D).map f)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj j).val) b).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj j).val) b }\n ii =\n Multifork.\u03b9 S ii \u226b NatTrans.app (NatTrans.app E.\u03c0 i) (op ii.Y) \u226b NatTrans.app (F.map f).val (op ii.Y)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\n\u22a2 \u2200 (E_1 : Multifork (GrothendieckTopology.Cover.index W E.pt)) (i : (GrothendieckTopology.Cover.index W E.pt).L),\n (fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n E_1 \u226b\n Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i =\n Multifork.\u03b9 E_1 i\n[PROOFSTEP]\nintro S i\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\n\u22a2 (fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S \u226b\n Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i =\n Multifork.\u03b9 S i\n[PROOFSTEP]\napply (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) hE).hom_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\n\u22a2 \u2200 (j : K),\n ((fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S \u226b\n Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i) \u226b\n NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op i.Y)).mapCone E).\u03c0 j =\n Multifork.\u03b9 S i \u226b NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op i.Y)).mapCone E).\u03c0 j\n[PROOFSTEP]\nintro k\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 ((fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S \u226b\n Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i) \u226b\n NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op i.Y)).mapCone E).\u03c0 k =\n Multifork.\u03b9 S i \u226b NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op i.Y)).mapCone E).\u03c0 k\n[PROOFSTEP]\ndsimp [Multifork.of\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 (IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S) \u226b\n Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y) =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nerw [Category.assoc, (E.\u03c0.app k).naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op X) \u226b ((F \u22d9 sheafToPresheaf J D).obj k).map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op X) \u226b (F.obj k).val.map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 (IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b\n (F.obj k).val.map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nerw [(isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac (multiforkEvaluationCone F E X W S)]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 NatTrans.app (multiforkEvaluationCone F E X W S).\u03c0 k \u226b (F.obj k).val.map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\ndsimp [multiforkEvaluationCone, Presheaf.isLimitOfIsSheaf]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj k).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n (F.obj k).val.map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nerw [Presheaf.IsSheaf.amalgamate_map]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\ni : (GrothendieckTopology.Cover.index W E.pt).L\nk : K\n\u22a2 Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n i =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\n\u22a2 \u2200 (E_1 : Multifork (GrothendieckTopology.Cover.index W E.pt))\n (m : E_1.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt),\n (\u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 E_1 i) \u2192\n m =\n (fun S =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n E_1\n[PROOFSTEP]\nintro S m hm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\n\u22a2 m =\n (fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S\n[PROOFSTEP]\napply (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).hom_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\n\u22a2 \u2200 (j : K),\n m \u226b NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op X)).mapCone E).\u03c0 j =\n (fun S =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S \u226b\n NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op X)).mapCone E).\u03c0 j\n[PROOFSTEP]\nintro k\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\n\u22a2 m \u226b NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op X)).mapCone E).\u03c0 k =\n (fun S => IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S))\n S \u226b\n NatTrans.app (((evaluation C\u1d52\u1d56 D).obj (op X)).mapCone E).\u03c0 k\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\n\u22a2 m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X) =\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE) (multiforkEvaluationCone F E X W S) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k) (op X)\n[PROOFSTEP]\nerw [(isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\n\u22a2 m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X) = NatTrans.app (multiforkEvaluationCone F E X W S).\u03c0 k\n[PROOFSTEP]\napply Presheaf.IsSheaf.hom_ext (F.obj k).2 W\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\n\u22a2 \u2200 (I : GrothendieckTopology.Cover.Arrow W),\n (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map I.f.op =\n NatTrans.app (multiforkEvaluationCone F E X W S).\u03c0 k \u226b (F.obj k).val.map I.f.op\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map i.f.op =\n NatTrans.app (multiforkEvaluationCone F E X W S).\u03c0 k \u226b (F.obj k).val.map i.f.op\n[PROOFSTEP]\ndsimp only [multiforkEvaluationCone, Presheaf.isLimitOfIsSheaf]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map i.f.op =\n Presheaf.IsSheaf.amalgamate (_ : Presheaf.IsSheaf J (F.obj k).val) W\n (fun I =>\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation W),\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) I =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) I) \u226b\n (F.obj k).val.map i.f.op\n[PROOFSTEP]\nrw [(F.obj k).cond.amalgamate_map]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map i.f.op =\n Multifork.\u03b9\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index W (F.obj k).val) S.pt\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (_ :\n \u2200 (i : (GrothendieckTopology.Cover.index W (F.obj k).val).R),\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) i =\n (fun i => Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y))\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index W (F.obj k).val) i) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index W (F.obj k).val) i))\n i\n[PROOFSTEP]\ndsimp [Multifork.of\u03b9]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map i.f.op =\n Multifork.\u03b9\n { pt := S.pt,\n \u03c0 :=\n NatTrans.mk fun x =>\n match x with\n | WalkingMulticospan.left a => Multifork.\u03b9 S a \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op a.Y)\n | WalkingMulticospan.right b =>\n (Multifork.\u03b9 S (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) b) \u226b\n NatTrans.app (NatTrans.app E.\u03c0 k)\n (op (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index W (F.obj k).val) b).Y)) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index W (F.obj k).val) b }\n i\n[PROOFSTEP]\nchange _ = S.\u03b9 i \u226b _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 (m \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op X)) \u226b (F.obj k).val.map i.f.op =\n Multifork.\u03b9 S i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nerw [\u2190 hm, Category.assoc, \u2190 (E.\u03c0.app k).naturality, Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nW : GrothendieckTopology.Cover J X\nS : Multifork (GrothendieckTopology.Cover.index W E.pt)\nm : S.pt \u27f6 (GrothendieckTopology.Cover.multifork W E.pt).pt\nhm :\n \u2200 (i : (GrothendieckTopology.Cover.index W E.pt).L),\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i = Multifork.\u03b9 S i\nk : K\ni : GrothendieckTopology.Cover.Arrow W\n\u22a2 m \u226b (((Functor.const K).obj E.pt).obj k).map i.f.op \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y) =\n m \u226b Multifork.\u03b9 (GrothendieckTopology.Cover.multifork W E.pt) i \u226b NatTrans.app (NatTrans.app E.\u03c0 k) (op i.Y)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\n\u22a2 Presheaf.IsSheaf J E.pt\n[PROOFSTEP]\nrw [Presheaf.isSheaf_iff_multifork]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\n\u22a2 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))\n[PROOFSTEP]\nintro X S\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))\n[PROOFSTEP]\nexact \u27e8isLimitMultiforkOfIsLimit _ _ hE _ _\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nj : K\n\u22a2 NatTrans.app\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).\u03c0\n j =\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)).hom \u226b\n NatTrans.app E.\u03c0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nj : K\n\u22a2 NatTrans.app E.\u03c0 j = \ud835\udfd9 E.pt \u226b NatTrans.app E.\u03c0 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nS : Cone F\nj : K\n\u22a2 (fun S => { val := IsLimit.lift hE ((sheafToPresheaf J D).mapCone S) }) S \u226b\n NatTrans.app\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.\u03c0\n j =\n NatTrans.app S.\u03c0 j\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nS : Cone F\nj : K\n\u22a2 ((fun S => { val := IsLimit.lift hE ((sheafToPresheaf J D).mapCone S) }) S \u226b\n NatTrans.app\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.\u03c0\n j).val =\n (NatTrans.app S.\u03c0 j).val\n[PROOFSTEP]\napply hE.fac ((sheafToPresheaf J D).mapCone S) j\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nS : Cone F\nm :\n S.pt \u27f6\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.pt\nhm :\n \u2200 (j : K),\n m \u226b\n NatTrans.app\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.\u03c0\n j =\n NatTrans.app S.\u03c0 j\n\u22a2 m = (fun S => { val := IsLimit.lift hE ((sheafToPresheaf J D).mapCone S) }) S\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nS : Cone F\nm :\n S.pt \u27f6\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.pt\nhm :\n \u2200 (j : K),\n m \u226b\n NatTrans.app\n {\n liftedCone :=\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } },\n validLift :=\n Cones.ext\n (eqToIso\n (_ :\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt =\n ((sheafToPresheaf J D).mapCone\n { pt := { val := E.pt, cond := (_ : Presheaf.IsSheaf J E.pt) },\n \u03c0 := NatTrans.mk fun t => { val := NatTrans.app E.\u03c0 t } }).pt)) }.liftedCone.\u03c0\n j =\n NatTrans.app S.\u03c0 j\n\u22a2 m.val = ((fun S => { val := IsLimit.lift hE ((sheafToPresheaf J D).mapCone S) }) S).val\n[PROOFSTEP]\nexact hE.uniq ((sheafToPresheaf J D).mapCone S) m.val fun j => congr_arg Hom.val (hm j)\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\ni j : K\nf : i \u27f6 j\n\u22a2 F.map f \u226b (fun k => { val := NatTrans.app E.\u03b9 k \u226b GrothendieckTopology.toSheafify J E.pt }) j =\n (fun k => { val := NatTrans.app E.\u03b9 k \u226b GrothendieckTopology.toSheafify J E.pt }) i \u226b\n ((Functor.const K).obj\n { val := GrothendieckTopology.sheafify J E.pt,\n cond :=\n (_ : Presheaf.IsSheaf J (GrothendieckTopology.plusObj J (GrothendieckTopology.plusObj J E.pt))) }).map\n f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\ni j : K\nf : i \u27f6 j\n\u22a2 (F.map f \u226b (fun k => { val := NatTrans.app E.\u03b9 k \u226b GrothendieckTopology.toSheafify J E.pt }) j).val =\n ((fun k => { val := NatTrans.app E.\u03b9 k \u226b GrothendieckTopology.toSheafify J E.pt }) i \u226b\n ((Functor.const K).obj\n { val := GrothendieckTopology.sheafify J E.pt,\n cond :=\n (_ : Presheaf.IsSheaf J (GrothendieckTopology.plusObj J (GrothendieckTopology.plusObj J E.pt))) }).map\n f).val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\ni j : K\nf : i \u27f6 j\n\u22a2 (F.map f).val \u226b NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt =\n (NatTrans.app E.\u03b9 i \u226b GrothendieckTopology.toSheafify J E.pt) \u226b \ud835\udfd9 (GrothendieckTopology.sheafify J E.pt)\n[PROOFSTEP]\nerw [Category.comp_id, \u2190 Category.assoc, E.w f]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\n\u22a2 \u2200 (s : Cocone F) (j : K),\n NatTrans.app (sheafifyCocone E).\u03b9 j \u226b\n (fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nintro S j\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nj : K\n\u22a2 NatTrans.app (sheafifyCocone E).\u03b9 j \u226b\n (fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n S =\n NatTrans.app S.\u03b9 j\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nj : K\n\u22a2 (NatTrans.app (sheafifyCocone E).\u03b9 j \u226b\n (fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n S).val =\n (NatTrans.app S.\u03b9 j).val\n[PROOFSTEP]\ndsimp [sheafifyCocone]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nj : K\n\u22a2 (NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt) \u226b\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) =\n (NatTrans.app S.\u03b9 j).val\n[PROOFSTEP]\nerw [Category.assoc, J.toSheafify_sheafifyLift, hE.fac]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nj : K\n\u22a2 NatTrans.app ((sheafToPresheaf J D).mapCocone S).\u03b9 j = (NatTrans.app S.\u03b9 j).val\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\n\u22a2 \u2200 (s : Cocone F) (m : (sheafifyCocone E).pt \u27f6 s.pt),\n (\u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192\n m =\n (fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n s\n[PROOFSTEP]\nintro S m hm\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m =\n (fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n S\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m.val =\n ((fun S =>\n {\n val :=\n GrothendieckTopology.sheafifyLift J (IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S))\n (_ : Presheaf.IsSheaf J S.pt.val) })\n S).val\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 GrothendieckTopology.toSheafify J E.pt \u226b m.val = IsColimit.desc hE ((sheafToPresheaf J D).mapCocone S)\n[PROOFSTEP]\napply hE.uniq ((sheafToPresheaf J D).mapCocone S)\n[GOAL]\ncase h.a.x\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 \u2200 (j : K),\n NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt \u226b m.val =\n NatTrans.app ((sheafToPresheaf J D).mapCocone S).\u03b9 j\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h.a.x\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nj : K\n\u22a2 NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt \u226b m.val =\n NatTrans.app ((sheafToPresheaf J D).mapCocone S).\u03b9 j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.a.x\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nj : K\n\u22a2 NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt \u226b m.val = (NatTrans.app S.\u03b9 j).val\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, \u2190 hm]\n -- Porting note: was `simpa only [...]`\n[GOAL]\ncase h.a.x\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\nK : Type (max v u)\ninst\u271d\u2076 : SmallCategory K\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF : K \u2964 Sheaf J D\nE : Cocone (F \u22d9 sheafToPresheaf J D)\nhE : IsColimit E\nS : Cocone F\nm : (sheafifyCocone E).pt \u27f6 S.pt\nhm : \u2200 (j : K), NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nj : K\n\u22a2 (NatTrans.app E.\u03b9 j \u226b GrothendieckTopology.toSheafify J E.pt) \u226b m.val = (NatTrans.app (sheafifyCocone E).\u03b9 j \u226b m).val\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Limits", "llama_tokens": 42002, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.2999748917119344}} {"text": "[GOAL]\nX Y Z : TopCat\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 (map (\ud835\udfd9 X)).op.obj U = U\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y Z : TopCat\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191Y\n\u22a2 (map f).obj (iSup U) = iSup ((map f).toPrefunctor.obj \u2218 U)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX Y Z : TopCat\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191Y\n\u22a2 \u2191((map f).obj (iSup U)) = \u2191(iSup ((map f).toPrefunctor.obj \u2218 U))\n[PROOFSTEP]\nrw [iSup_def, iSup_def, map_obj]\n[GOAL]\ncase h\nX Y Z : TopCat\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191Y\n\u22a2 \u2191{ carrier := \u2191f \u207b\u00b9' \u22c3 (i : \u03b9), \u2191(U i), is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' \u22c3 (i : \u03b9), \u2191(U i))) } =\n \u2191{ carrier := \u22c3 (i : \u03b9), \u2191(((map f).toPrefunctor.obj \u2218 U) i),\n is_open' := (_ : IsOpen (\u22c3 (i : \u03b9), \u2191(((map f).toPrefunctor.obj \u2218 U) i))) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nX Y Z : TopCat\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191Y\n\u22a2 \u2191f \u207b\u00b9' \u22c3 (i : \u03b9), \u2191(U i) = \u22c3 (i : \u03b9), \u2191((map f).obj (U i))\n[PROOFSTEP]\nrw [Set.preimage_iUnion]\n[GOAL]\ncase h\nX Y Z : TopCat\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191Y\n\u22a2 \u22c3 (i : \u03b9), \u2191f \u207b\u00b9' \u2191(U i) = \u22c3 (i : \u03b9), \u2191((map f).obj (U i))\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : TopCat\n\u22a2 map (\ud835\udfd9 X) = \ud835\udfed (Opens \u2191X)\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : TopCat\nf g : X \u27f6 Y\nh : f = g\nU : Opens \u2191Y\n\u22a2 (map f).obj U = (map g).obj U\n[PROOFSTEP]\nrw [congr_arg map h]\n[GOAL]\nX Y Z : TopCat\nf g : X \u27f6 Y\nh : f = g\n\u22a2 map f = map g\n[PROOFSTEP]\nsubst h\n[GOAL]\nX Y Z : TopCat\nf : X \u27f6 Y\n\u22a2 map f = map f\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : TopCat\nf g : X \u27f6 Y\nh : f = g\nU : Opens \u2191Y\n\u22a2 (map f).obj U = (map g).obj U\n[PROOFSTEP]\nrw [h]\n[GOAL]\nX Y Z : TopCat\nf g : X \u27f6 Y\nh : f = g\nU : Opens \u2191Y\n\u22a2 (map g).obj U = (map f).obj U\n[PROOFSTEP]\nrw [h]\n[GOAL]\nX\u271d Y\u271d Z X Y : TopCat\nH : X \u2245 Y\nU : Opens \u2191Y\n\u22a2 (\ud835\udfed (Opens \u2191Y)).obj U = (map H.hom \u22d9 map H.inv).obj U\n[PROOFSTEP]\nsimp [map, Set.preimage_preimage]\n[GOAL]\nX\u271d Y\u271d Z X Y : TopCat\nH : X \u2245 Y\nU : Opens \u2191X\n\u22a2 (map H.inv \u22d9 map H.hom).obj U = (\ud835\udfed (Opens \u2191X)).obj U\n[PROOFSTEP]\nsimp [map, Set.preimage_preimage]\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nH : Mono f\nX\u271d Y\u271d : Opens \u2191X\ni : (functor hf).obj X\u271d \u27f6 (functor hf).obj Y\u271d\nx : \u2191X\nhx : x \u2208 \u2191X\u271d\n\u22a2 x \u2208 \u2191Y\u271d\n[PROOFSTEP]\nobtain \u27e8y, hy, eq\u27e9 := i.le \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nH : Mono f\nX\u271d Y\u271d : Opens \u2191X\ni : (functor hf).obj X\u271d \u27f6 (functor hf).obj Y\u271d\nx : \u2191X\nhx : x \u2208 \u2191X\u271d\ny : (forget TopCat).obj X\nhy : y \u2208 \u2191Y\u271d\neq : \u2191f y = \u2191f x\n\u22a2 x \u2208 \u2191Y\u271d\n[PROOFSTEP]\nexact (TopCat.mono_iff_injective f).mp H eq \u25b8 hy\n[GOAL]\nX : TopCat\nU : Opens \u2191X\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4 = U\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX : TopCat\nU : Opens \u2191X\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4) = \u2191U\n[PROOFSTEP]\nexact Set.image_univ.trans Subtype.range_coe\n[GOAL]\nX : TopCat\nU : Opens \u2191X\n\u22a2 (map (inclusion U)).obj U = \u22a4\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX : TopCat\nU : Opens \u2191X\n\u22a2 \u2191((map (inclusion U)).obj U) = \u2191\u22a4\n[PROOFSTEP]\nexact Subtype.coe_preimage_self _\n[GOAL]\nX : TopCat\nU : Opens \u2191X\n\u22a2 (map (inclusion U) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj U = (\ud835\udfed (Opens \u2191X)).obj U\n[PROOFSTEP]\nsimp\n[GOAL]\nX : TopCat\n\u22a2 IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4)) = map (inclusionTopIso X).inv\n[PROOFSTEP]\nrefine' CategoryTheory.Functor.ext _ _\n[GOAL]\ncase refine'_1\nX : TopCat\n\u22a2 \u2200 (X_1 : Opens \u2191((toTopCat X).obj \u22a4)),\n (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1\n[PROOFSTEP]\nintro U\n[GOAL]\ncase refine'_1\nX : TopCat\nU : Opens \u2191((toTopCat X).obj \u22a4)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj U = (map (inclusionTopIso X).inv).obj U\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_1.h.h\nX : TopCat\nU : Opens \u2191((toTopCat X).obj \u22a4)\nx : \u2191X\n\u22a2 x \u2208 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj U) \u2194 x \u2208 \u2191((map (inclusionTopIso X).inv).obj U)\n[PROOFSTEP]\nexact \u27e8fun \u27e8\u27e8_, _\u27e9, h, rfl\u27e9 => h, fun h => \u27e8\u27e8x, trivial\u27e9, h, rfl\u27e9\u27e9\n[GOAL]\ncase refine'_2\nX : TopCat\n\u22a2 \u2200 (X_1 Y : Opens \u2191((toTopCat X).obj \u22a4)) (f : X_1 \u27f6 Y),\n (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).map f =\n eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj X_1 = (map (inclusionTopIso X).inv).obj X_1) \u226b\n (map (inclusionTopIso X).inv).map f \u226b\n eqToHom (_ : (map (inclusionTopIso X).inv).obj Y = (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj Y)\n[PROOFSTEP]\nintros U V f\n[GOAL]\ncase refine'_2\nX : TopCat\nU V : Opens \u2191((toTopCat X).obj \u22a4)\nf : U \u27f6 V\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).map f =\n eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj U = (map (inclusionTopIso X).inv).obj U) \u226b\n (map (inclusionTopIso X).inv).map f \u226b\n eqToHom (_ : (map (inclusionTopIso X).inv).obj V = (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion \u22a4))).obj V)\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\n\u22a2 (IsOpenMap.functor hf).obj ((map f).obj U) = (IsOpenMap.functor hf).obj \u22a4 \u2293 U\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\nx\u271d : \u2191Y\n\u22a2 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj ((map f).obj U)) \u2194 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj \u22a4 \u2293 U)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\nx\u271d : \u2191Y\n\u22a2 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj ((map f).obj U)) \u2192 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj \u22a4 \u2293 U)\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase h.h.mp.intro.intro\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\nx : (forget TopCat).obj X\nhx : x \u2208 \u2191((map f).obj U)\n\u22a2 \u2191f x \u2208 \u2191((IsOpenMap.functor hf).obj \u22a4 \u2293 U)\n[PROOFSTEP]\nexact \u27e8\u27e8x, trivial, rfl\u27e9, hx\u27e9\n[GOAL]\ncase h.h.mpr\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\nx\u271d : \u2191Y\n\u22a2 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj \u22a4 \u2293 U) \u2192 x\u271d \u2208 \u2191((IsOpenMap.functor hf).obj ((map f).obj U))\n[PROOFSTEP]\nrintro \u27e8\u27e8x, -, rfl\u27e9, hx\u27e9\n[GOAL]\ncase h.h.mpr.intro.intro.intro\nX Y : TopCat\nf : X \u27f6 Y\nhf : IsOpenMap \u2191f\nU : Opens \u2191Y\nx : (forget TopCat).obj X\nhx : \u2191f x \u2208 \u2191U\n\u22a2 \u2191f x \u2208 \u2191((IsOpenMap.functor hf).obj ((map f).obj U))\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nX : TopCat\nU : Opens \u2191X\n\u22a2 Set.range ((forget TopCat).map (inclusion U)) = \u2191U\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nX : TopCat\nU : Opens \u2191X\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map (inclusion U)) \u2194 x \u2208 \u2191U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nX : TopCat\nU : Opens \u2191X\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map (inclusion U)) \u2192 x \u2208 \u2191U\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase h.mp.intro\nX : TopCat\nU : Opens \u2191X\nx : (forget TopCat).obj ((toTopCat X).obj U)\n\u22a2 (forget TopCat).map (inclusion U) x \u2208 \u2191U\n[PROOFSTEP]\nexact x.2\n[GOAL]\ncase h.mpr\nX : TopCat\nU : Opens \u2191X\nx : (forget TopCat).obj X\n\u22a2 x \u2208 \u2191U \u2192 x \u2208 Set.range ((forget TopCat).map (inclusion U))\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nX : TopCat\nU : Opens \u2191X\nx : (forget TopCat).obj X\nh : x \u2208 \u2191U\n\u22a2 x \u2208 Set.range ((forget TopCat).map (inclusion U))\n[PROOFSTEP]\nexact \u27e8\u27e8x, h\u27e9, rfl\u27e9\n[GOAL]\nX : TopCat\nU V : Opens \u2191X\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj ((map (inclusion U)).obj V) = V \u2293 U\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX : TopCat\nU V : Opens \u2191X\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj ((map (inclusion U)).obj V)) = \u2191(V \u2293 U)\n[PROOFSTEP]\nrefine' Set.image_preimage_eq_inter_range.trans _\n[GOAL]\ncase h\nX : TopCat\nU V : Opens \u2191X\n\u22a2 (V.1 \u2229 Set.range fun x => \u2191(inclusion U) x) = \u2191(V \u2293 U)\n[PROOFSTEP]\nerw [set_range_forget_map_inclusion U]\n[GOAL]\ncase h\nX : TopCat\nU V : Opens \u2191X\n\u22a2 V.1 \u2229 \u2191U = \u2191(V \u2293 U)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : TopCat\nU : Opens \u2191X\nV : Opens { x // x \u2208 U }\n\u22a2 (map (inclusion U) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V) =\n (\ud835\udfed (Opens \u2191X)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V)\n[PROOFSTEP]\ndsimp\n[GOAL]\nX : TopCat\nU : Opens \u2191X\nV : Opens { x // x \u2208 U }\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj\n ((map (inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V)) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V\n[PROOFSTEP]\nrw [map_functor_eq V]\n[GOAL]\nX : TopCat\nU : Opens \u2191X\nV : Opens { x // x \u2208 U }\n\u22a2 NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V) =\n eqToHom\n (_ :\n (map (inclusion U) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V) =\n (\ud835\udfed (Opens \u2191X)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj V))\n[PROOFSTEP]\napply Subsingleton.elim\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.TopCat.Opens", "llama_tokens": 4438, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.2997249254562815}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : PreservesFiniteLimits G\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 HasKernel f\n[PROOFSTEP]\nhave := NatIso.naturality_1 i f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : PreservesFiniteLimits G\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis : NatTrans.app i.inv X\u271d \u226b (F \u22d9 G).map f \u226b NatTrans.app i.hom Y\u271d = (\ud835\udfed C).map f\n\u22a2 HasKernel f\n[PROOFSTEP]\nsimp at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : PreservesFiniteLimits G\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\n\u22a2 HasKernel f\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : PreservesFiniteLimits G\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\n\u22a2 HasKernel (NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n[PROOFSTEP]\nhaveI : HasKernel (G.map (F.map f) \u226b i.hom.app _) := Limits.hasKernel_comp_mono _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : PreservesFiniteLimits G\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis\u271d : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\nthis : HasKernel (G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n\u22a2 HasKernel (NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n[PROOFSTEP]\napply Limits.hasKernel_iso_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\n\u22a2 HasCokernel f\n[PROOFSTEP]\nhave : PreservesColimits G := adj.leftAdjointPreservesColimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis : PreservesColimits G\n\u22a2 HasCokernel f\n[PROOFSTEP]\nhave := NatIso.naturality_1 i f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis\u271d : PreservesColimits G\nthis : NatTrans.app i.inv X\u271d \u226b (F \u22d9 G).map f \u226b NatTrans.app i.hom Y\u271d = (\ud835\udfed C).map f\n\u22a2 HasCokernel f\n[PROOFSTEP]\nsimp at this \n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis\u271d : PreservesColimits G\nthis : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\n\u22a2 HasCokernel f\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis\u271d : PreservesColimits G\nthis : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\n\u22a2 HasCokernel (NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n[PROOFSTEP]\nhaveI : HasCokernel (G.map (F.map f) \u226b i.hom.app _) := Limits.hasCokernel_comp_iso _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v, u\u2081} C\ninst\u271d\u00b3 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\ninst\u271d\u00b9 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nX\u271d Y\u271d : C\nf : X\u271d \u27f6 Y\u271d\nthis\u271d\u00b9 : PreservesColimits G\nthis\u271d : NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d = f\nthis : HasCokernel (G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n\u22a2 HasCokernel (NatTrans.app i.inv X\u271d \u226b G.map (F.map f) \u226b NatTrans.app i.hom Y\u271d)\n[PROOFSTEP]\napply Limits.hasCokernel_epi_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 G.obj (cokernel (F.map f)) \u2245 cokernel f\n[PROOFSTEP]\nhave : PreservesColimits G := adj.leftAdjointPreservesColimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\nthis : PreservesColimits G\n\u22a2 G.obj (cokernel (F.map f)) \u2245 cokernel f\n[PROOFSTEP]\nhave : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v, u\u2081} C\ninst\u271d\u2074 : Preadditive C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v, u\u2082} D\ninst\u271d\u00b2 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d : HasCokernels C\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesColimits G\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 G.obj (cokernel (F.map f)) \u2245 cokernel f\n[PROOFSTEP]\ncalc\n G.obj (cokernel (F.map f)) \u2245 cokernel (G.map (F.map f)) := (asIso (cokernelComparison _ G)).symm\n _ \u2245 cokernel (i.hom.app X \u226b f \u226b i.inv.app Y) := (cokernelIsoOfEq (NatIso.naturality_2 i f).symm)\n _ \u2245 cokernel (f \u226b i.inv.app Y) := (cokernelEpiComp (i.hom.app X) (f \u226b i.inv.app Y))\n _ \u2245 cokernel f := cokernelCompIsIso f (i.inv.app Y)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v, u\u2082} D\ninst\u271d\u2074 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b3 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : PreservesFiniteLimits G\nX Y : C\nf : X \u27f6 Y\n\u22a2 kernel (G.map (cokernel.\u03c0 (F.map f))) \u2245 kernel (cokernel.\u03c0 f)\n[PROOFSTEP]\nhave : PreservesColimits G := adj.leftAdjointPreservesColimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v, u\u2082} D\ninst\u271d\u2074 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b3 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : PreservesFiniteLimits G\nX Y : C\nf : X \u27f6 Y\nthis : PreservesColimits G\n\u22a2 kernel (G.map (cokernel.\u03c0 (F.map f))) \u2245 kernel (cokernel.\u03c0 f)\n[PROOFSTEP]\nhave : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v, u\u2082} D\ninst\u271d\u2074 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b3 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : PreservesFiniteLimits G\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesColimits G\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 kernel (G.map (cokernel.\u03c0 (F.map f))) \u2245 kernel (cokernel.\u03c0 f)\n[PROOFSTEP]\ncalc\n kernel (G.map (cokernel.\u03c0 (F.map f))) \u2245 kernel (cokernel.\u03c0 (G.map (F.map f)) \u226b cokernelComparison (F.map f) G) :=\n kernelIsoOfEq (\u03c0_comp_cokernelComparison _ _).symm\n _ \u2245 kernel (cokernel.\u03c0 (G.map (F.map f))) := (kernelCompMono _ _)\n _ \u2245 kernel (cokernel.\u03c0 (_ \u226b f \u226b _) \u226b (cokernelIsoOfEq _).hom) :=\n (kernelIsoOfEq (\u03c0_comp_cokernelIsoOfEq_hom (NatIso.naturality_2 i f)).symm)\n _ \u2245 kernel (cokernel.\u03c0 (_ \u226b f \u226b _)) := (kernelCompMono _ _)\n _ \u2245 kernel (cokernel.\u03c0 (f \u226b i.inv.app Y) \u226b (cokernelEpiComp (i.hom.app X) _).inv) :=\n (kernelIsoOfEq (by simp only [cokernel.\u03c0_desc, cokernelEpiComp_inv]))\n _ \u2245 kernel (cokernel.\u03c0 (f \u226b _)) := (kernelCompMono _ _)\n _ \u2245 kernel (inv (i.inv.app Y) \u226b cokernel.\u03c0 f \u226b (cokernelCompIsIso f (i.inv.app Y)).inv) :=\n (kernelIsoOfEq\n (by simp only [cokernel.\u03c0_desc, cokernelCompIsIso_inv, Iso.hom_inv_id_app_assoc, NatIso.inv_inv_app]))\n _ \u2245 kernel (cokernel.\u03c0 f \u226b _) := (kernelIsIsoComp _ _)\n _ \u2245 kernel (cokernel.\u03c0 f) := kernelCompMono _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v, u\u2082} D\ninst\u271d\u2074 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b3 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : PreservesFiniteLimits G\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesColimits G\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) \u226b (cokernelEpiComp (NatTrans.app i.hom X) (f \u226b NatTrans.app i.inv Y)).inv\n[PROOFSTEP]\nsimp only [cokernel.\u03c0_desc, cokernelEpiComp_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v, u\u2082} D\ninst\u271d\u2074 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u00b3 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b2 : HasCokernels C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : PreservesFiniteLimits G\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesColimits G\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) =\n inv (NatTrans.app i.inv Y) \u226b cokernel.\u03c0 f \u226b (cokernelCompIsIso f (NatTrans.app i.inv Y)).inv\n[PROOFSTEP]\nsimp only [cokernel.\u03c0_desc, cokernelCompIsIso_inv, Iso.hom_inv_id_app_assoc, NatIso.inv_inv_app]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\n\u22a2 Abelian.coimage f \u2245 Abelian.image f\n[PROOFSTEP]\nhave : PreservesLimits F := adj.rightAdjointPreservesLimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis : PreservesLimits F\n\u22a2 Abelian.coimage f \u2245 Abelian.image f\n[PROOFSTEP]\nhaveI : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesLimits F\nthis : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 Abelian.coimage f \u2245 Abelian.image f\n[PROOFSTEP]\ncalc\n Abelian.coimage f \u2245 cokernel (kernel.\u03b9 f) := Iso.refl _\n _ \u2245 G.obj (cokernel (F.map (kernel.\u03b9 f))) := (cokernelIso _ _ i adj _).symm\n _ \u2245 G.obj (cokernel (kernelComparison f F \u226b kernel.\u03b9 (F.map f))) := (G.mapIso (cokernelIsoOfEq (by simp)))\n _ \u2245 G.obj (cokernel (kernel.\u03b9 (F.map f))) := (G.mapIso (cokernelEpiComp _ _))\n _ \u2245 G.obj (Abelian.coimage (F.map f)) := (Iso.refl _)\n _ \u2245 G.obj (Abelian.image (F.map f)) := (G.mapIso (Abelian.coimageIsoImage _))\n _ \u2245 G.obj (kernel (cokernel.\u03c0 (F.map f))) := (Iso.refl _)\n _ \u2245 kernel (G.map (cokernel.\u03c0 (F.map f))) := (PreservesKernel.iso _ _)\n _ \u2245 kernel (cokernel.\u03c0 f) := (coimageIsoImageAux F G i adj f)\n _ \u2245 Abelian.image f := Iso.refl _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d : PreservesLimits F\nthis : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 F.map (kernel.\u03b9 f) = kernelComparison f F \u226b kernel.\u03b9 (F.map f)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\n\u22a2 (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f\n[PROOFSTEP]\nhave : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f\n[PROOFSTEP]\nhave : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasKernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\nthis : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasKernel f'\n\u22a2 (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f\n[PROOFSTEP]\nhave : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d\u00b9 : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\nthis\u271d : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasKernel f'\nthis : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f'\n\u22a2 (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f\n[PROOFSTEP]\nhave : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasKernel f' := inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d\u00b2 : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\nthis\u271d\u00b9 : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasKernel f'\nthis\u271d : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f'\nthis : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasKernel f'\n\u22a2 (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f\n[PROOFSTEP]\ndsimp only [coimageIsoImage, Iso.instTransIso_trans, Iso.refl, Iso.trans, Iso.symm, Functor.mapIso, cokernelEpiComp,\n cokernelIso, cokernelCompIsIso_inv, asIso, coimageIsoImageAux, kernelCompMono]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v, u\u2081} C\ninst\u271d\u2077 : Preadditive C\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v, u\u2082} D\ninst\u271d\u2075 : Abelian D\nF : C \u2964 D\nG : D \u2964 C\ninst\u271d\u2074 : Functor.PreservesZeroMorphisms G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : HasKernels C\ninst\u271d\u00b9 : PreservesFiniteLimits G\ninst\u271d : Functor.PreservesZeroMorphisms F\nX Y : C\nf : X \u27f6 Y\nthis\u271d\u00b2 : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasCokernel f'\nthis\u271d\u00b9 : \u2200 (X' Y' : C) (f' : X' \u27f6 Y'), HasKernel f'\nthis\u271d : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasCokernel f'\nthis : \u2200 (X' Y' : D) (f' : X' \u27f6 Y'), HasKernel f'\n\u22a2 ((((((((\ud835\udfd9 (Abelian.coimage f) \u226b\n cokernel.desc (kernel.\u03b9 f) (NatTrans.app i.inv X \u226b cokernel.\u03c0 (kernel.\u03b9 f \u226b NatTrans.app i.inv X))\n (_ : kernel.\u03b9 f \u226b NatTrans.app i.inv X \u226b cokernel.\u03c0 (kernel.\u03b9 f \u226b NatTrans.app i.inv X) = 0) \u226b\n cokernel.desc (kernel.\u03b9 f \u226b NatTrans.app i.inv X)\n (cokernel.\u03c0 (NatTrans.app i.hom (kernel f) \u226b kernel.\u03b9 f \u226b NatTrans.app i.inv X))\n (_ :\n (kernel.\u03b9 f \u226b NatTrans.app i.inv X) \u226b\n cokernel.\u03c0 (NatTrans.app i.hom (kernel f) \u226b kernel.\u03b9 f \u226b NatTrans.app i.inv X) =\n 0) \u226b\n (cokernelIsoOfEq\n (_ :\n (F \u22d9 G).map (kernel.\u03b9 f) =\n NatTrans.app i.hom (kernel f) \u226b\n (\ud835\udfed C).map (kernel.\u03b9 f) \u226b NatTrans.app i.inv X)).inv \u226b\n cokernelComparison (F.map (kernel.\u03b9 f)) G) \u226b\n G.map (cokernelIsoOfEq (_ : F.map (kernel.\u03b9 f) = kernelComparison f F \u226b kernel.\u03b9 (F.map f))).hom) \u226b\n G.map\n (cokernel.desc (kernelComparison f F \u226b kernel.\u03b9 (F.map f)) (cokernel.\u03c0 (kernel.\u03b9 (F.map f)))\n (_ : (kernelComparison f F \u226b kernel.\u03b9 (F.map f)) \u226b cokernel.\u03c0 (kernel.\u03b9 (F.map f)) = 0))) \u226b\n \ud835\udfd9 (G.obj (cokernel (kernel.\u03b9 (F.map f))))) \u226b\n G.map (Abelian.coimageIsoImage (F.map f)).hom) \u226b\n \ud835\udfd9 (G.obj (Abelian.image (F.map f)))) \u226b\n (PreservesKernel.iso G (cokernel.\u03c0 (F.map f))).hom) \u226b\n ((((((((kernelIsoOfEq\n (_ :\n G.map (cokernel.\u03c0 (F.map f)) =\n cokernel.\u03c0 (G.map (F.map f)) \u226b cokernelComparison (F.map f) G)).hom \u226b\n kernel.lift (cokernel.\u03c0 (G.map (F.map f)))\n (kernel.\u03b9 (cokernel.\u03c0 (G.map (F.map f)) \u226b cokernelComparison (F.map f) G))\n (_ :\n kernel.\u03b9 (cokernel.\u03c0 (G.map (F.map f)) \u226b cokernelComparison (F.map f) G) \u226b\n cokernel.\u03c0 (G.map (F.map f)) =\n 0)) \u226b\n (kernelIsoOfEq\n (_ :\n cokernel.\u03c0 ((F \u22d9 G).map f) =\n cokernel.\u03c0 (NatTrans.app i.hom X \u226b (\ud835\udfed C).map f \u226b NatTrans.app i.inv Y) \u226b\n (cokernelIsoOfEq\n (_ :\n NatTrans.app i.hom X \u226b (\ud835\udfed C).map f \u226b NatTrans.app i.inv Y =\n (F \u22d9 G).map f)).hom)).hom) \u226b\n kernel.lift (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y))\n (kernel.\u03b9\n (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) \u226b\n (cokernelIsoOfEq\n (_ : NatTrans.app i.hom X \u226b (\ud835\udfed C).map f \u226b NatTrans.app i.inv Y = (F \u22d9 G).map f)).hom))\n (_ :\n kernel.\u03b9\n (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) \u226b\n (cokernelIsoOfEq\n (_ :\n NatTrans.app i.hom X \u226b (\ud835\udfed C).map f \u226b NatTrans.app i.inv Y = (F \u22d9 G).map f)).hom) \u226b\n cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n 0)) \u226b\n (kernelIsoOfEq\n (_ :\n cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) \u226b\n cokernel.desc (f \u226b NatTrans.app i.inv Y)\n (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y))\n (_ :\n (f \u226b NatTrans.app i.inv Y) \u226b\n cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n 0))).hom) \u226b\n kernel.lift (cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y))\n (kernel.\u03b9\n (cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) \u226b\n cokernel.desc (f \u226b NatTrans.app i.inv Y)\n (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y))\n (_ :\n (f \u226b NatTrans.app i.inv Y) \u226b cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n 0)))\n (_ :\n kernel.\u03b9\n (cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) \u226b\n cokernel.desc (f \u226b NatTrans.app i.inv Y)\n (cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y))\n (_ :\n (f \u226b NatTrans.app i.inv Y) \u226b\n cokernel.\u03c0 (NatTrans.app i.hom X \u226b f \u226b NatTrans.app i.inv Y) =\n 0)) \u226b\n cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) =\n 0)) \u226b\n (kernelIsoOfEq\n (_ :\n cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) =\n inv (NatTrans.app i.inv Y) \u226b\n cokernel.\u03c0 f \u226b\n cokernel.desc f (NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y))\n (_ : f \u226b NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) = 0))).hom) \u226b\n (kernelIsIsoComp (inv (NatTrans.app i.inv Y))\n (cokernel.\u03c0 f \u226b\n cokernel.desc f (NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y))\n (_ : f \u226b NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) = 0))).hom) \u226b\n kernel.lift (cokernel.\u03c0 f)\n (kernel.\u03b9\n (cokernel.\u03c0 f \u226b\n cokernel.desc f (NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y))\n (_ : f \u226b NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) = 0)))\n (_ :\n kernel.\u03b9\n (cokernel.\u03c0 f \u226b\n cokernel.desc f (NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y))\n (_ : f \u226b NatTrans.app i.inv Y \u226b cokernel.\u03c0 (f \u226b NatTrans.app i.inv Y) = 0)) \u226b\n cokernel.\u03c0 f =\n 0)) \u226b\n \ud835\udfd9 (kernel (cokernel.\u03c0 f)) =\n Abelian.coimageImageComparison f\n[PROOFSTEP]\nsimpa only [\u2190 cancel_mono (Abelian.image.\u03b9 f), \u2190 cancel_epi (Abelian.coimage.\u03c0 f), Category.assoc, Category.id_comp,\n cokernel.\u03c0_desc_assoc, \u03c0_comp_cokernelIsoOfEq_inv_assoc, PreservesKernel.iso_hom, \u03c0_comp_cokernelComparison_assoc, \u2190\n G.map_comp_assoc, kernel.lift_\u03b9, Abelian.coimage_image_factorisation, lift_comp_kernelIsoOfEq_hom_assoc,\n kernelIsIsoComp_hom, kernel.lift_\u03b9_assoc, kernelIsoOfEq_hom_comp_\u03b9_assoc, kernelComparison_comp_\u03b9_assoc,\n \u03c0_comp_cokernelIsoOfEq_hom_assoc, asIso_hom, NatIso.inv_inv_app] using NatIso.naturality_1 i f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\n\u22a2 Abelian C\n[PROOFSTEP]\nhaveI := hasKernels F G i\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis : HasKernels C\n\u22a2 Abelian C\n[PROOFSTEP]\nhaveI := hasCokernels F G i adj\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis\u271d : HasKernels C\nthis : HasCokernels C\n\u22a2 Abelian C\n[PROOFSTEP]\nhave : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (Abelian.coimageImageComparison f) :=\n by\n intro X Y f\n rw [\u2190 coimageIsoImage_hom F G i adj f]\n infer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis\u271d : HasKernels C\nthis : HasCokernels C\n\u22a2 \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (Abelian.coimageImageComparison f)\n[PROOFSTEP]\nintro X Y f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis\u271d : HasKernels C\nthis : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 IsIso (Abelian.coimageImageComparison f)\n[PROOFSTEP]\nrw [\u2190 coimageIsoImage_hom F G i adj f]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis\u271d : HasKernels C\nthis : HasCokernels C\nX Y : C\nf : X \u27f6 Y\n\u22a2 IsIso (coimageIsoImage F G i adj f).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2077 : Category.{v, u\u2081} C\ninst\u271d\u2076 : Preadditive C\ninst\u271d\u2075 : HasFiniteProducts C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v, u\u2082} D\ninst\u271d\u00b3 : Abelian D\nF : C \u2964 D\ninst\u271d\u00b2 : Functor.PreservesZeroMorphisms F\nG : D \u2964 C\ninst\u271d\u00b9 : Functor.PreservesZeroMorphisms G\ninst\u271d : PreservesFiniteLimits G\ni : F \u22d9 G \u2245 \ud835\udfed C\nadj : G \u22a3 F\nthis\u271d\u00b9 : HasKernels C\nthis\u271d : HasCokernels C\nthis : \u2200 {X Y : C} (f : X \u27f6 Y), IsIso (Abelian.coimageImageComparison f)\n\u22a2 Abelian C\n[PROOFSTEP]\napply Abelian.ofCoimageImageComparisonIsIso\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.Transfer", "llama_tokens": 13010, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.29960686320547475}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\nx y : \u03b1\ne : encode x = encode y\n\u22a2 some x = some y\n[PROOFSTEP]\nrw [\u2190 encodek, e, encodek]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nx : \u03b1\n\u22a2 (fun n => iget (decode n)) (encode x) = x\n[PROOFSTEP]\nsimp_rw [Encodable.encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\nf : \u03b2 \u2192 \u03b1\nfinv : \u03b1 \u2192 Option \u03b2\nlinv : \u2200 (b : \u03b2), finv (f b) = some b\nb : \u03b2\n\u22a2 (fun n => Option.bind (decode n) finv) ((fun b => encode (f b)) b) = some b\n[PROOFSTEP]\nsimp [Encodable.encodek, linv]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : Encodable \u03b1\ne : \u03b2 \u2243 \u03b1\nn : \u2115\n\u22a2 Option.bind (decode n) (some \u2218 \u2191e.symm) = Option.map (\u2191e.symm) (decode n)\n[PROOFSTEP]\nrw [Option.map_eq_bind]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx\u271d : PUnit\n\u22a2 (fun n => Nat.casesOn n (some PUnit.unit) fun x => none) ((fun x => 0) x\u271d) = some x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nh : Encodable \u03b1\no : Option \u03b1\n\u22a2 (fun n => Nat.casesOn n (some none) fun m => Option.map some (decode m))\n ((fun o => Option.casesOn o zero fun a => succ (encode a)) o) =\n some o\n[PROOFSTEP]\ncases o\n[GOAL]\ncase none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nh : Encodable \u03b1\n\u22a2 (fun n => Nat.casesOn n (some none) fun m => Option.map some (decode m))\n ((fun o => Option.casesOn o zero fun a => succ (encode a)) none) =\n some none\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nh : Encodable \u03b1\nval\u271d : \u03b1\n\u22a2 (fun n => Nat.casesOn n (some none) fun m => Option.map some (decode m))\n ((fun o => Option.casesOn o zero fun a => succ (encode a)) (some val\u271d)) =\n some (some val\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nh : Encodable \u03b1\nval\u271d : \u03b1\n\u22a2 Option.map some (decode (encode val\u271d)) = some (some val\u271d)\n[PROOFSTEP]\nsimp [encodek, Nat.succ_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2208 decode\u2082 \u03b1 n \u2194 a \u2208 decode n \u2227 encode a = n\n[PROOFSTEP]\nsimp [decode\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\nn : \u2115\na : \u03b1\n\u22a2 (\u2203 a_1, decode n = some a_1 \u2227 a_1 = a \u2227 encode a_1 = n) \u2194 decode n = some a \u2227 encode a = n\n[PROOFSTEP]\nexact \u27e8fun \u27e8_, h\u2081, rfl, h\u2082\u27e9 => \u27e8h\u2081, h\u2082\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8_, h\u2081, rfl, h\u2082\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\na : \u03b1\n\u22a2 decode\u2082 \u03b1 (encode a) = some a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 decode\u2082 \u03b1 (encode a) \u2194 a\u271d \u2208 some a\n[PROOFSTEP]\nsimp [mem_decode\u2082, eq_comm, decode\u2082_eq_some]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Encodable \u03b1\nn : \u2115\n\u22a2 decode\u2082 \u03b1 n \u2260 none \u2194 n \u2208 Set.range encode\n[PROOFSTEP]\nsimp_rw [Set.range, Set.mem_setOf_eq, Ne.def, Option.eq_none_iff_forall_not_mem, Encodable.mem_decode\u2082, not_forall,\n not_not]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx : \u2115\nh : isSome (decode\u2082 \u03b1 x) = true\n\u22a2 encode (Option.get (decode\u2082 \u03b1 x) h) = x\n[PROOFSTEP]\nrw [\u2190 decode\u2082_is_partial_inv (Option.get _ h), Option.some_get]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx : \u2115\nx\u271d : (fun x => x \u2208 Set.range encode) x\nn : \u03b1\nhn : encode n = x\n\u22a2 isSome (decode\u2082 \u03b1 x) = true\n[PROOFSTEP]\nrw [\u2190 hn, encodek\u2082]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx : \u2115\nx\u271d : (fun x => x \u2208 Set.range encode) x\nn : \u03b1\nhn : encode n = x\n\u22a2 isSome (some n) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nn : \u2191(Set.range encode)\n\u22a2 isSome (decode\u2082 \u03b1 \u2191n) = true\n[PROOFSTEP]\ncases' n.2 with x hx\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nn : \u2191(Set.range encode)\nx : \u03b1\nhx : encode x = \u2191n\n\u22a2 isSome (decode\u2082 \u03b1 \u2191n) = true\n[PROOFSTEP]\nrw [\u2190 hx, encodek\u2082]\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nn : \u2191(Set.range encode)\nx : \u03b1\nhx : encode x = \u2191n\n\u22a2 isSome (some x) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\na : \u03b1\n\u22a2 (fun n => Option.get (decode\u2082 \u03b1 \u2191n) (_ : isSome (decode\u2082 \u03b1 \u2191n) = true))\n ((fun a => { val := encode a, property := (_ : encode a \u2208 Set.range encode) }) a) =\n a\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\na : \u03b1\n\u22a2 Option.get (decode\u2082 \u03b1 (encode a)) (_ : isSome (decode\u2082 \u03b1 (encode a)) = true) = a\n[PROOFSTEP]\nrw [\u2190 Option.some_inj, Option.some_get, encodek\u2082]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n\u22a2 (fun a => { val := encode a, property := (_ : encode a \u2208 Set.range encode) })\n ((fun n => Option.get (decode\u2082 \u03b1 \u2191n) (_ : isSome (decode\u2082 \u03b1 \u2191n) = true))\n { val := n, property := (_ : \u2203 y, encode y = n) }) =\n { val := n, property := (_ : \u2203 y, encode y = n) }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n\u22a2 \u2191((fun a => { val := encode a, property := (_ : encode a \u2208 Set.range encode) })\n ((fun n => Option.get (decode\u2082 \u03b1 \u2191n) (_ : isSome (decode\u2082 \u03b1 \u2191n) = true))\n { val := n, property := (_ : \u2203 y, encode y = n) })) =\n \u2191{ val := n, property := (_ : \u2203 y, encode y = n) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n\u22a2 encode (Option.get (decode\u2082 \u03b1 n) (_ : isSome (decode\u2082 \u03b1 n) = true)) = n\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 hx]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n| encode (Option.get (decode\u2082 \u03b1 n) (_ : isSome (decode\u2082 \u03b1 n) = true)) = n\n[PROOFSTEP]\n rhs\n rw [\u2190 hx]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n| encode (Option.get (decode\u2082 \u03b1 n) (_ : isSome (decode\u2082 \u03b1 n) = true)) = n\n[PROOFSTEP]\n rhs\n rw [\u2190 hx]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n| encode (Option.get (decode\u2082 \u03b1 n) (_ : isSome (decode\u2082 \u03b1 n) = true)) = n\n[PROOFSTEP]\nrhs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n| n\n[PROOFSTEP]\nrw [\u2190 hx]\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d : Encodable \u03b1\nx\u271d : \u2191(Set.range encode)\nn : \u2115\nx : \u03b1\nhx : encode x = n\n\u22a2 encode (Option.get (decode\u2082 \u03b1 n) (_ : isSome (decode\u2082 \u03b1 n) = true)) = encode x\n[PROOFSTEP]\nrw [encode_injective.eq_iff, \u2190 Option.some_inj, Option.some_get, \u2190 hx, encodek\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ns : \u03b1 \u2295 \u03b2\n\u22a2 decodeSum (encodeSum s) = some s\n[PROOFSTEP]\ncases s\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\nval\u271d : \u03b1\n\u22a2 decodeSum (encodeSum (Sum.inl val\u271d)) = some (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp [encodeSum, div2_val, decodeSum, encodek]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\nval\u271d : \u03b2\n\u22a2 decodeSum (encodeSum (Sum.inr val\u271d)) = some (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp [encodeSum, div2_val, decodeSum, encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\n\u22a2 decode n = none\n[PROOFSTEP]\nsuffices decodeSum n = none by\n change (decodeSum n).bind _ = none\n rw [this]\n rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : decodeSum n = none\n\u22a2 decode n = none\n[PROOFSTEP]\nchange (decodeSum n).bind _ = none\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : decodeSum n = none\n\u22a2 Option.bind (decodeSum n) (some \u2218 \u2191Equiv.boolEquivPUnitSumPUnit.symm) = none\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : decodeSum n = none\n\u22a2 Option.bind none (some \u2218 \u2191Equiv.boolEquivPUnitSumPUnit.symm) = none\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\n\u22a2 decodeSum n = none\n[PROOFSTEP]\nhave : 1 \u2264 n / 2 := by\n rw [Nat.le_div_iff_mul_le]\n exacts [h, by decide]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\n\u22a2 1 \u2264 n / 2\n[PROOFSTEP]\nrw [Nat.le_div_iff_mul_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\n\u22a2 1 * 2 \u2264 n\n\u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 h : 2 \u2264 n \u22a2 0 < 2\n[PROOFSTEP]\nexacts [h, by decide]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : 1 \u2264 n / 2\n\u22a2 decodeSum n = none\n[PROOFSTEP]\ncases' exists_eq_succ_of_ne_zero (_root_.ne_of_gt this) with m e\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : 1 \u2264 n / 2\nm : \u2115\ne : n / 2 = succ m\n\u22a2 decodeSum n = none\n[PROOFSTEP]\nsimp [decodeSum, div2_val]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : 1 \u2264 n / 2\nm : \u2115\ne : n / 2 = succ m\n\u22a2 (match (bodd n, n / 2) with\n | (false, m) => Option.map Sum.inl (decode m)\n | (fst, m) => Option.map Sum.inr (decode m)) =\n none\n[PROOFSTEP]\ncases bodd n\n[GOAL]\ncase intro.false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : 1 \u2264 n / 2\nm : \u2115\ne : n / 2 = succ m\n\u22a2 (match (false, n / 2) with\n | (false, m) => Option.map Sum.inl (decode m)\n | (fst, m) => Option.map Sum.inr (decode m)) =\n none\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase intro.true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nh : 2 \u2264 n\nthis : 1 \u2264 n / 2\nm : \u2115\ne : n / 2 = succ m\n\u22a2 (match (true, n / 2) with\n | (false, m) => Option.map Sum.inl (decode m)\n | (fst, m) => Option.map Sum.inr (decode m)) =\n none\n[PROOFSTEP]\nsimp [e]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : \u03b1 \u2192 Type u_3\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b3 a)\nx\u271d : Sigma \u03b3\na : \u03b1\nb : \u03b3 a\n\u22a2 decodeSigma (encodeSigma { fst := a, snd := b }) = some { fst := a, snd := b }\n[PROOFSTEP]\nsimp [encodeSigma, decodeSigma, unpair_pair, encodek]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\n\u22a2 decode n = Option.bind (decode (unpair n).fst) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)\n[PROOFSTEP]\nsimp only [decode_ofEquiv, Equiv.symm_symm, decode_sigma_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2))\n (Option.bind (decode (unpair n).fst) fun a => Option.map (Sigma.mk a) (decode (unpair n).snd)) =\n Option.bind (decode (unpair n).fst) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)\n[PROOFSTEP]\ncases (decode n.unpair.1 : Option \u03b1)\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2)) (Option.bind none fun a => Option.map (Sigma.mk a) (decode (unpair n).snd)) =\n Option.bind none fun a => Option.map (Prod.mk a) (decode (unpair n).snd)\n[PROOFSTEP]\ncases (decode n.unpair.2 : Option \u03b2)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\nval\u271d : \u03b1\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2))\n (Option.bind (some val\u271d) fun a => Option.map (Sigma.mk a) (decode (unpair n).snd)) =\n Option.bind (some val\u271d) fun a => Option.map (Prod.mk a) (decode (unpair n).snd)\n[PROOFSTEP]\ncases (decode n.unpair.2 : Option \u03b2)\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2)) (Option.bind none fun a => Option.map (Sigma.mk a) none) =\n Option.bind none fun a => Option.map (Prod.mk a) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\nval\u271d : \u03b2\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2)) (Option.bind none fun a => Option.map (Sigma.mk a) (some val\u271d)) =\n Option.bind none fun a => Option.map (Prod.mk a) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\nval\u271d : \u03b1\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2)) (Option.bind (some val\u271d) fun a => Option.map (Sigma.mk a) none) =\n Option.bind (some val\u271d) fun a => Option.map (Prod.mk a) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Encodable \u03b2\ni : Encodable \u03b1\nn : \u2115\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map (\u2191(Equiv.sigmaEquivProd \u03b1 \u03b2)) (Option.bind (some val\u271d\u00b9) fun a => Option.map (Sigma.mk a) (some val\u271d)) =\n Option.bind (some val\u271d\u00b9) fun a => Option.map (Prod.mk a) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nP : \u03b1 \u2192 Prop\nencA : Encodable \u03b1\ndecP : DecidablePred P\nx\u271d : { a // P a }\nv : \u03b1\nh : P v\n\u22a2 decodeSubtype (encodeSubtype { val := v, property := h }) = some { val := v, property := h }\n[PROOFSTEP]\nsimp [encodeSubtype, decodeSubtype, encodek, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nP : \u03b1 \u2192 Prop\nencA : Encodable \u03b1\ndecP : DecidablePred P\na : Subtype P\n\u22a2 encode a = encode \u2191a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nP : \u03b1 \u2192 Prop\nencA : Encodable \u03b1\ndecP : DecidablePred P\nval\u271d : \u03b1\nproperty\u271d : P val\u271d\n\u22a2 encode { val := val\u271d, property := property\u271d } = encode \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 Countable \u2115+\n[PROOFSTEP]\ndelta PNat\n[GOAL]\n\u22a2 Countable { n // 0 < n }\n[PROOFSTEP]\ninfer_instance\n -- short-circuit instance search\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 DecidableEq (ULower \u03b1)\n[PROOFSTEP]\ndelta ULower\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 DecidableEq \u2191(Set.range Encodable.encode)\n[PROOFSTEP]\nexact Encodable.decidableEqOfEncodable _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 Encodable (ULower \u03b1)\n[PROOFSTEP]\ndelta ULower\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\n\u22a2 Encodable \u2191(Set.range Encodable.encode)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Encodable \u03b1\na : \u03b1\n\u22a2 up (down a) = a\n[PROOFSTEP]\nsimp [up, down, Equiv.left_inv _ _, Equiv.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\nn : Option \u03b1\n\u22a2 Decidable (Encodable.good p n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase none\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\n\u22a2 Decidable (Encodable.good p none)\n[PROOFSTEP]\nunfold good\n[GOAL]\ncase some\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\nval\u271d : \u03b1\n\u22a2 Decidable (Encodable.good p (some val\u271d))\n[PROOFSTEP]\nunfold good\n[GOAL]\ncase none\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\n\u22a2 Decidable\n (match none with\n | some a => p a\n | none => False)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\nval\u271d : \u03b1\n\u22a2 Decidable\n (match some val\u271d with\n | some a => p a\n | none => False)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\n\u22a2 Decidable False\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase some\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\nval\u271d : \u03b1\n\u22a2 Decidable (p val\u271d)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : DecidablePred p\nh : \u2203 x, p x\nw : \u03b1\npw : p w\n\u22a2 Encodable.good p (decode (encode w))\n[PROOFSTEP]\nsimp [good, encodek, pw]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\nn : \u2115\n\u22a2 r (f (Directed.sequence f hf n)) (f (Directed.sequence f hf (n + 1)))\n[PROOFSTEP]\ndsimp [Directed.sequence]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\nn : \u2115\n\u22a2 r (f (Directed.sequence f hf n))\n (f\n (match decode n with\n | none =>\n Classical.choose (_ : \u2203 z, r (f (Directed.sequence f hf n)) (f z) \u2227 r (f (Directed.sequence f hf n)) (f z))\n | some a => Classical.choose (_ : \u2203 z, r (f (Directed.sequence f hf n)) (f z) \u2227 r (f a) (f z))))\n[PROOFSTEP]\ngeneralize hf.sequence f n = p\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\nn : \u2115\np : \u03b1\n\u22a2 r (f p)\n (f\n (match decode n with\n | none => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f p) (f z))\n | some a => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f a) (f z))))\n[PROOFSTEP]\ncases' h : (decode n : Option \u03b1) with a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\nn : \u2115\np : \u03b1\nh : decode n = none\n\u22a2 r (f p)\n (f\n (match none with\n | none => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f p) (f z))\n | some a => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f a) (f z))))\n[PROOFSTEP]\nexact (Classical.choose_spec (hf p p)).1\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\nn : \u2115\np a : \u03b1\nh : decode n = some a\n\u22a2 r (f p)\n (f\n (match some a with\n | none => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f p) (f z))\n | some a => Classical.choose (_ : \u2203 z, r (f p) (f z) \u2227 r (f a) (f z))))\n[PROOFSTEP]\nexact (Classical.choose_spec (hf p a)).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\na : \u03b1\n\u22a2 r (f a) (f (Directed.sequence f hf (encode a + 1)))\n[PROOFSTEP]\nsimp only [Directed.sequence, add_eq, add_zero, encodek, and_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b1\ninst\u271d : Inhabited \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Directed r f\na : \u03b1\n\u22a2 r (f a) (f (Classical.choose (_ : \u2203 x, (fun x => r (f (Directed.sequence f hf (encode a))) (f x) \u2227 r (f a) (f x)) x)))\n[PROOFSTEP]\nexact (Classical.choose_spec (hf _ a)).2\n[GOAL]\n\u03b1 : Type u_1\ns : Setoid \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2248 x_1\ninst\u271d : Encodable \u03b1\n\u22a2 \u2200 (a : Quotient s), (fun n => Quotient.mk'' <$> decode n) ((fun q => encode (rep q)) a) = some a\n[PROOFSTEP]\nrintro \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ns : Setoid \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2248 x_1\ninst\u271d : Encodable \u03b1\na\u271d : Quotient s\nl : \u03b1\n\u22a2 (fun n => Quotient.mk'' <$> decode n) ((fun q => encode (rep q)) (Quot.mk Setoid.r l)) = some (Quot.mk Setoid.r l)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ns : Setoid \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2248 x_1\ninst\u271d : Encodable \u03b1\na\u271d : Quotient s\nl : \u03b1\n\u22a2 Option.map Quotient.mk'' (decode (encode (rep (Quot.mk Setoid.r l)))) = some (Quot.mk Setoid.r l)\n[PROOFSTEP]\nrw [encodek]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ns : Setoid \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2248 x_1\ninst\u271d : Encodable \u03b1\na\u271d : Quotient s\nl : \u03b1\n\u22a2 Option.map Quotient.mk'' (some (rep (Quot.mk Setoid.r l))) = some (Quot.mk Setoid.r l)\n[PROOFSTEP]\nexact congr_arg some \u27e6l\u27e7.rep_spec\n", "meta": {"mathlib_filename": "Mathlib.Logic.Encodable.Basic", "llama_tokens": 8932, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.29960684905953566}} {"text": "[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\n\u22a2 (x \u25c3 y) \u25c3 x = x \u25c3 y\n[PROOFSTEP]\nhave h : (x \u25c3 y) \u25c3 x = (x \u25c3 y) \u25c3 (x \u25c3 1) := by rw [act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\n\u22a2 (x \u25c3 y) \u25c3 x = (x \u25c3 y) \u25c3 x \u25c3 1\n[PROOFSTEP]\nrw [act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\nh : (x \u25c3 y) \u25c3 x = (x \u25c3 y) \u25c3 x \u25c3 1\n\u22a2 (x \u25c3 y) \u25c3 x = x \u25c3 y\n[PROOFSTEP]\nrw [h, \u2190 Shelf.self_distrib, act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx : S\n\u22a2 x \u25c3 x = x\n[PROOFSTEP]\nrw [\u2190 act_one x, \u2190 Shelf.self_distrib, act_one, act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\n\u22a2 x \u25c3 x \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nhave h : x \u25c3 (x \u25c3 y) = (x \u25c3 1) \u25c3 (x \u25c3 y) := by rw [act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\n\u22a2 x \u25c3 x \u25c3 y = (x \u25c3 1) \u25c3 x \u25c3 y\n[PROOFSTEP]\nrw [act_one]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y : S\nh : x \u25c3 x \u25c3 y = (x \u25c3 1) \u25c3 x \u25c3 y\n\u22a2 x \u25c3 x \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nrw [h, \u2190 Shelf.self_distrib, one_act]\n[GOAL]\nS : Type u_1\ninst\u271d : UnitalShelf S\nx y z : S\n\u22a2 (x \u25c3 y) \u25c3 z = x \u25c3 y \u25c3 z\n[PROOFSTEP]\nrw [self_distrib, self_distrib, act_act_self_eq, act_self_act_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 x \u25c3 y = x \u25c3 y' \u2194 y = y'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 x \u25c3 y = x \u25c3 y' \u2192 y = y'\ncase mpr R : Type u_1 inst\u271d : Rack R x y y' : R \u22a2 y = y' \u2192 x \u25c3 y = x \u25c3 y'\n[PROOFSTEP]\napply (act' x).injective\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 y = y' \u2192 x \u25c3 y = x \u25c3 y'\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y' \u2194 y = y'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y' \u2192 y = y'\ncase mpr R : Type u_1 inst\u271d : Rack R x y y' : R \u22a2 y = y' \u2192 x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y'\n[PROOFSTEP]\napply (act' x).symm.injective\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx y y' : R\n\u22a2 y = y' \u2192 x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y'\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = (x \u25c3\u207b\u00b9 y) \u25c3\u207b\u00b9 x \u25c3\u207b\u00b9 z\n[PROOFSTEP]\nrw [\u2190 left_cancel (x \u25c3\u207b\u00b9 y), right_inv, \u2190 left_cancel x, right_inv, self_distrib]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 (x \u25c3 x \u25c3\u207b\u00b9 y) \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z\n[PROOFSTEP]\nrepeat' rw [right_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 (x \u25c3 x \u25c3\u207b\u00b9 y) \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z\n[PROOFSTEP]\nrw [right_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 y \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z\n[PROOFSTEP]\nrw [right_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 y \u25c3 y \u25c3\u207b\u00b9 z = z\n[PROOFSTEP]\nrw [right_inv]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx y : R\n\u22a2 act' (x \u25c3 y) = act' x * act' y * (act' x)\u207b\u00b9\n[PROOFSTEP]\nrw [eq_mul_inv_iff_mul_eq]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx y : R\n\u22a2 act' (x \u25c3 y) * act' x = act' x * act' y\n[PROOFSTEP]\next z\n[GOAL]\ncase H\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx y z : R\n\u22a2 \u2191(act' (x \u25c3 y) * act' x) z = \u2191(act' x * act' y) z\n[PROOFSTEP]\napply self_distrib.symm\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 \u2200 {x y z : R\u1d50\u1d52\u1d56},\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) y z) =\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x y)\n ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x z)\n[PROOFSTEP]\nintro x y z\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y z : R\u1d50\u1d52\u1d56\n\u22a2 (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) y z) =\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x y)\n ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) x z)\n[PROOFSTEP]\ninduction x using MulOpposite.rec'\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : Rack R\ny z : R\u1d50\u1d52\u1d56\nX\u271d : R\n\u22a2 (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) y z) =\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d) y)\n ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d) z)\n[PROOFSTEP]\ninduction y using MulOpposite.rec'\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d : Rack R\nz : R\u1d50\u1d52\u1d56\nX\u271d\u00b9 X\u271d : R\n\u22a2 (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b9) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d) z) =\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b9) (op X\u271d))\n ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b9) z)\n[PROOFSTEP]\ninduction z using MulOpposite.rec'\n[GOAL]\ncase h.h.h\nR : Type u_1\ninst\u271d : Rack R\nX\u271d\u00b2 X\u271d\u00b9 X\u271d : R\n\u22a2 (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b2) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b9) (op X\u271d)) =\n (fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b2) (op X\u271d\u00b9))\n ((fun x y => op (unop x \u25c3\u207b\u00b9 unop y)) (op X\u271d\u00b2) (op X\u271d))\n[PROOFSTEP]\nsimp only [op_inj, unop_op, op_unop]\n[GOAL]\ncase h.h.h\nR : Type u_1\ninst\u271d : Rack R\nX\u271d\u00b2 X\u271d\u00b9 X\u271d : R\n\u22a2 X\u271d\u00b2 \u25c3\u207b\u00b9 X\u271d\u00b9 \u25c3\u207b\u00b9 X\u271d = (X\u271d\u00b2 \u25c3\u207b\u00b9 X\u271d\u00b9) \u25c3\u207b\u00b9 X\u271d\u00b2 \u25c3\u207b\u00b9 X\u271d\n[PROOFSTEP]\nrw [self_distrib_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (fun x y => op (unop x \u25c3 unop y)) (op x) (op x \u25c3 op y) = op y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 op x \u25c3 (fun x y => op (unop x \u25c3 unop y)) (op x) (op y) = op y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (x \u25c3 x) \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nrw [\u2190 right_inv x y, \u2190 self_distrib]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (x \u25c3\u207b\u00b9 x) \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nhave h := @self_act_act_eq _ _ (op x) (op y)\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : (op x \u25c3 op x) \u25c3 op y = op x \u25c3 op y\n\u22a2 (x \u25c3\u207b\u00b9 x) \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (x \u25c3 x) \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nrw [\u2190 left_cancel (x \u25c3 x)]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (x \u25c3 x) \u25c3 (x \u25c3 x) \u25c3\u207b\u00b9 y = (x \u25c3 x) \u25c3 x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nrw [right_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 y = (x \u25c3 x) \u25c3 x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nrw [self_act_act_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 y = x \u25c3 x \u25c3\u207b\u00b9 y\n[PROOFSTEP]\nrw [right_inv]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 (x \u25c3\u207b\u00b9 x) \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nhave h := @self_act_invAct_eq _ _ (op x) (op y)\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : (op x \u25c3 op x) \u25c3\u207b\u00b9 op y = op x \u25c3\u207b\u00b9 op y\n\u22a2 (x \u25c3\u207b\u00b9 x) \u25c3 y = x \u25c3 y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3 x = y \u25c3 y \u2194 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3 x = y \u25c3 y \u2192 x = y\ncase mpr R : Type u_1 inst\u271d : Rack R x y : R \u22a2 x = y \u2192 x \u25c3 x = y \u25c3 y\n[PROOFSTEP]\nswap\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x = y \u2192 x \u25c3 x = y \u25c3 y\ncase mp R : Type u_1 inst\u271d : Rack R x y : R \u22a2 x \u25c3 x = y \u25c3 y \u2192 x = y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : Rack R\nx : R\n\u22a2 x \u25c3 x = x \u25c3 x\ncase mp R : Type u_1 inst\u271d : Rack R x y : R \u22a2 x \u25c3 x = y \u25c3 y \u2192 x = y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3 x = y \u25c3 y \u2192 x = y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : x \u25c3 x = y \u25c3 y\n\u22a2 x = y\n[PROOFSTEP]\ntrans (x \u25c3 x) \u25c3\u207b\u00b9 x \u25c3 x\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : x \u25c3 x = y \u25c3 y\n\u22a2 x = (x \u25c3 x) \u25c3\u207b\u00b9 x \u25c3 x\nR : Type u_1 inst\u271d : Rack R x y : R h : x \u25c3 x = y \u25c3 y \u22a2 (x \u25c3 x) \u25c3\u207b\u00b9 x \u25c3 x = y\n[PROOFSTEP]\nrw [\u2190 left_cancel (x \u25c3 x), right_inv, self_act_act_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : x \u25c3 x = y \u25c3 y\n\u22a2 (x \u25c3 x) \u25c3\u207b\u00b9 x \u25c3 x = y\n[PROOFSTEP]\nrw [h, \u2190 left_cancel (y \u25c3 y), right_inv, self_act_act_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 x \u25c3\u207b\u00b9 x = y \u25c3\u207b\u00b9 y \u2194 x = y\n[PROOFSTEP]\nhave h := @self_act_eq_iff_eq _ _ (op x) (op y)\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\nh : op x \u25c3 op x = op y \u25c3 op y \u2194 op x = op y\n\u22a2 x \u25c3\u207b\u00b9 x = y \u25c3\u207b\u00b9 y \u2194 x = y\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx : R\n\u22a2 (fun x => x \u25c3\u207b\u00b9 x) ((fun x => x \u25c3 x) x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx : R\n\u22a2 (fun x => x \u25c3 x) ((fun x => x \u25c3\u207b\u00b9 x) x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nh : IsInvolutory R\nx y : R\n\u22a2 x \u25c3\u207b\u00b9 y = x \u25c3 y\n[PROOFSTEP]\nrw [\u2190 left_cancel x, right_inv, h x]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx y z : R\n\u22a2 x \u25c3 y \u25c3 z = (x \u25c3 y) \u25c3 z \u2194 x \u25c3 z = z\n[PROOFSTEP]\nrw [self_distrib]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Rack R\u271d\nR : Type u_2\ninst\u271d : Rack R\nx y z : R\n\u22a2 (x \u25c3 y) \u25c3 x \u25c3 z = (x \u25c3 y) \u25c3 z \u2194 x \u25c3 z = z\n[PROOFSTEP]\nrw [left_cancel]\n[GOAL]\nS\u2081 : Type u_1\nS\u2082 : Type u_2\nS\u2083 : Type u_3\ninst\u271d\u00b3 : Shelf S\u2081\ninst\u271d\u00b2 : Shelf S\u2082\ninst\u271d\u00b9 : Shelf S\u2083\nS : Type u_4\ninst\u271d : Shelf S\n\u22a2 \u2200 {x y : S}, (fun x => x) (x \u25c3 y) = (fun x => x) x \u25c3 (fun x => x) y\n[PROOFSTEP]\nsimp\n[GOAL]\nS\u2081 : Type u_1\nS\u2082 : Type u_2\nS\u2083 : Type u_3\ninst\u271d\u00b2 : Shelf S\u2081\ninst\u271d\u00b9 : Shelf S\u2082\ninst\u271d : Shelf S\u2083\ng : S\u2082 \u2192\u25c3 S\u2083\nf : S\u2081 \u2192\u25c3 S\u2082\n\u22a2 \u2200 {x y : S\u2081}, (g.toFun \u2218 f.toFun) (x \u25c3 y) = (g.toFun \u2218 f.toFun) x \u25c3 (g.toFun \u2218 f.toFun) y\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nx : Q\n\u22a2 x \u25c3\u207b\u00b9 x = x\n[PROOFSTEP]\nrw [\u2190 left_cancel x]\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nx : Q\n\u22a2 x \u25c3 x \u25c3\u207b\u00b9 x = x \u25c3 x\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\n\u22a2 \u2200 {x : Q\u1d50\u1d52\u1d56}, x \u25c3 x = x\n[PROOFSTEP]\nintro x\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nx : Q\u1d50\u1d52\u1d56\n\u22a2 x \u25c3 x = x\n[PROOFSTEP]\ninduction' x using MulOpposite.rec'\n[GOAL]\ncase h\nQ : Type u_1\ninst\u271d : Quandle Q\nX\u271d : Q\n\u22a2 op X\u271d \u25c3 op X\u271d = op X\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\n\u22a2 \u2200 {x y z : Conj G},\n (fun x => \u2191(\u2191MulAut.conj x)) x ((fun x => \u2191(\u2191MulAut.conj x)) y z) =\n (fun x => \u2191(\u2191MulAut.conj x)) ((fun x => \u2191(\u2191MulAut.conj x)) x y) ((fun x => \u2191(\u2191MulAut.conj x)) x z)\n[PROOFSTEP]\nintro x y z\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y z : Conj G\n\u22a2 (fun x => \u2191(\u2191MulAut.conj x)) x ((fun x => \u2191(\u2191MulAut.conj x)) y z) =\n (fun x => \u2191(\u2191MulAut.conj x)) ((fun x => \u2191(\u2191MulAut.conj x)) x y) ((fun x => \u2191(\u2191MulAut.conj x)) x z)\n[PROOFSTEP]\ndsimp only [MulAut.conj_apply]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y z : Conj G\n\u22a2 x * (y * z * y\u207b\u00b9) * x\u207b\u00b9 = x * y * x\u207b\u00b9 * (x * z * x\u207b\u00b9) * (x * y * x\u207b\u00b9)\u207b\u00b9\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 (fun x => \u2191(MulEquiv.symm (\u2191MulAut.conj x))) x (x \u25c3 y) = y\n[PROOFSTEP]\nsimp [act', mul_assoc]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 x \u25c3 (fun x => \u2191(MulEquiv.symm (\u2191MulAut.conj x))) x y = y\n[PROOFSTEP]\nsimp [act', mul_assoc]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\n\u22a2 \u2200 {x : Conj G}, x \u25c3 x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 x \u25c3 y = y \u2194 y \u25c3 x = x\n[PROOFSTEP]\ndsimp [Conj] at *\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 x * y * x\u207b\u00b9 = y \u2194 y * x * y\u207b\u00b9 = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 x * y * x\u207b\u00b9 = y \u2192 y * x * y\u207b\u00b9 = x\ncase mpr Q : Type u_1 inst\u271d\u00b9 : Quandle Q G : Type u_2 inst\u271d : Group G x y : Conj G \u22a2 y * x * y\u207b\u00b9 = x \u2192 x * y * x\u207b\u00b9 = y\n[PROOFSTEP]\nrepeat' intro h; conv_rhs => rw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\ncase mp\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 x * y * x\u207b\u00b9 = y \u2192 y * x * y\u207b\u00b9 = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : x * y * x\u207b\u00b9 = y\n\u22a2 y * x * y\u207b\u00b9 = x\n[PROOFSTEP]\nconv_rhs => rw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : x * y * x\u207b\u00b9 = y\n| x\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : x * y * x\u207b\u00b9 = y\n| x\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : x * y * x\u207b\u00b9 = y\n| x\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : x * y * x\u207b\u00b9 = y\n| y * x\u207b\u00b9\u207b\u00b9 * y\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\n\u22a2 y * x * y\u207b\u00b9 = x \u2192 x * y * x\u207b\u00b9 = y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : y * x * y\u207b\u00b9 = x\n\u22a2 x * y * x\u207b\u00b9 = y\n[PROOFSTEP]\nconv_rhs => rw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : y * x * y\u207b\u00b9 = x\n| y\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : y * x * y\u207b\u00b9 = x\n| y\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]; simp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : y * x * y\u207b\u00b9 = x\n| y\n[PROOFSTEP]\nrw [eq_mul_inv_of_mul_eq (eq_mul_inv_of_mul_eq h)]\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b9 : Quandle Q\nG : Type u_2\ninst\u271d : Group G\nx y : Conj G\nh : y * x * y\u207b\u00b9 = x\n| x * y\u207b\u00b9\u207b\u00b9 * x\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d\u00b2 : Quandle Q\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\n\u22a2 \u2200 {x y : Conj G}, \u2191f (x \u25c3 y) = \u2191f x \u25c3 \u2191f y\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\na : ZMod n\n\u22a2 Function.Involutive (dihedralAct n a)\n[PROOFSTEP]\nintro b\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\na b : ZMod n\n\u22a2 dihedralAct n a (dihedralAct n a b) = b\n[PROOFSTEP]\ndsimp only [dihedralAct]\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\na b : ZMod n\n\u22a2 2 * a - (2 * a - b) = b\n[PROOFSTEP]\nsimp\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\n\u22a2 \u2200 {x y z : Dihedral n}, dihedralAct n x (dihedralAct n y z) = dihedralAct n (dihedralAct n x y) (dihedralAct n x z)\n[PROOFSTEP]\nintro x y z\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\nx y z : Dihedral n\n\u22a2 dihedralAct n x (dihedralAct n y z) = dihedralAct n (dihedralAct n x y) (dihedralAct n x z)\n[PROOFSTEP]\nsimp only [dihedralAct]\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\nx y z : Dihedral n\n\u22a2 2 * x - (2 * y - z) = 2 * (2 * x - y) - (2 * x - z)\n[PROOFSTEP]\nring_nf\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\n\u22a2 \u2200 {x : Dihedral n}, x \u25c3 x = x\n[PROOFSTEP]\nintro x\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\nx : Dihedral n\n\u22a2 x \u25c3 x = x\n[PROOFSTEP]\nsimp only [dihedralAct]\n[GOAL]\nQ : Type u_1\ninst\u271d : Quandle Q\nn : \u2115\nx : Dihedral n\n\u22a2 2 * x - x = x\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 \u2200 {x y : R}, act' (x \u25c3 y) = act' x \u25c3 act' y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\nx y : R\n\u22a2 act' (x \u25c3 y) = act' x \u25c3 act' y\n[PROOFSTEP]\nexact ad_conj x y\n[GOAL]\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 Equivalence (PreEnvelGroupRel R)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refl\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 \u2200 (x : PreEnvelGroup R), PreEnvelGroupRel R x x\n[PROOFSTEP]\napply PreEnvelGroupRel.refl\n[GOAL]\ncase symm\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 \u2200 {x y : PreEnvelGroup R}, PreEnvelGroupRel R x y \u2192 PreEnvelGroupRel R y x\n[PROOFSTEP]\napply PreEnvelGroupRel.symm\n[GOAL]\ncase trans\nR : Type u_1\ninst\u271d : Rack R\n\u22a2 \u2200 {x y z : PreEnvelGroup R}, PreEnvelGroupRel R x y \u2192 PreEnvelGroupRel R y z \u2192 PreEnvelGroupRel R x z\n[PROOFSTEP]\napply PreEnvelGroupRel.trans\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na\u271d b\u271d a'\u271d b'\u271d : PreEnvelGroup R\nha : PreEnvelGroupRel' R a\u271d a'\u271d\nhb : PreEnvelGroupRel' R b\u271d b'\u271d\n\u22a2 mapAux f (PreEnvelGroup.mul a\u271d b\u271d) = mapAux f (PreEnvelGroup.mul a'\u271d b'\u271d)\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux, well_def f ha, well_def f hb]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na\u271d a'\u271d : PreEnvelGroup R\nha : PreEnvelGroupRel' R a\u271d a'\u271d\n\u22a2 mapAux f (PreEnvelGroup.inv a\u271d) = mapAux f (PreEnvelGroup.inv a'\u271d)\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux, well_def f ha]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na b c : PreEnvelGroup R\n\u22a2 mapAux f (PreEnvelGroup.mul (PreEnvelGroup.mul a b) c) = mapAux f (PreEnvelGroup.mul a (PreEnvelGroup.mul b c))\n[PROOFSTEP]\napply mul_assoc\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na : PreEnvelGroup R\n\u22a2 mapAux f (PreEnvelGroup.mul unit a) = mapAux f a\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na : PreEnvelGroup R\n\u22a2 mapAux f (PreEnvelGroup.mul a unit) = mapAux f a\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\na : PreEnvelGroup R\n\u22a2 mapAux f (PreEnvelGroup.mul (PreEnvelGroup.inv a) a) = mapAux f unit\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\nx y : R\n\u22a2 mapAux f (PreEnvelGroup.mul (PreEnvelGroup.mul (incl x) (incl y)) (PreEnvelGroup.inv (incl x))) =\n mapAux f (incl (x \u25c3 y))\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\n\u22a2 (fun x => Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b)) 1 = 1\n[PROOFSTEP]\nchange Quotient.liftOn \u27e6Rack.PreEnvelGroup.unit\u27e7 (toEnvelGroup.mapAux f) _ = 1\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\n\u22a2 Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1\n[PROOFSTEP]\nsimp only [Quotient.lift_mk, mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\nx\u271d y\u271d : EnvelGroup R\nx y : PreEnvelGroup R\n\u22a2 OneHom.toFun\n { toFun := fun x => Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (Quotient.mk (setoid R) x * Quotient.mk (setoid R) y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (Quotient.mk (setoid R) x) *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (Quotient.mk (setoid R) y)\n[PROOFSTEP]\nsimp only [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\nx\u271d y\u271d : EnvelGroup R\nx y : PreEnvelGroup R\n\u22a2 Quotient.liftOn (Quotient.mk (setoid R) x * Quotient.mk (setoid R) y) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n Quotient.liftOn (Quotient.mk (setoid R) x) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) *\n Quotient.liftOn (Quotient.mk (setoid R) y) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b)\n[PROOFSTEP]\nchange Quotient.liftOn \u27e6mul x y\u27e7 (toEnvelGroup.mapAux f) _ = _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\nx\u271d y\u271d : EnvelGroup R\nx y : PreEnvelGroup R\n\u22a2 Quotient.liftOn (Quotient.mk (setoid R) (PreEnvelGroup.mul x y)) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n Quotient.liftOn (Quotient.mk (setoid R) x) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) *\n Quotient.liftOn (Quotient.mk (setoid R) y) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b)\n[PROOFSTEP]\nsimp [toEnvelGroup.mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\n\u22a2 (fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R))\n ((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase toFun.h\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nf : R \u2192\u25c3 Quandle.Conj G\nx\u271d : R\n\u22a2 ShelfHom.toFun\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R))\n ((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n f))\n x\u271d =\n ShelfHom.toFun f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx : PreEnvelGroup R\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\n[PROOFSTEP]\ninduction' x with _ x y ih_x ih_y x ih_x\n[GOAL]\ncase unit\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx : EnvelGroup R\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) unit) =\n \u2191F (Quotient.mk (setoid R) unit)\n[PROOFSTEP]\nexact F.map_one.symm\n[GOAL]\ncase incl\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx : EnvelGroup R\nx\u271d : R\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) (incl x\u271d)) =\n \u2191F (Quotient.mk (setoid R) (incl x\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mul\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nih_y :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) y) =\n \u2191F (Quotient.mk (setoid R) y)\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) (PreEnvelGroup.mul x y)) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.mul x y))\n[PROOFSTEP]\nhave hm : \u27e6x.mul y\u27e7 = @Mul.mul (EnvelGroup R) _ \u27e6x\u27e7 \u27e6y\u27e7 := rfl\n[GOAL]\ncase mul\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nih_y :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) y) =\n \u2191F (Quotient.mk (setoid R) y)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.mul x y) = Mul.mul (Quotient.mk (setoid R) x) (Quotient.mk (setoid R) y)\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) (PreEnvelGroup.mul x y)) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.mul x y))\n[PROOFSTEP]\nsimp only [MonoidHom.coe_mk, OneHom.coe_mk, Quotient.lift_mk]\n[GOAL]\ncase mul\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nih_y :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) y) =\n \u2191F (Quotient.mk (setoid R) y)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.mul x y) = Mul.mul (Quotient.mk (setoid R) x) (Quotient.mk (setoid R) y)\n\u22a2 mapAux (ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) (PreEnvelGroup.mul x y) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.mul x y))\n[PROOFSTEP]\nsuffices \u2200 x y, F (Mul.mul x y) = F (x) * F (y)\n by\n simp_all only [MonoidHom.coe_mk, OneHom.coe_mk, Quotient.lift_mk, hm]\n rw [\u2190 ih_x, \u2190 ih_y, mapAux]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nih_y :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) y) =\n \u2191F (Quotient.mk (setoid R) y)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.mul x y) = Mul.mul (Quotient.mk (setoid R) x) (Quotient.mk (setoid R) y)\nthis : \u2200 (x y : EnvelGroup R), \u2191F (Mul.mul x y) = \u2191F x * \u2191F y\n\u22a2 mapAux (ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) (PreEnvelGroup.mul x y) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.mul x y))\n[PROOFSTEP]\nsimp_all only [MonoidHom.coe_mk, OneHom.coe_mk, Quotient.lift_mk, hm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x : mapAux (ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) x = \u2191F (Quotient.mk (setoid R) x)\nih_y : mapAux (ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) y = \u2191F (Quotient.mk (setoid R) y)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.mul x y) = Mul.mul (Quotient.mk (setoid R) x) (Quotient.mk (setoid R) y)\nthis : \u2200 (x y : EnvelGroup R), \u2191F (Mul.mul x y) = \u2191F x * \u2191F y\n\u22a2 mapAux (ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) (PreEnvelGroup.mul x y) =\n \u2191F (Quotient.mk (setoid R) x) * \u2191F (Quotient.mk (setoid R) y)\n[PROOFSTEP]\nrw [\u2190 ih_x, \u2190 ih_y, mapAux]\n[GOAL]\ncase mul\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx y : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nih_y :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) y) =\n \u2191F (Quotient.mk (setoid R) y)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.mul x y) = Mul.mul (Quotient.mk (setoid R) x) (Quotient.mk (setoid R) y)\n\u22a2 \u2200 (x y : EnvelGroup R), \u2191F (Mul.mul x y) = \u2191F x * \u2191F y\n[PROOFSTEP]\nexact F.map_mul\n[GOAL]\ncase inv\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) (PreEnvelGroup.inv x)) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.inv x))\n[PROOFSTEP]\nhave hm : \u27e6x.inv\u27e7 = @Inv.inv (EnvelGroup R) _ \u27e6x\u27e7 := rfl\n[GOAL]\ncase inv\nR : Type u_1\ninst\u271d\u00b9 : Rack R\nG : Type u_2\ninst\u271d : Group G\nF : EnvelGroup R \u2192* G\nx\u271d : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) x) =\n \u2191F (Quotient.mk (setoid R) x)\nhm : Quotient.mk (setoid R) (PreEnvelGroup.inv x) = (Quotient.mk (setoid R) x)\u207b\u00b9\n\u22a2 \u2191((fun f =>\n {\n toOneHom :=\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f) (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : EnvelGroup R),\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun x =>\n Quotient.liftOn x (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b),\n map_one' :=\n (_ :\n Quotient.liftOn (Quotient.mk (setoid R) unit) (mapAux f)\n (_ : \u2200 (a b : PreEnvelGroup R), a \u2248 b \u2192 mapAux f a = mapAux f b) =\n 1) }\n y) })\n ((fun F => ShelfHom.comp (Quandle.Conj.map F) (toEnvelGroup R)) F))\n (Quotient.mk (setoid R) (PreEnvelGroup.inv x)) =\n \u2191F (Quotient.mk (setoid R) (PreEnvelGroup.inv x))\n[PROOFSTEP]\nrw [hm, F.map_inv, MonoidHom.map_inv, ih_x]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Quandle", "llama_tokens": 29215, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318194686359, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.2996068490595356}} {"text": "[GOAL]\nC : Type ?u.30361\ninst\u271d : Category.{?u.30362, ?u.30361} C\nX Y : C\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 X \u226b f = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type ?u.30529\ninst\u271d : Category.{?u.30530, ?u.30529} C\nX Y : C\nf : X \u27f6 Y\n\u22a2 f \u226b \ud835\udfd9 Y = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\nw : f = g\nh : Y \u27f6 Z\n\u22a2 f \u226b h = g \u226b h\n[PROOFSTEP]\nrw [w]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : g = h\n\u22a2 f \u226b g = f \u226b h\n[PROOFSTEP]\nrw [w]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\nw : \u2200 {Z : C} (h : Y \u27f6 Z), f \u226b h = g \u226b h\n\u22a2 f = g\n[PROOFSTEP]\nconvert w (\ud835\udfd9 Y)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\nw : \u2200 {Z : C} (h : Y \u27f6 Z), f \u226b h = g \u226b h\n\u22a2 f = f \u226b \ud835\udfd9 Y\n[PROOFSTEP]\naesop\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : X \u27f6 Y\nw : \u2200 {Z : C} (h : Y \u27f6 Z), f \u226b h = g \u226b h\n\u22a2 g = g \u226b \ud835\udfd9 Y\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : Y \u27f6 Z\nw : \u2200 {X : C} (h : X \u27f6 Y), h \u226b f = h \u226b g\n\u22a2 f = g\n[PROOFSTEP]\nconvert w (\ud835\udfd9 Y)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : Y \u27f6 Z\nw : \u2200 {X : C} (h : X \u27f6 Y), h \u226b f = h \u226b g\n\u22a2 f = \ud835\udfd9 Y \u226b f\n[PROOFSTEP]\naesop\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf g : Y \u27f6 Z\nw : \u2200 {X : C} (h : X \u27f6 Y), h \u226b f = h \u226b g\n\u22a2 g = \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z\u271d : C\nf g : X \u27f6 Y\nw : (fun {Z} h => f \u226b h) = fun {Z} h => g \u226b h\nZ : C\nh : Y \u27f6 Z\n\u22a2 f \u226b h = g \u226b h\n[PROOFSTEP]\nconvert congr_fun (congr_fun w Z) h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y Z : C\nf g : Y \u27f6 Z\nw : (fun {X} h => h \u226b f) = fun {X} h => h \u226b g\nX : C\nh : X \u27f6 Y\n\u22a2 h \u226b f = h \u226b g\n[PROOFSTEP]\nconvert congr_fun (congr_fun w X) h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 X\nw : \u2200 {Y : C} (g : X \u27f6 Y), f \u226b g = g\n\u22a2 f = \ud835\udfd9 X\n[PROOFSTEP]\nconvert w (\ud835\udfd9 X)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 X\nw : \u2200 {Y : C} (g : X \u27f6 Y), f \u226b g = g\n\u22a2 f = f \u226b \ud835\udfd9 X\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 X\nw : \u2200 {Y : C} (g : Y \u27f6 X), g \u226b f = g\n\u22a2 f = \ud835\udfd9 X\n[PROOFSTEP]\nconvert w (\ud835\udfd9 X)\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 X\nw : \u2200 {Y : C} (g : Y \u27f6 X), g \u226b f = g\n\u22a2 f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nX Y Z : C\nf : X \u27f6 Y\ng g' : Y \u27f6 Z\n\u22a2 (f \u226b if P then g else g') = if P then f \u226b g else f \u226b g'\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nX Y Z : C\nf f' : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 (if P then f else f') \u226b g = if P then f \u226b g else f' \u226b g\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nX Y Z : C\nf : X \u27f6 Y\ng : P \u2192 (Y \u27f6 Z)\ng' : \u00acP \u2192 (Y \u27f6 Z)\n\u22a2 (f \u226b if h : P then g h else g' h) = if h : P then f \u226b g h else f \u226b g' h\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d : C\nP : Prop\ninst\u271d : Decidable P\nX Y Z : C\nf : P \u2192 (X \u27f6 Y)\nf' : \u00acP \u2192 (X \u27f6 Y)\ng : Y \u27f6 Z\n\u22a2 (if h : P then f h else f' h) \u226b g = if h : P then f h \u226b g else f' h \u226b g\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y Z X Z\u271d : C\ng h : X \u27f6 Z\u271d\nw : \ud835\udfd9 X \u226b g = \ud835\udfd9 X \u226b h\n\u22a2 g = h\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y Z X Z\u271d : C\ng h : Z\u271d \u27f6 X\nw : g \u226b \ud835\udfd9 X = h \u226b \ud835\udfd9 X\n\u22a2 g = h\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\nh : Y \u27f6 Y\n\u22a2 f \u226b h = f \u2194 h = \ud835\udfd9 Y\n[PROOFSTEP]\nconvert cancel_epi f\n[GOAL]\ncase h.e'_1.h.e'_3.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Epi f\nh : Y \u27f6 Y\n\u22a2 f = f \u226b \ud835\udfd9 Y\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 : Mono f\ng : X \u27f6 X\n\u22a2 g \u226b f = f \u2194 g = \ud835\udfd9 X\n[PROOFSTEP]\nconvert cancel_mono f\n[GOAL]\ncase h.e'_1.h.e'_3.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : X \u27f6 X\n\u22a2 f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\ng : Y \u27f6 Z\ninst\u271d : Epi g\n\u22a2 Epi (f \u226b g)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\ng : Y \u27f6 Z\ninst\u271d : Epi g\n\u22a2 \u2200 {Z_1 : C} (g_1 h : Z \u27f6 Z_1), (f \u226b g) \u226b g_1 = (f \u226b g) \u226b h \u2192 g_1 = h\n[PROOFSTEP]\nintro Z a b w\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi g\nZ : C\na b : Z\u271d \u27f6 Z\nw : (f \u226b g) \u226b a = (f \u226b g) \u226b b\n\u22a2 a = b\n[PROOFSTEP]\napply (cancel_epi g).1\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi g\nZ : C\na b : Z\u271d \u27f6 Z\nw : (f \u226b g) \u226b a = (f \u226b g) \u226b b\n\u22a2 g \u226b a = g \u226b b\n[PROOFSTEP]\napply (cancel_epi f).1\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Epi f\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi g\nZ : C\na b : Z\u271d \u27f6 Z\nw : (f \u226b g) \u226b a = (f \u226b g) \u226b b\n\u22a2 f \u226b g \u226b a = f \u226b g \u226b b\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ng : Y \u27f6 Z\ninst\u271d : Mono g\n\u22a2 Mono (f \u226b g)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ng : Y \u27f6 Z\ninst\u271d : Mono g\n\u22a2 \u2200 {Z_1 : C} (g_1 h : Z_1 \u27f6 X), g_1 \u226b f \u226b g = h \u226b f \u226b g \u2192 g_1 = h\n[PROOFSTEP]\nintro Z a b w\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono g\nZ : C\na b : Z \u27f6 X\nw : a \u226b f \u226b g = b \u226b f \u226b g\n\u22a2 a = b\n[PROOFSTEP]\napply (cancel_mono f).1\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono g\nZ : C\na b : Z \u27f6 X\nw : a \u226b f \u226b g = b \u226b f \u226b g\n\u22a2 a \u226b f = b \u226b f\n[PROOFSTEP]\napply (cancel_mono g).1\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : Mono f\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono g\nZ : C\na b : Z \u27f6 X\nw : a \u226b f \u226b g = b \u226b f \u226b g\n\u22a2 (a \u226b f) \u226b g = (b \u226b f) \u226b g\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Mono (f \u226b g)\n\u22a2 Mono f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Mono (f \u226b g)\n\u22a2 \u2200 {Z : C} (g h : Z \u27f6 X), g \u226b f = h \u226b f \u2192 g = h\n[PROOFSTEP]\nintro Z a b w\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono (f \u226b g)\nZ : C\na b : Z \u27f6 X\nw : a \u226b f = b \u226b f\n\u22a2 a = b\n[PROOFSTEP]\nreplace w := congr_arg (fun k => k \u226b g) w\n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono (f \u226b g)\nZ : C\na b : Z \u27f6 X\nw : (fun k => k \u226b g) (a \u226b f) = (fun k => k \u226b g) (b \u226b f)\n\u22a2 a = b\n[PROOFSTEP]\ndsimp at w \n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono (f \u226b g)\nZ : C\na b : Z \u27f6 X\nw : (a \u226b f) \u226b g = (b \u226b f) \u226b g\n\u22a2 a = b\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc] at w \n[GOAL]\ncase right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Mono (f \u226b g)\nZ : C\na b : Z \u27f6 X\nw : a \u226b f \u226b g = b \u226b f \u226b g\n\u22a2 a = b\n[PROOFSTEP]\nexact (cancel_mono _).1 w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : X \u27f6 Z\ninst\u271d : Mono h\nw : f \u226b g = h\n\u22a2 Mono f\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Mono (f \u226b g)\n\u22a2 Mono f\n[PROOFSTEP]\nexact mono_of_mono f g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Epi (f \u226b g)\n\u22a2 Epi g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Epi (f \u226b g)\n\u22a2 \u2200 {Z_1 : C} (g_1 h : Z \u27f6 Z_1), g \u226b g_1 = g \u226b h \u2192 g_1 = h\n[PROOFSTEP]\nintro Z a b w\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi (f \u226b g)\nZ : C\na b : Z\u271d \u27f6 Z\nw : g \u226b a = g \u226b b\n\u22a2 a = b\n[PROOFSTEP]\nreplace w := congr_arg (fun k => f \u226b k) w\n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi (f \u226b g)\nZ : C\na b : Z\u271d \u27f6 Z\nw : (fun k => f \u226b k) (g \u226b a) = (fun k => f \u226b k) (g \u226b b)\n\u22a2 a = b\n[PROOFSTEP]\ndsimp at w \n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi (f \u226b g)\nZ : C\na b : Z\u271d \u27f6 Z\nw : f \u226b g \u226b a = f \u226b g \u226b b\n\u22a2 a = b\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Category.assoc] at w \n[GOAL]\ncase left_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d\u00b9 X Y Z\u271d : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\u271d\ninst\u271d : Epi (f \u226b g)\nZ : C\na b : Z\u271d \u27f6 Z\nw : (f \u226b g) \u226b a = (f \u226b g) \u226b b\n\u22a2 a = b\n[PROOFSTEP]\nexact (cancel_epi _).1 w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : X \u27f6 Z\ninst\u271d : Epi h\nw : f \u226b g = h\n\u22a2 Epi g\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Epi (f \u226b g)\n\u22a2 Epi g\n[PROOFSTEP]\nexact epi_of_epi f g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\ninst\u271d : SmallCategory D\n\u22a2 LargeCategory (ULift D)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Category.Basic", "llama_tokens": 5861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6548947425132315, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.2993764324600862}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\nha : IsLeast {g | g \u2208 H \u2227 0 < g} a\n\u22a2 H = closure {a}\n[PROOFSTEP]\nobtain \u27e8\u27e8a_in, a_pos\u27e9, a_min\u27e9 := ha\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H = closure {a}\n[PROOFSTEP]\nrefine' le_antisymm _ (H.closure_le.mpr <| by simp [a_in])\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 {a} \u2286 \u2191H\n[PROOFSTEP]\nsimp [a_in]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H \u2264 closure {a}\n[PROOFSTEP]\nintro g g_in\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\n\u22a2 g \u2208 closure {a}\n[PROOFSTEP]\nobtain \u27e8k, \u27e8nonneg, lt\u27e9, _\u27e9 := existsUnique_zsmul_near_of_pos' a_pos g\n[GOAL]\ncase intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\n\u22a2 g \u2208 closure {a}\n[PROOFSTEP]\nhave h_zero : g - k \u2022 a = 0 := by\n by_contra h\n have h : a \u2264 g - k \u2022 a := by\n refine' a_min \u27e8_, _\u27e9\n \u00b7 exact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k)\n \u00b7 exact lt_of_le_of_ne nonneg (Ne.symm h)\n have h' : \u00aca \u2264 g - k \u2022 a := not_le.mpr lt\n contradiction\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\n\u22a2 g - k \u2022 a = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 False\n[PROOFSTEP]\nhave h : a \u2264 g - k \u2022 a := by\n refine' a_min \u27e8_, _\u27e9\n \u00b7 exact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k)\n \u00b7 exact lt_of_le_of_ne nonneg (Ne.symm h)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 a \u2264 g - k \u2022 a\n[PROOFSTEP]\nrefine' a_min \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 g - k \u2022 a \u2208 H\n[PROOFSTEP]\nexact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k)\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 0 < g - k \u2022 a\n[PROOFSTEP]\nexact lt_of_le_of_ne nonneg (Ne.symm h)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh\u271d : \u00acg - k \u2022 a = 0\nh : a \u2264 g - k \u2022 a\n\u22a2 False\n[PROOFSTEP]\nhave h' : \u00aca \u2264 g - k \u2022 a := not_le.mpr lt\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh\u271d : \u00acg - k \u2022 a = 0\nh : a \u2264 g - k \u2022 a\nh' : \u00aca \u2264 g - k \u2022 a\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh_zero : g - k \u2022 a = 0\n\u22a2 g \u2208 closure {a}\n[PROOFSTEP]\nsimp [sub_eq_zero.mp h_zero, AddSubgroup.mem_closure_singleton]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n[PROOFSTEP]\nhave hex : \u2200 g > 0, \u2203 n : \u2115, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a) := fun g hg =>\n by\n rcases existsUnique_add_zsmul_mem_Ico h\u2080 0 (g - a) with \u27e8m, \u27e8hm, hm'\u27e9, -\u27e9\n simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, \u2190 add_one_zsmul] at hm hm' \n lift m to \u2115\n \u00b7 rw [\u2190 Int.lt_add_one_iff, \u2190 zsmul_lt_zsmul_iff h\u2080, zero_zsmul]\n exact hg.trans_le hm\n \u00b7 simp only [\u2190 Nat.cast_succ, coe_nat_zsmul] at hm hm' \n exact \u27e8m, hm', hm\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n[PROOFSTEP]\nrcases existsUnique_add_zsmul_mem_Ico h\u2080 0 (g - a) with \u27e8m, \u27e8hm, hm'\u27e9, -\u27e9\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g - a \u2264 0 + m \u2022 a\nhm' : 0 + m \u2022 a < g - a + a\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n[PROOFSTEP]\nsimp only [zero_add, sub_le_iff_le_add, sub_add_cancel, \u2190 add_one_zsmul] at hm hm' \n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n[PROOFSTEP]\nlift m to \u2115\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 \u2264 m\n[PROOFSTEP]\nrw [\u2190 Int.lt_add_one_iff, \u2190 zsmul_lt_zsmul_iff h\u2080, zero_zsmul]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 < (m + 1) \u2022 a\n[PROOFSTEP]\nexact hg.trans_le hm\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 (\u2191m + 1) \u2022 a\nhm' : \u2191m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n[PROOFSTEP]\nsimp only [\u2190 Nat.cast_succ, coe_nat_zsmul] at hm hm' \n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 Nat.succ m \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n[PROOFSTEP]\nexact \u27e8m, hm', hm\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n[PROOFSTEP]\nhave : \u2203 n : \u2115, Set.Nonempty (H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n[PROOFSTEP]\nrcases(bot_or_exists_ne_zero H).resolve_left hbot with \u27e8g, hgH, hg\u2080\u27e9\n[GOAL]\ncase this.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n[PROOFSTEP]\nrcases hex |g| (abs_pos.2 hg\u2080) with \u27e8n, hn\u27e9\n[GOAL]\ncase this.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\nn : \u2115\nhn : |g| \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n[PROOFSTEP]\nexact \u27e8n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n[PROOFSTEP]\nclassical rcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n[PROOFSTEP]\nrcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n[PROOFSTEP]\nby_contra hxmin\n[GOAL]\ncase mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u00ac\u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n\u22a2 False\n[PROOFSTEP]\nsimp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall, not_le] at hxmin \n[GOAL]\ncase mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\n\u22a2 False\n[PROOFSTEP]\nrcases hxmin x \u27e8hxH, (nsmul_nonneg h\u2080.le _).trans_lt hnx\u27e9 with \u27e8y, \u27e8hyH, hy\u2080\u27e9, hxy\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\n\u22a2 False\n[PROOFSTEP]\nrcases hex y hy\u2080 with \u27e8m, hm\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\n\u22a2 False\n[PROOFSTEP]\ncases' lt_or_le m n with hmn hnm\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhmn : m < n\n\u22a2 False\n[PROOFSTEP]\nexact hmin m hmn \u27e8y, hyH, hm\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 False\n[PROOFSTEP]\nrefine disjoint_left.1 hd (sub_mem hxH hyH) \u27e8sub_pos.2 hxy, sub_lt_iff_lt_add'.2 ?_\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 x < y + a\n[PROOFSTEP]\ncalc\n x \u2264 (n + 1) \u2022 a := hxn\n _ \u2264 (m + 1) \u2022 a := (nsmul_le_nsmul h\u2080.le (add_le_add_right hnm _))\n _ = m \u2022 a + a := (succ_nsmul' _ _)\n _ < y + a := add_lt_add_right hm.1 _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\n\u22a2 \u2203 b, H = closure {b}\n[PROOFSTEP]\nrcases eq_or_ne H \u22a5 with rfl | hbot\n[GOAL]\ncase inl\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191\u22a5) (Ioo 0 a)\n\u22a2 \u2203 b, \u22a5 = closure {b}\n[PROOFSTEP]\nexact \u27e80, closure_singleton_zero.symm\u27e9\n[GOAL]\ncase inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhbot : H \u2260 \u22a5\n\u22a2 \u2203 b, H = closure {b}\n[PROOFSTEP]\nexact (exists_isLeast_pos hbot h\u2080 hd).imp fun _ => cyclic_of_min\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup \u2124\nthis : Ioo 0 1 = \u2205\n\u22a2 Disjoint (\u2191H) (Ioo 0 1)\n[PROOFSTEP]\nsimp [this]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Archimedean", "llama_tokens": 9053, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5, "lm_q1q2_score": 0.29821657313231276}} {"text": "[GOAL]\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\n\u22a2 Irreducible (factor f)\n[PROOFSTEP]\nrw [factor]\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\nf : K[X]\n\u22a2 Irreducible (if H : \u2203 g, Irreducible g \u2227 g \u2223 f then Classical.choose H else X)\n[PROOFSTEP]\nsplit_ifs with H\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nH : \u2203 g, Irreducible g \u2227 g \u2223 f\n\u22a2 Irreducible (Classical.choose H)\n[PROOFSTEP]\nexact (Classical.choose_spec H).1\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nH : \u00ac\u2203 g, Irreducible g \u2227 g \u2223 f\n\u22a2 Irreducible X\n[PROOFSTEP]\nexact irreducible_X\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\nf : K[X]\nhf1 : \u00acIsUnit f\n\u22a2 factor f \u2223 f\n[PROOFSTEP]\nby_cases hf2 : f = 0\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nhf1 : \u00acIsUnit f\nhf2 : f = 0\n\u22a2 factor f \u2223 f\n[PROOFSTEP]\nrw [hf2]\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nhf1 : \u00acIsUnit f\nhf2 : f = 0\n\u22a2 factor 0 \u2223 0\n[PROOFSTEP]\nexact dvd_zero _\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nhf1 : \u00acIsUnit f\nhf2 : \u00acf = 0\n\u22a2 factor f \u2223 f\n[PROOFSTEP]\nrw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\nhf1 : \u00acIsUnit f\nhf2 : \u00acf = 0\n\u22a2 Classical.choose (_ : \u2203 i, Irreducible i \u2227 i \u2223 f) \u2223 f\n[PROOFSTEP]\nexact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).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\nf : K[X]\nhf : natDegree f \u2260 0\n\u22a2 (X - \u2191C (AdjoinRoot.root (factor f))) * removeFactor f = map (AdjoinRoot.of (factor f)) f\n[PROOFSTEP]\nlet \u27e8g, hg\u27e9 := factor_dvd_of_natDegree_ne_zero hf\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\nf : K[X]\nhf : natDegree f \u2260 0\ng : K[X]\nhg : f = factor f * g\n\u22a2 (X - \u2191C (AdjoinRoot.root (factor f))) * removeFactor f = map (AdjoinRoot.of (factor f)) f\n[PROOFSTEP]\napply (mul_divByMonic_eq_iff_isRoot (R := AdjoinRoot f.factor) (a := AdjoinRoot.root f.factor)).mpr\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\nf : K[X]\nhf : natDegree f \u2260 0\ng : K[X]\nhg : f = factor f * g\n\u22a2 IsRoot (map (AdjoinRoot.of (factor f)) f) (AdjoinRoot.root (factor f))\n[PROOFSTEP]\nrw [IsRoot.def, eval_map, hg, eval\u2082_mul, \u2190 hg, AdjoinRoot.eval\u2082_root, zero_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\nf : K[X]\n\u22a2 natDegree (removeFactor f) = natDegree f - 1\n[PROOFSTEP]\nrw [removeFactor, natDegree_divByMonic (map (AdjoinRoot.of f.factor) f) (monic_X_sub_C _), natDegree_map,\n natDegree_X_sub_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\nf : K[X]\nn : \u2115\nhfn : natDegree f = n + 1\n\u22a2 natDegree (removeFactor f) = n\n[PROOFSTEP]\nrw [natDegree_removeFactor, hfn, n.add_sub_cancel]\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Splits (algebraMap K (SplittingFieldAux n f)) f)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhf : natDegree f = Nat.succ n\n\u22a2 Splits (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f\n[PROOFSTEP]\nrw [\u2190 splits_id_iff_splits, algebraMap_succ, \u2190 map_map, splits_id_iff_splits, \u2190\n X_sub_C_mul_removeFactor f fun h => by rw [h] at hf ; cases hf]\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Splits (algebraMap K (SplittingFieldAux n f)) f)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhf : natDegree f = Nat.succ n\nh : natDegree f = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hf \n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Splits (algebraMap K (SplittingFieldAux n f)) f)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhf : 0 = Nat.succ n\nh : natDegree f = 0\n\u22a2 False\n[PROOFSTEP]\ncases hf\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Splits (algebraMap K (SplittingFieldAux n f)) f)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhf : natDegree f = Nat.succ n\n\u22a2 Splits (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n ((X - \u2191C (AdjoinRoot.root (factor f))) * removeFactor f)\n[PROOFSTEP]\nexact splits_mul _ (splits_X_sub_C _) (ih _ (natDegree_removeFactor' hf))\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\n\u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f))) = \u22a4\n[PROOFSTEP]\nhave hndf : f.natDegree \u2260 0 := by intro h; rw [h] at hfn ; cases hfn\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\n\u22a2 natDegree f \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nh : natDegree f = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hfn \n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : 0 = Nat.succ n\nh : natDegree f = 0\n\u22a2 False\n[PROOFSTEP]\ncases hfn\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\n\u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f))) = \u22a4\n[PROOFSTEP]\nhave hfn0 : f \u2260 0 := by intro h; rw [h] at hndf ; exact hndf rfl\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\n\u22a2 f \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nh : f = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hndf \n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree 0 \u2260 0\nh : f = 0\n\u22a2 False\n[PROOFSTEP]\nexact hndf rfl\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\n\u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f))) = \u22a4\n[PROOFSTEP]\nhave hmf0 : map (algebraMap K (SplittingFieldAux n.succ f)) f \u2260 0 := map_ne_zero hfn0\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\nhmf0 : map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f \u2260 0\n\u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f))) = \u22a4\n[PROOFSTEP]\nrw [algebraMap_succ, \u2190 map_map, \u2190 X_sub_C_mul_removeFactor _ hndf, Polynomial.map_mul] at hmf0 \u22a2\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\nhmf0 :\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n (X - \u2191C (AdjoinRoot.root (factor f))) *\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260\n 0\n\u22a2 Algebra.adjoin K\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n (X - \u2191C (AdjoinRoot.root (factor f))) *\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f)))) =\n \u22a4\n[PROOFSTEP]\nrw [roots_mul hmf0, Polynomial.map_sub, map_X, map_C, roots_X_sub_C, Multiset.toFinset_add, Finset.coe_union,\n Multiset.toFinset_singleton, Finset.coe_singleton, Algebra.adjoin_union_eq_adjoin_adjoin, \u2190 Set.image_singleton,\n Algebra.adjoin_algebraMap K (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor), AdjoinRoot.adjoinRoot_eq_top,\n Algebra.map_top]\n /- Porting note: was `rw [IsScalarTower.adjoin_range_toAlgHom K (AdjoinRoot f.factor)\n (SplittingFieldAux n f.removeFactor)]` -/\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\nhmf0 :\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n (X - \u2191C (AdjoinRoot.root (factor f))) *\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260\n 0\n\u22a2 Subalgebra.restrictScalars K\n (Algebra.adjoin\n { x //\n x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) }\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) =\n \u22a4\n[PROOFSTEP]\nhave :=\n IsScalarTower.adjoin_range_toAlgHom K (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)\n (\u2191(f.removeFactor.map <| algebraMap (AdjoinRoot f.factor) <| SplittingFieldAux n f.removeFactor).roots.toFinset :\n Set (SplittingFieldAux n f.removeFactor))\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\nhmf0 :\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n (X - \u2191C (AdjoinRoot.root (factor f))) *\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260\n 0\nthis :\n Subalgebra.restrictScalars K\n (Algebra.adjoin\n { x //\n x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) }\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) =\n Subalgebra.restrictScalars K\n (Algebra.adjoin (AdjoinRoot (factor f))\n \u2191(Multiset.toFinset\n (roots (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f)))))\n\u22a2 Subalgebra.restrictScalars K\n (Algebra.adjoin\n { x //\n x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) }\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) =\n \u22a4\n[PROOFSTEP]\nrefine this.trans ?_\n[GOAL]\nF : Type u\nK\u271d\u00b9 : Type v\nL : Type w\ninst\u271d\u00b3 : Field K\u271d\u00b9\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Field F\nn\u271d : \u2115\nK\u271d : Type u\ninst\u271d : Field K\u271d\nn : \u2115\nih :\n (fun n =>\n \u2200 {K : Type u} [inst : Field K] (f : K[X]),\n natDegree f = n \u2192\n Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux n f)) f))) = \u22a4)\n n\nK : Type u\nx\u271d : Field K\nf : K[X]\nhfn : natDegree f = Nat.succ n\nhndf : natDegree f \u2260 0\nhfn0 : f \u2260 0\nhmf0 :\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f)))\n (X - \u2191C (AdjoinRoot.root (factor f))) *\n map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260\n 0\nthis :\n Subalgebra.restrictScalars K\n (Algebra.adjoin\n { x //\n x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) }\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) =\n Subalgebra.restrictScalars K\n (Algebra.adjoin (AdjoinRoot (factor f))\n \u2191(Multiset.toFinset\n (roots (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f)))))\n\u22a2 Subalgebra.restrictScalars K\n (Algebra.adjoin (AdjoinRoot (factor f))\n \u2191(Multiset.toFinset\n (roots\n (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) =\n \u22a4\n[PROOFSTEP]\nrw [ih _ (natDegree_removeFactor' hfn), Subalgebra.restrictScalars_top]\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\nf : K[X]\n\u22a2 Function.Surjective \u2191(ofMvPolynomial f)\n[PROOFSTEP]\nsuffices AlgHom.range (ofMvPolynomial f) = \u22a4 by rw [\u2190 Set.range_iff_surjective]; rwa [SetLike.ext'_iff] at this \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\nf : K[X]\nthis : AlgHom.range (ofMvPolynomial f) = \u22a4\n\u22a2 Function.Surjective \u2191(ofMvPolynomial f)\n[PROOFSTEP]\nrw [\u2190 Set.range_iff_surjective]\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\nf : K[X]\nthis : AlgHom.range (ofMvPolynomial f) = \u22a4\n\u22a2 Set.range \u2191(ofMvPolynomial f) = Set.univ\n[PROOFSTEP]\nrwa [SetLike.ext'_iff] at this \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\nf : K[X]\n\u22a2 AlgHom.range (ofMvPolynomial f) = \u22a4\n[PROOFSTEP]\nrw [ofMvPolynomial, \u2190 Algebra.adjoin_range_eq_range_aeval K, eq_top_iff, \u2190 adjoin_rootSet _ _ rfl]\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\nf : K[X]\n\u22a2 Algebra.adjoin K (rootSet f (SplittingFieldAux (natDegree f) f)) \u2264 Algebra.adjoin K (Set.range fun i => \u2191i)\n[PROOFSTEP]\napply Algebra.adjoin_le\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\nf : K[X]\n\u22a2 rootSet f (SplittingFieldAux (natDegree f) f) \u2286 \u2191(Algebra.adjoin K (Set.range fun i => \u2191i))\n[PROOFSTEP]\nintro \u03b1 h\u03b1\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\nf : K[X]\n\u03b1 : SplittingFieldAux (natDegree f) f\nh\u03b1 : \u03b1 \u2208 rootSet f (SplittingFieldAux (natDegree f) f)\n\u22a2 \u03b1 \u2208 \u2191(Algebra.adjoin K (Set.range fun i => \u2191i))\n[PROOFSTEP]\napply Algebra.subset_adjoin\n[GOAL]\ncase H.a\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\nf : K[X]\n\u03b1 : SplittingFieldAux (natDegree f) f\nh\u03b1 : \u03b1 \u2208 rootSet f (SplittingFieldAux (natDegree f) f)\n\u22a2 \u03b1 \u2208 Set.range fun i => \u2191i\n[PROOFSTEP]\nexact \u27e8\u27e8\u03b1, h\u03b1\u27e9, rfl\u27e9\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\napply_fun e\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\n\u22a2 \u2191e (a * a\u207b\u00b9) = \u2191e 1\n[PROOFSTEP]\nhave : e a \u2260 0 := fun w' => by\n apply w\n simp at w' \n exact w'\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nw' : \u2191e a = 0\n\u22a2 False\n[PROOFSTEP]\napply w\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nw' : \u2191e a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nsimp at w' \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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nw' : a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nexact w'\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nthis : \u2191e a \u2260 0\n\u22a2 \u2191e (a * a\u207b\u00b9) = \u2191e 1\n[PROOFSTEP]\nsimp only [map_mul, AlgEquiv.apply_symm_apply, ne_eq, AddEquivClass.map_eq_zero_iff, map_one]\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nthis : \u2191e a \u2260 0\n\u22a2 \u2191(algEquivSplittingFieldAux f) a * (\u2191(algEquivSplittingFieldAux f) a)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [mul_inv_cancel]\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : SplittingField f\nw : a \u2260 0\nthis : \u2191e a \u2260 0\n\u22a2 \u2191(algEquivSplittingFieldAux f) a \u2260 0\n[PROOFSTEP]\nassumption\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\n\u22a2 0\u207b\u00b9 = 0\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\napply_fun e\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191e \u2191(Rat.mk' a b) = \u2191e (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nchange e (algebraMap K _ _) = _\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191e (\u2191(algebraMap K (SplittingField f)) \u2191(Rat.mk' a b)) = \u2191e (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [map_ratCast, map_natCast, map_mul, map_intCast, AlgEquiv.commutes, AlgEquiv.apply_symm_apply]\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\napply Field.ratCast_mk\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u211a\nx : SplittingField f\np : MvPolynomial (\u2191(rootSet f (SplittingFieldAux (natDegree f) f))) K\n\u22a2 (fun x x_1 => x \u2022 x_1) a p =\n (fun x x_1 => x * x_1) (\u2191(algebraMap K (MvPolynomial (\u2191(rootSet f (SplittingFieldAux (natDegree f) f))) K)) \u2191a) p\n[PROOFSTEP]\next\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\nf : K[X]\ne : SplittingField f \u2243\u2090[K] SplittingFieldAux (natDegree f) f := algEquivSplittingFieldAux f\na : \u211a\nx : SplittingField f\np : MvPolynomial (\u2191(rootSet f (SplittingFieldAux (natDegree f) f))) K\nm\u271d : \u2191(rootSet f (SplittingFieldAux (natDegree f) f)) \u2192\u2080 \u2115\n\u22a2 MvPolynomial.coeff m\u271d ((fun x x_1 => x \u2022 x_1) a p) =\n MvPolynomial.coeff m\u271d\n ((fun x x_1 => x * x_1) (\u2191(algebraMap K (MvPolynomial (\u2191(rootSet f (SplittingFieldAux (natDegree f) f))) K)) \u2191a)\n p)\n[PROOFSTEP]\nsimp [MvPolynomial.algebraMap_eq, Rat.smul_def]\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\n\u22a2 L \u2243\u2090[K] SplittingField f\n[PROOFSTEP]\nrefine'\n AlgEquiv.ofBijective (lift L f <| splits (SplittingField f) f)\n \u27e8RingHom.injective (lift L f <| splits (SplittingField f) f).toRingHom, _\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\n\u22a2 Function.Surjective \u2191(lift L f (_ : Splits (algebraMap K (SplittingField f)) f))\n[PROOFSTEP]\nhaveI := finiteDimensional (SplittingField f) f\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\nthis : FiniteDimensional K (SplittingField f)\n\u22a2 Function.Surjective \u2191(lift L f (_ : Splits (algebraMap K (SplittingField f)) f))\n[PROOFSTEP]\nhaveI := finiteDimensional L f\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\nthis\u271d : FiniteDimensional K (SplittingField f)\nthis : FiniteDimensional K L\n\u22a2 Function.Surjective \u2191(lift L f (_ : Splits (algebraMap K (SplittingField f)) f))\n[PROOFSTEP]\nhave : FiniteDimensional.finrank K L = FiniteDimensional.finrank K (SplittingField f) :=\n le_antisymm\n (LinearMap.finrank_le_finrank_of_injective\n (show Function.Injective (lift L f <| splits (SplittingField f) f).toLinearMap from\n RingHom.injective (lift L f <| splits (SplittingField f) f : L \u2192+* f.SplittingField)))\n (LinearMap.finrank_le_finrank_of_injective\n (show Function.Injective (lift (SplittingField f) f <| splits L f).toLinearMap from\n RingHom.injective (lift (SplittingField f) f <| splits L f : f.SplittingField \u2192+* L)))\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\nthis\u271d\u00b9 : FiniteDimensional K (SplittingField f)\nthis\u271d : FiniteDimensional K L\nthis : FiniteDimensional.finrank K L = FiniteDimensional.finrank K (SplittingField f)\n\u22a2 Function.Surjective \u2191(lift L f (_ : Splits (algebraMap K (SplittingField f)) f))\n[PROOFSTEP]\nchange Function.Surjective (lift L f <| splits (SplittingField f) f).toLinearMap\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\nthis\u271d\u00b9 : FiniteDimensional K (SplittingField f)\nthis\u271d : FiniteDimensional K L\nthis : FiniteDimensional.finrank K L = FiniteDimensional.finrank K (SplittingField f)\n\u22a2 Function.Surjective \u2191(AlgHom.toLinearMap (lift L f (_ : Splits (algebraMap K (SplittingField f)) f)))\n[PROOFSTEP]\nrefine' (LinearMap.injective_iff_surjective_of_finrank_eq_finrank this).1 _\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\nthis\u271d\u00b9 : FiniteDimensional K (SplittingField f)\nthis\u271d : FiniteDimensional K L\nthis : FiniteDimensional.finrank K L = FiniteDimensional.finrank K (SplittingField f)\n\u22a2 Function.Injective \u2191(AlgHom.toLinearMap (lift L f (_ : Splits (algebraMap K (SplittingField f)) f)))\n[PROOFSTEP]\nexact RingHom.injective (lift L f <| splits (SplittingField f) f : L \u2192+* f.SplittingField)\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.SplittingField.Construction", "llama_tokens": 12843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.29746244384854564}} {"text": "[GOAL]\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\n\u22a2 run (f <$> x) = Option.map f <$> run x\n[PROOFSTEP]\nrw [\u2190 bind_pure_comp _ x.run]\n[GOAL]\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\n\u22a2 run (f <$> x) = do\n let a \u2190 run x\n pure (Option.map f a)\n[PROOFSTEP]\nchange\n x.run >>=\n (fun\n | some a => OptionT.run (pure (f a))\n | none => pure none) =\n _\n[GOAL]\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\n\u22a2 (do\n let x \u2190 run x\n match x with\n | some a => run (pure (f a))\n | none => pure none) =\n do\n let a \u2190 run x\n pure (Option.map f a)\n[PROOFSTEP]\napply bind_congr\n[GOAL]\ncase h\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\n\u22a2 \u2200 (a : Option \u03b1),\n (match a with\n | some a => run (pure (f a))\n | none => pure none) =\n pure (Option.map f a)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\na : Option \u03b1\n\u22a2 (match a with\n | some a => run (pure (f a))\n | none => pure none) =\n pure (Option.map f a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase h.none\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\n\u22a2 (match none with\n | some a => run (pure (f a))\n | none => pure none) =\n pure (Option.map f none)\n[PROOFSTEP]\nsimp [Option.map, Option.bind]\n[GOAL]\ncase h.some\n\u03b1 \u03b2 : Type u\nm : Type u \u2192 Type v\nx : OptionT m \u03b1\ninst\u271d\u00b9 : Monad m\nf : \u03b1 \u2192 \u03b2\ninst\u271d : LawfulMonad m\nval\u271d : \u03b1\n\u22a2 (match some val\u271d with\n | some a => run (pure (f a))\n | none => pure none) =\n pure (Option.map f (some val\u271d))\n[PROOFSTEP]\nsimp [Option.map, Option.bind]\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u22a2 \u2200 {\u03b1 : Type u} (x : OptionT m \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\napply OptionT.ext\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\n\u22a2 OptionT.run (id <$> x\u271d) = OptionT.run x\u271d\n[PROOFSTEP]\nsimp only [OptionT.run_map]\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\n\u22a2 Option.map id <$> OptionT.run x\u271d = OptionT.run x\u271d\n[PROOFSTEP]\nrw [map_congr, id_map]\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\n\u22a2 \u2200 (a : Option \u03b1\u271d), Option.map id a = id a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\na : Option \u03b1\u271d\n\u22a2 Option.map id a = id a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase h.none\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\n\u22a2 Option.map id none = id none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.some\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nval\u271d : \u03b1\u271d\n\u22a2 Option.map id (some val\u271d) = id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u} (x : \u03b1) (f : \u03b1 \u2192 OptionT m \u03b2), pure x >>= f = f x\n[PROOFSTEP]\nintros\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d : Type u\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\n\u22a2 pure x\u271d >>= f\u271d = f\u271d x\u271d\n[PROOFSTEP]\napply OptionT.ext\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d : Type u\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\n\u22a2 OptionT.run (pure x\u271d >>= f\u271d) = OptionT.run (f\u271d x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : OptionT m \u03b1) (f : \u03b1 \u2192 OptionT m \u03b2) (g : \u03b2 \u2192 OptionT 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\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\n\u22a2 x\u271d >>= f\u271d >>= g\u271d = x\u271d >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\napply OptionT.ext\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\n\u22a2 OptionT.run (x\u271d >>= f\u271d >>= g\u271d) = OptionT.run (x\u271d >>= fun x => f\u271d x >>= g\u271d)\n[PROOFSTEP]\nsimp only [OptionT.run_bind, bind_assoc]\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\n\u22a2 (do\n let x \u2190 OptionT.run x\u271d\n let x \u2190\n match x with\n | some a => OptionT.run (f\u271d a)\n | none => pure none\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none) =\n do\n let x \u2190 OptionT.run x\u271d\n match x with\n | some a => do\n let x \u2190 OptionT.run (f\u271d a)\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none\n | none => pure none\n[PROOFSTEP]\nrw [bind_congr]\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\n\u22a2 \u2200 (a : Option \u03b1\u271d),\n (do\n let x \u2190\n match a with\n | some a => OptionT.run (f\u271d a)\n | none => pure none\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none) =\n match a with\n | some a => do\n let x \u2190 OptionT.run (f\u271d a)\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none\n | none => pure none\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\na : Option \u03b1\u271d\n\u22a2 (do\n let x \u2190\n match a with\n | some a => OptionT.run (f\u271d a)\n | none => pure none\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none) =\n match a with\n | some a => do\n let x \u2190 OptionT.run (f\u271d a)\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none\n | none => pure none\n[PROOFSTEP]\ncases a\n[GOAL]\ncase h.none\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\n\u22a2 (do\n let x \u2190\n match none with\n | some a => OptionT.run (f\u271d a)\n | none => pure none\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none) =\n match none with\n | some a => do\n let x \u2190 OptionT.run (f\u271d a)\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none\n | none => pure none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.some\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nx\u271d : OptionT m \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 OptionT m \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 OptionT m \u03b3\u271d\nval\u271d : \u03b1\u271d\n\u22a2 (do\n let x \u2190\n match some val\u271d with\n | some a => OptionT.run (f\u271d a)\n | none => pure none\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none) =\n match some val\u271d with\n | some a => do\n let x \u2190 OptionT.run (f\u271d a)\n match x with\n | some a => OptionT.run (g\u271d a)\n | none => pure none\n | none => pure none\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Init.Control.Lawful", "llama_tokens": 3649, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.29731079805152366}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g\u271d : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Filter \u03b1\nh\u2081 : NeBot f\nh\u2082 : \u2200 (g : Filter \u03b1), NeBot g \u2192 g \u2264 f \u2192 f \u2264 g\ng : Filter \u03b1\nneBot'\u271d : NeBot g\nle_of_le\u271d : \u2200 (g_1 : Filter \u03b1), NeBot g_1 \u2192 g_1 \u2264 g \u2192 g \u2264 g_1\nx\u271d : \u2191{ toFilter := f, neBot' := h\u2081, le_of_le := h\u2082 } = \u2191{ toFilter := g, neBot' := neBot'\u271d, le_of_le := le_of_le\u271d }\n\u22a2 { toFilter := f, neBot' := h\u2081, le_of_le := h\u2082 } = { toFilter := g, neBot' := neBot'\u271d, le_of_le := le_of_le\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g\u271d : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b1\ng : Filter \u03b1\nhg : NeBot (g \u2293 \u2191f)\n\u22a2 NeBot (\u2191f \u2293 g)\n[PROOFSTEP]\nrwa [inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g\u271d : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b1\ng : Filter \u03b1\n\u22a2 Disjoint (\u2191f) g \u2194 \u00ac\u2191f \u2264 g\n[PROOFSTEP]\nrw [\u2190 inf_neBot_iff, neBot_iff, Ne.def, not_not, disjoint_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nhsc : \u00acs\u1d9c \u2208 f\nh : \u2191f \u2293 \ud835\udcdf s = \u22a5\n\u22a2 \u2191f \u2293 \ud835\udcdf s\u1d9c\u1d9c = \u22a5\n[PROOFSTEP]\nrwa [compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\n\u22a2 s\u1d9c \u2208 f \u2194 \u00acs \u2208 f\n[PROOFSTEP]\nrw [\u2190 compl_not_mem_iff, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Filter \u03b1\nh : \u2200 (s : Set \u03b1), \u00acs\u1d9c \u2208 f \u2194 s \u2208 f\nhf : f = \u22a5\n\u22a2 False\n[PROOFSTEP]\nsimp [hf] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g\u271d : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nu : Ultrafilter \u03b1\nf g : Filter \u03b1\n\u22a2 \u00ac\u2191u \u2264 f \u2294 g \u2194 \u00ac(\u2191u \u2264 f \u2228 \u2191u \u2264 g)\n[PROOFSTEP]\nsimp only [\u2190 disjoint_iff_not_le, not_or, disjoint_sup_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\n\u22a2 s \u222a t \u2208 f \u2194 s \u2208 f \u2228 t \u2208 f\n[PROOFSTEP]\nsimp only [\u2190 mem_coe, \u2190 le_principal_iff, \u2190 sup_principal, le_sup_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in \u2191f, p x \u2192 q x) \u2194 (\u2200\u1da0 (x : \u03b1) in \u2191f, p x) \u2192 \u2200\u1da0 (x : \u03b1) in \u2191f, q x\n[PROOFSTEP]\nsimp only [imp_iff_not_or, eventually_or, eventually_not]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns\u271d t : Set \u03b1\np q : \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nhs : Set.Finite s\n\u22a2 \u22c3\u2080 \u2205 \u2208 f \u2194 \u2203 t, t \u2208 \u2205 \u2227 t \u2208 f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns\u271d\u00b9 t : Set \u03b1\np q : \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nhs : Set.Finite s\na\u271d : Set \u03b1\ns\u271d : Set (Set \u03b1)\nx\u271d\u00b9 : \u00aca\u271d \u2208 s\u271d\nx\u271d : Set.Finite s\u271d\nhis : \u22c3\u2080 s\u271d \u2208 f \u2194 \u2203 t, t \u2208 s\u271d \u2227 t \u2208 f\n\u22a2 \u22c3\u2080 insert a\u271d s\u271d \u2208 f \u2194 \u2203 t, t \u2208 insert a\u271d s\u271d \u2227 t \u2208 f\n[PROOFSTEP]\nsimp [union_mem_iff, his, or_and_right, exists_or]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns\u271d t : Set \u03b1\np q : \u03b1 \u2192 Prop\nis : Set \u03b2\ns : \u03b2 \u2192 Set \u03b1\nhis : Set.Finite is\n\u22a2 \u22c3 (i : \u03b2) (_ : i \u2208 is), s i \u2208 f \u2194 \u2203 i, i \u2208 is \u2227 s i \u2208 f\n[PROOFSTEP]\nsimp only [\u2190 sUnion_image, finite_sUnion_mem_iff (his.image s), bex_image_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g\u271d : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nm : \u03b1 \u2192 \u03b2\nu : Ultrafilter \u03b2\ninj : Injective m\nlarge : range m \u2208 u\ng : Filter \u03b1\nhg : NeBot g\nhgu : g \u2264 Filter.comap m \u2191u\n\u22a2 Filter.comap m \u2191u \u2264 g\n[PROOFSTEP]\nsimp only [\u2190 u.unique (map_le_iff_le_comap.2 hgu), comap_map inj, le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b1\nh\u2080 : optParam (Injective id) (_ : Injective id)\n\u22a2 range id \u2208 f\n[PROOFSTEP]\nrw [range_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b1\nh\u2080 : optParam (Injective id) (_ : Injective id)\n\u22a2 univ \u2208 f\n[PROOFSTEP]\nexact univ_mem\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b3\nm : \u03b1 \u2192 \u03b2\nn : \u03b2 \u2192 \u03b3\ninj\u2080 : Injective n\nlarge\u2080 : range n \u2208 f\ninj\u2081 : Injective m\nlarge\u2081 : range m \u2208 comap f inj\u2080 large\u2080\ninj\u2082 : optParam (Injective (n \u2218 m)) (_ : Injective (n \u2218 m))\n\u22a2 range (n \u2218 m) \u2208 f\n[PROOFSTEP]\nrw [range_comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b3\nm : \u03b1 \u2192 \u03b2\nn : \u03b2 \u2192 \u03b3\ninj\u2080 : Injective n\nlarge\u2080 : range n \u2208 f\ninj\u2081 : Injective m\nlarge\u2081 : range m \u2208 comap f inj\u2080 large\u2080\ninj\u2082 : optParam (Injective (n \u2218 m)) (_ : Injective (n \u2218 m))\n\u22a2 n '' range m \u2208 f\n[PROOFSTEP]\nexact image_mem_of_mem_comap large\u2080 large\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns\u271d t : Set \u03b1\np q : \u03b1 \u2192 Prop\n\u03b1\u271d : Type ?u.19487\na : \u03b1\u271d\ns : Set \u03b1\u271d\n\u22a2 \u00acs\u1d9c \u2208 pure a \u2194 s \u2208 pure a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nm : \u03b1 \u2192 \u03b2\na : \u03b1\ninj : Injective m\nlarge : range m \u2208 pure (m a)\n\u22a2 \ud835\udcdf (m \u207b\u00b9' {m a}) = \u2191(pure a)\n[PROOFSTEP]\nrw [coe_pure, \u2190 principal_singleton, \u2190 image_singleton, preimage_image_eq _ inj]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nh : Set.Finite s\nh' : s \u2208 f\n\u22a2 \u2203 x, x \u2208 s \u2227 f = pure x\n[PROOFSTEP]\nrw [\u2190 biUnion_of_singleton s] at h' \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nh : Set.Finite s\nh' : \u22c3 (x : \u03b1) (_ : x \u2208 s), {x} \u2208 f\n\u22a2 \u2203 x, x \u2208 s \u2227 f = pure x\n[PROOFSTEP]\nrcases(Ultrafilter.finite_biUnion_mem_iff h).mp h' with \u27e8a, has, haf\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\nh : Set.Finite s\nh' : \u22c3 (x : \u03b1) (_ : x \u2208 s), {x} \u2208 f\na : \u03b1\nhas : a \u2208 s\nhaf : {a} \u2208 f\n\u22a2 \u2203 x, x \u2208 s \u2227 f = pure x\n[PROOFSTEP]\nexact \u27e8a, has, eq_of_le (Filter.le_pure_iff.2 haf)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d g : Ultrafilter \u03b1\ns\u271d t : Set \u03b1\np q : \u03b1 \u2192 Prop\nf : Ultrafilter \u03b1\nm : \u03b1 \u2192 Ultrafilter \u03b2\ns : Set \u03b2\n\u22a2 (\u00acs\u1d9c \u2208 Filter.bind \u2191f fun x => \u2191(m x)) \u2194 s \u2208 Filter.bind \u2191f fun x => \u2191(m x)\n[PROOFSTEP]\nsimp only [mem_bind', mem_coe, \u2190 compl_mem_iff_not_mem, compl_setOf, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 s \u2208 f \u2194 \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 f \u2192 s \u2208 g\n[PROOFSTEP]\nrefine' \u27e8fun hf g hg => hg hf, fun H => by_contra fun hf => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nH : \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 f \u2192 s \u2208 g\nhf : \u00acs \u2208 f\n\u22a2 False\n[PROOFSTEP]\nset g : Filter (s\u1d9c : Set \u03b1) := comap (\u2191) f\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nH : \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 f \u2192 s \u2208 g\nhf : \u00acs \u2208 f\ng : Filter \u2191s\u1d9c := comap Subtype.val f\n\u22a2 False\n[PROOFSTEP]\nhaveI : NeBot g := comap_neBot_iff_compl_range.2 (by simpa [compl_setOf])\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nH : \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 f \u2192 s \u2208 g\nhf : \u00acs \u2208 f\ng : Filter \u2191s\u1d9c := comap Subtype.val f\n\u22a2 \u00ac(range Subtype.val)\u1d9c \u2208 f\n[PROOFSTEP]\nsimpa [compl_setOf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nH : \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 f \u2192 s \u2208 g\nhf : \u00acs \u2208 f\ng : Filter \u2191s\u1d9c := comap Subtype.val f\nthis : NeBot g\n\u22a2 False\n[PROOFSTEP]\nsimpa using H ((of g).map (\u2191)) (map_le_iff_le_comap.mpr (of_le g))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nf f' : Filter \u03b1\n\u22a2 \u2a06 (g : Ultrafilter \u03b1) (_ : \u2191g \u2264 f), \u2191g \u2264 f' \u2194 f \u2264 f'\n[PROOFSTEP]\nsimp only [iSup_le_iff, \u2190 le_iff_ultrafilter]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : Filter \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nl\u2081 : Filter \u03b1\nl\u2082 : Filter \u03b2\n\u22a2 Tendsto f l\u2081 l\u2082 \u2194 \u2200 (g : Ultrafilter \u03b1), \u2191g \u2264 l\u2081 \u2192 Tendsto f (\u2191g) l\u2082\n[PROOFSTEP]\nsimpa only [tendsto_iff_comap] using le_iff_ultrafilter\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\ng : Filter \u03b1\np : Filter \u03b1 \u2192 Prop\nhp : Monotone p\n\u22a2 (\u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 g \u2192 p f) \u2194 \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 g \u2192 p \u2191f\n[PROOFSTEP]\nrefine' \u27e8fun H f hf => H f f.neBot hf, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : Filter \u03b1\ns : Set \u03b1\na : \u03b1\ng : Filter \u03b1\np : Filter \u03b1 \u2192 Prop\nhp : Monotone p\n\u22a2 (\u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 g \u2192 p \u2191f) \u2192 \u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 g \u2192 p f\n[PROOFSTEP]\nintro H f hf hfg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : Filter \u03b1\ns : Set \u03b1\na : \u03b1\ng : Filter \u03b1\np : Filter \u03b1 \u2192 Prop\nhp : Monotone p\nH : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 g \u2192 p \u2191f\nf : Filter \u03b1\nhf : NeBot f\nhfg : f \u2264 g\n\u22a2 p f\n[PROOFSTEP]\nexact hp (of_le f) (H _ ((of_le f).trans hfg))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\n\u22a2 s \u2208 ofComapInfPrincipal h\n[PROOFSTEP]\nlet f := Filter.comap m g \u2293 \ud835\udcdf s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\n\u22a2 s \u2208 ofComapInfPrincipal h\n[PROOFSTEP]\nhaveI : f.NeBot := comap_inf_principal_neBot_of_image_mem h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\nthis : NeBot f\n\u22a2 s \u2208 ofComapInfPrincipal h\n[PROOFSTEP]\nhave : s \u2208 f := mem_inf_of_right (mem_principal_self s)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\nthis\u271d : NeBot f\nthis : s \u2208 f\n\u22a2 s \u2208 ofComapInfPrincipal h\n[PROOFSTEP]\nexact le_def.mp (of_le _) s this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\n\u22a2 map m (ofComapInfPrincipal h) = g\n[PROOFSTEP]\nlet f := Filter.comap m g \u2293 \ud835\udcdf s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\n\u22a2 map m (ofComapInfPrincipal h) = g\n[PROOFSTEP]\nhaveI : f.NeBot := comap_inf_principal_neBot_of_image_mem h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\nthis : NeBot f\n\u22a2 map m (ofComapInfPrincipal h) = g\n[PROOFSTEP]\napply eq_of_le\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\nthis : NeBot f\n\u22a2 \u2191(map m (ofComapInfPrincipal h)) \u2264 \u2191g\n[PROOFSTEP]\ncalc\n Filter.map m (of f) \u2264 Filter.map m f := map_mono (of_le _)\n _ \u2264 (Filter.map m <| Filter.comap m g) \u2293 Filter.map m (\ud835\udcdf s) := map_inf_le\n _ = (Filter.map m <| Filter.comap m g) \u2293 (\ud835\udcdf <| m '' s) := by rw [map_principal]\n _ \u2264 \u2191g \u2293 (\ud835\udcdf <| m '' s) := (inf_le_inf_right _ map_comap_le)\n _ = \u2191g := inf_of_le_left (le_principal_iff.mpr h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\ng : Ultrafilter \u03b2\nh : m '' s \u2208 g\nf : Filter \u03b1 := Filter.comap m \u2191g \u2293 \ud835\udcdf s\nthis : NeBot f\n\u22a2 Filter.map m (Filter.comap m \u2191g) \u2293 Filter.map m (\ud835\udcdf s) = Filter.map m (Filter.comap m \u2191g) \u2293 \ud835\udcdf (m '' s)\n[PROOFSTEP]\nrw [map_principal]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Ultrafilter", "llama_tokens": 5558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6039318479832805, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.29724809036130573}} {"text": "[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d f : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nx y : M\n\u22a2 Pi.prod (\u2191f) (\u2191g) (x + y) = Pi.prod (\u2191f) (\u2191g) x + Pi.prod (\u2191f) (\u2191g) y\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.mk_add_mk, map_add]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d f : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nc : R\nx : M\n\u22a2 AddHom.toFun\n { toFun := Pi.prod \u2191f \u2191g, map_add' := (_ : \u2200 (x y : M), (\u2191f (x + y), \u2191g (x + y)) = (\u2191f x + \u2191f y, \u2191g x + \u2191g y)) }\n (c \u2022 x) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := Pi.prod \u2191f \u2191g, map_add' := (_ : \u2200 (x y : M), (\u2191f (x + y), \u2191g (x + y)) = (\u2191f x + \u2191f y, \u2191g x + \u2191g y)) }\n x\n[PROOFSTEP]\nsimp only [Pi.prod, Prod.smul_mk, map_smul, RingHom.id_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)\n\u22a2 (fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)\nx\u271d : M\n\u22a2 \u2191((fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n f)).fst\n x\u271d =\n \u2191f.fst x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)\nx\u271d : M\n\u22a2 \u2191((fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n f)).snd\n x\u271d =\n \u2191f.snd x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : M \u2192\u2097[R] M\u2082 \u00d7 M\u2083\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n ((fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f)) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\u2081\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : M \u2192\u2097[R] M\u2082 \u00d7 M\u2083\nx\u271d : M\n\u22a2 (\u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n ((fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f)) f))\n x\u271d).fst =\n (\u2191f x\u271d).fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h\u2082\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b3 : Module S M\u2082\ninst\u271d\u00b2 : Module S M\u2083\ninst\u271d\u00b9 : SMulCommClass R S M\u2082\ninst\u271d : SMulCommClass R S M\u2083\nf : M \u2192\u2097[R] M\u2082 \u00d7 M\u2083\nx\u271d : M\n\u22a2 (\u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a) =\n AddHom.toFun\n { toFun := fun f => prod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2082) \u00d7 (M \u2192\u2097[R] M\u2083)),\n (fun f => prod f.fst f.snd) (a + b) = (fun f => prod f.fst f.snd) (a + b)) }\n (r \u2022 a)) }.toAddHom\n ((fun f => (comp (fst R M\u2082 M\u2083) f, comp (snd R M\u2082 M\u2083) f)) f))\n x\u271d).snd =\n (\u2191f x\u271d).snd\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 range (inl R M M\u2082) = ker (snd R M M\u2082)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 x \u2208 range (inl R M M\u2082) \u2194 x \u2208 ker (snd R M M\u2082)\n[PROOFSTEP]\nsimp only [mem_ker, mem_range]\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, \u2191(inl R M M\u2082) y = x) \u2194 \u2191(snd R M M\u2082) x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, \u2191(inl R M M\u2082) y = x) \u2192 \u2191(snd R M M\u2082) x = 0\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase h.mp.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ny : M\n\u22a2 \u2191(snd R M M\u2082) (\u2191(inl R M M\u2082) y) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 \u2191(snd R M M\u2082) x = 0 \u2192 \u2203 y, \u2191(inl R M M\u2082) y = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\nh : \u2191(snd R M M\u2082) x = 0\n\u22a2 \u2203 y, \u2191(inl R M M\u2082) y = x\n[PROOFSTEP]\nexact \u27e8x.fst, Prod.ext rfl h.symm\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 range (inr R M M\u2082) = ker (fst R M M\u2082)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 x \u2208 range (inr R M M\u2082) \u2194 x \u2208 ker (fst R M M\u2082)\n[PROOFSTEP]\nsimp only [mem_ker, mem_range]\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, \u2191(inr R M M\u2082) y = x) \u2194 \u2191(fst R M M\u2082) x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 (\u2203 y, \u2191(inr R M M\u2082) y = x) \u2192 \u2191(fst R M M\u2082) x = 0\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase h.mp.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ny : M\u2082\n\u22a2 \u2191(fst R M M\u2082) (\u2191(inr R M M\u2082) y) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\n\u22a2 \u2191(fst R M M\u2082) x = 0 \u2192 \u2203 y, \u2191(inr R M M\u2082) y = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx : M \u00d7 M\u2082\nh : \u2191(fst R M M\u2082) x = 0\n\u22a2 \u2203 y, \u2191(inr R M M\u2082) y = x\n[PROOFSTEP]\nexact \u27e8x.snd, Prod.ext h.symm rfl\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M\n\u22a2 \u2200 \u2983a\u2082 : M\u2984, \u2191(inl R M M\u2082) x\u271d = \u2191(inl R M M\u2082) a\u2082 \u2192 x\u271d = a\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M\u2082\n\u22a2 \u2200 \u2983a\u2082 : M\u2082\u2984, \u2191(inr R M M\u2082) x\u271d = \u2191(inr R M M\u2082) a\u2082 \u2192 x\u271d = a\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\n\u22a2 comp (coprod f g) (inl R M M\u2082) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nx\u271d : M\n\u22a2 \u2191(comp (coprod f g) (inl R M M\u2082)) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp only [map_zero, add_zero, coprod_apply, inl_apply, comp_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\n\u22a2 comp (coprod f g) (inr R M M\u2082) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nx\u271d : M\u2082\n\u22a2 \u2191(comp (coprod f g) (inr R M M\u2082)) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nsimp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 coprod (inl R M M\u2082) (inr R M M\u2082) = id\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\u2081\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M \u00d7 M\u2082\n\u22a2 (\u2191(coprod (inl R M M\u2082) (inr R M M\u2082)) x\u271d).fst = (\u2191id x\u271d).fst\n[PROOFSTEP]\nsimp only [Prod.mk_add_mk, add_zero, id_apply, coprod_apply, inl_apply, inr_apply, zero_add]\n[GOAL]\ncase h.h\u2082\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M \u00d7 M\u2082\n\u22a2 (\u2191(coprod (inl R M M\u2082) (inr R M M\u2082)) x\u271d).snd = (\u2191id x\u271d).snd\n[PROOFSTEP]\nsimp only [Prod.mk_add_mk, add_zero, id_apply, coprod_apply, inl_apply, inr_apply, zero_add]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 fst R M M\u2082 = coprod id 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M \u00d7 M\u2082\n\u22a2 \u2191(fst R M M\u2082) x\u271d = \u2191(coprod id 0) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\n\u22a2 snd R M M\u2082 = coprod 0 id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nx\u271d : M \u00d7 M\u2082\n\u22a2 \u2191(snd R M M\u2082) x\u271d = \u2191(coprod 0 id) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 \u2191(Submodule.map (coprod f g) (Submodule.prod S S')) = \u2191(Submodule.map f S \u2294 Submodule.map g S')\n[PROOFSTEP]\nsimp only [LinearMap.coprod_apply, Submodule.coe_sup, Submodule.map_coe]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.fst + \u2191g a.snd) '' \u2191(Submodule.prod S S') = (fun a => \u2191f a) '' \u2191S + (fun a => \u2191g a) '' \u2191S'\n[PROOFSTEP]\nrw [\u2190 Set.image2_add, Set.image2_image_left, Set.image2_image_right]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS\u271d : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\u271d\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nS : Submodule R M\nS' : Submodule R M\u2082\n\u22a2 (fun a => \u2191f a.fst + \u2191g a.snd) '' \u2191(Submodule.prod S S') = Set.image2 (fun a b => \u2191f a + \u2191g b) \u2191S \u2191S'\n[PROOFSTEP]\nexact Set.image_prod fun m m\u2082 => f m + g m\u2082\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\na b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\n\u22a2 (fun f => coprod f.fst f.snd) (a + b) = (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\na b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\nx\u271d : M \u00d7 M\u2082\n\u22a2 \u2191((fun f => coprod f.fst f.snd) (a + b)) x\u271d = \u2191((fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) x\u271d\n[PROOFSTEP]\nsimp only [Prod.snd_add, add_apply, coprod_apply, Prod.fst_add, add_add_add_comm]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\nr : S\na : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\n\u22a2 AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n (r \u2022 a) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n a\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\nr : S\na : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\n\u22a2 coprod (r \u2022 a.fst) (r \u2022 a.snd) = r \u2022 coprod a.fst a.snd\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\nr : S\na : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\nx\u271d : M \u00d7 M\u2082\n\u22a2 \u2191(coprod (r \u2022 a.fst) (r \u2022 a.snd)) x\u271d = \u2191(r \u2022 coprod a.fst a.snd) x\u271d\n[PROOFSTEP]\nsimp only [smul_add, smul_apply, Prod.smul_snd, Prod.smul_fst, coprod_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\nf : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)\n\u22a2 (fun f => (comp f (inl R M M\u2082), comp f (inr R M M\u2082)))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n (r \u2022 a) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n a) }.toAddHom\n f) =\n f\n[PROOFSTEP]\nsimp only [Prod.mk.eta, coprod_inl, coprod_inr]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : Semiring S\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : AddCommMonoid M\u2085\ninst\u271d\u2078 : AddCommMonoid M\u2086\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : Module R M\u2083\ninst\u271d\u2074 : Module R M\u2084\ninst\u271d\u00b3 : Module R M\u2085\ninst\u271d\u00b2 : Module R M\u2086\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\u2083\ninst\u271d : SMulCommClass R S M\u2083\nf : M \u00d7 M\u2082 \u2192\u2097[R] M\u2083\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) },\n map_smul' :=\n (_ :\n \u2200 (r : S) (a : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n (r \u2022 a) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n { toFun := fun f => coprod f.fst f.snd,\n map_add' :=\n (_ :\n \u2200 (a b : (M \u2192\u2097[R] M\u2083) \u00d7 (M\u2082 \u2192\u2097[R] M\u2083)),\n (fun f => coprod f.fst f.snd) (a + b) =\n (fun f => coprod f.fst f.snd) a + (fun f => coprod f.fst f.snd) b) }\n a) }.toAddHom\n ((fun f => (comp f (inl R M M\u2082), comp f (inr R M M\u2082))) f) =\n f\n[PROOFSTEP]\nsimp only [\u2190 comp_coprod, comp_id, coprod_inl_inr]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d f : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\n\u22a2 ker (prodMap f g) = Submodule.prod (ker f) (ker g)\n[PROOFSTEP]\ndsimp only [ker]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : AddCommMonoid M\u2085\ninst\u271d\u2076 : AddCommMonoid M\u2086\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M\u2084\ninst\u271d\u00b9 : Module R M\u2085\ninst\u271d : Module R M\u2086\nf\u271d f : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\n\u22a2 Submodule.comap (prodMap f g) \u22a5 = Submodule.prod (Submodule.comap f \u22a5) (Submodule.comap g \u22a5)\n[PROOFSTEP]\nrw [\u2190 prodMap_comap_prod, Submodule.prod_bot]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nA : Type u_4\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring A\ninst\u271d\u00b2 : Module R A\nB : Type u_5\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring B\ninst\u271d : Module R B\na\u2081 a\u2082 : A\n\u22a2 (\u2191(inl R A B) (a\u2081 * a\u2082)).snd = (\u2191(inl R A B) a\u2081 * \u2191(inl R A B) a\u2082).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring S\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2085\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2086\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : Module R M\u2085\ninst\u271d\u2074 : Module R M\u2086\nf : M \u2192\u2097[R] M\u2082\nA : Type u_4\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring A\ninst\u271d\u00b2 : Module R A\nB : Type u_5\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring B\ninst\u271d : Module R B\nb\u2081 b\u2082 : B\n\u22a2 (\u2191(inr R A B) (b\u2081 * b\u2082)).fst = (\u2191(inr R A B) b\u2081 * \u2191(inr R A B) b\u2082).fst\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nx : M\u2083\n\u22a2 x \u2208 range (coprod f g) \u2194 x \u2208 range f \u2294 range g\n[PROOFSTEP]\nsimp [mem_sup]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 IsCompl (range (inl R M M\u2082)) (range (inr R M M\u2082))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Disjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))\n[PROOFSTEP]\nrw [disjoint_def]\n[GOAL]\ncase disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u2200 (x : M \u00d7 M\u2082), x \u2208 range (inl R M M\u2082) \u2192 x \u2208 range (inr R M M\u2082) \u2192 x = 0\n[PROOFSTEP]\nrintro \u27e8_, _\u27e9 \u27e8x, hx\u27e9 \u27e8y, hy\u27e9\n[GOAL]\ncase disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\nhx : \u2191(inl R M M\u2082) x = (fst\u271d, snd\u271d)\ny : M\u2082\nhy : \u2191(inr R M M\u2082) y = (fst\u271d, snd\u271d)\n\u22a2 (fst\u271d, snd\u271d) = 0\n[PROOFSTEP]\nsimp only [Prod.ext_iff, inl_apply, inr_apply, mem_bot] at hx hy \u22a2\n[GOAL]\ncase disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\ny : M\u2082\nhx : x = fst\u271d \u2227 0 = snd\u271d\nhy : 0 = fst\u271d \u2227 y = snd\u271d\n\u22a2 fst\u271d = 0.fst \u2227 snd\u271d = 0.snd\n[PROOFSTEP]\nexact \u27e8hy.1.symm, hx.2.symm\u27e9\n[GOAL]\ncase codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Codisjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))\n[PROOFSTEP]\nrw [codisjoint_iff_le_sup]\n[GOAL]\ncase codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u22a4 \u2264 range (inl R M M\u2082) \u2294 range (inr R M M\u2082)\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 -\n[GOAL]\ncase codisjoint.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 range (inl R M M\u2082) \u2294 range (inr R M M\u2082)\n[PROOFSTEP]\nsimp only [mem_sup, mem_range, exists_prop]\n[GOAL]\ncase codisjoint.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nx : M\ny : M\u2082\n\u22a2 \u2203 y_1, (\u2203 y, \u2191(inl R M M\u2082) y = y_1) \u2227 \u2203 z, (\u2203 y, \u2191(inr R M M\u2082) y = z) \u2227 y_1 + z = (x, y)\n[PROOFSTEP]\nrefine' \u27e8(x, 0), \u27e8x, rfl\u27e9, (0, y), \u27e8y, rfl\u27e9, _\u27e9\n[GOAL]\ncase codisjoint.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nx : M\ny : M\u2082\n\u22a2 (x, 0) + (0, y) = (x, y)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Disjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))\n[PROOFSTEP]\nsimp (config := { contextual := true }) [disjoint_def, @eq_comm M 0, @eq_comm M\u2082 0]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (coprod f g) (Submodule.prod p q) = map f p \u2294 map g q\n[PROOFSTEP]\nrefine' le_antisymm _ (sup_le (map_le_iff_le_comap.2 _) (map_le_iff_le_comap.2 _))\n[GOAL]\ncase refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (coprod f g) (Submodule.prod p q) \u2264 map f p \u2294 map g q\n[PROOFSTEP]\nrw [SetLike.le_def]\n[GOAL]\ncase refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 \u2983x : M\u2083\u2984, x \u2208 map (coprod f g) (Submodule.prod p q) \u2192 x \u2208 map f p \u2294 map g q\n[PROOFSTEP]\nrintro _ \u27e8x, \u27e8h\u2081, h\u2082\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M \u00d7 M\u2082\nh\u2081 : x.fst \u2208 \u2191p\nh\u2082 : x.snd \u2208 \u2191q\n\u22a2 \u2191(coprod f g) x \u2208 map f p \u2294 map g q\n[PROOFSTEP]\nexact mem_sup.2 \u27e8_, \u27e8_, h\u2081, rfl\u27e9, _, \u27e8_, h\u2082, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 p \u2264 comap f (map (coprod f g) (Submodule.prod p q))\n[PROOFSTEP]\nexact fun x hx => \u27e8(x, 0), by simp [hx]\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M\nhx : x \u2208 p\n\u22a2 (x, 0) \u2208 \u2191(Submodule.prod p q) \u2227 \u2191(coprod f g) (x, 0) = \u2191f x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase refine'_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 q \u2264 comap g (map (coprod f g) (Submodule.prod p q))\n[PROOFSTEP]\nexact fun x hx => \u27e8(0, x), by simp [hx]\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\nhx : x \u2208 q\n\u22a2 (0, x) \u2208 \u2191(Submodule.prod p q) \u2227 \u2191(coprod f g) (0, x) = \u2191g x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 Submodule.prod p q = map (inl R M M\u2082) p \u2294 map (inr R M M\u2082) q\n[PROOFSTEP]\nrw [\u2190 map_coprod_prod, coprod_inl_inr, map_id]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\ns : Set M\nt : Set M\u2082\n\u22a2 span R (\u2191(inl R M M\u2082) '' s \u222a \u2191(inr R M M\u2082) '' t) = Submodule.prod (span R s) (span R t)\n[PROOFSTEP]\nrw [span_union, prod_eq_sup_map, \u2190 span_image, \u2190 span_image]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\n\u22a2 ker (prod f g) = ker f \u2293 ker g\n[PROOFSTEP]\nrw [ker, \u2190 prod_bot, comap_prod_prod]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\n\u22a2 comap f \u22a5 \u2293 comap g \u22a5 = ker f \u2293 ker g\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\n\u22a2 range (prod f g) \u2264 Submodule.prod (range f) (range g)\n[PROOFSTEP]\nsimp only [SetLike.le_def, prod_apply, mem_range, SetLike.mem_coe, mem_prod, exists_imp]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\n\u22a2 \u2200 \u2983x : M\u2082 \u00d7 M\u2083\u2984 (x_1 : M), Pi.prod (\u2191f) (\u2191g) x_1 = x \u2192 (\u2203 y, \u2191f y = x.fst) \u2227 \u2203 y, \u2191g y = x.snd\n[PROOFSTEP]\nrintro _ x rfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nx : M\n\u22a2 (\u2203 y, \u2191f y = (Pi.prod (\u2191f) (\u2191g) x).fst) \u2227 \u2203 y, \u2191g y = (Pi.prod (\u2191f) (\u2191g) x).snd\n[PROOFSTEP]\nexact \u27e8\u27e8x, rfl\u27e9, \u27e8x, rfl\u27e9\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\n\u22a2 Submodule.prod (ker f) (ker g) \u2264 ker (coprod f g)\n[PROOFSTEP]\nrintro \u27e8y, z\u27e9\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\ny : M\nz : M\u2082\n\u22a2 (y, z) \u2208 Submodule.prod (ker f) (ker g) \u2192 (y, z) \u2208 ker (coprod f g)\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\n\u22a2 ker (coprod f g) = Submodule.prod (ker f) (ker g)\n[PROOFSTEP]\napply le_antisymm _ (ker_prod_ker_le_ker_coprod f g)\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\n\u22a2 ker (coprod f g) \u2264 Submodule.prod (ker f) (ker g)\n[PROOFSTEP]\nrintro \u27e8y, z\u27e9 h\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : (y, z) \u2208 ker (coprod f g)\n\u22a2 (y, z) \u2208 Submodule.prod (ker f) (ker g)\n[PROOFSTEP]\nsimp only [mem_ker, mem_prod, coprod_apply] at h \u22a2\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\n\u22a2 \u2191f y = 0 \u2227 \u2191g z = 0\n[PROOFSTEP]\nhave : f y \u2208 (range f) \u2293 (range g) :=\n by\n simp only [true_and_iff, mem_range, mem_inf, exists_apply_eq_apply]\n use-z\n rwa [eq_comm, map_neg, \u2190 sub_eq_zero, sub_neg_eq_add]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\n\u22a2 \u2191f y \u2208 range f \u2293 range g\n[PROOFSTEP]\nsimp only [true_and_iff, mem_range, mem_inf, exists_apply_eq_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\n\u22a2 \u2203 y_1, \u2191g y_1 = \u2191f y\n[PROOFSTEP]\nuse-z\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\n\u22a2 \u2191g (-z) = \u2191f y\n[PROOFSTEP]\nrwa [eq_comm, map_neg, \u2190 sub_eq_zero, sub_neg_eq_add]\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\nthis : \u2191f y \u2208 range f \u2293 range g\n\u22a2 \u2191f y = 0 \u2227 \u2191g z = 0\n[PROOFSTEP]\nrw [hd.eq_bot, mem_bot] at this \n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : \u2191f y + \u2191g z = 0\nthis : \u2191f y = 0\n\u22a2 \u2191f y = 0 \u2227 \u2191g z = 0\n[PROOFSTEP]\nrw [this] at h \n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082\u271d : Type w\nV\u2082 : Type w'\nM\u2083\u271d : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\u271d\ninst\u271d\u2079 : AddCommMonoid M\u2083\u271d\ninst\u271d\u2078 : AddCommMonoid M\u2084\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M\u2082\u271d\ninst\u271d\u2075 : Module R M\u2083\u271d\ninst\u271d\u2074 : Module R M\u2084\nM\u2082 : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\nM\u2083 : Type u_4\ninst\u271d\u00b9 : AddCommGroup M\u2083\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nhd : Disjoint (range f) (range g)\ny : M\nz : M\u2082\nh : 0 + \u2191g z = 0\nthis : \u2191f y = 0\n\u22a2 \u2191f y = 0 \u2227 \u2191g z = 0\n[PROOFSTEP]\nsimpa [this] using h\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np q : Submodule R M\nx : M\n\u22a2 x \u2208 p \u2294 q \u2194 x \u2208 range (coprod (Submodule.subtype p) (Submodule.subtype q))\n[PROOFSTEP]\nsimp [Submodule.mem_sup, SetLike.exists]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (inl R M M\u2082) p = prod p \u22a5\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 map (inl R M M\u2082) p \u2194 (x, y) \u2208 prod p \u22a5\n[PROOFSTEP]\nsimp only [and_left_comm, eq_comm, mem_map, Prod.mk.inj_iff, inl_apply, mem_bot, exists_eq_left', mem_prod]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (inr R M M\u2082) q = prod \u22a5 q\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 map (inr R M M\u2082) q \u2194 (x, y) \u2208 prod \u22a5 q\n[PROOFSTEP]\nsimp [and_left_comm, eq_comm, and_comm]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 comap (fst R M M\u2082) p = prod p \u22a4\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 comap (fst R M M\u2082) p \u2194 (x, y) \u2208 prod p \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 comap (snd R M M\u2082) q = prod \u22a4 q\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 comap (snd R M M\u2082) q \u2194 (x, y) \u2208 prod \u22a4 q\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 comap (inl R M M\u2082) (prod p q) = p\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx\u271d : M\n\u22a2 x\u271d \u2208 comap (inl R M M\u2082) (prod p q) \u2194 x\u271d \u2208 p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 comap (inr R M M\u2082) (prod p q) = q\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx\u271d : M\u2082\n\u22a2 x\u271d \u2208 comap (inr R M M\u2082) (prod p q) \u2194 x\u271d \u2208 q\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (fst R M M\u2082) (prod p q) = p\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (fst R M M\u2082) (prod p q) \u2194 x \u2208 p\n[PROOFSTEP]\nsimp [(\u27e80, zero_mem _\u27e9 : \u2203 x, x \u2208 q)]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (snd R M M\u2082) (prod p q) = q\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\n\u22a2 x \u2208 map (snd R M M\u2082) (prod p q) \u2194 x \u2208 q\n[PROOFSTEP]\nsimp [(\u27e80, zero_mem _\u27e9 : \u2203 x, x \u2208 p)]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 ker (inl R M M\u2082) = \u22a5\n[PROOFSTEP]\nrw [ker, \u2190 prod_bot, prod_comap_inl]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 ker (inr R M M\u2082) = \u22a5\n[PROOFSTEP]\nrw [ker, \u2190 prod_bot, prod_comap_inr]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 range (fst R M M\u2082) = \u22a4\n[PROOFSTEP]\nrw [range_eq_map, \u2190 prod_top, prod_map_fst]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 range (snd R M M\u2082) = \u22a4\n[PROOFSTEP]\nrw [range_eq_map, \u2190 prod_top, prod_map_snd]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 (x y : { x // x \u2208 fst R M M\u2082 }), (fun x => (\u2191x).fst) (x + y) = (fun x => (\u2191x).fst) x + (fun x => (\u2191x).fst) y\n[PROOFSTEP]\nsimp only [AddSubmonoid.coe_add, coe_toAddSubmonoid, Prod.fst_add, Subtype.forall, implies_true, Prod.forall,\n forall_const]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 (r : R) (x : { x // x \u2208 fst R M M\u2082 }),\n AddHom.toFun\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) }\n x\n[PROOFSTEP]\nsimp only [SetLike.val_smul, Prod.smul_fst, RingHom.id_apply, Subtype.forall, implies_true, Prod.forall, forall_const]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\n\u22a2 (m, 0) \u2208 fst R M M\u2082\n[PROOFSTEP]\nsimp only [fst, comap_bot, mem_ker, snd_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 Function.LeftInverse (fun m => { val := (m, 0), property := (_ : (m, 0) \u2208 ker (snd R M M\u2082)) })\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 fst R M M\u2082 }), a \u2022 (\u2191a_1).fst = a \u2022 (\u2191a_1).fst) }.toAddHom.toFun\n[PROOFSTEP]\nrintro \u27e8\u27e8x, y\u27e9, hy\u27e9\n[GOAL]\ncase mk.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\nhy : (x, y) \u2208 fst R M M\u2082\n\u22a2 (fun m => { val := (m, 0), property := (_ : (m, 0) \u2208 ker (snd R M M\u2082)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 fst R M M\u2082 }), a \u2022 (\u2191a_1).fst = a \u2022 (\u2191a_1).fst) }.toAddHom\n { val := (x, y), property := hy }) =\n { val := (x, y), property := hy }\n[PROOFSTEP]\nsimp only [fst, comap_bot, mem_ker, snd_apply] at hy \n[GOAL]\ncase mk.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\nhy\u271d : (x, y) \u2208 fst R M M\u2082\nhy : y = 0\n\u22a2 (fun m => { val := (m, 0), property := (_ : (m, 0) \u2208 ker (snd R M M\u2082)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 fst R M M\u2082 }), a \u2022 (\u2191a_1).fst = a \u2022 (\u2191a_1).fst) }.toAddHom\n { val := (x, y), property := hy\u271d }) =\n { val := (x, y), property := hy\u271d }\n[PROOFSTEP]\nsimpa only [Subtype.mk.injEq, Prod.mk.injEq, true_and] using hy.symm\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 Function.RightInverse (fun m => { val := (m, 0), property := (_ : (m, 0) \u2208 ker (snd R M M\u2082)) })\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 fst R M M\u2082 }), a \u2022 (\u2191a_1).fst = a \u2022 (\u2191a_1).fst) }.toAddHom.toFun\n[PROOFSTEP]\nrintro x\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).fst,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 fst R M M\u2082 }), (\u2191a).fst + (\u2191a_1).fst = (\u2191a).fst + (\u2191a_1).fst) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 fst R M M\u2082 }), a \u2022 (\u2191a_1).fst = a \u2022 (\u2191a_1).fst) }.toAddHom\n ((fun m => { val := (m, 0), property := (_ : (m, 0) \u2208 ker (snd R M M\u2082)) }) x) =\n x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) (fst R M M\u2082) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u22a4 \u2264 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)\n[PROOFSTEP]\nrintro x -\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)\n[PROOFSTEP]\nsimp only [fst, comap_bot, mem_map, mem_ker, snd_apply, fst_apply, Prod.exists, exists_eq_left, exists_eq]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (snd R M M\u2082) (fst R M M\u2082) = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (snd R M M\u2082) (fst R M M\u2082) \u2264 \u22a5\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\n\u22a2 x \u2208 map (snd R M M\u2082) (fst R M M\u2082) \u2192 x \u2208 \u22a5\n[PROOFSTEP]\nsimp only [fst, comap_bot, mem_map, mem_ker, snd_apply, eq_comm, Prod.exists, exists_eq_left, exists_const, mem_bot,\n imp_self]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 (x y : { x // x \u2208 snd R M M\u2082 }), (fun x => (\u2191x).snd) (x + y) = (fun x => (\u2191x).snd) x + (fun x => (\u2191x).snd) y\n[PROOFSTEP]\nsimp only [AddSubmonoid.coe_add, coe_toAddSubmonoid, Prod.snd_add, Subtype.forall, implies_true, Prod.forall,\n forall_const]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 (r : R) (x : { x // x \u2208 snd R M M\u2082 }),\n AddHom.toFun\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) }\n x\n[PROOFSTEP]\nsimp only [SetLike.val_smul, Prod.smul_snd, RingHom.id_apply, Subtype.forall, implies_true, Prod.forall, forall_const]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nn : M\u2082\n\u22a2 (0, n) \u2208 snd R M M\u2082\n[PROOFSTEP]\nsimp only [snd, comap_bot, mem_ker, fst_apply]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 Function.LeftInverse (fun n => { val := (0, n), property := (_ : (0, n) \u2208 ker (LinearMap.fst R M M\u2082)) })\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 snd R M M\u2082 }), a \u2022 (\u2191a_1).snd = a \u2022 (\u2191a_1).snd) }.toAddHom.toFun\n[PROOFSTEP]\nrintro \u27e8\u27e8x, y\u27e9, hx\u27e9\n[GOAL]\ncase mk.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\nhx : (x, y) \u2208 snd R M M\u2082\n\u22a2 (fun n => { val := (0, n), property := (_ : (0, n) \u2208 ker (LinearMap.fst R M M\u2082)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 snd R M M\u2082 }), a \u2022 (\u2191a_1).snd = a \u2022 (\u2191a_1).snd) }.toAddHom\n { val := (x, y), property := hx }) =\n { val := (x, y), property := hx }\n[PROOFSTEP]\nsimp only [snd, comap_bot, mem_ker, fst_apply] at hx \n[GOAL]\ncase mk.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\nhx\u271d : (x, y) \u2208 snd R M M\u2082\nhx : x = 0\n\u22a2 (fun n => { val := (0, n), property := (_ : (0, n) \u2208 ker (LinearMap.fst R M M\u2082)) })\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 snd R M M\u2082 }), a \u2022 (\u2191a_1).snd = a \u2022 (\u2191a_1).snd) }.toAddHom\n { val := (x, y), property := hx\u271d }) =\n { val := (x, y), property := hx\u271d }\n[PROOFSTEP]\nsimpa only [Subtype.mk.injEq, Prod.mk.injEq, and_true] using hx.symm\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 Function.RightInverse (fun n => { val := (0, n), property := (_ : (0, n) \u2208 ker (LinearMap.fst R M M\u2082)) })\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 snd R M M\u2082 }), a \u2022 (\u2191a_1).snd = a \u2022 (\u2191a_1).snd) }.toAddHom.toFun\n[PROOFSTEP]\nrintro x\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun x => (\u2191x).snd,\n map_add' := (_ : \u2200 (a a_1 : { x // x \u2208 snd R M M\u2082 }), (\u2191a).snd + (\u2191a_1).snd = (\u2191a).snd + (\u2191a_1).snd) },\n map_smul' := (_ : \u2200 (a : R) (a_1 : { x // x \u2208 snd R M M\u2082 }), a \u2022 (\u2191a_1).snd = a \u2022 (\u2191a_1).snd) }.toAddHom\n ((fun n => { val := (0, n), property := (_ : (0, n) \u2208 ker (LinearMap.fst R M M\u2082)) }) x) =\n x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) (snd R M M\u2082) = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) (snd R M M\u2082) \u2264 \u22a5\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (LinearMap.fst R M M\u2082) (snd R M M\u2082) \u2192 x \u2208 \u22a5\n[PROOFSTEP]\nsimp only [snd, comap_bot, mem_map, mem_ker, fst_apply, eq_comm, Prod.exists, exists_eq_left, exists_const, mem_bot,\n imp_self]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (LinearMap.snd R M M\u2082) (snd R M M\u2082) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u22a4 \u2264 map (LinearMap.snd R M M\u2082) (snd R M M\u2082)\n[PROOFSTEP]\nrintro x -\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\n\u22a2 x \u2208 map (LinearMap.snd R M M\u2082) (snd R M M\u2082)\n[PROOFSTEP]\nsimp only [snd, comap_bot, mem_map, mem_ker, snd_apply, fst_apply, Prod.exists, exists_eq_right, exists_eq]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 fst R M M\u2082 \u2294 snd R M M\u2082 = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u22a4 \u2264 fst R M M\u2082 \u2294 snd R M M\u2082\n[PROOFSTEP]\nrintro \u27e8m, n\u27e9 -\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 (m, n) \u2208 fst R M M\u2082 \u2294 snd R M M\u2082\n[PROOFSTEP]\nrw [show (m, n) = (m, 0) + (0, n) by simp]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 (m, n) = (m, 0) + (0, n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 (m, 0) + (0, n) \u2208 fst R M M\u2082 \u2294 snd R M M\u2082\n[PROOFSTEP]\napply Submodule.add_mem (Submodule.fst R M M\u2082 \u2294 Submodule.snd R M M\u2082)\n[GOAL]\ncase mk.h\u2081\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 (m, 0) \u2208 fst R M M\u2082 \u2294 snd R M M\u2082\n[PROOFSTEP]\nexact Submodule.mem_sup_left (Submodule.mem_comap.mpr (by simp))\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 \u2191(LinearMap.snd R M M\u2082) (m, 0) \u2208 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.h\u2082\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 (0, n) \u2208 fst R M M\u2082 \u2294 snd R M M\u2082\n[PROOFSTEP]\nexact Submodule.mem_sup_right (Submodule.mem_comap.mpr (by simp))\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nm : M\nn : M\u2082\n\u22a2 \u2191(LinearMap.fst R M M\u2082) (0, n) \u2208 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 fst R M M\u2082 \u2293 snd R M M\u2082 = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 fst R M M\u2082 \u2293 snd R M M\u2082 \u2264 \u22a5\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 fst R M M\u2082 \u2293 snd R M M\u2082 \u2192 (x, y) \u2208 \u22a5\n[PROOFSTEP]\nsimp only [fst, comap_bot, snd, ge_iff_le, mem_inf, mem_ker, snd_apply, fst_apply, mem_bot, Prod.mk_eq_zero, and_comm,\n imp_self]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 q \u2264 prod p\u2081 p\u2082 \u2194 map (LinearMap.fst R M M\u2082) q \u2264 p\u2081 \u2227 map (LinearMap.snd R M M\u2082) q \u2264 p\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 q \u2264 prod p\u2081 p\u2082 \u2192 map (LinearMap.fst R M M\u2082) q \u2264 p\u2081 \u2227 map (LinearMap.snd R M M\u2082) q \u2264 p\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : q \u2264 prod p\u2081 p\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) q \u2264 p\u2081 \u2227 map (LinearMap.snd R M M\u2082) q \u2264 p\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : q \u2264 prod p\u2081 p\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) q \u2264 p\u2081\n[PROOFSTEP]\nrintro x \u27e8\u27e8y1, y2\u27e9, \u27e8hy1, rfl\u27e9\u27e9\n[GOAL]\ncase mp.left.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : q \u2264 prod p\u2081 p\u2082\ny1 : M\ny2 : M\u2082\nhy1 : (y1, y2) \u2208 \u2191q\n\u22a2 \u2191(LinearMap.fst R M M\u2082) (y1, y2) \u2208 p\u2081\n[PROOFSTEP]\nexact (h hy1).1\n[GOAL]\ncase mp.right\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : q \u2264 prod p\u2081 p\u2082\n\u22a2 map (LinearMap.snd R M M\u2082) q \u2264 p\u2082\n[PROOFSTEP]\nrintro x \u27e8\u27e8y1, y2\u27e9, \u27e8hy1, rfl\u27e9\u27e9\n[GOAL]\ncase mp.right.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : q \u2264 prod p\u2081 p\u2082\ny1 : M\ny2 : M\u2082\nhy1 : (y1, y2) \u2208 \u2191q\n\u22a2 \u2191(LinearMap.snd R M M\u2082) (y1, y2) \u2208 p\u2082\n[PROOFSTEP]\nexact (h hy1).2\n[GOAL]\ncase mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 map (LinearMap.fst R M M\u2082) q \u2264 p\u2081 \u2227 map (LinearMap.snd R M M\u2082) q \u2264 p\u2082 \u2192 q \u2264 prod p\u2081 p\u2082\n[PROOFSTEP]\nrintro \u27e8hH, hK\u27e9 \u27e8x1, x2\u27e9 h\n[GOAL]\ncase mpr.intro.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (LinearMap.fst R M M\u2082) q \u2264 p\u2081\nhK : map (LinearMap.snd R M M\u2082) q \u2264 p\u2082\nx1 : M\nx2 : M\u2082\nh : (x1, x2) \u2208 q\n\u22a2 (x1, x2) \u2208 prod p\u2081 p\u2082\n[PROOFSTEP]\nexact \u27e8hH \u27e8_, h, rfl\u27e9, hK \u27e8_, h, rfl\u27e9\u27e9\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 prod p\u2081 p\u2082 \u2264 q \u2194 map (inl R M M\u2082) p\u2081 \u2264 q \u2227 map (inr R M M\u2082) p\u2082 \u2264 q\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 prod p\u2081 p\u2082 \u2264 q \u2192 map (inl R M M\u2082) p\u2081 \u2264 q \u2227 map (inr R M M\u2082) p\u2082 \u2264 q\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\n\u22a2 map (inl R M M\u2082) p\u2081 \u2264 q \u2227 map (inr R M M\u2082) p\u2082 \u2264 q\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\n\u22a2 map (inl R M M\u2082) p\u2081 \u2264 q\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mp.left.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\nx : M\nhx : x \u2208 \u2191p\u2081\n\u22a2 \u2191(inl R M M\u2082) x \u2208 q\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.left.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\nx : M\nhx : x \u2208 \u2191p\u2081\n\u22a2 \u2191(inl R M M\u2082) x \u2208 prod p\u2081 p\u2082\n[PROOFSTEP]\nexact \u27e8hx, zero_mem p\u2082\u27e9\n[GOAL]\ncase mp.right\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\n\u22a2 map (inr R M M\u2082) p\u2082 \u2264 q\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase mp.right.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\nx : M\u2082\nhx : x \u2208 \u2191p\u2082\n\u22a2 \u2191(inr R M M\u2082) x \u2208 q\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.right.intro.intro.a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nh : prod p\u2081 p\u2082 \u2264 q\nx : M\u2082\nhx : x \u2208 \u2191p\u2082\n\u22a2 \u2191(inr R M M\u2082) x \u2208 prod p\u2081 p\u2082\n[PROOFSTEP]\nexact \u27e8zero_mem p\u2081, hx\u27e9\n[GOAL]\ncase mpr\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\n\u22a2 map (inl R M M\u2082) p\u2081 \u2264 q \u2227 map (inr R M M\u2082) p\u2082 \u2264 q \u2192 prod p\u2081 p\u2082 \u2264 q\n[PROOFSTEP]\nrintro \u27e8hH, hK\u27e9 \u27e8x1, x2\u27e9 \u27e8h1, h2\u27e9\n[GOAL]\ncase mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\n\u22a2 (x1, x2) \u2208 q\n[PROOFSTEP]\nhave h1' : (LinearMap.inl R _ _) x1 \u2208 q := by\n apply hH\n simpa using h1\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\n\u22a2 \u2191(inl R M M\u2082) x1 \u2208 q\n[PROOFSTEP]\napply hH\n[GOAL]\ncase a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\n\u22a2 \u2191(inl R M M\u2082) x1 \u2208 map (inl R M M\u2082) p\u2081\n[PROOFSTEP]\nsimpa using h1\n[GOAL]\ncase mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\nh1' : \u2191(inl R M M\u2082) x1 \u2208 q\n\u22a2 (x1, x2) \u2208 q\n[PROOFSTEP]\nhave h2' : (LinearMap.inr R _ _) x2 \u2208 q := by\n apply hK\n simpa using h2\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\nh1' : \u2191(inl R M M\u2082) x1 \u2208 q\n\u22a2 \u2191(inr R M M\u2082) x2 \u2208 q\n[PROOFSTEP]\napply hK\n[GOAL]\ncase a\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\nh1' : \u2191(inl R M M\u2082) x1 \u2208 q\n\u22a2 \u2191(inr R M M\u2082) x2 \u2208 map (inr R M M\u2082) p\u2082\n[PROOFSTEP]\nsimpa using h2\n[GOAL]\ncase mpr.intro.mk.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq\u271d : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\nq : Submodule R (M \u00d7 M\u2082)\nhH : map (inl R M M\u2082) p\u2081 \u2264 q\nhK : map (inr R M M\u2082) p\u2082 \u2264 q\nx1 : M\nx2 : M\u2082\nh1 : (x1, x2).fst \u2208 \u2191p\u2081\nh2 : (x1, x2).snd \u2208 \u2191p\u2082\nh1' : \u2191(inl R M M\u2082) x1 \u2208 q\nh2' : \u2191(inr R M M\u2082) x2 \u2208 q\n\u22a2 (x1, x2) \u2208 q\n[PROOFSTEP]\nsimpa using add_mem h1' h2'\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\n\u22a2 prod p\u2081 p\u2082 = \u22a5 \u2194 p\u2081 = \u22a5 \u2227 p\u2082 = \u22a5\n[PROOFSTEP]\nsimp only [eq_bot_iff, prod_le_iff, (gc_map_comap _).le_iff_le, comap_bot, ker_inl, ker_inr]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\np\u2081 : Submodule R M\np\u2082 : Submodule R M\u2082\n\u22a2 prod p\u2081 p\u2082 = \u22a4 \u2194 p\u2081 = \u22a4 \u2227 p\u2082 = \u22a4\n[PROOFSTEP]\nsimp only [eq_top_iff, le_prod_iff, \u2190 (gc_map_comap _).le_iff_le, map_top, range_fst, range_snd]\n[GOAL]\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\n\u22a2 LinearMap.comp (LinearMap.fst R N M) \u2191(prodComm R M N) = LinearMap.snd R M N\n[PROOFSTEP]\next\n[GOAL]\ncase hl.h\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx\u271d : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (LinearMap.fst R N M) \u2191(prodComm R M N)) (LinearMap.inl R M N)) x\u271d =\n \u2191(LinearMap.comp (LinearMap.snd R M N) (LinearMap.inl R M N)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hr.h\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx\u271d : N\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (LinearMap.fst R N M) \u2191(prodComm R M N)) (LinearMap.inr R M N)) x\u271d =\n \u2191(LinearMap.comp (LinearMap.snd R M N) (LinearMap.inr R M N)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\n\u22a2 LinearMap.comp (LinearMap.snd R N M) \u2191(prodComm R M N) = LinearMap.fst R M N\n[PROOFSTEP]\next\n[GOAL]\ncase hl.h\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx\u271d : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (LinearMap.snd R N M) \u2191(prodComm R M N)) (LinearMap.inl R M N)) x\u271d =\n \u2191(LinearMap.comp (LinearMap.fst R M N) (LinearMap.inl R M N)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hr.h\nN : Type u_3\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx\u271d : N\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (LinearMap.snd R N M) \u2191(prodComm R M N)) (LinearMap.inr R M N)) x\u271d =\n \u2191(LinearMap.comp (LinearMap.fst R M N) (LinearMap.inr R M N)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommGroup M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R M\u2082\nmodule_M\u2083 : Module R M\u2083\nmodule_M\u2084 : Module R M\u2084\ne\u2081 : M \u2243\u2097[R] M\u2082\ne\u2082 : M\u2083 \u2243\u2097[R] M\u2084\nf : M \u2192\u2097[R] M\u2084\nsrc\u271d : M \u00d7 M\u2083 \u2192\u2097[R] M\u2082 \u00d7 M\u2084 :=\n LinearMap.prod (LinearMap.comp (\u2191e\u2081) (LinearMap.fst R M M\u2083))\n (LinearMap.comp (\u2191e\u2082) (LinearMap.snd R M M\u2083) + LinearMap.comp f (LinearMap.fst R M M\u2083))\np : M \u00d7 M\u2083\n\u22a2 (fun p => (\u2191(symm e\u2081) p.fst, \u2191(symm e\u2082) (p.snd - \u2191f (\u2191(symm e\u2081) p.fst))))\n (AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : M \u00d7 M\u2083),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n p) =\n p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommGroup M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R M\u2082\nmodule_M\u2083 : Module R M\u2083\nmodule_M\u2084 : Module R M\u2084\ne\u2081 : M \u2243\u2097[R] M\u2082\ne\u2082 : M\u2083 \u2243\u2097[R] M\u2084\nf : M \u2192\u2097[R] M\u2084\nsrc\u271d : M \u00d7 M\u2083 \u2192\u2097[R] M\u2082 \u00d7 M\u2084 :=\n LinearMap.prod (LinearMap.comp (\u2191e\u2081) (LinearMap.fst R M M\u2083))\n (LinearMap.comp (\u2191e\u2082) (LinearMap.snd R M M\u2083) + LinearMap.comp f (LinearMap.fst R M M\u2083))\np : M\u2082 \u00d7 M\u2084\n\u22a2 AddHom.toFun\n { toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : R) (x : M \u00d7 M\u2083),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n ((fun p => (\u2191(symm e\u2081) p.fst, \u2191(symm e\u2082) (p.snd - \u2191f (\u2191(symm e\u2081) p.fst)))) p) =\n p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\n\u22a2 range (prod f g) = Submodule.prod (range f) (range g)\n[PROOFSTEP]\nrefine' le_antisymm (f.range_prod_le g) _\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\n\u22a2 Submodule.prod (range f) (range g) \u2264 range (prod f g)\n[PROOFSTEP]\nsimp only [SetLike.le_def, prod_apply, mem_range, SetLike.mem_coe, mem_prod, exists_imp, and_imp, Prod.forall, Pi.prod]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\n\u22a2 \u2200 (a : M\u2082) (b : M\u2083) (x : M), \u2191f x = a \u2192 \u2200 (x : M), \u2191g x = b \u2192 \u2203 y, (\u2191f y, \u2191g y) = (a, b)\n[PROOFSTEP]\nrintro _ _ x rfl y rfl\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\n\u22a2 \u2203 y_1, (\u2191f y_1, \u2191g y_1) = (\u2191f x, \u2191g y)\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff, \u2190 sub_mem_ker_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\n\u22a2 \u2203 y_1, y_1 - x \u2208 ker f \u2227 y_1 - y \u2208 ker g\n[PROOFSTEP]\nhave : y - x \u2208 ker f \u2294 ker g := by simp only [h, mem_top]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\n\u22a2 y - x \u2208 ker f \u2294 ker g\n[PROOFSTEP]\nsimp only [h, mem_top]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\nthis : y - x \u2208 ker f \u2294 ker g\n\u22a2 \u2203 y_1, y_1 - x \u2208 ker f \u2227 y_1 - y \u2208 ker g\n[PROOFSTEP]\nrcases mem_sup.1 this with \u27e8x', hx', y', hy', H\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\nthis : y - x \u2208 ker f \u2294 ker g\nx' : M\nhx' : x' \u2208 ker f\ny' : M\nhy' : y' \u2208 ker g\nH : x' + y' = y - x\n\u22a2 \u2203 y_1, y_1 - x \u2208 ker f \u2227 y_1 - y \u2208 ker g\n[PROOFSTEP]\nrefine' \u27e8x' + x, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\nthis : y - x \u2208 ker f \u2294 ker g\nx' : M\nhx' : x' \u2208 ker f\ny' : M\nhy' : y' \u2208 ker g\nH : x' + y' = y - x\n\u22a2 x' + x - x \u2208 ker f\n[PROOFSTEP]\nrwa [add_sub_cancel]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M \u2192\u2097[R] M\u2083\nh : ker f \u2294 ker g = \u22a4\nx y : M\nthis : y - x \u2208 ker f \u2294 ker g\nx' : M\nhx' : x' \u2208 ker f\ny' : M\nhy' : y' \u2208 ker g\nH : x' + y' = y - x\n\u22a2 x' + x - y \u2208 ker g\n[PROOFSTEP]\nsimp [\u2190 eq_sub_iff_add_eq.1 H, map_add, add_left_inj, self_eq_add_right, mem_ker.mp hy']\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 \u2191OrderDual.toDual (tunnel' f i n).fst \u2264 \u2191OrderDual.toDual (tunnel' f i (n + 1)).fst\n[PROOFSTEP]\ndsimp [tunnel', tunnelAux]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 \u2191OrderDual.toDual (tunnel' f i n).fst \u2264\n \u2191OrderDual.toDual\n (Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) \u2191(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.fst R M N))\n[PROOFSTEP]\nrw [Submodule.map_comp, Submodule.map_comp]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 \u2191OrderDual.toDual (tunnel' f i n).fst \u2264\n \u2191OrderDual.toDual\n (Submodule.map (Submodule.subtype (tunnel' f i n).fst)\n (Submodule.map (\u2191(LinearEquiv.symm (tunnel' f i n).snd)) (Submodule.map f (Submodule.fst R M N))))\n[PROOFSTEP]\napply Submodule.map_subtype_le\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 tailing f i n \u2264 \u2191OrderDual.ofDual (\u2191(tunnel f i) n)\n[PROOFSTEP]\ndsimp [tailing, tunnelAux]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) \u2191(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.snd R M N) \u2264\n \u2191OrderDual.ofDual (\u2191(tunnel f i) n)\n[PROOFSTEP]\nrw [Submodule.map_comp, Submodule.map_comp]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Submodule.map (Submodule.subtype (tunnel' f i n).fst)\n (Submodule.map (\u2191(LinearEquiv.symm (tunnel' f i n).snd)) (Submodule.map f (Submodule.snd R M N))) \u2264\n \u2191OrderDual.ofDual (\u2191(tunnel f i) n)\n[PROOFSTEP]\napply Submodule.map_subtype_le\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Disjoint (tailing f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n[PROOFSTEP]\nrw [disjoint_iff]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 tailing f i n \u2293 \u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)) = \u22a5\n[PROOFSTEP]\ndsimp [tailing, tunnel, tunnel']\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Submodule.map (tunnelAux f (tunnel' f i n)) (Submodule.snd R M N) \u2293\n Submodule.map (tunnelAux f (tunnel' f i n)) (Submodule.fst R M N) =\n \u22a5\n[PROOFSTEP]\nerw [Submodule.map_inf_eq_map_inf_comap, Submodule.comap_map_eq_of_injective (tunnelAux_injective _ i _), inf_comm,\n Submodule.fst_inf_snd, Submodule.map_bot]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 tailing f i n \u2294 \u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)) \u2264 \u2191OrderDual.ofDual (\u2191(tunnel f i) n)\n[PROOFSTEP]\ndsimp [tailing, tunnel, tunnel', tunnelAux]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) \u2191(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.snd R M N) \u2294\n Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) \u2191(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.fst R M N) \u2264\n (tunnel' f i n).fst\n[PROOFSTEP]\nerw [\u2190 Submodule.map_sup, sup_comm, Submodule.fst_sup_snd, Submodule.map_comp, Submodule.map_comp]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Submodule.map (Submodule.subtype (tunnel' f i n).fst)\n (Submodule.map (\u2191(LinearEquiv.symm (tunnel' f i n).snd)) (Submodule.map f \u22a4)) \u2264\n (tunnel' f i n).fst\n[PROOFSTEP]\napply Submodule.map_subtype_le\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\n\u22a2 tailings f i 0 = tailing f i 0\n[PROOFSTEP]\nsimp [tailings]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 tailings f i (n + 1) = tailings f i n \u2294 tailing f i (n + 1)\n[PROOFSTEP]\nsimp [tailings]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\n\u22a2 Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\n\u22a2 Disjoint (tailings f i Nat.zero) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.zero + 1)))\n[PROOFSTEP]\nsimp only [tailings_zero]\n[GOAL]\ncase zero\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\n\u22a2 Disjoint (tailing f i 0) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.zero + 1)))\n[PROOFSTEP]\napply tailing_disjoint_tunnel_succ\n[GOAL]\ncase succ\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 Disjoint (tailings f i (Nat.succ n)) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)))\n[PROOFSTEP]\nsimp only [tailings_succ]\n[GOAL]\ncase succ\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 Disjoint (tailings f i n \u2294 tailing f i (n + 1)) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)))\n[PROOFSTEP]\nrefine' Disjoint.disjoint_sup_left_of_disjoint_sup_right _ _\n[GOAL]\ncase succ.refine'_1\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 Disjoint (tailing f i (n + 1)) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)))\ncase succ.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 Disjoint (tailings f i n) (tailing f i (n + 1) \u2294 \u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)))\n[PROOFSTEP]\napply tailing_disjoint_tunnel_succ\n[GOAL]\ncase succ.refine'_2\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 Disjoint (tailings f i n) (tailing f i (n + 1) \u2294 \u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)))\n[PROOFSTEP]\napply Disjoint.mono_right _ ih\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Ring R\nN : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\nf : M \u00d7 N \u2192\u2097[R] M\ni : Injective \u2191f\nn : \u2115\nih : Disjoint (tailings f i n) (\u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1)))\n\u22a2 tailing f i (n + 1) \u2294 \u2191OrderDual.ofDual (\u2191(tunnel f i) (Nat.succ n + 1)) \u2264 \u2191OrderDual.ofDual (\u2191(tunnel f i) (n + 1))\n[PROOFSTEP]\napply tailing_sup_tunnel_succ_le_tunnel\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\na\u271d b\u271d : M \u00d7 M\u2082\nha : a\u271d.snd = \u2191f a\u271d.fst\nhb : b\u271d.snd = \u2191f b\u271d.fst\n\u22a2 a\u271d + b\u271d \u2208 {p | p.snd = \u2191f p.fst}\n[PROOFSTEP]\nchange _ + _ = f (_ + _)\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\na\u271d b\u271d : M \u00d7 M\u2082\nha : a\u271d.snd = \u2191f a\u271d.fst\nhb : b\u271d.snd = \u2191f b\u271d.fst\n\u22a2 a\u271d.snd + b\u271d.snd = \u2191f (a\u271d.fst + b\u271d.fst)\n[PROOFSTEP]\nrw [map_add, ha, hb]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\nc : R\nx : M \u00d7 M\u2082\nhx : x.snd = \u2191f x.fst\n\u22a2 c \u2022 x \u2208\n {\n toAddSubsemigroup :=\n { carrier := {p | p.snd = \u2191f p.fst},\n add_mem' :=\n (_ : \u2200 {a b : M \u00d7 M\u2082}, a.snd = \u2191f a.fst \u2192 b.snd = \u2191f b.fst \u2192 a.snd + b.snd = \u2191f (a.fst + b.fst)) },\n zero_mem' := (_ : 0.snd = \u2191f 0.fst) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nchange _ \u2022 _ = f (_ \u2022 _)\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\nc : R\nx : M \u00d7 M\u2082\nhx : x.snd = \u2191f x.fst\n\u22a2 c \u2022 x.snd = \u2191f (c \u2022 x.fst)\n[PROOFSTEP]\nrw [map_smul, hx]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\n\u22a2 graph g = ker (coprod (-g) id)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\nx : M\u2083 \u00d7 M\u2084\n\u22a2 x \u2208 graph g \u2194 x \u2208 ker (coprod (-g) id)\n[PROOFSTEP]\nchange _ = _ \u2194 -g x.1 + x.2 = _\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\nx : M\u2083 \u00d7 M\u2084\n\u22a2 x.snd = \u2191g x.fst \u2194 -\u2191g x.fst + x.snd = 0\n[PROOFSTEP]\nrw [add_comm, add_neg_eq_zero]\n[GOAL]\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\n\u22a2 graph f = range (prod id f)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : AddCommGroup M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2082\ng : M\u2083 \u2192\u2097[R] M\u2084\nx : M \u00d7 M\u2082\n\u22a2 x \u2208 graph f \u2194 x \u2208 range (prod id f)\n[PROOFSTEP]\nexact \u27e8fun hx => \u27e8x.1, Prod.ext rfl hx.symm\u27e9, fun \u27e8u, hu\u27e9 => hu \u25b8 rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Prod", "llama_tokens": 69300, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.2969507681797326}} {"text": "[GOAL]\nR : Type u_1\n\ud835\udd5c\u2081 : Type u_2\n\ud835\udd5c\u2082 : Type u_3\nE : Type u_4\nF : Type u_5\ninst\u271d\u00b9\u00b3 : AddCommGroup E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : AddCommGroup F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : OrderedSemiring R\ninst\u271d\u2077 : NormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : NormedField \ud835\udd5c\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : ContinuousConstSMul R F\ninst\u271d\u00b9 : LocallyConvexSpace R F\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 R 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\n\u22a2 LocallyConvexSpace R (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nletI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n[GOAL]\nR : Type u_1\n\ud835\udd5c\u2081 : Type u_2\n\ud835\udd5c\u2082 : Type u_3\nE : Type u_4\nF : Type u_5\ninst\u271d\u00b9\u00b3 : AddCommGroup E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : AddCommGroup F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : OrderedSemiring R\ninst\u271d\u2077 : NormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : NormedField \ud835\udd5c\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : ContinuousConstSMul R F\ninst\u271d\u00b9 : LocallyConvexSpace R F\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 R 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\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 LocallyConvexSpace R (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\nhaveI : TopologicalAddGroup (E \u2192SL[\u03c3] F) := strongTopology.topologicalAddGroup _ _ _\n[GOAL]\nR : Type u_1\n\ud835\udd5c\u2081 : Type u_2\n\ud835\udd5c\u2082 : Type u_3\nE : Type u_4\nF : Type u_5\ninst\u271d\u00b9\u00b3 : AddCommGroup E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : AddCommGroup F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : OrderedSemiring R\ninst\u271d\u2077 : NormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : NormedField \ud835\udd5c\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : ContinuousConstSMul R F\ninst\u271d\u00b9 : LocallyConvexSpace R F\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 R 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\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : TopologicalAddGroup (E \u2192SL[\u03c3] F)\n\u22a2 LocallyConvexSpace R (E \u2192SL[\u03c3] F)\n[PROOFSTEP]\napply\n LocallyConvexSpace.ofBasisZero _ _ _ _\n (strongTopology.hasBasis_nhds_zero_of_basis _ _ _ h\ud835\udd16\u2081 h\ud835\udd16\u2082 (LocallyConvexSpace.convex_basis_zero R F)) _\n[GOAL]\nR : Type u_1\n\ud835\udd5c\u2081 : Type u_2\n\ud835\udd5c\u2082 : Type u_3\nE : Type u_4\nF : Type u_5\ninst\u271d\u00b9\u00b3 : AddCommGroup E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : AddCommGroup F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : OrderedSemiring R\ninst\u271d\u2077 : NormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : NormedField \ud835\udd5c\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : ContinuousConstSMul R F\ninst\u271d\u00b9 : LocallyConvexSpace R F\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 R 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\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : TopologicalAddGroup (E \u2192SL[\u03c3] F)\n\u22a2 \u2200 (i : Set E \u00d7 Set F),\n i.fst \u2208 \ud835\udd16 \u2227 i.snd \u2208 \ud835\udcdd 0 \u2227 Convex R i.snd \u2192 Convex R {f | \u2200 (x : E), x \u2208 i.fst \u2192 \u2191f x \u2208 _root_.id i.snd}\n[PROOFSTEP]\nrintro \u27e8S, V\u27e9 \u27e8_, _, hVconvex\u27e9 f hf g hg a b ha hb hab x hx\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\n\ud835\udd5c\u2081 : Type u_2\n\ud835\udd5c\u2082 : Type u_3\nE : Type u_4\nF : Type u_5\ninst\u271d\u00b9\u00b3 : AddCommGroup E\ninst\u271d\u00b9\u00b2 : TopologicalSpace E\ninst\u271d\u00b9\u00b9 : AddCommGroup F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : OrderedSemiring R\ninst\u271d\u2077 : NormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : NormedField \ud835\udd5c\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : ContinuousConstSMul R F\ninst\u271d\u00b9 : LocallyConvexSpace R F\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 R 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\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : TopologicalAddGroup (E \u2192SL[\u03c3] F)\nS : Set E\nV : Set F\nleft\u271d\u00b9 : (S, V).fst \u2208 \ud835\udd16\nleft\u271d : (S, V).snd \u2208 \ud835\udcdd 0\nhVconvex : Convex R (S, V).snd\nf : E \u2192SL[\u03c3] F\nhf : f \u2208 {f | \u2200 (x : E), x \u2208 (S, V).fst \u2192 \u2191f x \u2208 _root_.id (S, V).snd}\ng : E \u2192SL[\u03c3] F\nhg : g \u2208 {f | \u2200 (x : E), x \u2208 (S, V).fst \u2192 \u2191f x \u2208 _root_.id (S, V).snd}\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nx : E\nhx : x \u2208 (S, V).fst\n\u22a2 \u2191(a \u2022 f + b \u2022 g) x \u2208 _root_.id (S, V).snd\n[PROOFSTEP]\nexact hVconvex (hf x hx) (hg x hx) ha hb hab\n", "meta": {"mathlib_filename": "Mathlib.Analysis.LocallyConvex.StrongTopology", "llama_tokens": 2223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.3960681662740417, "lm_q1q2_score": 0.2966896033148437}} {"text": "[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\n\u22a2 InjOn (\u2191(embeddingPiTangent f)) s\n[PROOFSTEP]\nintro x hx y _ h\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh : \u2191(embeddingPiTangent f) x = \u2191(embeddingPiTangent f) y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [embeddingPiTangent_coe, funext_iff] at h \n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh :\n \u2200 (a : \u03b9),\n (\u2191(toFun s f a) x \u2022 \u2191(extChartAt I (c s f a)) x, \u2191(toFun s f a) x) =\n (\u2191(toFun s f a) y \u2022 \u2191(extChartAt I (c s f a)) y, \u2191(toFun s f a) y)\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8h\u2081, h\u2082\u27e9 := Prod.mk.inj_iff.1 (h (f.ind x hx))\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh :\n \u2200 (a : \u03b9),\n (\u2191(toFun s f a) x \u2022 \u2191(extChartAt I (c s f a)) x, \u2191(toFun s f a) x) =\n (\u2191(toFun s f a) y \u2022 \u2191(extChartAt I (c s f a)) y, \u2191(toFun s f a) y)\nh\u2081 :\n \u2191(toFun s f (ind f x hx)) x \u2022 \u2191(extChartAt I (c s f (ind f x hx))) x =\n \u2191(toFun s f (ind f x hx)) y \u2022 \u2191(extChartAt I (c s f (ind f x hx))) y\nh\u2082 : \u2191(toFun s f (ind f x hx)) x = \u2191(toFun s f (ind f x hx)) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [f.apply_ind x hx] at h\u2082 \n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh :\n \u2200 (a : \u03b9),\n (\u2191(toFun s f a) x \u2022 \u2191(extChartAt I (c s f a)) x, \u2191(toFun s f a) x) =\n (\u2191(toFun s f a) y \u2022 \u2191(extChartAt I (c s f a)) y, \u2191(toFun s f a) y)\nh\u2081 :\n \u2191(toFun s f (ind f x hx)) x \u2022 \u2191(extChartAt I (c s f (ind f x hx))) x =\n \u2191(toFun s f (ind f x hx)) y \u2022 \u2191(extChartAt I (c s f (ind f x hx))) y\nh\u2082 : 1 = \u2191(toFun s f (ind f x hx)) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 h\u2082, f.apply_ind x hx, one_smul, one_smul] at h\u2081 \n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh :\n \u2200 (a : \u03b9),\n (\u2191(toFun s f a) x \u2022 \u2191(extChartAt I (c s f a)) x, \u2191(toFun s f a) x) =\n (\u2191(toFun s f a) y \u2022 \u2191(extChartAt I (c s f a)) y, \u2191(toFun s f a) y)\nh\u2081 : \u2191(extChartAt I (c s f (ind f x hx))) x = \u2191(extChartAt I (c s f (ind f x hx))) y\nh\u2082 : 1 = \u2191(toFun s f (ind f x hx)) y\n\u22a2 x = y\n[PROOFSTEP]\nhave := f.mem_extChartAt_source_of_eq_one h\u2082.symm\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\ny : M\na\u271d : y \u2208 s\nh :\n \u2200 (a : \u03b9),\n (\u2191(toFun s f a) x \u2022 \u2191(extChartAt I (c s f a)) x, \u2191(toFun s f a) x) =\n (\u2191(toFun s f a) y \u2022 \u2191(extChartAt I (c s f a)) y, \u2191(toFun s f a) y)\nh\u2081 : \u2191(extChartAt I (c s f (ind f x hx))) x = \u2191(extChartAt I (c s f (ind f x hx))) y\nh\u2082 : 1 = \u2191(toFun s f (ind f x hx)) y\nthis : y \u2208 (extChartAt I (c s f (ind f x hx))).source\n\u22a2 x = y\n[PROOFSTEP]\nexact (extChartAt I (f.c _)).injOn (f.mem_extChartAt_ind_source x hx) this h\u2081\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\n\u22a2 ContinuousLinearMap.comp\n (ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx)))\n (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) =\n mfderiv I I (\u2191(chartAt H (c s f (ind f x hx)))) x\n[PROOFSTEP]\nset L :=\n (ContinuousLinearMap.fst \u211d E \u211d).comp\n (@ContinuousLinearMap.proj \u211d _ \u03b9 (fun _ => E \u00d7 \u211d) _ _ (fun _ => inferInstance) (f.ind x hx))\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\n\u22a2 ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) =\n mfderiv I I (\u2191(chartAt H (c s f (ind f x hx)))) x\n[PROOFSTEP]\nhave := L.hasMFDerivAt.comp x f.embeddingPiTangent.smooth.mdifferentiableAt.hasMFDerivAt\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\nthis :\n HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x\n (ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\n\u22a2 ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) =\n mfderiv I I (\u2191(chartAt H (c s f (ind f x hx)))) x\n[PROOFSTEP]\nconvert hasMFDerivAt_unique this _\n[GOAL]\ncase convert_2\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\nthis :\n HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x\n (ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\n\u22a2 HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x (mfderiv I I (\u2191(chartAt H (c s f (ind f x hx)))) x)\n[PROOFSTEP]\nrefine' (hasMFDerivAt_extChartAt I (f.mem_chartAt_ind_source x hx)).congr_of_eventuallyEq _\n[GOAL]\ncase convert_2\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\nthis :\n HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x\n (ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\n\u22a2 \u2191L \u2218 \u2191(embeddingPiTangent f) =\u1da0[\ud835\udcdd x] \u2191(extChartAt I (c s f (ind f x hx)))\n[PROOFSTEP]\nrefine' (f.eventuallyEq_one x hx).mono fun y hy => _\n[GOAL]\ncase convert_2\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\nthis :\n HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x\n (ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\ny : M\nhy : \u2191(toFun s f (ind f x hx)) y = OfNat.ofNat 1 y\n\u22a2 (\u2191L \u2218 \u2191(embeddingPiTangent f)) y = \u2191(extChartAt I (c s f (ind f x hx))) y\n[PROOFSTEP]\nsimp only [embeddingPiTangent_coe, ContinuousLinearMap.coe_comp', (\u00b7 \u2218 \u00b7), ContinuousLinearMap.coe_fst',\n ContinuousLinearMap.proj_apply]\n[GOAL]\ncase convert_2\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\nL : (\u03b9 \u2192 E \u00d7 \u211d) \u2192L[\u211d] E :=\n ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx))\nthis :\n HasMFDerivAt I \ud835\udcd8(\u211d, E) (\u2191L \u2218 \u2191(embeddingPiTangent f)) x\n (ContinuousLinearMap.comp L (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\ny : M\nhy : \u2191(toFun s f (ind f x hx)) y = OfNat.ofNat 1 y\n\u22a2 \u2191(toFun s f (ind f x hx)) y \u2022 \u2191(extChartAt I (c s f (ind f x hx))) y = \u2191(extChartAt I (c s f (ind f x hx))) y\n[PROOFSTEP]\nrw [hy, Pi.one_apply, one_smul]\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\n\u22a2 LinearMap.ker (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) = \u22a5\n[PROOFSTEP]\napply bot_unique\n[GOAL]\ncase h\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\n\u22a2 LinearMap.ker (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) \u2264 \u22a5\n[PROOFSTEP]\nrw [\u2190 (mdifferentiable_chart I (f.c (f.ind x hx))).ker_mfderiv_eq_bot (f.mem_chartAt_ind_source x hx), \u2190\n comp_embeddingPiTangent_mfderiv]\n[GOAL]\ncase h\n\u03b9 : Type u\u03b9\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\ninst\u271d : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf : SmoothBumpCovering \u03b9 I M s\nx : M\nhx : x \u2208 s\n\u22a2 LinearMap.ker (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x) \u2264\n LinearMap.ker\n (ContinuousLinearMap.comp\n (ContinuousLinearMap.comp (ContinuousLinearMap.fst \u211d E \u211d) (ContinuousLinearMap.proj (ind f x hx)))\n (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\n[PROOFSTEP]\nexact LinearMap.ker_le_ker_comp _ _\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nset F := EuclideanSpace \u211d (Fin <| finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d))\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nletI : IsNoetherian \u211d (E \u00d7 \u211d) := IsNoetherian.iff_fg.2 inferInstance\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\nthis : IsNoetherian \u211d (E \u00d7 \u211d) := Iff.mpr IsNoetherian.iff_fg inferInstance\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nletI : FiniteDimensional \u211d (\u03b9 \u2192 E \u00d7 \u211d) := IsNoetherian.iff_fg.1 inferInstance\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\nthis\u271d : IsNoetherian \u211d (E \u00d7 \u211d) := Iff.mpr IsNoetherian.iff_fg inferInstance\nthis : FiniteDimensional \u211d (\u03b9 \u2192 E \u00d7 \u211d) := Iff.mp IsNoetherian.iff_fg inferInstance\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nset eEF : (\u03b9 \u2192 E \u00d7 \u211d) \u2243L[\u211d] F := ContinuousLinearEquiv.ofFinrankEq finrank_euclideanSpace_fin.symm\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\nthis\u271d : IsNoetherian \u211d (E \u00d7 \u211d) := Iff.mpr IsNoetherian.iff_fg inferInstance\nthis : FiniteDimensional \u211d (\u03b9 \u2192 E \u00d7 \u211d) := Iff.mp IsNoetherian.iff_fg inferInstance\neEF : (\u03b9 \u2192 E \u00d7 \u211d) \u2243L[\u211d] F :=\n ContinuousLinearEquiv.ofFinrankEq\n (_ : finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d) = finrank \u211d (EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))))\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n Injective e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nrefine\n \u27e8_, eEF \u2218 f.embeddingPiTangent, eEF.toDiffeomorph.smooth.comp f.embeddingPiTangent.smooth,\n eEF.injective.comp f.embeddingPiTangent_injective, fun x => ?_\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\nthis\u271d : IsNoetherian \u211d (E \u00d7 \u211d) := Iff.mpr IsNoetherian.iff_fg inferInstance\nthis : FiniteDimensional \u211d (\u03b9 \u2192 E \u00d7 \u211d) := Iff.mp IsNoetherian.iff_fg inferInstance\neEF : (\u03b9 \u2192 E \u00d7 \u211d) \u2243L[\u211d] F :=\n ContinuousLinearEquiv.ofFinrankEq\n (_ : finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d) = finrank \u211d (EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))))\nx : M\n\u22a2 Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))) (\u2191eEF \u2218 \u2191(embeddingPiTangent f)) x)\n[PROOFSTEP]\nrw [mfderiv_comp _ eEF.differentiableAt.mdifferentiableAt f.embeddingPiTangent.smooth.mdifferentiableAt, eEF.mfderiv_eq]\n[GOAL]\ncase intro\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\nhi : Fintype \u03b9\ns : Set M\nf\u271d : SmoothBumpCovering \u03b9 I M s\ninst\u271d : Finite \u03b9\nf : SmoothBumpCovering \u03b9 I M univ\nval\u271d : Fintype \u03b9\nF : Type := EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))\nthis\u271d : IsNoetherian \u211d (E \u00d7 \u211d) := Iff.mpr IsNoetherian.iff_fg inferInstance\nthis : FiniteDimensional \u211d (\u03b9 \u2192 E \u00d7 \u211d) := Iff.mp IsNoetherian.iff_fg inferInstance\neEF : (\u03b9 \u2192 E \u00d7 \u211d) \u2243L[\u211d] F :=\n ContinuousLinearEquiv.ofFinrankEq\n (_ : finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d) = finrank \u211d (EuclideanSpace \u211d (Fin (finrank \u211d (\u03b9 \u2192 E \u00d7 \u211d)))))\nx : M\n\u22a2 Injective \u2191(ContinuousLinearMap.comp (\u2191eEF) (mfderiv I \ud835\udcd8(\u211d, \u03b9 \u2192 E \u00d7 \u211d) (\u2191(embeddingPiTangent f)) x))\n[PROOFSTEP]\nexact eEF.injective.comp (f.embeddingPiTangent_injective_mfderiv _ trivial)\n[GOAL]\n\u03b9 : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\ninst\u271d : CompactSpace M\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n ClosedEmbedding e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nrcases SmoothBumpCovering.exists_isSubordinate I isClosed_univ fun (x : M) _ => univ_mem with \u27e8\u03b9, f, -\u27e9\n[GOAL]\ncase intro.intro\n\u03b9\u271d : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\ninst\u271d : CompactSpace M\n\u03b9 : Type uM\nf : SmoothBumpCovering \u03b9 I M univ\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n ClosedEmbedding e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nhaveI := f.fintype\n[GOAL]\ncase intro.intro\n\u03b9\u271d : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\ninst\u271d : CompactSpace M\n\u03b9 : Type uM\nf : SmoothBumpCovering \u03b9 I M univ\nthis : Fintype \u03b9\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n ClosedEmbedding e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nrcases f.exists_immersion_euclidean with \u27e8n, e, hsmooth, hinj, hinj_mfderiv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9\u271d : Type u\u03b9\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\ninst\u271d\u00b9 : T2Space M\ninst\u271d : CompactSpace M\n\u03b9 : Type uM\nf : SmoothBumpCovering \u03b9 I M univ\nthis : Fintype \u03b9\nn : \u2115\ne : M \u2192 EuclideanSpace \u211d (Fin n)\nhsmooth : Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e\nhinj : Injective e\nhinj_mfderiv : \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n\u22a2 \u2203 n e,\n Smooth I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e \u2227\n ClosedEmbedding e \u2227 \u2200 (x : M), Injective \u2191(mfderiv I \ud835\udcd8(\u211d, EuclideanSpace \u211d (Fin n)) e x)\n[PROOFSTEP]\nexact \u27e8n, e, hsmooth, hsmooth.continuous.closedEmbedding hinj, hinj_mfderiv\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.WhitneyEmbedding", "llama_tokens": 10852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.2959585594837405}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\n\u22a2 \u2200 (y : { x // x \u2208 nonZeroDivisors \u2124 }), IsUnit (\u2191(algebraMap \u2124 \u211a) \u2191y)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx : \u2124\nhx : x \u2208 nonZeroDivisors \u2124\n\u22a2 IsUnit (\u2191(algebraMap \u2124 \u211a) \u2191{ val := x, property := hx })\n[PROOFSTEP]\nrw [mem_nonZeroDivisors_iff_ne_zero] at hx \n[GOAL]\ncase mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx : \u2124\nhx\u271d : x \u2208 nonZeroDivisors \u2124\nhx : x \u2260 0\n\u22a2 IsUnit (\u2191(algebraMap \u2124 \u211a) \u2191{ val := x, property := hx\u271d })\n[PROOFSTEP]\nsimpa only [eq_intCast, isUnit_iff_ne_zero, Int.cast_eq_zero, Ne.def, Subtype.coe_mk] using hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\n\u22a2 \u2200 (z : \u211a), \u2203 x, z * \u2191(algebraMap \u2124 \u211a) \u2191x.snd = \u2191(algebraMap \u2124 \u211a) x.fst\n[PROOFSTEP]\nrintro \u27e8n, d, hd, h\u27e9\n[GOAL]\ncase mk'\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nn : \u2124\nd : \u2115\nhd : d \u2260 0\nh : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2203 x, mk' n d * \u2191(algebraMap \u2124 \u211a) \u2191x.snd = \u2191(algebraMap \u2124 \u211a) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8n, \u27e8d, _\u27e9\u27e9, Rat.mul_den_eq_num\u27e9\n[GOAL]\ncase mk'\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nn : \u2124\nd : \u2115\nhd : d \u2260 0\nh : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2191d \u2208 nonZeroDivisors \u2124\n[PROOFSTEP]\nrw [mem_nonZeroDivisors_iff_ne_zero, Int.coe_nat_ne_zero_iff_pos]\n[GOAL]\ncase mk'\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nn : \u2124\nd : \u2115\nhd : d \u2260 0\nh : Nat.coprime (Int.natAbs n) d\n\u22a2 0 < d\n[PROOFSTEP]\nexact Nat.zero_lt_of_ne_zero hd\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\n\u22a2 \u2200 {x y : \u2124}, \u2191(algebraMap \u2124 \u211a) x = \u2191(algebraMap \u2124 \u211a) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nintro x y\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y : \u2124\n\u22a2 \u2191(algebraMap \u2124 \u211a) x = \u2191(algebraMap \u2124 \u211a) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [eq_intCast, eq_intCast, Int.cast_inj]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y : \u2124\n\u22a2 x = y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\napply Iff.intro\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y : \u2124\n\u22a2 x = y \u2192 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx : \u2124\n\u22a2 \u2203 c, \u2191c * x = \u2191c * x\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y : \u2124\n\u22a2 (\u2203 c, \u2191c * x = \u2191c * y) \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8\u27e8c, hc\u27e9, h\u27e9\n[GOAL]\ncase mpr.intro.mk\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y c : \u2124\nhc : c \u2208 nonZeroDivisors \u2124\nh : \u2191{ val := c, property := hc } * x = \u2191{ val := c, property := hc } * y\n\u22a2 x = y\n[PROOFSTEP]\napply mul_left_cancel\u2080 _ h\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nx y c : \u2124\nhc : c \u2208 nonZeroDivisors \u2124\nh : \u2191{ val := c, property := hc } * x = \u2191{ val := c, property := hc } * y\n\u22a2 \u2191{ val := c, property := hc } \u2260 0\n[PROOFSTEP]\nrwa [mem_nonZeroDivisors_iff_ne_zero] at hc \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nhx : x \u2260 0\n\u22a2 x * IsFractionRing.inv A x = 1\n[PROOFSTEP]\nrw [IsFractionRing.inv, dif_neg hx, \u2190\n IsUnit.mul_left_inj\n (map_units K\n \u27e8(sec _ x).1,\n mem_nonZeroDivisors_iff_ne_zero.2 fun h0 => hx <| eq_zero_of_fst_eq_zero (sec_spec (nonZeroDivisors A) x) h0\u27e9),\n one_mul, mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nhx : x \u2260 0\n\u22a2 x *\n (mk' K \u2191(sec (nonZeroDivisors A) x).snd\n { val := (sec (nonZeroDivisors A) x).fst,\n property := (_ : (sec (nonZeroDivisors A) x).fst \u2208 nonZeroDivisors A) } *\n \u2191(algebraMap A K)\n \u2191{ val := (sec (nonZeroDivisors A) x).fst,\n property := (_ : (sec (nonZeroDivisors A) x).fst \u2208 nonZeroDivisors A) }) =\n \u2191(algebraMap A K)\n \u2191{ val := (sec (nonZeroDivisors A) x).fst, property := (_ : (sec (nonZeroDivisors A) x).fst \u2208 nonZeroDivisors A) }\n[PROOFSTEP]\nrw [mk'_spec, \u2190 eq_mk'_iff_mul_eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nhx : x \u2260 0\n\u22a2 x =\n mk'\n ((fun x => K)\n \u2191{ val := (sec (nonZeroDivisors A) x).fst,\n property := (_ : (sec (nonZeroDivisors A) x).fst \u2208 nonZeroDivisors A) })\n (\u2191{ val := (sec (nonZeroDivisors A) x).fst,\n property := (_ : (sec (nonZeroDivisors A) x).fst \u2208 nonZeroDivisors A) })\n (sec (nonZeroDivisors A) x).snd\n[PROOFSTEP]\nexact (mk'_sec _ x).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nsrc\u271d\u00b9 : IsDomain K := IsFractionRing.isDomain A\nsrc\u271d : CommRing K := inferInstanceAs (CommRing K)\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nchange IsFractionRing.inv A (0 : K) = 0\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nsrc\u271d\u00b9 : IsDomain K := IsFractionRing.isDomain A\nsrc\u271d : CommRing K := inferInstanceAs (CommRing K)\n\u22a2 IsFractionRing.inv A 0 = 0\n[PROOFSTEP]\nrw [IsFractionRing.inv]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommRing R\nM : 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\nA : Type u_4\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : IsDomain A\nK : Type u_5\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsFractionRing R K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nsrc\u271d\u00b9 : IsDomain K := IsFractionRing.isDomain A\nsrc\u271d : CommRing K := inferInstanceAs (CommRing K)\n\u22a2 (if h : 0 = 0 then 0\n else\n mk' K \u2191(sec (nonZeroDivisors A) 0).snd\n { val := (sec (nonZeroDivisors A) 0).fst,\n property := (_ : (sec (nonZeroDivisors A) 0).fst \u2208 nonZeroDivisors A) }) =\n 0\n[PROOFSTEP]\nexact dif_pos rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nz : K\nx y : A\nhy : y \u2208 nonZeroDivisors A\nh : mk' K x { val := y, property := hy } = z\n\u22a2 \u2191(algebraMap A K) x / \u2191(algebraMap A K) y = z\n[PROOFSTEP]\nrwa [mk'_eq_div] at h \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx : R\ny : { x // x \u2208 nonZeroDivisors R }\n\u22a2 mk' K x y = 0 \u2194 x = 0\n[PROOFSTEP]\nrefine' \u27e8fun hxy => _, fun h => by rw [h, mk'_zero]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx : R\ny : { x // x \u2208 nonZeroDivisors R }\nh : x = 0\n\u22a2 mk' K x y = 0\n[PROOFSTEP]\nrw [h, mk'_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx : R\ny : { x // x \u2208 nonZeroDivisors R }\nhxy : mk' K x y = 0\n\u22a2 x = 0\n[PROOFSTEP]\nsimp_rw [mk'_eq_zero_iff, mul_left_coe_nonZeroDivisors_eq_zero_iff] at hxy \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx : R\ny : { x // x \u2208 nonZeroDivisors R }\nhxy : \u2203 m, x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact (exists_const _).mp hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\n\u22a2 mk' K x y = 1 \u2194 x = \u2191y\n[PROOFSTEP]\nrefine' \u27e8_, fun hxy => by rw [hxy, mk'_self']\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\nhxy : x = \u2191y\n\u22a2 mk' K x y = 1\n[PROOFSTEP]\nrw [hxy, mk'_self']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\n\u22a2 mk' K x y = 1 \u2192 x = \u2191y\n[PROOFSTEP]\nintro hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\nhxy : mk' K x y = 1\n\u22a2 x = \u2191y\n[PROOFSTEP]\nhave hy : (algebraMap A K) \u2191y \u2260 (0 : K) := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors y.property\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\nhxy : mk' K x y = 1\nhy : \u2191(algebraMap A K) \u2191y \u2260 0\n\u22a2 x = \u2191y\n[PROOFSTEP]\nrw [IsFractionRing.mk'_eq_div, div_eq_one_iff_eq hy] at hxy \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\nhxy : \u2191(algebraMap A K) x = \u2191(algebraMap A K) \u2191y\nhy : \u2191(algebraMap A K) \u2191y \u2260 0\n\u22a2 x = \u2191y\n[PROOFSTEP]\nexact IsFractionRing.injective A K hxy\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nhg : Injective \u2191g\nx : A\ny : { x // x \u2208 nonZeroDivisors A }\n\u22a2 \u2191(lift hg) (mk' K x y) = \u2191g x / \u2191g \u2191y\n[PROOFSTEP]\nsimp only [mk'_eq_div, map_div\u2080, lift_algebraMap]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra B L\ninst\u271d : IsFractionRing B L\nh : A \u2243+* B\n\u22a2 Submonoid.map (RingEquiv.toMonoidHom h) (?m.392285 h) = ?m.392287 h\n[PROOFSTEP]\next b\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra B L\ninst\u271d : IsFractionRing B L\nh : A \u2243+* B\nb : B\n\u22a2 b \u2208 Submonoid.map (RingEquiv.toMonoidHom h) (?m.392285 h) \u2194 b \u2208 ?m.392287 h\n[PROOFSTEP]\nshow b \u2208 h.toEquiv '' _ \u2194 _\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra B L\ninst\u271d : IsFractionRing B L\nh : A \u2243+* B\nb : B\n\u22a2 b \u2208 \u2191h.toEquiv '' \u2191(?m.392285 h) \u2194 b \u2208 ?m.392287 h\n[PROOFSTEP]\nerw [h.toEquiv.image_eq_preimage, Set.preimage, Set.mem_setOf_eq, mem_nonZeroDivisors_iff_ne_zero,\n mem_nonZeroDivisors_iff_ne_zero]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9\u2070 : CommRing P\nA : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : IsDomain B\ninst\u271d\u2075 : Field K\nL : Type u_7\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ng : A \u2192+* L\ninst\u271d\u00b9 : Algebra B L\ninst\u271d : IsFractionRing B L\nh : A \u2243+* B\nb : B\n\u22a2 \u2191h.symm b \u2260 0 \u2194 b \u2260 0\n[PROOFSTEP]\nexact h.symm.map_ne_zero_iff\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 IsFractionRing R S \u2194 IsFractionRing P S\n[PROOFSTEP]\ndelta IsFractionRing\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 IsLocalization (nonZeroDivisors R) S \u2194 IsLocalization (nonZeroDivisors P) S\n[PROOFSTEP]\nconvert isLocalization_iff_of_base_ringEquiv (nonZeroDivisors R) S h\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 nonZeroDivisors P = Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194 x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)\n[PROOFSTEP]\nerw [Submonoid.map_equiv_eq_comap_symm]\n[GOAL]\ncase h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194\n x \u2208 Submonoid.comap (MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv h))) (nonZeroDivisors R)\n[PROOFSTEP]\nsimp only [MulEquiv.coe_toMonoidHom, RingEquiv.toMulEquiv_eq_coe, Submonoid.mem_comap]\n[GOAL]\ncase h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194 \u2191(MulEquiv.symm \u2191h) x \u2208 nonZeroDivisors R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_2.h.e'_3.h.mp\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2192 \u2191(MulEquiv.symm \u2191h) x \u2208 nonZeroDivisors R\n[PROOFSTEP]\nrintro hx z (hz : z * h.symm x = 0)\n[GOAL]\ncase h.e'_2.h.e'_3.h.mp\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : x \u2208 nonZeroDivisors P\nz : R\nhz : z * \u2191(RingEquiv.symm h) x = 0\n\u22a2 z = 0\n[PROOFSTEP]\nrw [\u2190 h.map_eq_zero_iff]\n[GOAL]\ncase h.e'_2.h.e'_3.h.mp\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : x \u2208 nonZeroDivisors P\nz : R\nhz : z * \u2191(RingEquiv.symm h) x = 0\n\u22a2 \u2191h z = 0\n[PROOFSTEP]\napply hx\n[GOAL]\ncase h.e'_2.h.e'_3.h.mp.a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : x \u2208 nonZeroDivisors P\nz : R\nhz : z * \u2191(RingEquiv.symm h) x = 0\n\u22a2 \u2191h z * x = 0\n[PROOFSTEP]\nsimpa only [h.map_zero, h.apply_symm_apply, h.map_mul] using congr_arg h hz\n[GOAL]\ncase h.e'_2.h.e'_3.h.mpr\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 \u2191(MulEquiv.symm \u2191h) x \u2208 nonZeroDivisors R \u2192 x \u2208 nonZeroDivisors P\n[PROOFSTEP]\nrintro (hx : h.symm x \u2208 _) z hz\n[GOAL]\ncase h.e'_2.h.e'_3.h.mpr\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : \u2191(RingEquiv.symm h) x \u2208 nonZeroDivisors R\nz : P\nhz : z * x = 0\n\u22a2 z = 0\n[PROOFSTEP]\nrw [\u2190 h.symm.map_eq_zero_iff]\n[GOAL]\ncase h.e'_2.h.e'_3.h.mpr\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : \u2191(RingEquiv.symm h) x \u2208 nonZeroDivisors R\nz : P\nhz : z * x = 0\n\u22a2 \u2191(RingEquiv.symm h) z = 0\n[PROOFSTEP]\napply hx\n[GOAL]\ncase h.e'_2.h.e'_3.h.mpr.a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_3\ninst\u271d\u2078 : CommRing P\nA : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type u_7\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : \u2191(RingEquiv.symm h) x \u2208 nonZeroDivisors R\nz : P\nhz : z * x = 0\n\u22a2 \u2191(RingEquiv.symm h) z * \u2191(RingEquiv.symm h) x = 0\n[PROOFSTEP]\nrw [\u2190 h.symm.map_mul, hz, h.symm.map_zero]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 Nontrivial S\n[PROOFSTEP]\napply nontrivial_of_ne\n[GOAL]\ncase h\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 ?x \u2260 ?y\ncase x\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\ncase y\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\nh : ?x = ?y\n\u22a2 False\ncase x\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\ncase y\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\n[PROOFSTEP]\napply @zero_ne_one R\n[GOAL]\ncase h.a\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\nh : ?x = ?y\n\u22a2 0 = 1\ncase x\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\ncase y\nR\u271d : Type u_1\ninst\u271d\u00b9\u2076 : CommRing R\u271d\nM : Submonoid R\u271d\nS\u271d : Type u_2\ninst\u271d\u00b9\u2075 : CommRing S\u271d\ninst\u271d\u00b9\u2074 : Algebra R\u271d S\u271d\nP : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing P\nA : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : IsDomain A\nK : Type u_5\nB : Type u_6\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : IsDomain B\ninst\u271d\u2078 : Field K\nL : Type u_7\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra A K\ninst\u271d\u2075 : IsFractionRing A K\ng : A \u2192+* L\nR : Type u_8\nS : Type u_9\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsFractionRing R S\n\u22a2 S\n[PROOFSTEP]\nexact\n IsLocalization.injective S (le_of_eq rfl) (((algebraMap R S).map_zero.trans h).trans (algebraMap R S).map_one.symm)\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\nA : Type u_4\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nK : Type u_5\nr : A\ns : { x // x \u2208 nonZeroDivisors A }\n\u22a2 Localization.mk r s = \u2191(algebraMap A (FractionRing A)) r / \u2191(algebraMap A (FractionRing A)) \u2191s\n[PROOFSTEP]\nrw [Localization.mk_eq_mk', IsFractionRing.mk'_eq_div]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\nK : Type u_5\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 NoZeroSMulDivisors R (FractionRing A)\n[PROOFSTEP]\napply NoZeroSMulDivisors.of_algebraMap_injective\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\nK : Type u_5\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 Function.Injective \u2191(algebraMap R (FractionRing A))\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq R A]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommRing P\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\nK : Type u_5\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 Function.Injective \u2191(RingHom.comp (algebraMap A (FractionRing A)) (algebraMap R A))\n[PROOFSTEP]\napply\n Function.Injective.comp (NoZeroSMulDivisors.algebraMap_injective A (FractionRing A))\n (NoZeroSMulDivisors.algebraMap_injective R A)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.FractionRing", "llama_tokens": 17327, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6150878696277512, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.29553660648204527}} {"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nht : TopologicalSpace \u03b1\nh : PolishSpace \u03b1\n\u22a2 CompleteSpace \u03b1\n[PROOFSTEP]\nconvert h.complete.choose_spec.2\n[GOAL]\ncase h.e'_2.h.e'_2.h.e'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\nht : TopologicalSpace \u03b1\nh : PolishSpace \u03b1\n\u22a2 polishSpaceMetric \u03b1 = Exists.choose (_ : \u2203 m, UniformSpace.toTopologicalSpace = ht \u2227 CompleteSpace \u03b1)\n[PROOFSTEP]\nexact MetricSpace.replaceTopology_eq _ _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\n\u22a2 T2Space \u03b1\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b1\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\n\u22a2 T2Space \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d : \u2200 (i : \u03b9), PolishSpace (E i)\n\u22a2 PolishSpace ((i : \u03b9) \u2192 E i)\n[PROOFSTEP]\ncases nonempty_encodable \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d : \u2200 (i : \u03b9), PolishSpace (E i)\nval\u271d : Encodable \u03b9\n\u22a2 PolishSpace ((i : \u03b9) \u2192 E i)\n[PROOFSTEP]\nletI := fun i => upgradePolishSpace (E i)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d : \u2200 (i : \u03b9), PolishSpace (E i)\nval\u271d : Encodable \u03b9\nthis : (i : \u03b9) \u2192 UpgradedPolishSpace (E i) := fun i => upgradePolishSpace (E i)\n\u22a2 PolishSpace ((i : \u03b9) \u2192 E i)\n[PROOFSTEP]\nletI : MetricSpace (\u2200 i, E i) := PiCountable.metricSpace\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : Countable \u03b9\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d : \u2200 (i : \u03b9), PolishSpace (E i)\nval\u271d : Encodable \u03b9\nthis\u271d : (i : \u03b9) \u2192 UpgradedPolishSpace (E i) := fun i => upgradePolishSpace (E i)\nthis : MetricSpace ((i : \u03b9) \u2192 E i) := PiCountable.metricSpace\n\u22a2 PolishSpace ((i : \u03b9) \u2192 E i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nletI : MetricSpace \u03b1 := hf.toEmbedding.comapMetricSpace f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhaveI : SecondCountableTopology \u03b1 := hf.toEmbedding.secondCountableTopology\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave : CompleteSpace \u03b1 :=\n by\n rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]\n exact hf.closed_range.isComplete\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 CompleteSpace \u03b1\n[PROOFSTEP]\nrw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 IsComplete (range f)\n[PROOFSTEP]\nexact hf.closed_range.isComplete\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d\u00b9 : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis\u271d : SecondCountableTopology \u03b1\nthis : CompleteSpace \u03b1\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nrcases isEmpty_or_nonempty \u03b9 with (h\u03b9 | h\u03b9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : IsEmpty \u03b9\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nexact \u27e8t, fun i => (IsEmpty.elim h\u03b9 i : _), le_rfl, p\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\ninhabit \u03b9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nletI : \u2200 n : \u03b9, TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nhaveI : \u2200 n : \u03b9, PolishSpace (AuxCopy \u03b1 n) := fun n => h'm n\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nletI T : TopologicalSpace (\u2200 n : \u03b9, AuxCopy \u03b1 n) := inferInstance\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nlet f : \u03b1 \u2192 \u2200 n : \u03b9, AuxCopy \u03b1 n := fun x _ => x\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nhave T_le_m : \u2200 n, T.induced f \u2264 m n := fun n \u21a6 by\n rw [induced_to_pi]\n exact iInf_le_of_le n (@induced_id _ (m n)).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nn : \u03b9\n\u22a2 induced f T \u2264 m n\n[PROOFSTEP]\nrw [induced_to_pi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nn : \u03b9\n\u22a2 \u2a05 (i : \u03b9), induced (fun x => f x i) inferInstance \u2264 m n\n[PROOFSTEP]\nexact iInf_le_of_le n (@induced_id _ (m n)).le\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\n\u22a2 \u2203 t', (\u2200 (n : \u03b9), t' \u2264 m n) \u2227 t' \u2264 t \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nrefine'\n \u27e8T.induced f, fun n => T_le_m n, (T_le_m default).trans (hm default), _\u27e9\n -- show that the new topology is Polish, as the pullback of a Polish topology under a closed\n -- embedding.\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave A : range f = \u22c2 n, {x | x n = x default} := by\n ext x\n constructor\n \u00b7 rintro \u27e8y, rfl\u27e9\n exact mem_iInter.2 fun n => by simp only [mem_setOf_eq]\n \u00b7 refine fun hx \u21a6 \u27e8x default, ?_\u27e9\n ext1 n\n symm\n exact mem_iInter.1 hx n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\n\u22a2 range f = \u22c2 (n : \u03b9), {x | x n = x default}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\n\u22a2 x \u2208 range f \u2194 x \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\n\u22a2 x \u2208 range f \u2192 x \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\ny : \u03b1\n\u22a2 f y \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\n[PROOFSTEP]\nexact mem_iInter.2 fun n => by simp only [mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\ny : \u03b1\nn : \u03b9\n\u22a2 f y \u2208 {x | x n = x default}\n[PROOFSTEP]\nsimp only [mem_setOf_eq]\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\n\u22a2 x \u2208 \u22c2 (n : \u03b9), {x | x n = x default} \u2192 x \u2208 range f\n[PROOFSTEP]\nrefine fun hx \u21a6 \u27e8x default, ?_\u27e9\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\nhx : x \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\n\u22a2 f (x default) = x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.mpr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\nhx : x \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\nn : \u03b9\n\u22a2 f (x default) n = x n\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.mpr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nx : (n : \u03b9) \u2192 AuxCopy \u03b1 n\nhx : x \u2208 \u22c2 (n : \u03b9), {x | x n = x default}\nn : \u03b9\n\u22a2 x n = f (x default) n\n[PROOFSTEP]\nexact mem_iInter.1 hx n\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave f_closed : IsClosed (range f) := by\n rw [A]\n refine isClosed_iInter fun n => ?_\n have C : \u2200 i : \u03b9, Continuous fun x : \u2200 n, AuxCopy \u03b1 n => (id (x i) : \u03b1) := fun i \u21a6\n have : Continuous (show AuxCopy \u03b1 i \u2192 \u03b1 from id) := continuous_id_of_le (hm i)\n this.comp (continuous_apply i)\n apply isClosed_eq (C n) (C default)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\n\u22a2 IsClosed (range f)\n[PROOFSTEP]\nrw [A]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\n\u22a2 IsClosed (\u22c2 (n : \u03b9), {x | x n = x default})\n[PROOFSTEP]\nrefine isClosed_iInter fun n => ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nn : \u03b9\n\u22a2 IsClosed {x | x n = x default}\n[PROOFSTEP]\nhave C : \u2200 i : \u03b9, Continuous fun x : \u2200 n, AuxCopy \u03b1 n => (id (x i) : \u03b1) := fun i \u21a6\n have : Continuous (show AuxCopy \u03b1 i \u2192 \u03b1 from id) := continuous_id_of_le (hm i)\n this.comp (continuous_apply i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nn : \u03b9\nC : \u2200 (i : \u03b9), Continuous fun x => id (x i)\n\u22a2 IsClosed {x | x n = x default}\n[PROOFSTEP]\napply isClosed_eq (C n) (C default)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave K : @_root_.Embedding _ _ (T.induced f) T f :=\n by\n refine Function.Injective.embedding_induced fun x y hxy \u21a6 ?_\n have : f x default = f y default := by rw [hxy]\n exact this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\n\u22a2 _root_.Embedding f\n[PROOFSTEP]\nrefine Function.Injective.embedding_induced fun x y hxy \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nhave : f x default = f y default := by rw [hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 f x default = f y default\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d\u00b9 : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis\u271d : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nx y : \u03b1\nhxy : f x = f y\nthis : f x default = f y default\n\u22a2 x = y\n[PROOFSTEP]\nexact this\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave L : @ClosedEmbedding _ _ (T.induced f) T f :=\n by\n refine @ClosedEmbedding.mk _ _ (T.induced f) T f ?_ ?_\n \u00b7 exact K\n \u00b7 exact f_closed\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\n\u22a2 ClosedEmbedding f\n[PROOFSTEP]\nrefine @ClosedEmbedding.mk _ _ (T.induced f) T f ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\n\u22a2 _root_.Embedding f\n[PROOFSTEP]\nexact K\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\n\u22a2 IsClosed (range f)\n[PROOFSTEP]\nexact f_closed\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\nL : ClosedEmbedding f\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nexact @ClosedEmbedding.polishSpace _ _ (T.induced f) T (by infer_instance) _ L\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\ninst\u271d : Countable \u03b9\nt : TopologicalSpace \u03b1\np : PolishSpace \u03b1\nm : \u03b9 \u2192 TopologicalSpace \u03b1\nhm : \u2200 (n : \u03b9), m n \u2264 t\nh'm : \u2200 (n : \u03b9), PolishSpace \u03b1\nh\u03b9 : Nonempty \u03b9\ninhabited_h : Inhabited \u03b9\nthis\u271d : (n : \u03b9) \u2192 TopologicalSpace (AuxCopy \u03b1 n) := fun n => m n\nthis : \u2200 (n : \u03b9), PolishSpace (AuxCopy \u03b1 n)\nT : TopologicalSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n) := inferInstance\nf : \u03b1 \u2192 (n : \u03b9) \u2192 AuxCopy \u03b1 n := fun x x_1 => x\nT_le_m : \u2200 (n : \u03b9), induced f T \u2264 m n\nA : range f = \u22c2 (n : \u03b9), {x | x n = x default}\nf_closed : IsClosed (range f)\nK : _root_.Embedding f\nL : ClosedEmbedding f\n\u22a2 PolishSpace ((n : \u03b9) \u2192 AuxCopy \u03b1 n)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\n\u22a2 MetricSpace (CompleteCopy s)\n[PROOFSTEP]\nrefine\n @MetricSpace.ofT0PseudoMetricSpace (CompleteCopy s)\n (.ofDistTopology dist (fun _ \u21a6 ?_) (fun _ _ \u21a6 ?_) (fun x y z \u21a6 ?_) fun t \u21a6 ?_) _\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nx\u271d : CompleteCopy s\n\u22a2 dist x\u271d x\u271d = 0\n[PROOFSTEP]\nsimp only [dist_eq, dist_self, one_div, sub_self, abs_zero, add_zero]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nx\u271d\u00b9 x\u271d : CompleteCopy s\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d x\u271d\u00b9\n[PROOFSTEP]\nsimp only [dist_eq, dist_comm, abs_sub_comm]\n[GOAL]\ncase refine_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nx y z : CompleteCopy s\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\ncalc\n dist x z = dist x.1 z.1 + |1 / infDist x.1 s\u1d9c - 1 / infDist z.1 s\u1d9c| := rfl\n _ \u2264\n dist x.1 y.1 + dist y.1 z.1 +\n (|1 / infDist x.1 s\u1d9c - 1 / infDist y.1 s\u1d9c| + |1 / infDist y.1 s\u1d9c - 1 / infDist z.1 s\u1d9c|) :=\n (add_le_add (dist_triangle _ _ _) (dist_triangle (1 / infDist _ _) _ _))\n _ = dist x y + dist y z := add_add_add_comm ..\n[GOAL]\ncase refine_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\n\u22a2 IsOpen t \u2194 \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\n[PROOFSTEP]\nrefine \u27e8fun h x hx \u21a6 ?_, fun h \u21a6 isOpen_iff_mem_nhds.2 fun x hx \u21a6 ?_\u27e9\n[GOAL]\ncase refine_4.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : IsOpen t\nx : CompleteCopy s\nhx : x \u2208 t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\n[PROOFSTEP]\nrcases(Metric.isOpen_iff (\u03b1 := s)).1 h x hx with \u27e8\u03b5, \u03b50, h\u03b5\u27e9\n[GOAL]\ncase refine_4.refine_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : IsOpen t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\n[PROOFSTEP]\nexact \u27e8\u03b5, \u03b50, fun y hy \u21a6 h\u03b5 <| (dist_comm _ _).trans_lt <| (dist_val_le_dist _ _).trans_lt hy\u27e9\n[GOAL]\ncase refine_4.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u22a2 t \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrcases h x hx with \u27e8\u03b5, \u03b50, h\u03b5\u27e9\n[GOAL]\ncase refine_4.refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\n\u22a2 t \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimp only [dist_eq, one_div] at h\u03b5 \n[GOAL]\ncase refine_4.refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\n\u22a2 t \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave :\n Tendsto (fun y : s \u21a6 dist x.1 y.1 + |(infDist x.1 s\u1d9c)\u207b\u00b9 - (infDist y.1 s\u1d9c)\u207b\u00b9|) (\ud835\udcdd x)\n (\ud835\udcdd (dist x.1 x.1 + |(infDist x.1 s\u1d9c)\u207b\u00b9 - (infDist x.1 s\u1d9c)\u207b\u00b9|))\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\n\u22a2 Tendsto (fun y => dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9|) (\ud835\udcdd x)\n (\ud835\udcdd (dist \u2191x \u2191x + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9|))\n[PROOFSTEP]\nrefine (tendsto_const_nhds.dist continuous_subtype_val.continuousAt).add (tendsto_const_nhds.sub <| ?_).abs\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\n\u22a2 Tendsto (fun y => (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9) (\ud835\udcdd x) (\ud835\udcdd (infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9)\n[PROOFSTEP]\nrefine (continuousAt_inv_infDist_pt ?_).comp continuous_subtype_val.continuousAt\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\n\u22a2 \u00ac\u2191x \u2208 closure (\u2191s)\u1d9c\n[PROOFSTEP]\nrw [s.isOpen.isClosed_compl.closure_eq, mem_compl_iff, not_not]\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\n\u22a2 \u2191x \u2208 \u2191s\n[PROOFSTEP]\nexact x.2\n[GOAL]\ncase refine_4.refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\nthis :\n Tendsto (fun y => dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9|) (\ud835\udcdd x)\n (\ud835\udcdd (dist \u2191x \u2191x + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9|))\n\u22a2 t \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimp only [dist_self, sub_self, abs_zero, zero_add] at this \n[GOAL]\ncase refine_4.refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MetricSpace \u03b1\ns : Opens \u03b1\nt : Set (CompleteCopy s)\nh : \u2200 (x : CompleteCopy s), x \u2208 t \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : CompleteCopy s), dist x y < \u03b5 \u2192 y \u2208 t\nx : CompleteCopy s\nhx : x \u2208 t\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : \u2200 (y : CompleteCopy s), dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9| < \u03b5 \u2192 y \u2208 t\nthis : Tendsto (fun y => dist \u2191x \u2191y + |(infDist (\u2191x) (\u2191s)\u1d9c)\u207b\u00b9 - (infDist (\u2191y) (\u2191s)\u1d9c)\u207b\u00b9|) (\ud835\udcdd x) (\ud835\udcdd 0)\n\u22a2 t \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact mem_of_superset (this <| gt_mem_nhds \u03b50) h\u03b5\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 CompleteSpace (CompleteCopy s)\n[PROOFSTEP]\nrefine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ \u00b7) (by simp) fun u hu \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 \u2200 (n : \u2115), 0 < (fun x => (1 / 2) ^ x) n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nhave A : CauchySeq fun n => (u n).1\n[GOAL]\ncase A\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\n\u22a2 CauchySeq fun n => \u2191(u n)\n[PROOFSTEP]\nrefine cauchySeq_of_le_tendsto_0 (fun n : \u2115 => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_\n[GOAL]\ncase A.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nn m N : \u2115\nhNn : N \u2264 n\nhNm : N \u2264 m\n\u22a2 dist \u2191(u n) \u2191(u m) \u2264 (fun n => (1 / 2) ^ n) N\n[PROOFSTEP]\nexact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le\n[GOAL]\ncase A.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\n\u22a2 Tendsto (fun n => (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nobtain \u27e8x, xlim\u27e9 : \u2203 x, Tendsto (fun n => (u n).1) atTop (\ud835\udcdd x) := cauchySeq_tendsto_of_complete A\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nby_cases xs : x \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : x \u2208 s\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nexact \u27e8\u27e8x, xs\u27e9, tendsto_subtype_rng.2 xlim\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nobtain \u27e8C, hC\u27e9 : \u2203 C, \u2200 n, 1 / infDist (u n).1 s\u1d9c < C\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\n\u22a2 \u2203 C, \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\n[PROOFSTEP]\nrefine \u27e8(1 / 2) ^ 0 + 1 / infDist (u 0).1 s\u1d9c, fun n \u21a6 ?_\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nn : \u2115\n\u22a2 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < (1 / 2) ^ 0 + 1 / infDist (\u2191(u 0)) (\u2191s)\u1d9c\n[PROOFSTEP]\nrw [\u2190 sub_lt_iff_lt_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nn : \u2115\n\u22a2 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c - 1 / infDist (\u2191(u 0)) (\u2191s)\u1d9c < (1 / 2) ^ 0\n[PROOFSTEP]\ncalc\n _ \u2264 |1 / infDist (u n).1 s\u1d9c - 1 / infDist (u 0).1 s\u1d9c| := le_abs_self _\n _ = |1 / infDist (u 0).1 s\u1d9c - 1 / infDist (u n).1 s\u1d9c| := (abs_sub_comm _ _)\n _ \u2264 dist (u 0) (u n) := (le_add_of_nonneg_left dist_nonneg)\n _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nhave Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nhave Hmem : \u2200 {y}, y \u2208 s \u2194 0 < infDist y s\u1d9c := fun {y} \u21a6 by\n rw [\u2190 s.isOpen.isClosed_compl.not_mem_iff_infDist_pos \u27e8x, xs\u27e9]; exact not_not.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\ny : \u03b1\n\u22a2 y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\n[PROOFSTEP]\nrw [\u2190 s.isOpen.isClosed_compl.not_mem_iff_infDist_pos \u27e8x, xs\u27e9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\ny : \u03b1\n\u22a2 y \u2208 s \u2194 \u00acy \u2208 (\u2191s)\u1d9c\n[PROOFSTEP]\nexact not_not.symm\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nhave I : \u2200 n, 1 / C \u2264 infDist (u n).1 s\u1d9c := fun n \u21a6\n by\n have : 0 < infDist (u n).1 s\u1d9c := Hmem.1 (u n).2\n rw [div_le_iff' Cpos]\n exact (div_le_iff this).1 (hC n).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\nn : \u2115\n\u22a2 1 / C \u2264 infDist (\u2191(u n)) (\u2191s)\u1d9c\n[PROOFSTEP]\nhave : 0 < infDist (u n).1 s\u1d9c := Hmem.1 (u n).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\nn : \u2115\nthis : 0 < infDist (\u2191(u n)) (\u2191s)\u1d9c\n\u22a2 1 / C \u2264 infDist (\u2191(u n)) (\u2191s)\u1d9c\n[PROOFSTEP]\nrw [div_le_iff' Cpos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\nn : \u2115\nthis : 0 < infDist (\u2191(u n)) (\u2191s)\u1d9c\n\u22a2 1 \u2264 C * infDist (\u2191(u n)) (\u2191s)\u1d9c\n[PROOFSTEP]\nexact (div_le_iff this).1 (hC n).le\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\nI : \u2200 (n : \u2115), 1 / C \u2264 infDist (\u2191(u n)) (\u2191s)\u1d9c\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nhave I' : 1 / C \u2264 infDist x s\u1d9c :=\n have : Tendsto (fun n => infDist (u n).1 s\u1d9c) atTop (\ud835\udcdd (infDist x s\u1d9c)) :=\n ((continuous_infDist_pt (s\u1d9c : Set \u03b1)).tendsto x).comp xlim\n ge_of_tendsto' this I\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : MetricSpace \u03b1\ns : Opens \u03b1\ninst\u271d : CompleteSpace \u03b1\nu : \u2115 \u2192 CompleteCopy s\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun x => (1 / 2) ^ x) N\nA : CauchySeq fun n => \u2191(u n)\nx : \u03b1\nxlim : Tendsto (fun n => \u2191(u n)) atTop (\ud835\udcdd x)\nxs : \u00acx \u2208 s\nC : \u211d\nhC : \u2200 (n : \u2115), 1 / infDist (\u2191(u n)) (\u2191s)\u1d9c < C\nCpos : 0 < C\nHmem : \u2200 {y : \u03b1}, y \u2208 s \u2194 0 < infDist y (\u2191s)\u1d9c\nI : \u2200 (n : \u2115), 1 / C \u2264 infDist (\u2191(u n)) (\u2191s)\u1d9c\nI' : 1 / C \u2264 infDist x (\u2191s)\u1d9c\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nexact absurd (Hmem.2 <| lt_of_lt_of_le (div_pos one_pos Cpos) I') xs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 PolishSpace \u2191s\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b1\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\n\u22a2 PolishSpace \u2191s\n[PROOFSTEP]\nlift s to Opens \u03b1 using hs\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\ns : Opens \u03b1\n\u22a2 PolishSpace \u2191\u2191s\n[PROOFSTEP]\nhave : SecondCountableTopology s.CompleteCopy := inferInstanceAs (SecondCountableTopology s)\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nthis\u271d : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\ns : Opens \u03b1\nthis : SecondCountableTopology (CompleteCopy s)\n\u22a2 PolishSpace \u2191\u2191s\n[PROOFSTEP]\nexact inferInstanceAs (PolishSpace s.CompleteCopy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 IsClopenable s\n[PROOFSTEP]\nhaveI : PolishSpace s := hs.polishSpace\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis : PolishSpace \u2191s\n\u22a2 IsClopenable s\n[PROOFSTEP]\nlet t : Set \u03b1 := s\u1d9c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\n\u22a2 IsClopenable s\n[PROOFSTEP]\nhaveI : PolishSpace t := hs.isOpen_compl.polishSpace\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\n\u22a2 IsClopenable s\n[PROOFSTEP]\nlet f : s \u2295 t \u2243 \u03b1 := Equiv.Set.sumCompl s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\n\u22a2 IsClopenable s\n[PROOFSTEP]\nhave hle : TopologicalSpace.coinduced f instTopologicalSpaceSum \u2264 \u2039_\u203a\n[GOAL]\ncase hle\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\n\u22a2 coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n[PROOFSTEP]\nsimp only [instTopologicalSpaceSum, coinduced_sup, coinduced_compose, sup_le_iff, \u2190 continuous_iff_coinduced_le]\n[GOAL]\ncase hle\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\n\u22a2 Continuous (\u2191(Equiv.Set.sumCompl s) \u2218 Sum.inl) \u2227 Continuous (\u2191(Equiv.Set.sumCompl s) \u2218 Sum.inr)\n[PROOFSTEP]\nexact \u27e8continuous_subtype_val, continuous_subtype_val\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 IsClopenable s\n[PROOFSTEP]\nrefine \u27e8.coinduced f instTopologicalSpaceSum, hle, ?_, hs.mono hle, ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nrw [\u2190 f.induced_symm]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nexact f.symm.polishSpace_induced\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 IsOpen s\n[PROOFSTEP]\nrw [isOpen_coinduced, isOpen_sum_iff]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 IsOpen (Sum.inl \u207b\u00b9' (\u2191f \u207b\u00b9' s)) \u2227 IsOpen (Sum.inr \u207b\u00b9' (\u2191f \u207b\u00b9' s))\n[PROOFSTEP]\nconvert And.intro (isOpen_univ (\u03b1 := s)) (isOpen_empty (\u03b1 := (s\u1d9c : Set \u03b1)))\n[GOAL]\ncase h.e'_1.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 Sum.inl \u207b\u00b9' (\u2191f \u207b\u00b9' s) = univ\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\n\u22a2 Sum.inr \u207b\u00b9' (\u2191f \u207b\u00b9' s) = \u2205\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.e'_1.h.e'_3.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\nx : \u03b1\nhx : x \u2208 s\n\u22a2 { val := x, property := hx } \u2208 Sum.inl \u207b\u00b9' (\u2191f \u207b\u00b9' s) \u2194 { val := x, property := hx } \u2208 univ\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase h.e'_2.h.e'_3.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis\u271d : PolishSpace \u2191s\nt : Set \u03b1 := s\u1d9c\nthis : PolishSpace \u2191t\nf : \u2191s \u2295 \u2191t \u2243 \u03b1 := Equiv.Set.sumCompl s\nhle : coinduced (\u2191f) instTopologicalSpaceSum \u2264 inst\u271d\u00b9\nx : \u03b1\nhx : x \u2208 t\n\u22a2 { val := x, property := hx } \u2208 Sum.inr \u207b\u00b9' (\u2191f \u207b\u00b9' s) \u2194 { val := x, property := hx } \u2208 \u2205\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nhs : IsClopenable s\n\u22a2 IsClopenable s\u1d9c\n[PROOFSTEP]\nrcases hs with \u27e8t, t_le, t_polish, h, h'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\nt : TopologicalSpace \u03b1\nt_le : t \u2264 inst\u271d\nt_polish : PolishSpace \u03b1\nh : IsClosed s\nh' : IsOpen s\n\u22a2 IsClopenable s\u1d9c\n[PROOFSTEP]\nexact \u27e8t, t_le, t_polish, @IsOpen.isClosed_compl \u03b1 t s h', @IsClosed.isOpen_compl \u03b1 t s h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsClopenable s\n[PROOFSTEP]\nsimpa using hs.isClosed_compl.isClopenable.compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), IsClopenable (s n)\n\u22a2 IsClopenable (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\nchoose m mt m_polish _ m_open using hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\n\u22a2 IsClopenable (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\nobtain \u27e8t', t'm, -, t'_polish\u27e9 : \u2203 t' : TopologicalSpace \u03b1, (\u2200 n : \u2115, t' \u2264 m n) \u2227 t' \u2264 t \u2227 @PolishSpace \u03b1 t' :=\n exists_polishSpace_forall_le m mt m_polish\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\n\u22a2 IsClopenable (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\nhave A : IsOpen[t'] (\u22c3 n, s n) := by\n apply isOpen_iUnion\n intro n\n apply t'm n\n exact m_open n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\n\u22a2 IsOpen (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\napply isOpen_iUnion\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\n\u22a2 \u2200 (i : \u2115), IsOpen (s i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\nn : \u2115\n\u22a2 IsOpen (s n)\n[PROOFSTEP]\napply t'm n\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\nn : \u2115\n\u22a2 IsOpen (s n)\n[PROOFSTEP]\nexact m_open n\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\nA : IsOpen (\u22c3 (n : \u2115), s n)\n\u22a2 IsClopenable (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\nobtain \u27e8t'', t''_le, t''_polish, h1, h2\u27e9 :\n \u2203 t'' : TopologicalSpace \u03b1, t'' \u2264 t' \u2227 @PolishSpace \u03b1 t'' \u2227 IsClosed[t''] (\u22c3 n, s n) \u2227 IsOpen[t''] (\u22c3 n, s n) :=\n @IsOpen.isClopenable \u03b1 t' t'_polish _ A\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nm : \u2115 \u2192 TopologicalSpace \u03b1\nmt : \u2200 (n : \u2115), m n \u2264 t\nm_polish : \u2200 (n : \u2115), PolishSpace \u03b1\nh\u271d : \u2200 (n : \u2115), IsClosed (s n)\nm_open : \u2200 (n : \u2115), IsOpen (s n)\nt' : TopologicalSpace \u03b1\nt'm : \u2200 (n : \u2115), t' \u2264 m n\nt'_polish : PolishSpace \u03b1\nA : IsOpen (\u22c3 (n : \u2115), s n)\nt'' : TopologicalSpace \u03b1\nt''_le : t'' \u2264 t'\nt''_polish : PolishSpace \u03b1\nh1 : IsClosed (\u22c3 (n : \u2115), s n)\nh2 : IsOpen (\u22c3 (n : \u2115), s n)\n\u22a2 IsClopenable (\u22c3 (n : \u2115), s n)\n[PROOFSTEP]\nexact \u27e8t'', t''_le.trans ((t'm 0).trans (mt 0)), t''_polish, h1, h2\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Polish", "llama_tokens": 27301, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6688802471698041, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.295426349197185}} {"text": "[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\ng : \u211d[X]\n\u22a2 comp (Polynomial.toContinuousMapOn g (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) (attachBound \u2191f) = \u2191(\u2191(Polynomial.aeval f) g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\ng : \u211d[X]\na\u271d : X\n\u22a2 \u2191(comp (Polynomial.toContinuousMapOn g (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) (attachBound \u2191f)) a\u271d = \u2191\u2191(\u2191(Polynomial.aeval f) g) a\u271d\n[PROOFSTEP]\nsimp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,\n Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply,\n Polynomial.toContinuousMap_apply]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\ng : \u211d[X]\n\u22a2 comp (Polynomial.toContinuousMapOn g (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) (attachBound \u2191f) \u2208 A\n[PROOFSTEP]\nrw [polynomial_comp_attachBound]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\ng : \u211d[X]\n\u22a2 \u2191(\u2191(Polynomial.aeval f) g) \u2208 A\n[PROOFSTEP]\napply SetLike.coe_mem\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\n\u22a2 comp p (attachBound \u2191f) \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nhave mem_closure : p \u2208 (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016)).topologicalClosure :=\n continuousMap_mem_polynomialFunctions_closure _ _ p\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\n\u22a2 comp p (attachBound \u2191f) \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nhave frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\nfrequently_mem_polynomials :\n \u2203\u1da0 (x : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)) in nhds p, x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\n\u22a2 comp p (attachBound \u2191f) \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\napply mem_closure_iff_frequently.mpr\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\nfrequently_mem_polynomials :\n \u2203\u1da0 (x : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)) in nhds p, x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\n\u22a2 \u2203\u1da0 (x : C(X, \u211d)) in nhds (comp p (attachBound \u2191f)), x \u2208 \u2191A\n[PROOFSTEP]\nrefine'\n ((compRightContinuousMap \u211d (attachBound (f : C(X, \u211d)))).continuousAt p).tendsto.frequently_map _ _\n frequently_mem_polynomials\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\nfrequently_mem_polynomials :\n \u2203\u1da0 (x : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)) in nhds p, x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\n\u22a2 \u2200 (x : C(\u2191(Set.Icc (-\u2016\u2191f\u2016) \u2016\u2191f\u2016), \u211d)),\n x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) \u2192 \u2191(compRightContinuousMap \u211d (attachBound \u2191f)) x \u2208 \u2191A\n[PROOFSTEP]\nrintro _ \u27e8g, \u27e8-, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\nfrequently_mem_polynomials :\n \u2203\u1da0 (x : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)) in nhds p, x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\ng : \u211d[X]\n\u22a2 \u2191(compRightContinuousMap \u211d (attachBound \u2191f)) (\u2191\u2191(Polynomial.toContinuousMapOnAlgHom (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) g) \u2208 \u2191A\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, AlgHom.coe_toRingHom, compRightContinuousMap_apply,\n Polynomial.toContinuousMapOnAlgHom_apply]\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\np : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)\nmem_closure : p \u2208 Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\nfrequently_mem_polynomials :\n \u2203\u1da0 (x : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d)) in nhds p, x \u2208 \u2191(polynomialFunctions (Set.Icc (-\u2016f\u2016) \u2016f\u2016))\ng : \u211d[X]\n\u22a2 comp (Polynomial.toContinuousMapOn g (Set.Icc (-\u2016f\u2016) \u2016f\u2016)) (attachBound \u2191f) \u2208 A\n[PROOFSTEP]\napply polynomial_comp_attachBound_mem\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\n\u22a2 abs \u2191f \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nlet f' := attachBound (f : C(X, \u211d))\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\nf' : C(X, \u2191(Set.Icc (-\u2016\u2191f\u2016) \u2016\u2191f\u2016)) := attachBound \u2191f\n\u22a2 abs \u2191f \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nlet abs : C(Set.Icc (-\u2016f\u2016) \u2016f\u2016, \u211d) := { toFun := fun x : Set.Icc (-\u2016f\u2016) \u2016f\u2016 => |(x : \u211d)| }\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\nf' : C(X, \u2191(Set.Icc (-\u2016\u2191f\u2016) \u2016\u2191f\u2016)) := attachBound \u2191f\nabs : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d) := mk fun x => |\u2191x|\n\u22a2 ContinuousMap.abs \u2191f \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nchange abs.comp f' \u2208 A.topologicalClosure\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf : { x // x \u2208 A }\nf' : C(X, \u2191(Set.Icc (-\u2016\u2191f\u2016) \u2016\u2191f\u2016)) := attachBound \u2191f\nabs : C(\u2191(Set.Icc (-\u2016f\u2016) \u2016f\u2016), \u211d) := mk fun x => |\u2191x|\n\u22a2 comp abs f' \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\napply comp_attachBound_mem_closure\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 \u2191f \u2293 \u2191g \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nrw [inf_eq_half_smul_add_sub_abs_sub' \u211d]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 2\u207b\u00b9 \u2022 (\u2191f + \u2191g - |\u2191g - \u2191f|) \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nrefine'\n A.topologicalClosure.smul_mem\n (A.topologicalClosure.sub_mem\n (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _)\n _\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 |\u2191g - \u2191f| \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nexact_mod_cast abs_mem_subalgebra_closure A _\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191f \u2293 \u2191g \u2208 A\n[PROOFSTEP]\nconvert inf_mem_subalgebra_closure A f g\n[GOAL]\ncase h.e'_5\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 A = Subalgebra.topologicalClosure A\n[PROOFSTEP]\napply SetLike.ext'\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191A = \u2191(Subalgebra.topologicalClosure A)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = \u2191A\n[PROOFSTEP]\nerw [closure_eq_iff_isClosed]\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 IsClosed \u2191A\n[PROOFSTEP]\nexact h\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 \u2191f \u2294 \u2191g \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nrw [sup_eq_half_smul_add_add_abs_sub' \u211d]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 2\u207b\u00b9 \u2022 (\u2191f + \u2191g + |\u2191g - \u2191f|) \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nrefine'\n A.topologicalClosure.smul_mem\n (A.topologicalClosure.add_mem\n (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property) (A.le_topologicalClosure g.property)) _)\n _\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nf g : { x // x \u2208 A }\n\u22a2 |\u2191g - \u2191f| \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nexact_mod_cast abs_mem_subalgebra_closure A _\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191f \u2294 \u2191g \u2208 A\n[PROOFSTEP]\nconvert sup_mem_subalgebra_closure A f g\n[GOAL]\ncase h.e'_5\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 A = Subalgebra.topologicalClosure A\n[PROOFSTEP]\napply SetLike.ext'\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191A = \u2191(Subalgebra.topologicalClosure A)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = \u2191A\n[PROOFSTEP]\nerw [closure_eq_iff_isClosed]\n[GOAL]\ncase h.e'_5.h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nh : IsClosed \u2191A\nf g : { x // x \u2208 A }\n\u22a2 IsClosed \u2191A\n[PROOFSTEP]\nexact h\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\n\u22a2 closure L = \u22a4\n[PROOFSTEP]\napply eq_top_iff.mpr\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\n\u22a2 \u22a4 \u2264 closure L\n[PROOFSTEP]\nrintro f -\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u22a2 f \u2208 closure L\n[PROOFSTEP]\nrefine' Filter.Frequently.mem_closure ((Filter.HasBasis.frequently_iff Metric.nhds_basis_ball).mpr fun \u03b5 pos => _)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 Metric.ball f \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nsimp only [exists_prop, Metric.mem_ball]\n -- It will be helpful to assume `X` is nonempty later,\n -- so we get that out of the way here.\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nby_cases nX : Nonempty X\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : \u00acNonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : \u00acNonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nexact\n \u27e8nA.some, (dist_lt_iff pos).mpr fun x => False.elim (nX \u27e8x\u27e9), nA.choose_spec\u27e9\n /-\n The strategy now is to pick a family of continuous functions `g x y` in `A`\n with the property that `g x y x = f x` and `g x y y = f y`\n (this is immediate from `h : SeparatesPointsStrongly`)\n then use continuity to see that `g x y` is close to `f` near both `x` and `y`,\n and finally using compactness to produce the desired function `h`\n as a maximum over finitely many `x` of a minimum over finitely many `y` of the `g x y`.\n -/\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : Set.SeparatesPointsStrongly L\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\ndsimp only [Set.SeparatesPointsStrongly] at sep \n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nchoose g hg w\u2081 w\u2082 using sep f\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet U : X \u2192 X \u2192 Set X := fun x y => {z | f z - \u03b5 < g x y z}\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave U_nhd_y : \u2200 x y, U x y \u2208 \ud835\udcdd y := by\n intro x y\n refine' IsOpen.mem_nhds _ _\n \u00b7 apply isOpen_lt <;> continuity\n \u00b7 rw [Set.mem_setOf_eq, w\u2082]\n exact sub_lt_self _ pos\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\n\u22a2 \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\n[PROOFSTEP]\nintro x y\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 U x y \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrefine' IsOpen.mem_nhds _ _\n[GOAL]\ncase refine'_1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 IsOpen (U x y)\n[PROOFSTEP]\napply isOpen_lt\n[GOAL]\ncase refine'_1.hf\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 Continuous fun b => \u2191f b - \u03b5\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_1.hg\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 Continuous fun b => \u2191(g x y) b\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 y \u2208 U x y\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, w\u2082]\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nx y : X\n\u22a2 \u2191f y - \u03b5 < \u2191f y\n[PROOFSTEP]\nexact sub_lt_self _ pos\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet ys : \u2200 _, Finset X := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet ys_w : \u2200 x, \u22c3 y \u2208 ys x, U x y = \u22a4 := fun x => (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x)).choose_spec\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave ys_nonempty : \u2200 x, (ys x).Nonempty := fun x =>\n Set.nonempty_of_union_eq_top_of_nonempty _ _ nX\n (ys_w x)\n -- Thus for each `x` we have the desired `h x : A` so `f z - \u03b5 < h x z` everywhere\n -- and `h x x = f x`.\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet h : \u2200 _, L := fun x =>\n \u27e8(ys x).sup' (ys_nonempty x) fun y => (g x y : C(X, \u211d)), Finset.sup'_mem _ sup_mem _ _ _ fun y _ => hg x y\u27e9\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave lt_h : \u2200 x z, f z - \u03b5 < (h x : X \u2192 \u211d) z := by\n intro x z\n obtain \u27e8y, ym, zm\u27e9 := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z\n dsimp\n simp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff]\n exact \u27e8y, ym, zm\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\n\u22a2 \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\n[PROOFSTEP]\nintro x z\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nx z : X\n\u22a2 \u2191f z - \u03b5 < \u2191\u2191(h x) z\n[PROOFSTEP]\nobtain \u27e8y, ym, zm\u27e9 := Set.exists_set_mem_of_union_eq_top _ _ (ys_w x) z\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nx z y : X\nym : y \u2208 fun i => i \u2208 (ys x).val\nzm : z \u2208 U x y\n\u22a2 \u2191f z - \u03b5 < \u2191\u2191(h x) z\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nx z y : X\nym : y \u2208 fun i => i \u2208 (ys x).val\nzm : z \u2208 U x y\n\u22a2 \u2191f z - \u03b5 <\n \u2191(Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, sup'_coe, Finset.sup'_apply, Finset.lt_sup'_iff]\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nx z y : X\nym : y \u2208 fun i => i \u2208 (ys x).val\nzm : z \u2208 U x y\n\u22a2 \u2203 b, b \u2208 Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4) \u2227 \u2191f z - \u03b5 < \u2191(g x b) z\n[PROOFSTEP]\nexact \u27e8y, ym, zm\u27e9\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave h_eq : \u2200 x, (h x : X \u2192 \u211d) x = f x := by intro x;\n simp [w\u2081]\n -- For each `x`, we define `W x` to be `{z | h x z < f z + \u03b5}`,\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\n\u22a2 \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nx : X\n\u22a2 \u2191\u2191(h x) x = \u2191f x\n[PROOFSTEP]\nsimp [w\u2081]\n -- For each `x`, we define `W x` to be `{z | h x z < f z + \u03b5}`,\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet W : \u2200 _, Set X := fun x =>\n {z | (h x : X \u2192 \u211d) z < f z + \u03b5}\n -- This is still a neighbourhood of `x`.\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave W_nhd : \u2200 x, W x \u2208 \ud835\udcdd x := by\n intro x\n refine' IsOpen.mem_nhds _ _\n \u00b7\n -- Porting note: mathlib3 `continuity` found `continuous_set_coe`\n apply isOpen_lt (continuous_set_coe _ _)\n continuity\n \u00b7 dsimp only [Set.mem_setOf_eq]\n rw [h_eq]\n exact lt_add_of_pos_right _ pos\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\n\u22a2 \u2200 (x : X), W x \u2208 \ud835\udcdd x\n[PROOFSTEP]\nintro x\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 W x \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrefine' IsOpen.mem_nhds _ _\n[GOAL]\ncase refine'_1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 IsOpen (W x)\n[PROOFSTEP]\napply isOpen_lt (continuous_set_coe _ _)\n[GOAL]\ncase refine'_1\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 Continuous fun b => \u2191f b + \u03b5\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 x \u2208 W x\n[PROOFSTEP]\ndsimp only [Set.mem_setOf_eq]\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 \u2191(Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n x <\n \u2191f x + \u03b5\n[PROOFSTEP]\nrw [h_eq]\n[GOAL]\ncase refine'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nx : X\n\u22a2 \u2191f x < \u2191f x + \u03b5\n[PROOFSTEP]\nexact lt_add_of_pos_right _ pos\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet xs : Finset X := (CompactSpace.elim_nhds_subcover W W_nhd).choose\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet xs_w : \u22c3 x \u2208 xs, W x = \u22a4 := (CompactSpace.elim_nhds_subcover W W_nhd).choose_spec\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nhave xs_nonempty : xs.Nonempty := Set.nonempty_of_union_eq_top_of_nonempty _ _ nX xs_w\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nlet k : (L : Type _) :=\n \u27e8xs.inf' xs_nonempty fun x => (h x : C(X, \u211d)), Finset.inf'_mem _ inf_mem _ _ _ fun x _ => (h x).2\u27e9\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\n\u22a2 \u2203 x, dist x f < \u03b5 \u2227 x \u2208 L\n[PROOFSTEP]\nrefine'\n \u27e8k.1, _, k.2\u27e9\n -- We just need to verify the bound, which we do pointwise.\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\n\u22a2 dist (\u2191k) f < \u03b5\n[PROOFSTEP]\nrw [dist_lt_iff pos]\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\n\u22a2 \u2200 (x : X), dist (\u2191\u2191k x) (\u2191f x) < \u03b5\n[PROOFSTEP]\nintro z\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 dist (\u2191\u2191k z) (\u2191f z) < \u03b5\n[PROOFSTEP]\nrw [show \u2200 a b \u03b5 : \u211d, dist a b < \u03b5 \u2194 a < b + \u03b5 \u2227 b - \u03b5 < a by intros;\n simp only [\u2190 Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2200 (a b \u03b5 : \u211d), dist a b < \u03b5 \u2194 a < b + \u03b5 \u2227 b - \u03b5 < a\n[PROOFSTEP]\nintros\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\na\u271d b\u271d \u03b5\u271d : \u211d\n\u22a2 dist a\u271d b\u271d < \u03b5\u271d \u2194 a\u271d < b\u271d + \u03b5\u271d \u2227 b\u271d - \u03b5\u271d < a\u271d\n[PROOFSTEP]\nsimp only [\u2190 Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm]\n[GOAL]\ncase pos\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2191\u2191k z < \u2191f z + \u03b5 \u2227 \u2191f z - \u03b5 < \u2191\u2191k z\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase pos.left\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2191\u2191k z < \u2191f z + \u03b5\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase pos.left\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2191(Finset.inf'\n (Exists.choose\n (_ :\n \u2203 t,\n \u22c3 (x : X) (_ : x \u2208 t),\n {z |\n \u2191(Finset.sup'\n (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z <\n \u2191f z + \u03b5} =\n \u22a4))\n xs_nonempty fun x =>\n Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z <\n \u2191f z + \u03b5\n[PROOFSTEP]\nsimp only [Finset.inf'_lt_iff, ContinuousMap.inf'_apply]\n[GOAL]\ncase pos.left\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2203 i,\n i \u2208\n Exists.choose\n (_ :\n \u2203 t,\n \u22c3 (x : X) (_ : x \u2208 t),\n {z |\n \u2191(Finset.sup'\n (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z <\n \u2191f z + \u03b5} =\n \u22a4) \u2227\n \u2191(Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g i x) z} = \u22a4))\n (_ : Finset.Nonempty (ys i)) fun y => g i y)\n z <\n \u2191f z + \u03b5\n[PROOFSTEP]\nexact Set.exists_set_mem_of_union_eq_top _ _ xs_w z\n[GOAL]\ncase pos.right\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2191f z - \u03b5 < \u2191\u2191k z\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase pos.right\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2191f z - \u03b5 <\n \u2191(Finset.inf'\n (Exists.choose\n (_ :\n \u2203 t,\n \u22c3 (x : X) (_ : x \u2208 t),\n {z |\n \u2191(Finset.sup'\n (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z <\n \u2191f z + \u03b5} =\n \u22a4))\n xs_nonempty fun x =>\n Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z\n[PROOFSTEP]\nsimp only [Finset.lt_inf'_iff, ContinuousMap.inf'_apply]\n[GOAL]\ncase pos.right\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz : X\n\u22a2 \u2200 (i : X),\n i \u2208\n Exists.choose\n (_ :\n \u2203 t,\n \u22c3 (x : X) (_ : x \u2208 t),\n {z |\n \u2191(Finset.sup'\n (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z <\n \u2191f z + \u03b5} =\n \u22a4) \u2192\n \u2191f z - \u03b5 <\n \u2191(Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g i x) z} = \u22a4))\n (_ : Finset.Nonempty (ys i)) fun y => g i y)\n z\n[PROOFSTEP]\nrintro x -\n[GOAL]\ncase pos.right\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nL : Set C(X, \u211d)\nnA : Set.Nonempty L\ninf_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2293 g \u2208 L\nsup_mem : \u2200 (f : C(X, \u211d)), f \u2208 L \u2192 \u2200 (g : C(X, \u211d)), g \u2208 L \u2192 f \u2294 g \u2208 L\nsep : \u2200 (v : X \u2192 \u211d) (x y : X), \u2203 f, f \u2208 L \u2227 \u2191f x = v x \u2227 \u2191f y = v y\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nnX : Nonempty X\ng : X \u2192 X \u2192 C(X, \u211d)\nhg : \u2200 (x y : X), g x y \u2208 L\nw\u2081 : \u2200 (x y : X), \u2191(g x y) x = \u2191f x\nw\u2082 : \u2200 (x y : X), \u2191(g x y) y = \u2191f y\nU : X \u2192 X \u2192 Set X := fun x y => {z | \u2191f z - \u03b5 < \u2191(g x y) z}\nU_nhd_y : \u2200 (x y : X), U x y \u2208 \ud835\udcdd y\nys : X \u2192 Finset X := fun x => Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), U x x_1 = \u22a4)\nys_w : \u2200 (x : X), \u22c3 (y : X) (_ : y \u2208 ys x), U x y = \u22a4 :=\n fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))\nys_nonempty : \u2200 (x : X), Finset.Nonempty (ys x)\nh : X \u2192 \u2191L :=\n fun x =>\n { val := Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y,\n property := (_ : (Finset.sup' (ys x) (_ : Finset.Nonempty (ys x)) fun y => g x y) \u2208 L) }\nlt_h : \u2200 (x z : X), \u2191f z - \u03b5 < \u2191\u2191(h x) z\nh_eq : \u2200 (x : X), \u2191\u2191(h x) x = \u2191f x\nW : X \u2192 Set X := fun x => {z | \u2191\u2191(h x) z < \u2191f z + \u03b5}\nW_nhd : \u2200 (x : X), W x \u2208 \ud835\udcdd x\nxs : Finset X := Exists.choose (_ : \u2203 t, \u22c3 (x : X) (_ : x \u2208 t), W x = \u22a4)\nxs_w : \u22c3 (x : X) (_ : x \u2208 xs), W x = \u22a4 := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhd)\nxs_nonempty : Finset.Nonempty xs\nk : \u2191L :=\n { val := Finset.inf' xs xs_nonempty fun x => \u2191(h x),\n property := (_ : (Finset.inf' xs xs_nonempty fun x => \u2191(h x)) \u2208 L) }\nz x : X\n\u22a2 \u2191f z - \u03b5 <\n \u2191(Finset.sup' (Exists.choose (_ : \u2203 t, \u22c3 (x_1 : X) (_ : x_1 \u2208 t), {z | \u2191f z - \u03b5 < \u2191(g x x_1) z} = \u22a4))\n (_ : Finset.Nonempty (ys x)) fun y => g x y)\n z\n[PROOFSTEP]\napply lt_h\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\n\u22a2 Subalgebra.topologicalClosure A = \u22a4\n[PROOFSTEP]\napply SetLike.ext'\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = \u2191\u22a4\n[PROOFSTEP]\nlet L := A.topologicalClosure\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nL : Subalgebra \u211d C(X, \u211d) := Subalgebra.topologicalClosure A\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = \u2191\u22a4\n[PROOFSTEP]\nhave n : Set.Nonempty (L : Set C(X, \u211d)) := \u27e8(1 : C(X, \u211d)), A.le_topologicalClosure A.one_mem\u27e9\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nL : Subalgebra \u211d C(X, \u211d) := Subalgebra.topologicalClosure A\nn : Set.Nonempty \u2191L\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = \u2191\u22a4\n[PROOFSTEP]\nconvert\n sublattice_closure_eq_top (L : Set C(X, \u211d)) n\n (fun f fm g gm => inf_mem_closed_subalgebra L A.isClosed_topologicalClosure \u27e8f, fm\u27e9 \u27e8g, gm\u27e9)\n (fun f fm g gm => sup_mem_closed_subalgebra L A.isClosed_topologicalClosure \u27e8f, fm\u27e9 \u27e8g, gm\u27e9)\n (Subalgebra.SeparatesPoints.strongly (Subalgebra.separatesPoints_monotone A.le_topologicalClosure w))\n[GOAL]\ncase h.e'_2\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nL : Subalgebra \u211d C(X, \u211d) := Subalgebra.topologicalClosure A\nn : Set.Nonempty \u2191L\n\u22a2 \u2191(Subalgebra.topologicalClosure A) = closure \u2191L\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u22a2 f \u2208 Subalgebra.topologicalClosure A\n[PROOFSTEP]\nrw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u22a2 f \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\n\u22a2 \u2203 g, \u2016\u2191g - f\u2016 < \u03b5\n[PROOFSTEP]\nhave w := mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw\u271d : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nw : \u2203\u1da0 (x : C(X, \u211d)) in \ud835\udcdd f, x \u2208 \u2191A\n\u22a2 \u2203 g, \u2016\u2191g - f\u2016 < \u03b5\n[PROOFSTEP]\nrw [Metric.nhds_basis_ball.frequently_iff] at w \n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw\u271d : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nw : \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.ball f i \u2227 x \u2208 \u2191A\n\u22a2 \u2203 g, \u2016\u2191g - f\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8g, H, m\u27e9 := w \u03b5 pos\n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw\u271d : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nw : \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.ball f i \u2227 x \u2208 \u2191A\ng : C(X, \u211d)\nH : g \u2208 Metric.ball f \u03b5\nm : g \u2208 \u2191A\n\u22a2 \u2203 g, \u2016\u2191g - f\u2016 < \u03b5\n[PROOFSTEP]\nrw [Metric.mem_ball, dist_eq_norm] at H \n[GOAL]\ncase intro.intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw\u271d : Subalgebra.SeparatesPoints A\nf : C(X, \u211d)\n\u03b5 : \u211d\npos : 0 < \u03b5\nw : \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.ball f i \u2227 x \u2208 \u2191A\ng : C(X, \u211d)\nH : \u2016g - f\u2016 < \u03b5\nm : g \u2208 \u2191A\n\u22a2 \u2203 g, \u2016\u2191g - f\u2016 < \u03b5\n[PROOFSTEP]\nexact \u27e8\u27e8g, m\u27e9, H\u27e9\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : X \u2192 \u211d\nc : Continuous f\n\u03b5 : \u211d\npos : 0 < \u03b5\n\u22a2 \u2203 g, \u2200 (x : X), \u2016\u2191\u2191g x - f x\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8g, b\u27e9 := exists_mem_subalgebra_near_continuousMap_of_separatesPoints A w \u27e8f, c\u27e9 \u03b5 pos\n[GOAL]\ncase intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : X \u2192 \u211d\nc : Continuous f\n\u03b5 : \u211d\npos : 0 < \u03b5\ng : { x // x \u2208 A }\nb : \u2016\u2191g - mk f\u2016 < \u03b5\n\u22a2 \u2203 g, \u2200 (x : X), \u2016\u2191\u2191g x - f x\u2016 < \u03b5\n[PROOFSTEP]\nuse g\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : Subalgebra \u211d C(X, \u211d)\nw : Subalgebra.SeparatesPoints A\nf : X \u2192 \u211d\nc : Continuous f\n\u03b5 : \u211d\npos : 0 < \u03b5\ng : { x // x \u2208 A }\nb : \u2016\u2191g - mk f\u2016 < \u03b5\n\u22a2 \u2200 (x : X), \u2016\u2191\u2191g x - f x\u2016 < \u03b5\n[PROOFSTEP]\nrwa [norm_lt_iff _ pos] at b \n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\n\u22a2 SeparatesPoints\n (comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra))\n[PROOFSTEP]\nintro x\u2081 x\u2082 hx\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\n\u22a2 \u2203 f,\n f \u2208\n (fun f => \u2191f) ''\n \u2191(comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra)) \u2227\n f x\u2081 \u2260 f x\u2082\n[PROOFSTEP]\nobtain \u27e8_, \u27e8f, hfA, rfl\u27e9, hf\u27e9 := hA hx\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\n\u22a2 \u2203 f,\n f \u2208\n (fun f => \u2191f) ''\n \u2191(comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra)) \u2227\n f x\u2081 \u2260 f x\u2082\n[PROOFSTEP]\nlet F : C(X, \ud835\udd5c) :=\n f -\n const _\n (f x\u2082)\n -- Subtract the constant `f x\u2082` from `f`; this is still an element of the subalgebra\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\n\u22a2 \u2203 f,\n f \u2208\n (fun f => \u2191f) ''\n \u2191(comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra)) \u2227\n f x\u2081 \u2260 f x\u2082\n[PROOFSTEP]\nhave hFA : F \u2208 A :=\n by\n refine' A.sub_mem hfA (@Eq.subst _ (\u00b7 \u2208 A) _ _ _ <| A.smul_mem A.one_mem <| f x\u2082)\n ext1\n simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one, const_apply]\n -- Consider now the function `fun x \u21a6 |f x - f x\u2082| ^ 2`\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\n\u22a2 F \u2208 A\n[PROOFSTEP]\nrefine' A.sub_mem hfA (@Eq.subst _ (\u00b7 \u2208 A) _ _ _ <| A.smul_mem A.one_mem <| f x\u2082)\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\n\u22a2 \u2191f x\u2082 \u2022 1 = const X (\u2191f x\u2082)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\na\u271d : X\n\u22a2 \u2191(\u2191f x\u2082 \u2022 1) a\u271d = \u2191(const X (\u2191f x\u2082)) a\u271d\n[PROOFSTEP]\nsimp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one, const_apply]\n -- Consider now the function `fun x \u21a6 |f x - f x\u2082| ^ 2`\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\n\u22a2 \u2203 f,\n f \u2208\n (fun f => \u2191f) ''\n \u2191(comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra)) \u2227\n f x\u2081 \u2260 f x\u2082\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8(\u27e8IsROrC.normSq, continuous_normSq\u27e9 : C(\ud835\udd5c, \u211d)).comp F, _, rfl\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\n\u22a2 comp (ContinuousMap.mk \u2191normSq) F \u2208\n \u2191(comap (AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (restrictScalars \u211d A.toSubalgebra))\n[PROOFSTEP]\nrw [SetLike.mem_coe, Subalgebra.mem_comap]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\n\u22a2 \u2191(AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk \u2191normSq) F) \u2208\n restrictScalars \u211d A.toSubalgebra\n[PROOFSTEP]\nconvert (A.restrictScalars \u211d).mul_mem hFA (star_mem hFA : star F \u2208 A)\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\n\u22a2 \u2191(AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk \u2191normSq) F) = F * star F\n[PROOFSTEP]\next1\n[GOAL]\ncase h.e'_4.h\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\na\u271d : X\n\u22a2 \u2191(\u2191(AlgHom.compLeftContinuous \u211d ofRealAm (_ : Continuous ofReal)) (comp (ContinuousMap.mk \u2191normSq) F)) a\u271d =\n \u2191(F * star F) a\u271d\n[PROOFSTEP]\nexact (IsROrC.mul_conj (K := \ud835\udd5c) _).symm\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\n\u22a2 (fun f => \u2191f) (comp (ContinuousMap.mk \u2191normSq) F) x\u2081 \u2260 (fun f => \u2191f) (comp (ContinuousMap.mk \u2191normSq) F) x\u2082\n[PROOFSTEP]\nhave : f x\u2081 - f x\u2082 \u2260 0 := sub_ne_zero.mpr hf\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : TopologicalSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : SeparatesPoints A.toSubalgebra\nx\u2081 x\u2082 : X\nhx : x\u2081 \u2260 x\u2082\nf : C(X, \ud835\udd5c)\nhfA : f \u2208 \u2191A.toSubalgebra\nhf : (fun f => \u2191f) f x\u2081 \u2260 (fun f => \u2191f) f x\u2082\nF : C(X, \ud835\udd5c) := f - const X (\u2191f x\u2082)\nhFA : F \u2208 A\nthis : \u2191f x\u2081 - \u2191f x\u2082 \u2260 0\n\u22a2 (fun f => \u2191f) (comp (ContinuousMap.mk \u2191normSq) F) x\u2081 \u2260 (fun f => \u2191f) (comp (ContinuousMap.mk \u2191normSq) F) x\u2082\n[PROOFSTEP]\nsimpa only [comp_apply, coe_sub, coe_const, sub_apply, coe_mk, sub_self, map_zero, Ne.def, normSq_eq_zero,\n const_apply] using this\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\n\u22a2 StarSubalgebra.topologicalClosure A = \u22a4\n[PROOFSTEP]\nrw [StarSubalgebra.eq_top_iff]\n -- Let `I` be the natural inclusion of `C(X, \u211d)` into `C(X, \ud835\udd5c)`\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\n\u22a2 \u2200 (x : C(X, \ud835\udd5c)), x \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nlet I : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ofRealClm.compLeftContinuous \u211d X\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\n\u22a2 \u2200 (x : C(X, \ud835\udd5c)), x \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nhave key : LinearMap.range I \u2264 (A.toSubmodule.restrictScalars \u211d).topologicalClosure := by\n -- Let `A\u2080` be the subalgebra of `C(X, \u211d)` consisting of `A`'s purely real elements; it is the\n -- preimage of `A` under `I`. In this argument we only need its submodule structure.\n let A\u2080 : Submodule \u211d C(X, \u211d) := (A.toSubmodule.restrictScalars \u211d).comap I\n have SW : A\u2080.topologicalClosure = \u22a4 :=\n haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real\n congr_arg Subalgebra.toSubmodule this\n rw [\u2190 Submodule.map_top, \u2190 SW]\n -- So it suffices to prove that the image under `I` of the closure of `A\u2080` is contained in the\n -- closure of `A`, which follows by abstract nonsense\n have h\u2081 := A\u2080.topologicalClosure_map ((@ofRealClm \ud835\udd5c _).compLeftContinuousCompact X)\n have h\u2082 := (A.toSubmodule.restrictScalars \u211d).map_comap_le I\n exact\n h\u2081.trans\n (Submodule.topologicalClosure_mono h\u2082)\n -- In particular, for a function `f` in `C(X, \ud835\udd5c)`, the real and imaginary parts of `f` are in the\n -- closure of `A`\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\n\u22a2 LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nlet A\u2080 : Submodule \u211d C(X, \u211d) := (A.toSubmodule.restrictScalars \u211d).comap I\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nA\u2080 : Submodule \u211d C(X, \u211d) := Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n\u22a2 LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nhave SW : A\u2080.topologicalClosure = \u22a4 :=\n haveI := subalgebra_topologicalClosure_eq_top_of_separatesPoints _ hA.isROrC_to_real\n congr_arg Subalgebra.toSubmodule this\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nA\u2080 : Submodule \u211d C(X, \u211d) := Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nSW : Submodule.topologicalClosure A\u2080 = \u22a4\n\u22a2 LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nrw [\u2190 Submodule.map_top, \u2190 SW]\n -- So it suffices to prove that the image under `I` of the closure of `A\u2080` is contained in the\n -- closure of `A`, which follows by abstract nonsense\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nA\u2080 : Submodule \u211d C(X, \u211d) := Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nSW : Submodule.topologicalClosure A\u2080 = \u22a4\n\u22a2 Submodule.map I (Submodule.topologicalClosure A\u2080) \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nhave h\u2081 := A\u2080.topologicalClosure_map ((@ofRealClm \ud835\udd5c _).compLeftContinuousCompact X)\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nA\u2080 : Submodule \u211d C(X, \u211d) := Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nSW : Submodule.topologicalClosure A\u2080 = \u22a4\nh\u2081 :\n Submodule.map (\u2191(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A\u2080) \u2264\n Submodule.topologicalClosure (Submodule.map (\u2191(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A\u2080)\n\u22a2 Submodule.map I (Submodule.topologicalClosure A\u2080) \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nhave h\u2082 := (A.toSubmodule.restrictScalars \u211d).map_comap_le I\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nA\u2080 : Submodule \u211d C(X, \u211d) := Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nSW : Submodule.topologicalClosure A\u2080 = \u22a4\nh\u2081 :\n Submodule.map (\u2191(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) (Submodule.topologicalClosure A\u2080) \u2264\n Submodule.topologicalClosure (Submodule.map (\u2191(ContinuousLinearMap.compLeftContinuousCompact X ofRealClm)) A\u2080)\nh\u2082 :\n Submodule.map I (Submodule.comap I (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))) \u2264\n Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra)\n\u22a2 Submodule.map I (Submodule.topologicalClosure A\u2080) \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n[PROOFSTEP]\nexact\n h\u2081.trans\n (Submodule.topologicalClosure_mono h\u2082)\n -- In particular, for a function `f` in `C(X, \ud835\udd5c)`, the real and imaginary parts of `f` are in the\n -- closure of `A`\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\n\u22a2 \u2200 (x : C(X, \ud835\udd5c)), x \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nintro f\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nlet f_re : C(X, \u211d) := (\u27e8IsROrC.re, IsROrC.reClm.continuous\u27e9 : C(\ud835\udd5c, \u211d)).comp f\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nlet f_im : C(X, \u211d) := (\u27e8IsROrC.im, IsROrC.imClm.continuous\u27e9 : C(\ud835\udd5c, \u211d)).comp f\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nhave h_f_re : I f_re \u2208 A.topologicalClosure := key \u27e8f_re, rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nhave h_f_im : I f_im \u2208 A.topologicalClosure :=\n key\n \u27e8f_im, rfl\u27e9\n -- So `f_re + I \u2022 f_im` is in the closure of `A`\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nhave := A.topologicalClosure.add_mem h_f_re (A.topologicalClosure.smul_mem h_f_im IsROrC.I)\n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\nthis : \u2191I f_re + IsROrC.I \u2022 \u2191I f_im \u2208 (StarSubalgebra.topologicalClosure A).toSubalgebra\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nrw [StarSubalgebra.mem_toSubalgebra] at this \n[GOAL]\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\nthis : \u2191I f_re + IsROrC.I \u2022 \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\n\u22a2 f \u2208 StarSubalgebra.topologicalClosure A\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\nthis : \u2191I f_re + IsROrC.I \u2022 \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\n\u22a2 f = \u2191I f_re + IsROrC.I \u2022 \u2191I f_im\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_4.h\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\nthis : \u2191I f_re + IsROrC.I \u2022 \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\na\u271d : X\n\u22a2 \u2191f a\u271d = \u2191(\u2191I f_re + IsROrC.I \u2022 \u2191I f_im) a\u271d\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase h.e'_4.h.h\n\ud835\udd5c : Type u_2\nX : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : CompactSpace X\nA : StarSubalgebra \ud835\udd5c C(X, \ud835\udd5c)\nhA : Subalgebra.SeparatesPoints A.toSubalgebra\nI : C(X, \u211d) \u2192\u2097[\u211d] C(X, \ud835\udd5c) := ContinuousLinearMap.compLeftContinuous \u211d X ofRealClm\nkey :\n LinearMap.range I \u2264\n Submodule.topologicalClosure (Submodule.restrictScalars \u211d (\u2191Subalgebra.toSubmodule A.toSubalgebra))\nf : C(X, \ud835\udd5c)\nf_re : C(X, \u211d) := comp (mk \u2191re) f\nf_im : C(X, \u211d) := comp (mk \u2191im) f\nh_f_re : \u2191I f_re \u2208 StarSubalgebra.topologicalClosure A\nh_f_im : \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\nthis : \u2191I f_re + IsROrC.I \u2022 \u2191I f_im \u2208 StarSubalgebra.topologicalClosure A\na\u271d : X\n\u22a2 \u2191(\u2191I f_re + IsROrC.I \u2022 \u2191I f_im) a\u271d = \u2191f a\u271d\n[PROOFSTEP]\nsimp [mul_comm IsROrC.I _]\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra \u211d A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \u211d\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \u211d) \u2192\u2090[\u211d] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nsuffices (\u22a4 : Subalgebra \u211d C(s, \u211d)) \u2264 AlgHom.equalizer \u03c6 \u03c8 from AlgHom.ext fun x => this (by trivial)\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra \u211d A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \u211d\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \u211d) \u2192\u2090[\u211d] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\nthis : \u22a4 \u2264 AlgHom.equalizer \u03c6 \u03c8\nx : C(\u2191s, \u211d)\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra \u211d A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \u211d\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \u211d) \u2192\u2090[\u211d] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 \u22a4 \u2264 AlgHom.equalizer \u03c6 \u03c8\n[PROOFSTEP]\nrw [\u2190 polynomialFunctions.topologicalClosure s]\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra \u211d A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \u211d\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \u211d) \u2192\u2090[\u211d] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 Subalgebra.topologicalClosure (polynomialFunctions s) \u2264 AlgHom.equalizer \u03c6 \u03c8\n[PROOFSTEP]\nexact\n Subalgebra.topologicalClosure_minimal (polynomialFunctions s) (polynomialFunctions.le_equalizer s \u03c6 \u03c8 h)\n (isClosed_eq h\u03c6 h\u03c8)\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : StarRing A\ninst\u271d\u00b3 : Algebra \ud835\udd5c A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \ud835\udd5c\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \ud835\udd5c) \u2192\u22c6\u2090[\ud835\udd5c] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\nsuffices (\u22a4 : StarSubalgebra \ud835\udd5c C(s, \ud835\udd5c)) \u2264 StarAlgHom.equalizer \u03c6 \u03c8 from StarAlgHom.ext fun x => this mem_top\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : StarRing A\ninst\u271d\u00b3 : Algebra \ud835\udd5c A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \ud835\udd5c\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \ud835\udd5c) \u2192\u22c6\u2090[\ud835\udd5c] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 \u22a4 \u2264 StarAlgHom.equalizer \u03c6 \u03c8\n[PROOFSTEP]\nrw [\u2190 polynomialFunctions.starClosure_topologicalClosure s]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : StarRing A\ninst\u271d\u00b3 : Algebra \ud835\udd5c A\ninst\u271d\u00b2 : TopologicalSpace A\ninst\u271d\u00b9 : T2Space A\ns : Set \ud835\udd5c\ninst\u271d : CompactSpace \u2191s\n\u03c6 \u03c8 : C(\u2191s, \ud835\udd5c) \u2192\u22c6\u2090[\ud835\udd5c] A\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh : \u2191\u03c6 (\u2191(toContinuousMapOnAlgHom s) X) = \u2191\u03c8 (\u2191(toContinuousMapOnAlgHom s) X)\n\u22a2 topologicalClosure (Subalgebra.starClosure (polynomialFunctions s)) \u2264 StarAlgHom.equalizer \u03c6 \u03c8\n[PROOFSTEP]\nexact\n StarSubalgebra.topologicalClosure_minimal (polynomialFunctions.starClosure_le_equalizer s \u03c6 \u03c8 h) (isClosed_eq h\u03c6 h\u03c8)\n", "meta": {"mathlib_filename": "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "llama_tokens": 56913, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.29483607469542883}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX : C\n\u22a2 AddCommGroup (End X)\n[PROOFSTEP]\ndsimp [End]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX : C\n\u22a2 AddCommGroup (X \u27f6 X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf : P \u27f6 Q\ng g' : Q \u27f6 R\n\u22a2 (fun g => f \u226b g) (g + g') = (fun g => f \u226b g) g + (fun g => f \u226b g) g'\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\ng : Q \u27f6 R\nf f' : P \u27f6 Q\n\u22a2 (fun f => f \u226b g) (f + f') = (fun f => f \u226b g) f + (fun f => f \u226b g) f'\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf f' : P \u27f6 Q\ng g' : Q \u27f6 R\n\u22a2 (-f) \u226b (-g) = f \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP\u271d Q\u271d R : C\nf\u271d f' : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\nP Q : C\nf : P \u27f6 Q\ninst\u271d : Epi f\nZ\u271d : C\ng g' : Q \u27f6 Z\u271d\nH : (-f) \u226b g = (-f) \u226b g'\n\u22a2 g = g'\n[PROOFSTEP]\nrwa [neg_comp, neg_comp, \u2190 comp_neg, \u2190 comp_neg, cancel_epi, neg_inj] at H \n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP\u271d Q\u271d R : C\nf\u271d f' : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\nP Q : C\nf : P \u27f6 Q\ninst\u271d : Mono f\nZ\u271d : C\ng g' : Z\u271d \u27f6 P\nH : g \u226b (-f) = g' \u226b (-f)\n\u22a2 g = g'\n[PROOFSTEP]\nrwa [comp_neg, comp_neg, \u2190 neg_comp, \u2190 neg_comp, cancel_mono, neg_inj] at H \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng g' : Q \u27f6 R\nX : C\nsrc\u271d : Monoid (End X) := End.monoid\nf : End X\n\u22a2 0 * f = 0\n[PROOFSTEP]\ndsimp [mul]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng g' : Q \u27f6 R\nX : C\nsrc\u271d : Monoid (End X) := End.monoid\nf : End X\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nexact HasZeroMorphisms.comp_zero f _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng g' : Q \u27f6 R\nX : C\nsrc\u271d : Monoid (End X) := End.monoid\nf : End X\n\u22a2 f * 0 = 0\n[PROOFSTEP]\ndsimp [mul]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng g' : Q \u27f6 R\nX : C\nsrc\u271d : Monoid (End X) := End.monoid\nf : End X\n\u22a2 0 \u226b f = 0\n[PROOFSTEP]\nexact HasZeroMorphisms.zero_comp _ f\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng\u271d g' : Q \u27f6 R\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasLimit (parallelPair f 0)\nw : kernel.\u03b9 f = 0\nP\u271d : C\ng : P\u271d \u27f6 X\nh : g \u226b f = 0\n\u22a2 g = 0\n[PROOFSTEP]\nrw [\u2190 kernel.lift_\u03b9 f g h, w, Limits.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP Q R : C\nf\u271d f' : P \u27f6 Q\ng\u271d g' : Q \u27f6 R\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasColimit (parallelPair f 0)\nw : cokernel.\u03c0 f = 0\nR\u271d : C\ng : Y \u27f6 R\u271d\nh : f \u226b g = 0\n\u22a2 g = 0\n[PROOFSTEP]\nrw [\u2190 cokernel.\u03c0_desc f g h, w, Limits.zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP Q R : C\nf f' : P \u27f6 Q\ng g' : Q \u27f6 R\ninst\u271d : IsIso f\n\u22a2 f \u226b g = 0 \u2194 g = 0\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp, Limits.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : Preadditive C\nP Q R : C\nf f' : P \u27f6 Q\ng g' : Q \u27f6 R\ninst\u271d : IsIso g\n\u22a2 f \u226b g = 0 \u2194 f = 0\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv, Limits.zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : KernelFork (f - g)\n\u22a2 Fork.\u03b9 c \u226b f = Fork.\u03b9 c \u226b g\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 comp_sub, c.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Fork f g\n\u22a2 Fork.\u03b9 c \u226b (f - g) = Fork.\u03b9 c \u226b 0\n[PROOFSTEP]\nrw [comp_sub, comp_zero, sub_eq_zero, c.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nP : C\n\u03b9 : P \u27f6 X\nw : \u03b9 \u226b f = \u03b9 \u226b g\n\u22a2 \u03b9 \u226b (f - g) = 0\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : KernelFork (f - g)\ni : IsLimit c\ns : Fork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (forkOfKernelFork c).pt).obj WalkingParallelPair.zero\nh : m\u271d \u226b Fork.\u03b9 (forkOfKernelFork c) = Fork.\u03b9 s\n\u22a2 m\u271d = IsLimit.lift i (kernelForkOfFork s)\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : KernelFork (f - g)\ni : IsLimit c\ns : Fork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (forkOfKernelFork c).pt).obj WalkingParallelPair.zero\nh : m\u271d \u226b Fork.\u03b9 (forkOfKernelFork c) = Fork.\u03b9 s\n\u22a2 m\u271d \u226b Fork.\u03b9 c = IsLimit.lift i (kernelForkOfFork s) \u226b Fork.\u03b9 c\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Fork f g\ni : IsLimit c\ns : Fork (f - g) 0\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (kernelForkOfFork c).pt).obj WalkingParallelPair.zero\nh : m\u271d \u226b Fork.\u03b9 (kernelForkOfFork c) = Fork.\u03b9 s\n\u22a2 m\u271d = IsLimit.lift i (forkOfKernelFork s)\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Fork f g\ni : IsLimit c\ns : Fork (f - g) 0\nm\u271d :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (kernelForkOfFork c).pt).obj WalkingParallelPair.zero\nh : m\u271d \u226b Fork.\u03b9 (kernelForkOfFork c) = Fork.\u03b9 s\n\u22a2 m\u271d \u226b Fork.\u03b9 c = IsLimit.lift i (forkOfKernelFork s) \u226b Fork.\u03b9 c\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : CokernelCofork (f - g)\n\u22a2 f \u226b Cofork.\u03c0 c = g \u226b Cofork.\u03c0 c\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 sub_comp, c.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Cofork f g\n\u22a2 (f - g) \u226b Cofork.\u03c0 c = 0 \u226b Cofork.\u03c0 c\n[PROOFSTEP]\nrw [sub_comp, zero_comp, sub_eq_zero, c.condition]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nP : C\n\u03c0 : Y \u27f6 P\nw : f \u226b \u03c0 = g \u226b \u03c0\n\u22a2 (f - g) \u226b \u03c0 = 0\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : CokernelCofork (f - g)\ni : IsColimit c\ns : Cofork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (coforkOfCokernelCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (coforkOfCokernelCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = IsColimit.desc i (cokernelCoforkOfCofork s)\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : CokernelCofork (f - g)\ni : IsColimit c\ns : Cofork f g\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (coforkOfCokernelCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (coforkOfCokernelCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 c \u226b IsColimit.desc i (cokernelCoforkOfCofork s)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Cofork f g\ni : IsColimit c\ns : Cofork (f - g) 0\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (cokernelCoforkOfCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (cokernelCoforkOfCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = IsColimit.desc i (coforkOfCokernelCofork s)\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext i\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX Y : C\nf g : X \u27f6 Y\nc : Cofork f g\ni : IsColimit c\ns : Cofork (f - g) 0\nm\u271d :\n ((Functor.const WalkingParallelPair).obj (cokernelCoforkOfCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (cokernelCoforkOfCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 c \u226b m\u271d = Cofork.\u03c0 c \u226b IsColimit.desc i (coforkOfCokernelCofork s)\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.Basic", "llama_tokens": 4266, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837478, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.2946291439800485}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d : C\n\u22a2 { obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n (\ud835\udfd9 x\u271d) =\n \ud835\udfd9\n ({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n x\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d\u00b2 : C\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n \u2191(({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n x\u271d\u00b2).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n ({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n (\ud835\udfd9 x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n (\ud835\udfd9\n ({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat)\n X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d\u00b2 : C\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n \u2191(({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n x\u271d\u00b2).obj\n x\u271d\u00b9)\n\u22a2 x\u271d \u226b \ud835\udfd9 x\u271d\u00b2 = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n (f \u226b g) =\n { obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n f \u226b\n { obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n g\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n \u2191(({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n X\u271d).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n ({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n (f \u226b g))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n ({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat)\n X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n f \u226b\n { obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat)\n X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.map\n g)\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n \u2191(({ obj := fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun X_1 =>\n { toZeroHom := { toFun := fun g => g \u226b f, map_zero' := (_ : 0 \u226b f = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun Y => preadditiveYonedaObj Y \u22d9 forget\u2082 (ModuleCat (End Y)) AddCommGroupCat) X).obj\n X_1)),\n (g + g') \u226b f = g \u226b f + g' \u226b f) } }.obj\n X\u271d).obj\n x\u271d\u00b9)\n\u22a2 x\u271d \u226b f \u226b g = (x\u271d \u226b f) \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d : C\u1d52\u1d56\n\u22a2 { obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n (\ud835\udfd9 x\u271d) =\n \ud835\udfd9\n ({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n x\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d\u00b2 : C\u1d52\u1d56\nx\u271d\u00b9 : C\nx\u271d :\n \u2191(({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n x\u271d\u00b2).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n ({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat)\n X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n (\ud835\udfd9 x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n (\ud835\udfd9\n ({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat)\n X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nx\u271d\u00b2 : C\u1d52\u1d56\nx\u271d\u00b9 : C\nx\u271d :\n \u2191(({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n x\u271d\u00b2).obj\n x\u271d\u00b9)\n\u22a2 \ud835\udfd9 x\u271d\u00b2.unop \u226b x\u271d = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n (f \u226b g) =\n { obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n f \u226b\n { obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n g\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx\u271d\u00b9 : C\nx\u271d :\n \u2191(({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n X\u271d).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n ({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat)\n X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n (f \u226b g))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n ({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat)\n X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n f \u226b\n { obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat)\n X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.map\n g)\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Preadditive C\nX\u271d Y\u271d Z\u271d : C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nx\u271d\u00b9 : C\nx\u271d :\n \u2191(({ obj := fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat,\n map := fun {X Y} f =>\n NatTrans.mk fun Y_1 =>\n { toZeroHom := { toFun := fun g => f.unop \u226b g, map_zero' := (_ : f.unop \u226b 0 = 0) },\n map_add' :=\n (_ :\n \u2200\n (g g' :\n \u2191(((fun X => preadditiveCoyonedaObj X \u22d9 forget\u2082 (ModuleCat (End X)) AddCommGroupCat) X).obj\n Y_1)),\n f.unop \u226b (g + g') = f.unop \u226b g + f.unop \u226b g') } }.obj\n X\u271d).obj\n x\u271d\u00b9)\n\u22a2 (g.unop \u226b f.unop) \u226b x\u271d = g.unop \u226b f.unop \u226b x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.Yoneda.Basic", "llama_tokens": 7268, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.38861802670584894, "lm_q1q2_score": 0.2944925532135503}} {"text": "[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\n\u22a2 Unbounded (fun x x_1 => x \u2264 x_1) s \u2194 \u2200 (a : \u03b1), \u2203 b, b \u2208 s \u2227 a < b\n[PROOFSTEP]\nsimp only [Unbounded, not_le]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) s \u2194 \u2200 (a : \u03b1), \u2203 b, b \u2208 s \u2227 a \u2264 b\n[PROOFSTEP]\nsimp only [Unbounded, not_lt]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) s \u2194 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nrefine' \u27e8fun h => _, bounded_le_of_bounded_lt\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh : Bounded (fun x x_1 => x \u2264 x_1) s\n\u22a2 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\ncases' h with a ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2208 s \u2192 (fun x x_1 => x \u2264 x_1) b a\n\u22a2 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\ncases' exists_gt a with b hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2208 s \u2192 (fun x x_1 => x \u2264 x_1) b a\nb : \u03b1\nhb : a < b\n\u22a2 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nexact \u27e8b, fun c hc => lt_of_le_of_lt (ha c hc) hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) s \u2194 Unbounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nsimp_rw [\u2190 not_bounded_iff, bounded_le_iff_bounded_lt]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x < x_1) (Iic a)\n[PROOFSTEP]\nsimp only [\u2190 bounded_le_iff_bounded_lt, bounded_le_Iic]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x > x_1) (Ici a)\n[PROOFSTEP]\nsimp only [\u2190 bounded_ge_iff_bounded_gt, bounded_ge_Ici]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na : \u03b1\n\u22a2 Bounded r (s \u2229 {b | \u00acr b a}) \u2194 Bounded r s\n[PROOFSTEP]\nrefine' \u27e8_, Bounded.mono (Set.inter_subset_left s _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na : \u03b1\n\u22a2 Bounded r (s \u2229 {b | \u00acr b a}) \u2192 Bounded r s\n[PROOFSTEP]\nrintro \u27e8b, hb\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na b : \u03b1\nhb : \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2229 {b | \u00acr b a} \u2192 r b_1 b\n\u22a2 Bounded r s\n[PROOFSTEP]\ncases' H a b with m hm\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na b : \u03b1\nhb : \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2229 {b | \u00acr b a} \u2192 r b_1 b\nm : \u03b1\nhm : \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\n\u22a2 Bounded r s\n[PROOFSTEP]\nexact \u27e8m, fun c hc => hm c (or_iff_not_imp_left.2 fun hca => hb c \u27e8hc, hca\u27e9)\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na : \u03b1\n\u22a2 Unbounded r (s \u2229 {b | \u00acr b a}) \u2194 Unbounded r s\n[PROOFSTEP]\nsimp_rw [\u2190 not_bounded_iff, bounded_inter_not H]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | \u00acb \u2264 a}) \u2194 Unbounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nrw [\u2190 not_bounded_iff, \u2190 not_bounded_iff, not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | \u00acb \u2264 a}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nexact bounded_le_inter_not_le a\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a < b}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nsimp_rw [\u2190 not_le, bounded_le_inter_not_le]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a < b}) \u2194 Unbounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nconvert @unbounded_le_inter_not_le _ s _ a\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na x\u271d : \u03b1\n\u22a2 a < x\u271d \u2194 \u00acx\u271d \u2264 a\n[PROOFSTEP]\nexact lt_iff_not_le\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a \u2264 b}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nrefine' \u27e8_, Bounded.mono (Set.inter_subset_left s _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a \u2264 b}) \u2192 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nrw [\u2190 @bounded_le_inter_lt _ s _ a]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a \u2264 b}) \u2192 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a < b})\n[PROOFSTEP]\nexact Bounded.mono fun x \u27e8hx, hx'\u27e9 => \u27e8hx, le_of_lt hx'\u27e9\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a \u2264 b}) \u2194 Unbounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nrw [\u2190 not_bounded_iff, \u2190 not_bounded_iff, not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a \u2264 b}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nexact bounded_le_inter_le a\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) (s \u2229 {b | \u00acb < a}) \u2194 Unbounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nrw [\u2190 not_bounded_iff, \u2190 not_bounded_iff, not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : SemilatticeSup \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x < x_1) (s \u2229 {b | \u00acb < a}) \u2194 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nexact bounded_lt_inter_not_lt a\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x < x_1) (s \u2229 {b | a \u2264 b}) \u2194 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nconvert @bounded_lt_inter_not_lt _ s _ a\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na x\u271d : \u03b1\n\u22a2 a \u2264 x\u271d \u2194 \u00acx\u271d < a\n[PROOFSTEP]\nexact not_lt.symm\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) (s \u2229 {b | a \u2264 b}) \u2194 Unbounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nconvert @unbounded_lt_inter_not_lt _ s _ a\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na x\u271d : \u03b1\n\u22a2 a \u2264 x\u271d \u2194 \u00acx\u271d < a\n[PROOFSTEP]\nexact not_lt.symm\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x < x_1) (s \u2229 {b | a < b}) \u2194 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nrw [\u2190 bounded_le_iff_bounded_lt, \u2190 bounded_le_iff_bounded_lt]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a < b}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s\n[PROOFSTEP]\nexact bounded_le_inter_lt a\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) (s \u2229 {b | a < b}) \u2194 Unbounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nrw [\u2190 not_bounded_iff, \u2190 not_bounded_iff, not_iff_not]\n[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x < x_1) (s \u2229 {b | a < b}) \u2194 Bounded (fun x x_1 => x < x_1) s\n[PROOFSTEP]\nexact bounded_lt_inter_lt a\n", "meta": {"mathlib_filename": "Mathlib.Order.Bounded", "llama_tokens": 3948, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.2942262473483473}} {"text": "[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\n\u22a2 Set.IsPwo (Function.support 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\nx : HahnSeries \u0393 R\n\u22a2 Set.Nonempty (support x) \u2194 x \u2260 0\n[PROOFSTEP]\nrw [support, support_nonempty_iff, Ne.def, coeff_fun_eq_zero_iff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na b : \u0393\nr : R\n\u22a2 coeff (\u2191(single a) r) b = if b = a then r else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na b : \u0393\nr : R\nh : b = a\n\u22a2 coeff (\u2191(single a) r) b = r\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na b : \u0393\nr : R\nh : \u00acb = a\n\u22a2 coeff (\u2191(single a) r) b = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr s : R\nrs : \u2191(single a) r = \u2191(single a) s\n\u22a2 r = s\n[PROOFSTEP]\nrw [\u2190 single_coeff_same a r, \u2190 single_coeff_same a s, rs]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 \u2191(single a) r = 0 \u2194 r = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 \u2191(single a) r = 0 \u2192 r = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r \u2260 0 \u2192 \u2191(single a) r \u2260 0\n[PROOFSTEP]\nexact single_ne_zero\n[GOAL]\ncase mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r = 0 \u2192 \u2191(single a) r = 0\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Zero R\na b : \u0393\nr : R\ninst\u271d\u00b9 : Nonempty \u0393\ninst\u271d : Nontrivial R\n\u22a2 \u2203 x y, x \u2260 y\n[PROOFSTEP]\nobtain \u27e8r, s, rs\u27e9 := exists_pair_ne R\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Zero R\na b : \u0393\nr\u271d : R\ninst\u271d\u00b9 : Nonempty \u0393\ninst\u271d : Nontrivial R\nr s : R\nrs : r \u2260 s\n\u22a2 \u2203 x y, x \u2260 y\n[PROOFSTEP]\ninhabit \u0393\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Zero R\na b : \u0393\nr\u271d : R\ninst\u271d\u00b9 : Nonempty \u0393\ninst\u271d : Nontrivial R\nr s : R\nrs : r \u2260 s\ninhabited_h : Inhabited \u0393\n\u22a2 \u2203 x y, x \u2260 y\n[PROOFSTEP]\nrefine' \u27e8single default r, single default s, fun con => rs _\u27e9\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Zero R\na b : \u0393\nr\u271d : R\ninst\u271d\u00b9 : Nonempty \u0393\ninst\u271d : Nontrivial R\nr s : R\nrs : r \u2260 s\ninhabited_h : Inhabited \u0393\ncon : \u2191(single default) r = \u2191(single default) s\n\u22a2 r = s\n[PROOFSTEP]\nrw [\u2190 single_coeff_same (default : \u0393) r, con, single_coeff_same]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\nhx : x \u2260 0\n\u22a2 coeff x (order x) \u2260 0\n[PROOFSTEP]\nrw [order_of_ne hx]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\nhx : x \u2260 0\n\u22a2 coeff x (Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x))) \u2260 0\n[PROOFSTEP]\nexact x.isWf_support.min_mem (support_nonempty_iff.2 hx)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\ni : \u0393\nhi : i < order x\n\u22a2 coeff x i = 0\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase inl\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\ni : \u0393\nhi : i < order 0\n\u22a2 coeff 0 i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\ni : \u0393\nhi : i < order x\nhx : x \u2260 0\n\u22a2 coeff x i = 0\n[PROOFSTEP]\ncontrapose! hi\n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\ni : \u0393\nhx : x \u2260 0\nhi : coeff x i \u2260 0\n\u22a2 \u00aci < order x\n[PROOFSTEP]\nrw [\u2190 mem_support] at hi \n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\ni : \u0393\nhx : x \u2260 0\nhi : i \u2208 support x\n\u22a2 \u00aci < order x\n[PROOFSTEP]\nrw [order_of_ne hx]\n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\ninst\u271d : Zero \u0393\nx : HahnSeries \u0393 R\ni : \u0393\nhx : x \u2260 0\nhi : i \u2208 support x\n\u22a2 \u00aci < Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x))\n[PROOFSTEP]\nexact Set.IsWf.not_lt_min _ _ hi\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b\u271d : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\nb : \u0393'\nhb : b \u2208 Function.support fun b => if h : b \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0\n\u22a2 b \u2208 \u2191f '' support x\n[PROOFSTEP]\ncontrapose! hb\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b\u271d : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\nb : \u0393'\nhb : \u00acb \u2208 (fun a => \u2191f a) '' support x\n\u22a2 \u00acb \u2208\n Function.support fun b =>\n if h : b \u2208 (fun a => \u2191f a) '' support x then coeff x (Classical.choose (_ : b \u2208 \u2191f '' support x)) else 0\n[PROOFSTEP]\nrw [Function.mem_support, dif_neg hb, Classical.not_not]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (embDomain f x) (\u2191f a) = coeff x a\n[PROOFSTEP]\nrw [embDomain]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff\n { coeff := fun b => if h : b \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0,\n isPwo_support' :=\n (_ :\n Set.IsPwo (Function.support fun b => if h : b \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0)) }\n (\u2191f a) =\n coeff x a\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 (if h : \u2191f a \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0) = coeff x a\n[PROOFSTEP]\nby_cases ha : a \u2208 x.support\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : a \u2208 support x\n\u22a2 (if h : \u2191f a \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0) = coeff x a\n[PROOFSTEP]\nrw [dif_pos (Set.mem_image_of_mem f ha)]\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : a \u2208 support x\n\u22a2 coeff x (Classical.choose (_ : \u2191f a \u2208 \u2191f '' support x)) = coeff x a\n[PROOFSTEP]\nexact congr rfl (f.injective (Classical.choose_spec (Set.mem_image_of_mem f ha)).2)\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : \u00aca \u2208 support x\n\u22a2 (if h : \u2191f a \u2208 \u2191f '' support x then coeff x (Classical.choose h) else 0) = coeff x a\n[PROOFSTEP]\nrw [dif_neg, Classical.not_not.1 fun c => ha ((mem_support _ _).2 c)]\n[GOAL]\ncase neg.hnc\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : \u00aca \u2208 support x\n\u22a2 \u00ac\u2191f a \u2208 \u2191f '' support x\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\ncase neg.hnc\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : \u2191f a \u2208 \u2191f '' support x\n\u22a2 a \u2208 support x\n[PROOFSTEP]\nobtain \u27e8b, hb1, hb2\u27e9 := (Set.mem_image _ _ _).1 ha\n[GOAL]\ncase neg.hnc.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na\u271d b\u271d : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\na : \u0393\nha : \u2191f a \u2208 \u2191f '' support x\nb : \u0393\nhb1 : b \u2208 support x\nhb2 : \u2191f b = \u2191f a\n\u22a2 a \u2208 support x\n[PROOFSTEP]\nrwa [f.injective hb2] at hb1 \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\n\u22a2 support (embDomain f x) \u2286 \u2191f '' support x\n[PROOFSTEP]\nintro g hg\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\ng : \u0393'\nhg : g \u2208 support (embDomain f x)\n\u22a2 g \u2208 \u2191f '' support x\n[PROOFSTEP]\ncontrapose! hg\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx : HahnSeries \u0393 R\ng : \u0393'\nhg : \u00acg \u2208 (fun a => \u2191f a) '' support x\n\u22a2 \u00acg \u2208 support (embDomain f x)\n[PROOFSTEP]\nrw [mem_support, embDomain_notin_image_support hg, Classical.not_not]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\n\u22a2 embDomain f 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx\u271d : \u0393'\n\u22a2 coeff (embDomain f 0) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\nsimp [embDomain_notin_image_support]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\n\u22a2 embDomain f (\u2191(single g) r) = \u2191(single (\u2191f g)) r\n[PROOFSTEP]\next g'\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\n\u22a2 coeff (embDomain f (\u2191(single g) r)) g' = coeff (\u2191(single (\u2191f g)) r) g'\n[PROOFSTEP]\nby_cases h : g' = f g\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\nh : g' = \u2191f g\n\u22a2 coeff (embDomain f (\u2191(single g) r)) g' = coeff (\u2191(single (\u2191f g)) r) g'\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\nh : \u00acg' = \u2191f g\n\u22a2 coeff (embDomain f (\u2191(single g) r)) g' = coeff (\u2191(single (\u2191f g)) r) g'\n[PROOFSTEP]\nrw [embDomain_notin_image_support, single_coeff_of_ne h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\nh : \u00acg' = \u2191f g\n\u22a2 \u00acg' \u2208 \u2191f '' support (\u2191(single g) r)\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\nh : \u00acg' = \u2191f g\nhr : r = 0\n\u22a2 \u00acg' \u2208 \u2191f '' support (\u2191(single g) r)\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr\u271d : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\ng : \u0393\nr : R\ng' : \u0393'\nh : \u00acg' = \u2191f g\nhr : \u00acr = 0\n\u22a2 \u00acg' \u2208 \u2191f '' support (\u2191(single g) r)\n[PROOFSTEP]\nrwa [support_single_of_ne hr, Set.image_singleton, Set.mem_singleton_iff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\nxy : embDomain f x = embDomain f y\n\u22a2 x = y\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\nxy : embDomain f x = embDomain f y\ng : \u0393\n\u22a2 coeff x g = coeff y g\n[PROOFSTEP]\nrw [HahnSeries.ext_iff, Function.funext_iff] at xy \n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\nxy : \u2200 (a : \u0393'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : \u0393\n\u22a2 coeff x g = coeff y g\n[PROOFSTEP]\nhave xyg := xy (f g)\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : Zero R\na b : \u0393\nr : R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\nxy : \u2200 (a : \u0393'), coeff (embDomain f x) a = coeff (embDomain f y) a\ng : \u0393\nxyg : coeff (embDomain f x) (\u2191f g) = coeff (embDomain f y) (\u2191f g)\n\u22a2 coeff x g = coeff y g\n[PROOFSTEP]\nrwa [embDomain_coeff, embDomain_coeff] at xyg \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y z : HahnSeries \u0393 R\n\u22a2 x + y + z = x + (y + z)\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y z : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x + y + z) x\u271d = coeff (x + (y + z)) x\u271d\n[PROOFSTEP]\napply add_assoc\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx : HahnSeries \u0393 R\n\u22a2 0 + x = x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (0 + x) x\u271d = coeff x x\u271d\n[PROOFSTEP]\napply zero_add\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx : HahnSeries \u0393 R\n\u22a2 x + 0 = x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x + 0) x\u271d = coeff x x\u271d\n[PROOFSTEP]\napply add_zero\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y : HahnSeries \u0393 R\na : \u0393\nha : a \u2208 support (x + y)\n\u22a2 a \u2208 support x \u222a support y\n[PROOFSTEP]\nrw [mem_support, add_coeff] at ha \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y : HahnSeries \u0393 R\na : \u0393\nha : coeff x a + coeff y a \u2260 0\n\u22a2 a \u2208 support x \u222a support y\n[PROOFSTEP]\nrw [Set.mem_union, mem_support, mem_support]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y : HahnSeries \u0393 R\na : \u0393\nha : coeff x a + coeff y a \u2260 0\n\u22a2 coeff x a \u2260 0 \u2228 coeff y a \u2260 0\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\nx y : HahnSeries \u0393 R\na : \u0393\nha : coeff x a = 0 \u2227 coeff y a = 0\n\u22a2 coeff x a + coeff y a = 0\n[PROOFSTEP]\nrw [ha.1, ha.2, add_zero]\n[GOAL]\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : x = 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : \u00acx = 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nrw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x)))\n (Set.IsWf.min (_ : Set.IsWf (support y)) (_ : Set.Nonempty (support y))) \u2264\n Set.IsWf.min (_ : Set.IsWf (support (x + y))) (_ : Set.Nonempty (support (x + y)))\n[PROOFSTEP]\nrefine' le_of_eq_of_le _ (Set.IsWf.min_le_min_of_subset (support_add_subset (x := x) (y := y)))\n[GOAL]\ncase neg.refine'_1\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\u271d\ninst\u271d\u00b9 : AddMonoid R\n\u0393 : Type u_3\ninst\u271d : LinearOrderedCancelAddCommMonoid \u0393\nx y : HahnSeries \u0393 R\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x)))\n (Set.IsWf.min (_ : Set.IsWf (support y)) (_ : Set.Nonempty (support y))) =\n Set.IsWf.min ?neg.refine'_2\u271d ?neg.refine'_3\u271d\n[PROOFSTEP]\nexact (Set.IsWf.min_union _ _ _ _).symm\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\na : \u0393\nsrc\u271d : ZeroHom R (HahnSeries \u0393 R) := single a\nx y : R\n\u22a2 ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } (x + y) =\n ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } x +\n ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } y\n[PROOFSTEP]\next b\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\na : \u0393\nsrc\u271d : ZeroHom R (HahnSeries \u0393 R) := single a\nx y : R\nb : \u0393\n\u22a2 coeff (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } (x + y)) b =\n coeff\n (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } x +\n ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } y)\n b\n[PROOFSTEP]\nby_cases h : b = a\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\na : \u0393\nsrc\u271d : ZeroHom R (HahnSeries \u0393 R) := single a\nx y : R\nb : \u0393\nh : b = a\n\u22a2 coeff (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } (x + y)) b =\n coeff\n (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } x +\n ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } y)\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddMonoid R\na : \u0393\nsrc\u271d : ZeroHom R (HahnSeries \u0393 R) := single a\nx y : R\nb : \u0393\nh : \u00acb = a\n\u22a2 coeff (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } (x + y)) b =\n coeff\n (ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } x +\n ZeroHom.toFun { toFun := src\u271d.toFun, map_zero' := (_ : ZeroHom.toFun src\u271d 0 = 0) } y)\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddMonoid R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\n\u22a2 embDomain f (x + y) = embDomain f x + embDomain f y\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddMonoid R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\ng : \u0393'\n\u22a2 coeff (embDomain f (x + y)) g = coeff (embDomain f x + embDomain f y) g\n[PROOFSTEP]\nby_cases hg : g \u2208 Set.range f\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddMonoid R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : g \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (x + y)) g = coeff (embDomain f x + embDomain f y) g\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hg\n[GOAL]\ncase pos.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddMonoid R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (embDomain f (x + y)) (\u2191f a) = coeff (embDomain f x + embDomain f y) (\u2191f a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddMonoid R\n\u0393' : Type u_3\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : \u00acg \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (x + y)) g = coeff (embDomain f x + embDomain f y) g\n[PROOFSTEP]\nsimp [embDomain_notin_range hg]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\n\u22a2 x + y = y + x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x + y) x\u271d = coeff (y + x) x\u271d\n[PROOFSTEP]\napply add_comm\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\n\u22a2 Set.IsPwo (Function.support fun a => -coeff x a)\n[PROOFSTEP]\nrw [Function.support_neg]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\n\u22a2 Set.IsPwo (Function.support fun a => coeff x a)\n[PROOFSTEP]\nexact x.isPwo_support\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\n\u22a2 -x + x = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nsrc\u271d : AddMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddMonoid (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (-x + x) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\napply add_left_neg\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nx : HahnSeries \u0393 R\n\u22a2 support (-x) = support x\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 x\u271d \u2208 support (-x) \u2194 x\u271d \u2208 support x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nx y : HahnSeries \u0393 R\n\u22a2 (x - y).coeff = x.coeff - y.coeff\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x - y) x\u271d = (x.coeff - y.coeff) x\u271d\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddGroup R\nx y : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (x - y) a = coeff x a - coeff y a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddGroup R\ninst\u271d : Zero \u0393\nf : HahnSeries \u0393 R\n\u22a2 order (-f) = order f\n[PROOFSTEP]\nby_cases hf : f = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddGroup R\ninst\u271d : Zero \u0393\nf : HahnSeries \u0393 R\nhf : f = 0\n\u22a2 order (-f) = order f\n[PROOFSTEP]\nsimp only [hf, neg_zero]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : PartialOrder \u0393\ninst\u271d\u00b9 : AddGroup R\ninst\u271d : Zero \u0393\nf : HahnSeries \u0393 R\nhf : \u00acf = 0\n\u22a2 order (-f) = order f\n[PROOFSTEP]\nsimp only [order, support_neg, neg_eq_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d : HahnSeries \u0393 V\n\u22a2 1 \u2022 x\u271d = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b9 : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff (1 \u2022 x\u271d\u00b9) x\u271d = coeff x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b2 x\u271d\u00b9 : R\nx\u271d : HahnSeries \u0393 V\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 coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b3 x\u271d\u00b2 : R\nx\u271d\u00b9 : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff ((x\u271d\u00b3 * x\u271d\u00b2) \u2022 x\u271d\u00b9) x\u271d = coeff (x\u271d\u00b3 \u2022 x\u271d\u00b2 \u2022 x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp [mul_smul]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d : R\n\u22a2 x\u271d \u2022 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b9 : R\nx\u271d : \u0393\n\u22a2 coeff (x\u271d\u00b9 \u2022 0) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b2 : R\nx\u271d\u00b9 x\u271d : HahnSeries \u0393 V\n\u22a2 x\u271d\u00b2 \u2022 (x\u271d\u00b9 + x\u271d) = x\u271d\u00b2 \u2022 x\u271d\u00b9 + x\u271d\u00b2 \u2022 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid V\ninst\u271d : DistribMulAction R V\nx\u271d\u00b3 : R\nx\u271d\u00b2 x\u271d\u00b9 : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff (x\u271d\u00b3 \u2022 (x\u271d\u00b2 + x\u271d\u00b9)) x\u271d = coeff (x\u271d\u00b3 \u2022 x\u271d\u00b2 + x\u271d\u00b3 \u2022 x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp [smul_add]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : AddMonoid V\ninst\u271d\u2074 : DistribMulAction R V\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : DistribMulAction S V\ninst\u271d\u00b9 : SMul R S\ninst\u271d : IsScalarTower R S V\nr : R\ns : S\na : HahnSeries \u0393 V\n\u22a2 (r \u2022 s) \u2022 a = r \u2022 s \u2022 a\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2077 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : AddMonoid V\ninst\u271d\u2074 : DistribMulAction R V\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : DistribMulAction S V\ninst\u271d\u00b9 : SMul R S\ninst\u271d : IsScalarTower R S V\nr : R\ns : S\na : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff ((r \u2022 s) \u2022 a) x\u271d = coeff (r \u2022 s \u2022 a) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u2075 : Monoid R\ninst\u271d\u2074 : AddMonoid V\ninst\u271d\u00b3 : DistribMulAction R V\nS : Type u_4\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S V\ninst\u271d : SMulCommClass R S V\nr : R\ns : S\na : HahnSeries \u0393 V\n\u22a2 r \u2022 s \u2022 a = s \u2022 r \u2022 a\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : PartialOrder \u0393\nV : Type u_3\ninst\u271d\u2075 : Monoid R\ninst\u271d\u2074 : AddMonoid V\ninst\u271d\u00b3 : DistribMulAction R V\nS : Type u_4\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S V\ninst\u271d : SMulCommClass R S V\nr : R\ns : S\na : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff (r \u2022 s \u2022 a) x\u271d = coeff (s \u2022 r \u2022 a) x\u271d\n[PROOFSTEP]\nsimp [smul_comm]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\nsrc\u271d : DistribMulAction R (HahnSeries \u0393 V) := inferInstanceAs (DistribMulAction R (HahnSeries \u0393 V))\nx\u271d\u00b2 x\u271d\u00b9 : R\nx\u271d : HahnSeries \u0393 V\n\u22a2 (x\u271d\u00b2 + x\u271d\u00b9) \u2022 x\u271d = x\u271d\u00b2 \u2022 x\u271d + x\u271d\u00b9 \u2022 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\nsrc\u271d : DistribMulAction R (HahnSeries \u0393 V) := inferInstanceAs (DistribMulAction R (HahnSeries \u0393 V))\nx\u271d\u00b3 x\u271d\u00b2 : R\nx\u271d\u00b9 : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff ((x\u271d\u00b3 + x\u271d\u00b2) \u2022 x\u271d\u00b9) x\u271d = coeff (x\u271d\u00b3 \u2022 x\u271d\u00b9 + x\u271d\u00b2 \u2022 x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp [add_smul]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\nsrc\u271d : DistribMulAction R (HahnSeries \u0393 V) := inferInstanceAs (DistribMulAction R (HahnSeries \u0393 V))\nx\u271d : HahnSeries \u0393 V\n\u22a2 0 \u2022 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\nsrc\u271d : DistribMulAction R (HahnSeries \u0393 V) := inferInstanceAs (DistribMulAction R (HahnSeries \u0393 V))\nx\u271d\u00b9 : HahnSeries \u0393 V\nx\u271d : \u0393\n\u22a2 coeff (0 \u2022 x\u271d\u00b9) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\na : \u0393\nsrc\u271d : R \u2192+ HahnSeries \u0393 R := addMonoidHom a\nr s : R\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n s\n[PROOFSTEP]\next b\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\na : \u0393\nsrc\u271d : R \u2192+ HahnSeries \u0393 R := addMonoidHom a\nr s : R\nb : \u0393\n\u22a2 coeff\n (AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 s))\n b =\n coeff\n (\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n s)\n b\n[PROOFSTEP]\nby_cases h : b = a\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\na : \u0393\nsrc\u271d : R \u2192+ HahnSeries \u0393 R := addMonoidHom a\nr s : R\nb : \u0393\nh : b = a\n\u22a2 coeff\n (AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 s))\n b =\n coeff\n (\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n s)\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : PartialOrder \u0393\ninst\u271d\u00b2 : Semiring R\nV : Type u_3\ninst\u271d\u00b9 : AddCommMonoid V\ninst\u271d : Module R V\na : \u0393\nsrc\u271d : R \u2192+ HahnSeries \u0393 R := addMonoidHom a\nr s : R\nb : \u0393\nh : \u00acb = a\n\u22a2 coeff\n (AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n (r \u2022 s))\n b =\n coeff\n (\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ : \u2200 (x y : R), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) }\n s)\n b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : PartialOrder \u0393\ninst\u271d\u00b3 : Semiring R\nV : Type u_3\ninst\u271d\u00b2 : AddCommMonoid V\ninst\u271d\u00b9 : Module R V\n\u0393' : Type u_4\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nr : R\nx : HahnSeries \u0393 R\n\u22a2 embDomain f (r \u2022 x) = r \u2022 embDomain f x\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : PartialOrder \u0393\ninst\u271d\u00b3 : Semiring R\nV : Type u_3\ninst\u271d\u00b2 : AddCommMonoid V\ninst\u271d\u00b9 : Module R V\n\u0393' : Type u_4\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nr : R\nx : HahnSeries \u0393 R\ng : \u0393'\n\u22a2 coeff (embDomain f (r \u2022 x)) g = coeff (r \u2022 embDomain f x) g\n[PROOFSTEP]\nby_cases hg : g \u2208 Set.range f\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : PartialOrder \u0393\ninst\u271d\u00b3 : Semiring R\nV : Type u_3\ninst\u271d\u00b2 : AddCommMonoid V\ninst\u271d\u00b9 : Module R V\n\u0393' : Type u_4\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nr : R\nx : HahnSeries \u0393 R\ng : \u0393'\nhg : g \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (r \u2022 x)) g = coeff (r \u2022 embDomain f x) g\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hg\n[GOAL]\ncase pos.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : PartialOrder \u0393\ninst\u271d\u00b3 : Semiring R\nV : Type u_3\ninst\u271d\u00b2 : AddCommMonoid V\ninst\u271d\u00b9 : Module R V\n\u0393' : Type u_4\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nr : R\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (embDomain f (r \u2022 x)) (\u2191f a) = coeff (r \u2022 embDomain f x) (\u2191f a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : PartialOrder \u0393\ninst\u271d\u00b3 : Semiring R\nV : Type u_3\ninst\u271d\u00b2 : AddCommMonoid V\ninst\u271d\u00b9 : Module R V\n\u0393' : Type u_4\ninst\u271d : PartialOrder \u0393'\nf : \u0393 \u21aao \u0393'\nr : R\nx : HahnSeries \u0393 R\ng : \u0393'\nhg : \u00acg \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (r \u2022 x)) g = coeff (r \u2022 embDomain f x) g\n[PROOFSTEP]\nsimp [embDomain_notin_range hg]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : MulZeroOneClass R\n\u22a2 order 1 = 0\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial R with h h\n[GOAL]\ncase inl\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : MulZeroOneClass R\nh : Subsingleton R\n\u22a2 order 1 = 0\n[PROOFSTEP]\nhaveI := h\n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : MulZeroOneClass R\nh : Nontrivial R\n\u22a2 order 1 = 0\n[PROOFSTEP]\nhaveI := h\n[GOAL]\ncase inl\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : MulZeroOneClass R\nh this : Subsingleton R\n\u22a2 order 1 = 0\n[PROOFSTEP]\nrw [Subsingleton.elim (1 : HahnSeries \u0393 R) 0, order_zero]\n[GOAL]\ncase inr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : MulZeroOneClass R\nh this : Nontrivial R\n\u22a2 order 1 = 0\n[PROOFSTEP]\nexact order_single one_ne_zero\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 {a |\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a,\n coeff x ij.fst * coeff y ij.snd \u2260\n 0} \u2286\n {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a)}\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\nha :\n a \u2208\n {a |\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a,\n coeff x ij.fst * coeff y ij.snd \u2260\n 0}\n\u22a2 a \u2208 {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a)}\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\nha : \u00aca \u2208 {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a)}\n\u22a2 \u00aca \u2208\n {a |\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a,\n coeff x ij.fst * coeff y ij.snd \u2260\n 0}\n[PROOFSTEP]\nsimp [not_nonempty_iff_eq_empty.1 ha]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhys : support y \u2286 s\n\u22a2 coeff (x * y) a = \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) hs a, coeff x ij.fst * coeff y ij.snd\n[PROOFSTEP]\nrw [mul_coeff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhys : support y \u2286 s\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd =\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) hs a, coeff x ij.fst * coeff y ij.snd\n[PROOFSTEP]\napply sum_subset_zero_on_sdiff (addAntidiagonal_mono_right hys) _ fun _ _ => rfl\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhys : support y \u2286 s\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208\n addAntidiagonal (_ : Set.IsPwo (support x)) hs a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a \u2192\n coeff x x_1.fst * coeff y x_1.snd = 0\n[PROOFSTEP]\nintro b hb\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhys : support y \u2286 s\nb : \u0393 \u00d7 \u0393\nhb :\n b \u2208\n addAntidiagonal (_ : Set.IsPwo (support x)) hs a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a\n\u22a2 coeff x b.fst * coeff y b.snd = 0\n[PROOFSTEP]\nsimp only [not_and, mem_sdiff, mem_addAntidiagonal, mem_support, not_imp_not] at hb \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhys : support y \u2286 s\nb : \u0393 \u00d7 \u0393\nhb : (coeff x b.fst \u2260 0 \u2227 b.snd \u2208 s \u2227 b.fst + b.snd = a) \u2227 (coeff x b.fst \u2260 0 \u2192 b.fst + b.snd = a \u2192 coeff y b.snd = 0)\n\u22a2 coeff x b.fst * coeff y b.snd = 0\n[PROOFSTEP]\nrw [hb.2 hb.1.1 hb.1.2.2, mul_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhxs : support x \u2286 s\n\u22a2 coeff (x * y) a = \u2211 ij in addAntidiagonal hs (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd\n[PROOFSTEP]\nrw [mul_coeff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhxs : support x \u2286 s\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd =\n \u2211 ij in addAntidiagonal hs (_ : Set.IsPwo (support y)) a, coeff x ij.fst * coeff y ij.snd\n[PROOFSTEP]\napply sum_subset_zero_on_sdiff (addAntidiagonal_mono_left hxs) _ fun _ _ => rfl\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhxs : support x \u2286 s\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a \u2192\n coeff x x_1.fst * coeff y x_1.snd = 0\n[PROOFSTEP]\nintro b hb\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhxs : support x \u2286 s\nb : \u0393 \u00d7 \u0393\nhb :\n b \u2208\n addAntidiagonal hs (_ : Set.IsPwo (support y)) a \\\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a\n\u22a2 coeff x b.fst * coeff y b.snd = 0\n[PROOFSTEP]\nsimp only [not_and', mem_sdiff, mem_addAntidiagonal, mem_support, not_ne_iff] at hb \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\na : \u0393\ns : Set \u0393\nhs : Set.IsPwo s\nhxs : support x \u2286 s\nb : \u0393 \u00d7 \u0393\nhb : (b.fst \u2208 s \u2227 coeff y b.snd \u2260 0 \u2227 b.fst + b.snd = a) \u2227 (coeff y b.snd \u2260 0 \u2227 b.fst + b.snd = a \u2192 coeff x b.fst = 0)\n\u22a2 coeff x b.fst * coeff y b.snd = 0\n[PROOFSTEP]\nrw [hb.2 \u27e8hb.1.2.1, hb.1.2.2\u27e9, zero_mul]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\n\u22a2 x * (y + z) = x * y + x * z\n[PROOFSTEP]\next a\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (x * (y + z)) a = coeff (x * y + x * z) a\n[PROOFSTEP]\nhave hwf := y.isPwo_support.union z.isPwo_support\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\n\u22a2 coeff (x * (y + z)) a = coeff (x * y + x * z) a\n[PROOFSTEP]\nrw [mul_coeff_right' hwf, add_coeff, mul_coeff_right' hwf (Set.subset_union_right _ _),\n mul_coeff_right' hwf (Set.subset_union_left _ _)]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) hwf a, coeff x ij.fst * coeff (y + z) ij.snd =\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) hwf a, coeff x ij.fst * coeff y ij.snd +\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) hwf a, coeff x ij.fst * coeff z ij.snd\n[PROOFSTEP]\nsimp only [add_coeff, mul_add, sum_add_distrib]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\n\u22a2 support (y + z) \u2286 support y \u222a support z\n[PROOFSTEP]\nintro b\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\nb : \u0393\n\u22a2 b \u2208 support (y + z) \u2192 b \u2208 support y \u222a support z\n[PROOFSTEP]\nsimp only [add_coeff, Ne.def, Set.mem_union, Set.mem_setOf_eq, mem_support]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\nb : \u0393\n\u22a2 \u00accoeff y b + coeff z b = 0 \u2192 \u00accoeff y b = 0 \u2228 \u00accoeff z b = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\nb : \u0393\n\u22a2 coeff y b = 0 \u2227 coeff z b = 0 \u2192 coeff y b + coeff z b = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support y \u222a support z)\nb : \u0393\nh : coeff y b = 0 \u2227 coeff z b = 0\n\u22a2 coeff y b + coeff z b = 0\n[PROOFSTEP]\nrw [h.1, h.2, add_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\n\u22a2 (x + y) * z = x * z + y * z\n[PROOFSTEP]\next a\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff ((x + y) * z) a = coeff (x * z + y * z) a\n[PROOFSTEP]\nhave hwf := x.isPwo_support.union y.isPwo_support\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\n\u22a2 coeff ((x + y) * z) a = coeff (x * z + y * z) a\n[PROOFSTEP]\nrw [mul_coeff_left' hwf, add_coeff, mul_coeff_left' hwf (Set.subset_union_right _ _),\n mul_coeff_left' hwf (Set.subset_union_left _ _)]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\n\u22a2 \u2211 ij in addAntidiagonal hwf (_ : Set.IsPwo (support z)) a, coeff (x + y) ij.fst * coeff z ij.snd =\n \u2211 ij in addAntidiagonal hwf (_ : Set.IsPwo (support z)) a, coeff x ij.fst * coeff z ij.snd +\n \u2211 ij in addAntidiagonal hwf (_ : Set.IsPwo (support z)) a, coeff y ij.fst * coeff z ij.snd\n[PROOFSTEP]\nsimp only [add_coeff, add_mul, sum_add_distrib]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\n\u22a2 support (x + y) \u2286 support x \u222a support y\n[PROOFSTEP]\nintro b\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\nb : \u0393\n\u22a2 b \u2208 support (x + y) \u2192 b \u2208 support x \u222a support y\n[PROOFSTEP]\nsimp only [add_coeff, Ne.def, Set.mem_union, Set.mem_setOf_eq, mem_support]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\nb : \u0393\n\u22a2 \u00accoeff x b + coeff y b = 0 \u2192 \u00accoeff x b = 0 \u2228 \u00accoeff y b = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\nb : \u0393\n\u22a2 coeff x b = 0 \u2227 coeff y b = 0 \u2192 coeff x b + coeff y b = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : Mul (HahnSeries \u0393 R) := inferInstanceAs (Mul (HahnSeries \u0393 R))\nsrc\u271d : Add (HahnSeries \u0393 R) := inferInstanceAs (Add (HahnSeries \u0393 R))\nx y z : HahnSeries \u0393 R\na : \u0393\nhwf : Set.IsPwo (support x \u222a support y)\nb : \u0393\nh : coeff x b = 0 \u2227 coeff y b = 0\n\u22a2 coeff x b + coeff y b = 0\n[PROOFSTEP]\nrw [h.1, h.2, add_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\n\u22a2 coeff (\u2191(single b) r * x) (a + b) = r * coeff x a\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : r = 0\n\u22a2 coeff (\u2191(single b) r * x) (a + b) = r * coeff x a\n[PROOFSTEP]\nsimp [hr, mul_coeff]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\n\u22a2 coeff (\u2191(single b) r * x) (a + b) = r * coeff x a\n[PROOFSTEP]\nsimp only [hr, smul_coeff, mul_coeff, support_single_of_ne, Ne.def, not_false_iff, smul_eq_mul]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b),\n coeff (\u2191(single b) r) x_1.fst * coeff x x_1.snd =\n r * coeff x a\n[PROOFSTEP]\nby_cases hx : x.coeff a = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b),\n coeff (\u2191(single b) r) x_1.fst * coeff x x_1.snd =\n r * coeff x a\n[PROOFSTEP]\nsimp only [hx, mul_zero]\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b),\n coeff (\u2191(single b) r) x_1.fst * coeff x x_1.snd =\n 0\n[PROOFSTEP]\nrw [sum_congr _ fun _ _ => rfl, sum_empty]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b) = \u2205\n[PROOFSTEP]\next \u27e8a1, a2\u27e9\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\n\u22a2 (a1, a2) \u2208 addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b) \u2194 (a1, a2) \u2208 \u2205\n[PROOFSTEP]\nsimp only [not_mem_empty, not_and, Set.mem_singleton_iff, Classical.not_not, mem_addAntidiagonal, Set.mem_setOf_eq,\n iff_false_iff]\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 = b \u2192 a2 \u2208 support x \u2192 \u00aca1 + a2 = a + b\n[PROOFSTEP]\nrintro rfl h2 h1\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\nh2 : a2 \u2208 support x\nh1 : a1 + a2 = a + a1\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm] at h1 \n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\nh2 : a2 \u2208 support x\nh1 : a2 + a1 = a + a1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 add_right_cancel h1] at hx \n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\na1 a2 : \u0393\nhx : coeff x a2 = 0\nh2 : a2 \u2208 support x\nh1 : a2 + a1 = a + a1\n\u22a2 False\n[PROOFSTEP]\nexact h2 hx\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b),\n coeff (\u2191(single b) r) x_1.fst * coeff x x_1.snd =\n r * coeff x a\n[PROOFSTEP]\ntrans \u2211 ij : \u0393 \u00d7 \u0393 in {(b, a)}, (single b r).coeff ij.fst * x.coeff ij.snd\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b),\n coeff (\u2191(single b) r) x_1.fst * coeff x x_1.snd =\n \u2211 ij in {(b, a)}, coeff (\u2191(single b) r) ij.fst * coeff x ij.snd\n[PROOFSTEP]\napply sum_congr _ fun _ _ => rfl\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b) = {(b, a)}\n[PROOFSTEP]\next \u27e8a1, a2\u27e9\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 (a1, a2) \u2208 addAntidiagonal (_ : Set.IsPwo {b}) (_ : Set.IsPwo (support x)) (a + b) \u2194 (a1, a2) \u2208 {(b, a)}\n[PROOFSTEP]\nsimp only [Set.mem_singleton_iff, Prod.mk.inj_iff, mem_addAntidiagonal, mem_singleton, Set.mem_setOf_eq]\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 = b \u2227 a2 \u2208 support x \u2227 a1 + a2 = a + b \u2194 a1 = b \u2227 a2 = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mk.mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 = b \u2227 a2 \u2208 support x \u2227 a1 + a2 = a + b \u2192 a1 = b \u2227 a2 = a\n[PROOFSTEP]\nrintro \u27e8rfl, _, h1\u27e9\n[GOAL]\ncase a.mk.mp.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\nleft\u271d : a2 \u2208 support x\nh1 : a1 + a2 = a + a1\n\u22a2 a1 = a1 \u2227 a2 = a\n[PROOFSTEP]\nrw [add_comm] at h1 \n[GOAL]\ncase a.mk.mp.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\nleft\u271d : a2 \u2208 support x\nh1 : a2 + a1 = a + a1\n\u22a2 a1 = a1 \u2227 a2 = a\n[PROOFSTEP]\nrefine' \u27e8rfl, add_right_cancel h1\u27e9\n[GOAL]\ncase a.mk.mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 = b \u2227 a2 = a \u2192 a1 = b \u2227 a2 \u2208 support x \u2227 a1 + a2 = a + b\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase a.mk.mpr.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\nhr : \u00acr = 0\na1 a2 : \u0393\nhx : \u00accoeff x a2 = 0\n\u22a2 a1 = a1 \u2227 a2 \u2208 support x \u2227 a1 + a2 = a2 + a1\n[PROOFSTEP]\nexact \u27e8rfl, by simp [hx], add_comm _ _\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\nhr : \u00acr = 0\na1 a2 : \u0393\nhx : \u00accoeff x a2 = 0\n\u22a2 a2 \u2208 support x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 ij in {(b, a)}, coeff (\u2191(single b) r) ij.fst * coeff x ij.snd = r * coeff x a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\n\u22a2 coeff (x * \u2191(single b) r) (a + b) = coeff x a * r\n[PROOFSTEP]\nby_cases hr : r = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : r = 0\n\u22a2 coeff (x * \u2191(single b) r) (a + b) = coeff x a * r\n[PROOFSTEP]\nsimp [hr, mul_coeff]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\n\u22a2 coeff (x * \u2191(single b) r) (a + b) = coeff x a * r\n[PROOFSTEP]\nsimp only [hr, smul_coeff, mul_coeff, support_single_of_ne, Ne.def, not_false_iff, smul_eq_mul]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b),\n coeff x x_1.fst * coeff (\u2191(single b) r) x_1.snd =\n coeff x a * r\n[PROOFSTEP]\nby_cases hx : x.coeff a = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b),\n coeff x x_1.fst * coeff (\u2191(single b) r) x_1.snd =\n coeff x a * r\n[PROOFSTEP]\nsimp only [hx, zero_mul]\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b),\n coeff x x_1.fst * coeff (\u2191(single b) r) x_1.snd =\n 0\n[PROOFSTEP]\nrw [sum_congr _ fun _ _ => rfl, sum_empty]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\n\u22a2 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b) = \u2205\n[PROOFSTEP]\next \u27e8a1, a2\u27e9\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\n\u22a2 (a1, a2) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b) \u2194 (a1, a2) \u2208 \u2205\n[PROOFSTEP]\nsimp only [not_mem_empty, not_and, Set.mem_singleton_iff, Classical.not_not, mem_addAntidiagonal, Set.mem_setOf_eq,\n iff_false_iff]\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 \u2208 support x \u2192 a2 = b \u2192 \u00aca1 + a2 = a + b\n[PROOFSTEP]\nrintro h2 rfl h1\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : coeff x a = 0\na1 a2 : \u0393\nh2 : a1 \u2208 support x\nh1 : a1 + a2 = a + a2\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 add_right_cancel h1] at hx \n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\na1 : \u0393\nhx : coeff x a1 = 0\na2 : \u0393\nh2 : a1 \u2208 support x\nh1 : a1 + a2 = a + a2\n\u22a2 False\n[PROOFSTEP]\nexact h2 hx\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b),\n coeff x x_1.fst * coeff (\u2191(single b) r) x_1.snd =\n coeff x a * r\n[PROOFSTEP]\ntrans \u2211 ij : \u0393 \u00d7 \u0393 in {(a, b)}, x.coeff ij.fst * (single b r).coeff ij.snd\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b),\n coeff x x_1.fst * coeff (\u2191(single b) r) x_1.snd =\n \u2211 ij in {(a, b)}, coeff x ij.fst * coeff (\u2191(single b) r) ij.snd\n[PROOFSTEP]\napply sum_congr _ fun _ _ => rfl\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b) = {(a, b)}\n[PROOFSTEP]\next \u27e8a1, a2\u27e9\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 (a1, a2) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo {b}) (a + b) \u2194 (a1, a2) \u2208 {(a, b)}\n[PROOFSTEP]\nsimp only [Set.mem_singleton_iff, Prod.mk.inj_iff, mem_addAntidiagonal, mem_singleton, Set.mem_setOf_eq]\n[GOAL]\ncase a.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 \u2208 support x \u2227 a2 = b \u2227 a1 + a2 = a + b \u2194 a1 = a \u2227 a2 = b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mk.mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 \u2208 support x \u2227 a2 = b \u2227 a1 + a2 = a + b \u2192 a1 = a \u2227 a2 = b\n[PROOFSTEP]\nrintro \u27e8_, rfl, h1\u27e9\n[GOAL]\ncase a.mk.mp.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\nleft\u271d : a1 \u2208 support x\nh1 : a1 + a2 = a + a2\n\u22a2 a1 = a \u2227 a2 = a2\n[PROOFSTEP]\nrefine' \u27e8add_right_cancel h1, rfl\u27e9\n[GOAL]\ncase a.mk.mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\na1 a2 : \u0393\n\u22a2 a1 = a \u2227 a2 = b \u2192 a1 \u2208 support x \u2227 a2 = b \u2227 a1 + a2 = a + b\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase a.mk.mpr.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\nhr : \u00acr = 0\na1 a2 : \u0393\nhx : \u00accoeff x a1 = 0\n\u22a2 a1 \u2208 support x \u2227 a2 = a2 \u2227 a1 + a2 = a1 + a2\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na b : \u0393\nhr : \u00acr = 0\nhx : \u00accoeff x a = 0\n\u22a2 \u2211 ij in {(a, b)}, coeff x ij.fst * coeff (\u2191(single b) r) ij.snd = coeff x a * r\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (x * \u2191(single 0) r) a = coeff x a * r\n[PROOFSTEP]\nrw [\u2190 add_zero a, mul_single_coeff_add, add_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nr : R\nx : HahnSeries \u0393 R\na : \u0393\n\u22a2 coeff (\u2191(single 0) r * x) a = r * coeff x a\n[PROOFSTEP]\nrw [\u2190 add_zero a, single_mul_coeff_add, add_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\nr : R\nx : HahnSeries \u0393 R\n\u22a2 \u2191(single 0) r * x = r \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\nr : R\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (\u2191(single 0) r * x) x\u271d = coeff (r \u2022 x) x\u271d\n[PROOFSTEP]\nexact single_zero_mul_coeff\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 support (x * y) \u2286 support x + support y\n[PROOFSTEP]\napply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 Set.IsPwo (support x)\n[PROOFSTEP]\nexact x.isPwo_support\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 Set.IsPwo (support y)\n[PROOFSTEP]\nexact y.isPwo_support\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 \u2200 (x_1 : \u0393),\n x_1 \u2208 support (x * y) \u2192\n x_1 \u2208 {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a)}\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx\u271d y : HahnSeries \u0393 R\nx : \u0393\nhx : x \u2208 support (x\u271d * y)\n\u22a2 x \u2208 {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x\u271d)) (_ : Set.IsPwo (support y)) a)}\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx\u271d y : HahnSeries \u0393 R\nx : \u0393\nhx : \u00acx \u2208 {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x\u271d)) (_ : Set.IsPwo (support y)) a)}\n\u22a2 \u00acx \u2208 support (x\u271d * y)\n[PROOFSTEP]\nsimp only [not_nonempty_iff_eq_empty, Ne.def, Set.mem_setOf_eq] at hx \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx\u271d y : HahnSeries \u0393 R\nx : \u0393\nhx : addAntidiagonal (_ : Set.IsPwo (support x\u271d)) (_ : Set.IsPwo (support y)) x = \u2205\n\u22a2 \u00acx \u2208 support (x\u271d * y)\n[PROOFSTEP]\nsimp [hx, mul_coeff]\n[GOAL]\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\n\u22a2 coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\nhx : x = 0\n\u22a2 coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)\n[PROOFSTEP]\nsimp [hx, mul_coeff]\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\n\u22a2 coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)\n[PROOFSTEP]\nsimp [hy, mul_coeff]\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b9 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)\n[PROOFSTEP]\nrw [order_of_ne hx, order_of_ne hy, mul_coeff, Finset.addAntidiagonal_min_add_min, Finset.sum_singleton]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\next b\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 coeff (x * y * z) b = coeff (x * (y * z)) b\n[PROOFSTEP]\nrw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,\n mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b,\n coeff (x * y) ij.fst * coeff z ij.snd =\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b,\n coeff x ij.fst * coeff (y * z) ij.snd\n[PROOFSTEP]\nsimp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2211 x_1 in\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst,\n coeff x x_1.snd.fst * coeff y x_1.snd.snd * coeff z x_1.fst.snd =\n \u2211 x_1 in\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.snd,\n coeff x x_1.fst.fst * (coeff y x_1.snd.fst * coeff z x_1.snd.snd)\n[PROOFSTEP]\nrefine' sum_bij_ne_zero (fun a _ _ => \u27e8\u27e8a.2.1, a.2.2 + a.1.2\u27e9, \u27e8a.2.2, a.1.2\u27e9\u27e9) _ _ _ _\n[GOAL]\ncase coeff.h.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2200 (a : (_ : \u0393 \u00d7 \u0393) \u00d7 \u0393 \u00d7 \u0393)\n (h\u2081 :\n a \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst)\n (h\u2082 : coeff x a.snd.fst * coeff y a.snd.snd * coeff z a.fst.snd \u2260 0),\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081 h\u2082 \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.snd\n[PROOFSTEP]\nrintro \u27e8\u27e8i, j\u27e9, \u27e8k, l\u27e9\u27e9 H1 H2\n[GOAL]\ncase coeff.h.refine'_1.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i j k l : \u0393\nH1 :\n { fst := (i, j), snd := (k, l) } \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst\nH2 :\n coeff x { fst := (i, j), snd := (k, l) }.snd.fst * coeff y { fst := (i, j), snd := (k, l) }.snd.snd *\n coeff z { fst := (i, j), snd := (k, l) }.fst.snd \u2260\n 0\n\u22a2 (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i, j), snd := (k, l) } H1 H2 \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.snd\n[PROOFSTEP]\nsimp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def, Set.image_prod, mem_sigma,\n Set.mem_setOf_eq] at H1 H2 \u22a2\n[GOAL]\ncase coeff.h.refine'_1.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i j k l : \u0393\nH2 : \u00accoeff x k * coeff y l * coeff z j = 0\nH1 : (i \u2208 support x + support y \u2227 j \u2208 support z \u2227 i + j = b) \u2227 k \u2208 support x \u2227 l \u2208 support y \u2227 k + l = i\n\u22a2 (k \u2208 support x \u2227 l + j \u2208 support y + support z \u2227 k + (l + j) = b) \u2227 l \u2208 support y \u2227 j \u2208 support z\n[PROOFSTEP]\nobtain \u27e8\u27e8H3, nz, rfl\u27e9, nx, ny, rfl\u27e9 := H1\n[GOAL]\ncase coeff.h.refine'_1.mk.mk.mk.intro.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nj k l : \u0393\nH2 : \u00accoeff x k * coeff y l * coeff z j = 0\nnz : j \u2208 support z\nnx : k \u2208 support x\nny : l \u2208 support y\nH3 : k + l \u2208 support x + support y\n\u22a2 (k \u2208 support x \u2227 l + j \u2208 support y + support z \u2227 k + (l + j) = k + l + j) \u2227 l \u2208 support y \u2227 j \u2208 support z\n[PROOFSTEP]\nexact \u27e8\u27e8nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm\u27e9, ny, nz\u27e9\n[GOAL]\ncase coeff.h.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2200 (a\u2081 a\u2082 : (_ : \u0393 \u00d7 \u0393) \u00d7 \u0393 \u00d7 \u0393)\n (h\u2081\u2081 :\n a\u2081 \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst)\n (h\u2081\u2082 : coeff x a\u2081.snd.fst * coeff y a\u2081.snd.snd * coeff z a\u2081.fst.snd \u2260 0)\n (h\u2082\u2081 :\n a\u2082 \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst)\n (h\u2082\u2082 : coeff x a\u2082.snd.fst * coeff y a\u2082.snd.snd * coeff z a\u2082.fst.snd \u2260 0),\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a\u2081 h\u2081\u2081 h\u2081\u2082 =\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a\u2082 h\u2082\u2081 h\u2082\u2082 \u2192\n a\u2081 = a\u2082\n[PROOFSTEP]\nrintro \u27e8\u27e8i1, j1\u27e9, k1, l1\u27e9 \u27e8\u27e8i2, j2\u27e9, k2, l2\u27e9 H1 H2 H3 H4 H5\n[GOAL]\ncase coeff.h.refine'_2.mk.mk.mk.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i1 j1 k1 l1 i2 j2 k2 l2 : \u0393\nH1 :\n { fst := (i1, j1), snd := (k1, l1) } \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst\nH2 :\n coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.fst * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.snd *\n coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.snd \u2260\n 0\nH3 :\n { fst := (i2, j2), snd := (k2, l2) } \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst\nH4 :\n coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.fst * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.snd *\n coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.snd \u2260\n 0\nH5 :\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i1, j1), snd := (k1, l1) } H1 H2 =\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i2, j2), snd := (k2, l2) } H3 H4\n\u22a2 { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j2), snd := (k2, l2) }\n[PROOFSTEP]\nsimp only [Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq,\n heq_iff_eq] at H1 H3 H5 \n[GOAL]\ncase coeff.h.refine'_2.mk.mk.mk.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i1 j1 k1 l1 i2 j2 k2 l2 : \u0393\nH2 :\n coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.fst * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.snd *\n coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.snd \u2260\n 0\nH4 :\n coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.fst * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.snd *\n coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.snd \u2260\n 0\nH5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k2, l2 + j2), snd := (l2, j2) }\nH1 : (i1 \u2208 support x + support y \u2227 j1 \u2208 support z \u2227 i1 + j1 = b) \u2227 k1 \u2208 support x \u2227 l1 \u2208 support y \u2227 k1 + l1 = i1\nH3 : (i2 \u2208 support x + support y \u2227 j2 \u2208 support z \u2227 i2 + j2 = b) \u2227 k2 \u2208 support x \u2227 l2 \u2208 support y \u2227 k2 + l2 = i2\n\u22a2 { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j2), snd := (k2, l2) }\n[PROOFSTEP]\nobtain (\u27e8\u27e8rfl, _\u27e9, rfl, rfl\u27e9 : (k1 = k2 \u2227 l1 + j1 = l2 + j2) \u2227 l1 = l2 \u2227 j1 = j2) := by simpa using H5\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i1 j1 k1 l1 i2 j2 k2 l2 : \u0393\nH2 :\n coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.fst * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.snd *\n coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.snd \u2260\n 0\nH4 :\n coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.fst * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.snd *\n coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.snd \u2260\n 0\nH5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k2, l2 + j2), snd := (l2, j2) }\nH1 : (i1 \u2208 support x + support y \u2227 j1 \u2208 support z \u2227 i1 + j1 = b) \u2227 k1 \u2208 support x \u2227 l1 \u2208 support y \u2227 k1 + l1 = i1\nH3 : (i2 \u2208 support x + support y \u2227 j2 \u2208 support z \u2227 i2 + j2 = b) \u2227 k2 \u2208 support x \u2227 l2 \u2208 support y \u2227 k2 + l2 = i2\n\u22a2 (k1 = k2 \u2227 l1 + j1 = l2 + j2) \u2227 l1 = l2 \u2227 j1 = j2\n[PROOFSTEP]\nsimpa using H5\n[GOAL]\ncase coeff.h.refine'_2.mk.mk.mk.mk.mk.mk.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i1 j1 k1 l1 i2 : \u0393\nH2 :\n coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.fst * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.snd *\n coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.snd \u2260\n 0\nH1 : (i1 \u2208 support x + support y \u2227 j1 \u2208 support z \u2227 i1 + j1 = b) \u2227 k1 \u2208 support x \u2227 l1 \u2208 support y \u2227 k1 + l1 = i1\nright\u271d : l1 + j1 = l1 + j1\nH4 :\n coeff x { fst := (i2, j1), snd := (k1, l1) }.snd.fst * coeff y { fst := (i2, j1), snd := (k1, l1) }.snd.snd *\n coeff z { fst := (i2, j1), snd := (k1, l1) }.fst.snd \u2260\n 0\nH5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k1, l1 + j1), snd := (l1, j1) }\nH3 : (i2 \u2208 support x + support y \u2227 j1 \u2208 support z \u2227 i2 + j1 = b) \u2227 k1 \u2208 support x \u2227 l1 \u2208 support y \u2227 k1 + l1 = i2\n\u22a2 { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j1), snd := (k1, l1) }\n[PROOFSTEP]\nsimp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, \u2190 H1.2.2.2, \u2190 H3.2.2.2]\n[GOAL]\ncase coeff.h.refine'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2200 (b_1 : (_ : \u0393 \u00d7 \u0393) \u00d7 \u0393 \u00d7 \u0393),\n (b_1 \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.snd) \u2192\n coeff x b_1.fst.fst * (coeff y b_1.snd.fst * coeff z b_1.snd.snd) \u2260 0 \u2192\n \u2203 a h\u2081 h\u2082,\n b_1 = (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081 h\u2082\n[PROOFSTEP]\nrintro \u27e8\u27e8i, j\u27e9, \u27e8k, l\u27e9\u27e9 H1 H2\n[GOAL]\ncase coeff.h.refine'_3.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i j k l : \u0393\nH1 :\n { fst := (i, j), snd := (k, l) } \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.snd\nH2 :\n coeff x { fst := (i, j), snd := (k, l) }.fst.fst *\n (coeff y { fst := (i, j), snd := (k, l) }.snd.fst * coeff z { fst := (i, j), snd := (k, l) }.snd.snd) \u2260\n 0\n\u22a2 \u2203 a h\u2081 h\u2082,\n { fst := (i, j), snd := (k, l) } =\n (fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081 h\u2082\n[PROOFSTEP]\nsimp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists, Ne.def, Set.image_prod,\n mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 \u22a2\n[GOAL]\ncase coeff.h.refine'_3.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i j k l : \u0393\nH2 : \u00accoeff x i * (coeff y k * coeff z l) = 0\nH1 : (i \u2208 support x \u2227 j \u2208 support y + support z \u2227 i + j = b) \u2227 k \u2208 support y \u2227 l \u2208 support z \u2227 k + l = j\n\u22a2 \u2203 a b_1 a_1 b_2,\n ((a \u2208 support x + support y \u2227 b_1 \u2208 support z \u2227 a + b_1 = b) \u2227 a_1 \u2208 support x \u2227 b_2 \u2208 support y \u2227 a_1 + b_2 = a) \u2227\n \u00accoeff x a_1 * coeff y b_2 * coeff z b_1 = 0 \u2227\n { fst := (i, j), snd := (k, l) } = { fst := (a_1, b_2 + b_1), snd := (b_2, b_1) }\n[PROOFSTEP]\nobtain \u27e8\u27e8nx, H, rfl\u27e9, ny, nz, rfl\u27e9 := H1\n[GOAL]\ncase coeff.h.refine'_3.mk.mk.mk.intro.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\ni k l : \u0393\nH2 : \u00accoeff x i * (coeff y k * coeff z l) = 0\nnx : i \u2208 support x\nny : k \u2208 support y\nnz : l \u2208 support z\nH : k + l \u2208 support y + support z\n\u22a2 \u2203 a b a_1 b_1,\n ((a \u2208 support x + support y \u2227 b \u2208 support z \u2227 a + b = i + (k + l)) \u2227\n a_1 \u2208 support x \u2227 b_1 \u2208 support y \u2227 a_1 + b_1 = a) \u2227\n \u00accoeff x a_1 * coeff y b_1 * coeff z b = 0 \u2227\n { fst := (i, k + l), snd := (k, l) } = { fst := (a_1, b_1 + b), snd := (b_1, b) }\n[PROOFSTEP]\nexact\n \u27e8i + k, l, i, k, \u27e8\u27e8Set.add_mem_add nx ny, nz, add_assoc _ _ _\u27e9, nx, ny, rfl\u27e9, fun h => H2 <| by rw [\u2190 h, mul_assoc],\n rfl\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\ni k l : \u0393\nH2 : \u00accoeff x i * (coeff y k * coeff z l) = 0\nnx : i \u2208 support x\nny : k \u2208 support y\nnz : l \u2208 support z\nH : k + l \u2208 support y + support z\nh : coeff x i * coeff y k * coeff z l = 0\n\u22a2 coeff x i * (coeff y k * coeff z l) = 0\n[PROOFSTEP]\nrw [\u2190 h, mul_assoc]\n[GOAL]\ncase coeff.h.refine'_4\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb : \u0393\n\u22a2 \u2200 (a : (_ : \u0393 \u00d7 \u0393) \u00d7 \u0393 \u00d7 \u0393)\n (h\u2081 :\n a \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst)\n (h\u2082 : coeff x a.snd.fst * coeff y a.snd.snd * coeff z a.fst.snd \u2260 0),\n coeff x a.snd.fst * coeff y a.snd.snd * coeff z a.fst.snd =\n coeff x\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081\n h\u2082).fst.fst *\n (coeff y\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081\n h\u2082).snd.fst *\n coeff z\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) }) a h\u2081\n h\u2082).snd.snd)\n[PROOFSTEP]\nrintro \u27e8\u27e8i, j\u27e9, \u27e8k, l\u27e9\u27e9 _ _\n[GOAL]\ncase coeff.h.refine'_4.mk.mk.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalSemiring R\nx y z : HahnSeries \u0393 R\nb i j k l : \u0393\nh\u2081\u271d :\n { fst := (i, j), snd := (k, l) } \u2208\n Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>\n addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.fst\nh\u2082\u271d :\n coeff x { fst := (i, j), snd := (k, l) }.snd.fst * coeff y { fst := (i, j), snd := (k, l) }.snd.snd *\n coeff z { fst := (i, j), snd := (k, l) }.fst.snd \u2260\n 0\n\u22a2 coeff x { fst := (i, j), snd := (k, l) }.snd.fst * coeff y { fst := (i, j), snd := (k, l) }.snd.snd *\n coeff z { fst := (i, j), snd := (k, l) }.fst.snd =\n coeff x\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i, j), snd := (k, l) } h\u2081\u271d h\u2082\u271d).fst.fst *\n (coeff y\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i, j), snd := (k, l) } h\u2081\u271d h\u2082\u271d).snd.fst *\n coeff z\n ((fun a x_1 x => { fst := (a.snd.fst, a.snd.snd + a.fst.snd), snd := (a.snd.snd, a.fst.snd) })\n { fst := (i, j), snd := (k, l) } h\u2081\u271d h\u2082\u271d).snd.snd)\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : AddCommMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddCommMonoid (HahnSeries \u0393 R))\nsrc\u271d : Distrib (HahnSeries \u0393 R) := inferInstanceAs (Distrib (HahnSeries \u0393 R))\nx\u271d : HahnSeries \u0393 R\n\u22a2 0 * x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : AddCommMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddCommMonoid (HahnSeries \u0393 R))\nsrc\u271d : Distrib (HahnSeries \u0393 R) := inferInstanceAs (Distrib (HahnSeries \u0393 R))\nx\u271d\u00b9 : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (0 * x\u271d\u00b9) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\nsimp [mul_coeff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : AddCommMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddCommMonoid (HahnSeries \u0393 R))\nsrc\u271d : Distrib (HahnSeries \u0393 R) := inferInstanceAs (Distrib (HahnSeries \u0393 R))\nx\u271d : HahnSeries \u0393 R\n\u22a2 x\u271d * 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\nsrc\u271d\u00b9 : AddCommMonoid (HahnSeries \u0393 R) := inferInstanceAs (AddCommMonoid (HahnSeries \u0393 R))\nsrc\u271d : Distrib (HahnSeries \u0393 R) := inferInstanceAs (Distrib (HahnSeries \u0393 R))\nx\u271d\u00b9 : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x\u271d\u00b9 * 0) x\u271d = coeff 0 x\u271d\n[PROOFSTEP]\nsimp [mul_coeff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nsrc\u271d\u00b9 : AddMonoidWithOne (HahnSeries \u0393 R) := AddMonoidWithOne.unary\nsrc\u271d : NonUnitalNonAssocSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\n\u22a2 1 * x = x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nsrc\u271d\u00b9 : AddMonoidWithOne (HahnSeries \u0393 R) := AddMonoidWithOne.unary\nsrc\u271d : NonUnitalNonAssocSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (1 * x) x\u271d = coeff x x\u271d\n[PROOFSTEP]\nexact single_zero_mul_coeff.trans (one_mul _)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nsrc\u271d\u00b9 : AddMonoidWithOne (HahnSeries \u0393 R) := AddMonoidWithOne.unary\nsrc\u271d : NonUnitalNonAssocSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\n\u22a2 x * 1 = x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nsrc\u271d\u00b9 : AddMonoidWithOne (HahnSeries \u0393 R) := AddMonoidWithOne.unary\nsrc\u271d : NonUnitalNonAssocSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x * 1) x\u271d = coeff x x\u271d\n[PROOFSTEP]\nexact mul_single_zero_coeff.trans (mul_one _)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalCommSemiring R\nsrc\u271d : NonUnitalSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalSemiring (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\n\u22a2 x * y = y * x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalCommSemiring R\nsrc\u271d : NonUnitalSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalSemiring (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 coeff (x * y) x\u271d = coeff (y * x) x\u271d\n[PROOFSTEP]\nsimp_rw [mul_coeff, mul_comm]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalCommSemiring R\nsrc\u271d : NonUnitalSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalSemiring (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) x\u271d, coeff x ij.fst * coeff y ij.snd =\n \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support x)) x\u271d,\n coeff x x_1.snd * coeff y x_1.fst\n[PROOFSTEP]\nrefine'\n sum_bij (fun a _ => a.swap) (fun a ha => _) (fun a _ => rfl) (fun _ _ _ _ => Prod.swap_inj.1) fun a ha =>\n \u27e8a.swap, _, a.swap_swap.symm\u27e9\n[GOAL]\ncase coeff.h.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalCommSemiring R\nsrc\u271d : NonUnitalSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalSemiring (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\na : \u0393 \u00d7 \u0393\nha : a \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) x\u271d\n\u22a2 (fun a x => Prod.swap a) a ha \u2208 addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support x)) x\u271d\n[PROOFSTEP]\nrwa [swap_mem_addAntidiagonal]\n[GOAL]\ncase coeff.h.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalCommSemiring R\nsrc\u271d : NonUnitalSemiring (HahnSeries \u0393 R) := inferInstanceAs (NonUnitalSemiring (HahnSeries \u0393 R))\nx y : HahnSeries \u0393 R\nx\u271d : \u0393\na : \u0393 \u00d7 \u0393\nha : a \u2208 addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support x)) x\u271d\n\u22a2 Prod.swap a \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) x\u271d\n[PROOFSTEP]\nrwa [swap_mem_addAntidiagonal]\n[GOAL]\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nxy : x * y = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nxy : x * y = 0\nhx : x = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nxy : x * y = 0\nhx : x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase neg\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nxy : x * y = 0\nhx : \u00acx = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nxy : x * y = 0\nhx : \u00acx = 0\n\u22a2 y = 0\n[PROOFSTEP]\ncontrapose! xy\n[GOAL]\ncase neg.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nxy : y \u2260 0\n\u22a2 x * y \u2260 0\n[PROOFSTEP]\nrw [Ne, HahnSeries.ext_iff, Function.funext_iff, not_forall]\n[GOAL]\ncase neg.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nxy : y \u2260 0\n\u22a2 \u2203 x_1, \u00accoeff (x * y) x_1 = coeff 0 x_1\n[PROOFSTEP]\nrefine' \u27e8x.order + y.order, _\u27e9\n[GOAL]\ncase neg.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nxy : y \u2260 0\n\u22a2 \u00accoeff (x * y) (order x + order y) = coeff 0 (order x + order y)\n[PROOFSTEP]\nrw [mul_coeff_order_add_order x y, zero_coeff, mul_eq_zero]\n[GOAL]\ncase neg.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type ?u.1364118\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nxy : y \u2260 0\n\u22a2 \u00ac(coeff x (order x) = 0 \u2228 coeff y (order y) = 0)\n[PROOFSTEP]\nsimp [coeff_order_ne_zero, hx, xy]\n[GOAL]\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 order (x * y) = order x + order y\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 order (x * y) \u2264 order x + order y\n[PROOFSTEP]\napply order_le_of_coeff_ne_zero\n[GOAL]\ncase a.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 coeff (x * y) (order x + order y) \u2260 0\n[PROOFSTEP]\nrw [mul_coeff_order_add_order x y]\n[GOAL]\ncase a.h\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 coeff x (order x) * coeff y (order y) \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero (coeff_order_ne_zero hx) (coeff_order_ne_zero hy)\n[GOAL]\ncase a\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 order x + order y \u2264 order (x * y)\n[PROOFSTEP]\nrw [order_of_ne hx, order_of_ne hy, order_of_ne (mul_ne_zero hx hy), \u2190 Set.IsWf.min_add]\n[GOAL]\ncase a\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring R\ninst\u271d : NoZeroDivisors R\nx y : HahnSeries \u0393 R\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Set.IsWf.min (_ : Set.IsWf (support x + support y)) (_ : Set.Nonempty (support x + support y)) \u2264\n Set.IsWf.min (_ : Set.IsWf (support (x * y))) (_ : Set.Nonempty (support (x * y)))\n[PROOFSTEP]\nexact Set.IsWf.min_le_min_of_subset support_mul_subset_add_support\n[GOAL]\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\nx : HahnSeries \u0393 R\nn : \u2115\n\u22a2 order (x ^ n) = n \u2022 order x\n[PROOFSTEP]\ninduction' n with h IH\n[GOAL]\ncase zero\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\nx : HahnSeries \u0393 R\n\u22a2 order (x ^ Nat.zero) = Nat.zero \u2022 order x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\nx : HahnSeries \u0393 R\nh : \u2115\nIH : order (x ^ h) = h \u2022 order x\n\u22a2 order (x ^ Nat.succ h) = Nat.succ h \u2022 order x\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase succ.inl\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\nh : \u2115\nIH : order (0 ^ h) = h \u2022 order 0\n\u22a2 order (0 ^ Nat.succ h) = Nat.succ h \u2022 order 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.inr\n\u0393\u271d : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\u271d\n\u0393 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\nx : HahnSeries \u0393 R\nh : \u2115\nIH : order (x ^ h) = h \u2022 order x\nhx : x \u2260 0\n\u22a2 order (x ^ Nat.succ h) = Nat.succ h \u2022 order x\n[PROOFSTEP]\nrw [pow_succ', order_mul (pow_ne_zero _ hx) hx, succ_nsmul', IH]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\n\u22a2 \u2191(single a) r * \u2191(single b) s = \u2191(single (a + b)) (r * s)\n[PROOFSTEP]\next x\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\n\u22a2 coeff (\u2191(single a) r * \u2191(single b) s) x = coeff (\u2191(single (a + b)) (r * s)) x\n[PROOFSTEP]\nby_cases h : x = a + b\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\nh : x = a + b\n\u22a2 coeff (\u2191(single a) r * \u2191(single b) s) x = coeff (\u2191(single (a + b)) (r * s)) x\n[PROOFSTEP]\nrw [h, mul_single_coeff_add]\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\nh : x = a + b\n\u22a2 coeff (\u2191(single a) r) a * s = coeff (\u2191(single (a + b)) (r * s)) (a + b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\nh : \u00acx = a + b\n\u22a2 coeff (\u2191(single a) r * \u2191(single b) s) x = coeff (\u2191(single (a + b)) (r * s)) x\n[PROOFSTEP]\nrw [single_coeff_of_ne h, mul_coeff, sum_eq_zero]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\nh : \u00acx = a + b\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support (\u2191(single a) r))) (_ : Set.IsPwo (support (\u2191(single b) s))) x \u2192\n coeff (\u2191(single a) r) x_1.fst * coeff (\u2191(single b) s) x_1.snd = 0\n[PROOFSTEP]\nsimp_rw [mem_addAntidiagonal]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\nx : \u0393\nh : \u00acx = a + b\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1.fst \u2208 support (\u2191(single a) r) \u2227 x_1.snd \u2208 support (\u2191(single b) s) \u2227 x_1.fst + x_1.snd = x \u2192\n coeff (\u2191(single a) r) x_1.fst * coeff (\u2191(single b) s) x_1.snd = 0\n[PROOFSTEP]\nrintro \u27e8y, z\u27e9 \u27e8hy, hz, rfl\u27e9\n[GOAL]\ncase neg.mk.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\ny z : \u0393\nhy : (y, z).fst \u2208 support (\u2191(single a) r)\nhz : (y, z).snd \u2208 support (\u2191(single b) s)\nh : \u00ac(y, z).fst + (y, z).snd = a + b\n\u22a2 coeff (\u2191(single a) r) (y, z).fst * coeff (\u2191(single b) s) (y, z).snd = 0\n[PROOFSTEP]\nrw [eq_of_mem_support_single hy, eq_of_mem_support_single hz] at h \n[GOAL]\ncase neg.mk.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonUnitalNonAssocSemiring R\na b : \u0393\nr s : R\ny z : \u0393\nhy : (y, z).fst \u2208 support (\u2191(single a) r)\nhz : (y, z).snd \u2208 support (\u2191(single b) s)\nh : \u00aca + b = a + b\n\u22a2 coeff (\u2191(single a) r) (y, z).fst * coeff (\u2191(single b) s) (y, z).snd = 0\n[PROOFSTEP]\nexact (h rfl).elim\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nx y : R\n\u22a2 OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y\n[PROOFSTEP]\nrw [single_mul_single, zero_add]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nx y : R\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n y\n[PROOFSTEP]\next a\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nx y : R\na : \u0393\n\u22a2 coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n (x + y))\n a =\n coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n y)\n a\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nx y : R\na : \u0393\nh : a = 0\n\u22a2 coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n (x + y))\n a =\n coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n y)\n a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nx y : R\na : \u0393\nh : \u00aca = 0\n\u22a2 coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n (x + y))\n a =\n coeff\n (OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : R),\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } (x * y) =\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } x *\n OneHom.toFun { toFun := \u2191(single 0), map_one' := (_ : \u2191(single 0) 1 = \u2191(single 0) 1) } y) })\n y)\n a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\n\u22a2 Injective \u2191C\n[PROOFSTEP]\nintro r s rs\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr s : R\nrs : \u2191C r = \u2191C s\n\u22a2 r = s\n[PROOFSTEP]\nrw [HahnSeries.ext_iff, Function.funext_iff] at rs \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr s : R\nrs : \u2200 (a : \u0393), coeff (\u2191C r) a = coeff (\u2191C s) a\n\u22a2 r = s\n[PROOFSTEP]\nhave h := rs 0\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr s : R\nrs : \u2200 (a : \u0393), coeff (\u2191C r) a = coeff (\u2191C s) a\nh : coeff (\u2191C r) 0 = coeff (\u2191C s) 0\n\u22a2 r = s\n[PROOFSTEP]\nrwa [C_apply, single_coeff_same, C_apply, single_coeff_same] at h \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : r \u2260 0\n\u22a2 \u2191C r \u2260 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nrw [\u2190 C_zero] at h \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = \u2191C 0\n\u22a2 r = 0\n[PROOFSTEP]\nexact C_injective h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\n\u22a2 order (\u2191C r) = 0\n[PROOFSTEP]\nby_cases h : r = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : r = 0\n\u22a2 order (\u2191C r) = 0\n[PROOFSTEP]\nrw [h, C_zero, order_zero]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u00acr = 0\n\u22a2 order (\u2191C r) = 0\n[PROOFSTEP]\nexact order_single h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\n\u22a2 embDomain f (x * y) = embDomain f x * embDomain f y\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393'\n\u22a2 coeff (embDomain f (x * y)) g = coeff (embDomain f x * embDomain f y) g\n[PROOFSTEP]\nby_cases hg : g \u2208 Set.range f\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : g \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (x * y)) g = coeff (embDomain f x * embDomain f y) g\n[PROOFSTEP]\nobtain \u27e8g, rfl\u27e9 := hg\n[GOAL]\ncase pos.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 coeff (embDomain f (x * y)) (\u2191f g) = coeff (embDomain f x * embDomain f y) (\u2191f g)\n[PROOFSTEP]\nsimp only [mul_coeff, embDomain_coeff]\n[GOAL]\ncase pos.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g, coeff x ij.fst * coeff y ij.snd =\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g),\n coeff (embDomain f x) ij.fst * coeff (embDomain f y) ij.snd\n[PROOFSTEP]\ntrans\n \u2211 ij in\n (addAntidiagonal x.isPwo_support y.isPwo_support g).map (Function.Embedding.prodMap f.toEmbedding f.toEmbedding),\n (embDomain f x).coeff ij.1 * (embDomain f y).coeff ij.2\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g, coeff x ij.fst * coeff y ij.snd =\n \u2211 ij in\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g),\n coeff (embDomain f x) ij.fst * coeff (embDomain f y) ij.snd\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2211 ij in\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g),\n coeff (embDomain f x) ij.fst * coeff (embDomain f y) ij.snd =\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g),\n coeff (embDomain f x) ij.fst * coeff (embDomain f y) ij.snd\n[PROOFSTEP]\napply sum_subset\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g) \u2286\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase h.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\ni j : \u0393'\nhij :\n (i, j) \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g)\n\u22a2 (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\n[PROOFSTEP]\nsimp only [exists_prop, mem_map, Prod.mk.inj_iff, mem_addAntidiagonal, Function.Embedding.coe_prodMap, mem_support,\n Prod.exists] at hij \n[GOAL]\ncase h.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\ni j : \u0393'\nhij : \u2203 a b, (coeff x a \u2260 0 \u2227 coeff y b \u2260 0 \u2227 a + b = g) \u2227 Prod.map \u2191f.toEmbedding \u2191f.toEmbedding (a, b) = (i, j)\n\u22a2 (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\n[PROOFSTEP]\nobtain \u27e8i, j, \u27e8hx, hy, rfl\u27e9, rfl, rfl\u27e9 := hij\n[GOAL]\ncase h.mk.intro.intro.intro.intro.intro.refl\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ni j : \u0393\nhx : coeff x i \u2260 0\nhy : coeff y j \u2260 0\n\u22a2 (\u2191f.toEmbedding i, \u2191f.toEmbedding j) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f (i + j))\n[PROOFSTEP]\nsimp [hx, hy, hf]\n[GOAL]\ncase hf\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2200 (x_1 : \u0393' \u00d7 \u0393'),\n x_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g) \u2192\n \u00acx_1 \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g) \u2192\n coeff (embDomain f x) x_1.fst * coeff (embDomain f y) x_1.snd = 0\n[PROOFSTEP]\nrintro \u27e8_, _\u27e9 h1 h2\n[GOAL]\ncase hf.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\nfst\u271d snd\u271d : \u0393'\nh1 :\n (fst\u271d, snd\u271d) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\nh2 :\n \u00ac(fst\u271d, snd\u271d) \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g)\n\u22a2 coeff (embDomain f x) (fst\u271d, snd\u271d).fst * coeff (embDomain f y) (fst\u271d, snd\u271d).snd = 0\n[PROOFSTEP]\ncontrapose! h2\n[GOAL]\ncase hf.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\nfst\u271d snd\u271d : \u0393'\nh1 :\n (fst\u271d, snd\u271d) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\nh2 : coeff (embDomain f x) fst\u271d * coeff (embDomain f y) snd\u271d \u2260 0\n\u22a2 (fst\u271d, snd\u271d) \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g)\n[PROOFSTEP]\nobtain \u27e8i, _, rfl\u27e9 := support_embDomain_subset (ne_zero_and_ne_zero_of_mul h2).1\n[GOAL]\ncase hf.mk.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393\nsnd\u271d : \u0393'\ni : \u0393\nleft\u271d : i \u2208 support x\nh1 :\n (\u2191f i, snd\u271d) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\nh2 : coeff (embDomain f x) (\u2191f i) * coeff (embDomain f y) snd\u271d \u2260 0\n\u22a2 (\u2191f i, snd\u271d) \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g)\n[PROOFSTEP]\nobtain \u27e8j, _, rfl\u27e9 := support_embDomain_subset (ne_zero_and_ne_zero_of_mul h2).2\n[GOAL]\ncase hf.mk.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng i : \u0393\nleft\u271d\u00b9 : i \u2208 support x\nj : \u0393\nleft\u271d : j \u2208 support y\nh1 :\n (\u2191f i, \u2191f j) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\nh2 : coeff (embDomain f x) (\u2191f i) * coeff (embDomain f y) (\u2191f j) \u2260 0\n\u22a2 (\u2191f i, \u2191f j) \u2208\n Finset.map (Embedding.prodMap f.toEmbedding f.toEmbedding)\n (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) g)\n[PROOFSTEP]\nsimp only [exists_prop, mem_map, Prod.mk.inj_iff, mem_addAntidiagonal, Function.Embedding.coe_prodMap, mem_support,\n Prod.exists]\n[GOAL]\ncase hf.mk.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng i : \u0393\nleft\u271d\u00b9 : i \u2208 support x\nj : \u0393\nleft\u271d : j \u2208 support y\nh1 :\n (\u2191f i, \u2191f j) \u2208\n addAntidiagonal (_ : Set.IsPwo (support (embDomain f x))) (_ : Set.IsPwo (support (embDomain f y))) (\u2191f g)\nh2 : coeff (embDomain f x) (\u2191f i) * coeff (embDomain f y) (\u2191f j) \u2260 0\n\u22a2 \u2203 a b, (coeff x a \u2260 0 \u2227 coeff y b \u2260 0 \u2227 a + b = g) \u2227 Prod.map \u2191f.toEmbedding \u2191f.toEmbedding (a, b) = (\u2191f i, \u2191f j)\n[PROOFSTEP]\nsimp only [mem_addAntidiagonal, embDomain_coeff, mem_support, \u2190 hf, OrderEmbedding.eq_iff_eq] at h1 \n[GOAL]\ncase hf.mk.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng i : \u0393\nleft\u271d\u00b9 : i \u2208 support x\nj : \u0393\nleft\u271d : j \u2208 support y\nh2 : coeff (embDomain f x) (\u2191f i) * coeff (embDomain f y) (\u2191f j) \u2260 0\nh1 : coeff x i \u2260 0 \u2227 coeff y j \u2260 0 \u2227 i + j = g\n\u22a2 \u2203 a b, (coeff x a \u2260 0 \u2227 coeff y b \u2260 0 \u2227 a + b = g) \u2227 Prod.map \u2191f.toEmbedding \u2191f.toEmbedding (a, b) = (\u2191f i, \u2191f j)\n[PROOFSTEP]\nexact \u27e8i, j, h1, rfl\u27e9\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : \u00acg \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f (x * y)) g = coeff (embDomain f x * embDomain f y) g\n[PROOFSTEP]\nrw [embDomain_notin_range hg, eq_comm]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : \u00acg \u2208 Set.range \u2191f\n\u22a2 coeff (embDomain f x * embDomain f y) g = 0\n[PROOFSTEP]\ncontrapose! hg\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ng : \u0393'\nhg : coeff (embDomain f x * embDomain f y) g \u2260 0\n\u22a2 g \u2208 Set.range \u2191f\n[PROOFSTEP]\nobtain \u27e8_, _, hi, hj, rfl\u27e9 := support_mul_subset_add_support ((mem_support _ _).2 hg)\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\nw\u271d\u00b9 w\u271d : \u0393'\nhi : w\u271d\u00b9 \u2208 support (embDomain f x)\nhj : w\u271d \u2208 support (embDomain f y)\nhg : coeff (embDomain f x * embDomain f y) ((fun x x_1 => x + x_1) w\u271d\u00b9 w\u271d) \u2260 0\n\u22a2 (fun x x_1 => x + x_1) w\u271d\u00b9 w\u271d \u2208 Set.range \u2191f\n[PROOFSTEP]\nobtain \u27e8i, _, rfl\u27e9 := support_embDomain_subset hi\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\nw\u271d : \u0393'\nhj : w\u271d \u2208 support (embDomain f y)\ni : \u0393\nleft\u271d : i \u2208 support x\nhi : \u2191f i \u2208 support (embDomain f x)\nhg : coeff (embDomain f x * embDomain f y) ((fun x x_1 => x + x_1) (\u2191f i) w\u271d) \u2260 0\n\u22a2 (fun x x_1 => x + x_1) (\u2191f i) w\u271d \u2208 Set.range \u2191f\n[PROOFSTEP]\nobtain \u27e8j, _, rfl\u27e9 := support_embDomain_subset hj\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonUnitalNonAssocSemiring R\nf : \u0393 \u21aao \u0393'\nhf : \u2200 (x y : \u0393), \u2191f (x + y) = \u2191f x + \u2191f y\nx y : HahnSeries \u0393 R\ni : \u0393\nleft\u271d\u00b9 : i \u2208 support x\nhi : \u2191f i \u2208 support (embDomain f x)\nj : \u0393\nleft\u271d : j \u2208 support y\nhj : \u2191f j \u2208 support (embDomain f y)\nhg : coeff (embDomain f x * embDomain f y) ((fun x x_1 => x + x_1) (\u2191f i) (\u2191f j)) \u2260 0\n\u22a2 (fun x x_1 => x + x_1) (\u2191f i) (\u2191f j) \u2208 Set.range \u2191f\n[PROOFSTEP]\nrefine' \u27e8i + j, hf i j\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\n\u0393' : Type u_3\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393'\ninst\u271d : NonAssocSemiring R\nf : \u0393 \u2192+ \u0393'\nhfi : Injective \u2191f\nhf : \u2200 (g g' : \u0393), \u2191f g \u2264 \u2191f g' \u2194 g \u2264 g'\nr : R\n\u22a2 \u2191(single (\u2191{ toEmbedding := { toFun := \u2191f, inj' := hfi }, map_rel_iff' := (_ : \u2200 {a b : \u0393}, \u2191f a \u2264 \u2191f b \u2194 a \u2264 b) } 0))\n r =\n \u2191C r\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : (fun x => HahnSeries \u0393 A) r\n\u22a2 \u2191(RingHom.comp C (algebraMap R A)) r * x = x * \u2191(RingHom.comp C (algebraMap R A)) r\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : (fun x => HahnSeries \u0393 A) r\nx\u271d : \u0393\n\u22a2 coeff (\u2191(RingHom.comp C (algebraMap R A)) r * x) x\u271d = coeff (x * \u2191(RingHom.comp C (algebraMap R A)) r) x\u271d\n[PROOFSTEP]\nsimp only [smul_coeff, single_zero_mul_eq_smul, RingHom.coe_comp, RingHom.toFun_eq_coe, C_apply, Function.comp_apply,\n algebraMap_smul, mul_single_zero_coeff]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : (fun x => HahnSeries \u0393 A) r\nx\u271d : \u0393\n\u22a2 r \u2022 coeff x x\u271d = coeff x x\u271d * \u2191(algebraMap R A) r\n[PROOFSTEP]\nrw [\u2190 Algebra.commutes, Algebra.smul_def]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : (fun x => HahnSeries \u0393 A) r\n\u22a2 r \u2022 x = \u2191(RingHom.comp C (algebraMap R A)) r * x\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : (fun x => HahnSeries \u0393 A) r\nx\u271d : \u0393\n\u22a2 coeff (r \u2022 x) x\u271d = coeff (\u2191(RingHom.comp C (algebraMap R A)) r * x) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\n\u22a2 \u22a5 \u2260 \u22a4\n[PROOFSTEP]\nrw [Ne.def, SetLike.ext_iff, not_forall]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\n\u22a2 \u2203 x, \u00ac(x \u2208 \u22a5 \u2194 x \u2208 \u22a4)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := exists_ne (0 : \u0393)\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\n\u22a2 \u2203 x, \u00ac(x \u2208 \u22a5 \u2194 x \u2208 \u22a4)\n[PROOFSTEP]\nrefine' \u27e8single a 1, _\u27e9\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\n\u22a2 \u00ac(\u2191(single a) 1 \u2208 \u22a5 \u2194 \u2191(single a) 1 \u2208 \u22a4)\n[PROOFSTEP]\nsimp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true_iff, Algebra.mem_top]\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\n\u22a2 \u2200 (x : R), \u00ac\u2191(algebraMap R (HahnSeries \u0393 R)) x = \u2191(single a) 1\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\nx : R\n\u22a2 \u00ac\u2191(algebraMap R (HahnSeries \u0393 R)) x = \u2191(single a) 1\n[PROOFSTEP]\nrw [HahnSeries.ext_iff, Function.funext_iff, not_forall]\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\nx : R\n\u22a2 \u2203 x_1, \u00accoeff (\u2191(algebraMap R (HahnSeries \u0393 R)) x) x_1 = coeff (\u2191(single a) 1) x_1\n[PROOFSTEP]\nrefine' \u27e8a, _\u27e9\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\nx : R\n\u22a2 \u00accoeff (\u2191(algebraMap R (HahnSeries \u0393 R)) x) a = coeff (\u2191(single a) 1) a\n[PROOFSTEP]\nrw [single_coeff_same, algebraMap_apply, C_apply, single_coeff_of_ne ha]\n[GOAL]\ncase intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u2074 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Nontrivial \u0393\ninst\u271d : Nontrivial R\na : \u0393\nha : a \u2260 0\nx : R\n\u22a2 \u00ac0 = 1\n[PROOFSTEP]\nexact zero_ne_one\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf : HahnSeries \u2115 R\n\u22a2 (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf : HahnSeries \u2115 R\nx\u271d : \u2115\n\u22a2 coeff\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f))\n x\u271d =\n coeff f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf : PowerSeries R\n\u22a2 (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf : PowerSeries R\nn\u271d : \u2115\n\u22a2 \u2191(PowerSeries.coeff R n\u271d)\n ((fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f)) =\n \u2191(PowerSeries.coeff R n\u271d) f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\n\u22a2 Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n (f * g) =\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n f *\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n g\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\n\u22a2 \u2191(PowerSeries.coeff R n)\n (Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n (f * g)) =\n \u2191(PowerSeries.coeff R n)\n (Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n f *\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n g)\n[PROOFSTEP]\nsimp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, mul_coeff, isPwo_support]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n, coeff f ij.fst * coeff g ij.snd =\n \u2211 x in Nat.antidiagonal n, coeff f x.fst * coeff g x.snd\n[PROOFSTEP]\nclassical\nrefine' sum_filter_ne_zero.symm.trans ((sum_congr _ fun _ _ => rfl).trans sum_filter_ne_zero)\next m\nsimp only [Nat.mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter, mem_support]\nrintro h\nrw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n, coeff f ij.fst * coeff g ij.snd =\n \u2211 x in Nat.antidiagonal n, coeff f x.fst * coeff g x.snd\n[PROOFSTEP]\nrefine' sum_filter_ne_zero.symm.trans ((sum_congr _ fun _ _ => rfl).trans sum_filter_ne_zero)\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\n\u22a2 filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0)\n (addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n) =\n filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0) (Nat.antidiagonal n)\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\nm : \u2115 \u00d7 \u2115\n\u22a2 m \u2208\n filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0)\n (addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n) \u2194\n m \u2208 filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0) (Nat.antidiagonal n)\n[PROOFSTEP]\nsimp only [Nat.mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter, mem_support]\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\nm : \u2115 \u00d7 \u2115\n\u22a2 coeff f m.fst * coeff g m.snd \u2260 0 \u2192 (coeff f m.fst \u2260 0 \u2227 coeff g m.snd \u2260 0 \u2227 m.fst + m.snd = n \u2194 m.fst + m.snd = n)\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn : \u2115\nm : \u2115 \u00d7 \u2115\nh : coeff f m.fst * coeff g m.snd \u2260 0\n\u22a2 coeff f m.fst \u2260 0 \u2227 coeff g m.snd \u2260 0 \u2227 m.fst + m.snd = n \u2194 m.fst + m.snd = n\n[PROOFSTEP]\nrw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\n\u22a2 Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n (f + g) =\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n f +\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nf g : HahnSeries \u2115 R\nn\u271d : \u2115\n\u22a2 \u2191(PowerSeries.coeff R n\u271d)\n (Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n (f + g)) =\n \u2191(PowerSeries.coeff R n\u271d)\n (Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n f +\n Equiv.toFun\n { toFun := fun f => PowerSeries.mk f.coeff,\n invFun := fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries \u2115 R),\n (fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n ((fun f => PowerSeries.mk f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : PowerSeries R),\n (fun f => PowerSeries.mk f.coeff)\n ((fun f =>\n { coeff := fun n => \u2191(PowerSeries.coeff R n) f,\n isPwo_support' := (_ : Set.IsPwo (Function.support fun n => \u2191(PowerSeries.coeff R n) f)) })\n f) =\n f) }\n g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nx : PowerSeries R\n\u22a2 \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b\n[PROOFSTEP]\nsimp only [Function.Embedding.coeFn_mk]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nx : PowerSeries R\n\u22a2 \u2200 {a b : \u2115}, \u2191a \u2264 \u2191b \u2194 a \u2264 b\n[PROOFSTEP]\nexact Nat.cast_le\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nx : PowerSeries R\nn : \u2115\n\u22a2 coeff (\u2191(ofPowerSeries \u0393 R) x) \u2191n = \u2191(PowerSeries.coeff R n) x\n[PROOFSTEP]\nsimp [ofPowerSeries_apply]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\n\u22a2 \u2191(ofPowerSeries \u0393 R) (\u2191(PowerSeries.C R) r) = \u2191C r\n[PROOFSTEP]\next n\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\n\u22a2 coeff (\u2191(ofPowerSeries \u0393 R) (\u2191(PowerSeries.C R) r)) n = coeff (\u2191C r) n\n[PROOFSTEP]\nsimp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq, single_coeff]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r)))\n n =\n if n = 0 then r else 0\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\nhn : n = 0\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r)))\n n =\n r\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r)))\n 0 =\n r\n[PROOFSTEP]\nconvert @embDomain_coeff \u2115 R _ _ \u0393 _ _ _ 0\n[GOAL]\ncase h.e'_2.h.e'_6\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\n\u22a2 0 =\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\n\u22a2 r = coeff (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r)) 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\nhn : \u00acn = 0\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r)))\n n =\n 0\n[PROOFSTEP]\nrw [embDomain_notin_image_support]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\nhn : \u00acn = 0\n\u22a2 \u00acn \u2208\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) } ''\n support (\u2191(RingEquiv.symm toPowerSeries) (\u2191(PowerSeries.C R) r))\n[PROOFSTEP]\nsimp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support, PowerSeries.coeff_C]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\nhn : \u00acn = 0\n\u22a2 \u2200 (x : \u2115),\n \u00ac((if x = 0 then r else 0) \u2260 0 \u2227\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n x =\n n)\n[PROOFSTEP]\nintro\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nr : R\nn : \u0393\nhn : \u00acn = 0\nx\u271d : \u2115\n\u22a2 \u00ac((if x\u271d = 0 then r else 0) \u2260 0 \u2227\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n x\u271d =\n n)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Ne.symm hn]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\n\u22a2 \u2191(ofPowerSeries \u0393 R) PowerSeries.X = \u2191(single 1) 1\n[PROOFSTEP]\next n\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\n\u22a2 coeff (\u2191(ofPowerSeries \u0393 R) PowerSeries.X) n = coeff (\u2191(single 1) 1) n\n[PROOFSTEP]\nsimp only [single_coeff, ofPowerSeries_apply, RingHom.coe_mk]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X))\n n =\n if n = 1 then 1 else 0\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : n = 1\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X))\n n =\n 1\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : n = 1\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X))\n 1 =\n 1\n[PROOFSTEP]\nconvert @embDomain_coeff \u2115 R _ _ \u0393 _ _ _ 1\n[GOAL]\ncase h.e'_2.h.e'_6\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : n = 1\n\u22a2 1 =\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : n = 1\n\u22a2 1 = coeff (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : \u00acn = 1\n\u22a2 coeff\n (embDomain\n { toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X))\n n =\n 0\n[PROOFSTEP]\nrw [embDomain_notin_image_support]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : \u00acn = 1\n\u22a2 \u00acn \u2208\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) } ''\n support (\u2191(RingEquiv.symm toPowerSeries) PowerSeries.X)\n[PROOFSTEP]\nsimp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support, PowerSeries.coeff_X]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : \u00acn = 1\n\u22a2 \u2200 (x : \u2115),\n \u00ac((if x = 1 then 1 else 0) \u2260 0 \u2227\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n x =\n n)\n[PROOFSTEP]\nintro\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : StrictOrderedSemiring \u0393\nn : \u0393\nhn : \u00acn = 1\nx\u271d : \u2115\n\u22a2 \u00ac((if x\u271d = 1 then 1 else 0) \u2260 0 \u2227\n \u2191{ toEmbedding := { toFun := Nat.cast, inj' := (_ : Injective Nat.cast) },\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u2115},\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } a \u2264\n \u2191{ toFun := Nat.cast, inj' := (_ : Injective Nat.cast) } b \u2194\n a \u2264 b) }\n x\u271d =\n n)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Ne.symm hn]\n[GOAL]\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\nR : Type u_3\ninst\u271d : CommSemiring R\nn : \u2115\n\u22a2 \u2191(ofPowerSeries \u0393 R) (PowerSeries.X ^ n) = \u2191(single \u2191n) 1\n[PROOFSTEP]\nrw [RingHom.map_pow]\n[GOAL]\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\nR : Type u_3\ninst\u271d : CommSemiring R\nn : \u2115\n\u22a2 \u2191(ofPowerSeries \u0393 R) PowerSeries.X ^ n = \u2191(single \u2191n) 1\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\nR : Type u_3\ninst\u271d : CommSemiring R\n\u22a2 \u2191(ofPowerSeries \u0393 R) PowerSeries.X ^ Nat.zero = \u2191(single \u2191Nat.zero) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\nR : Type u_3\ninst\u271d : CommSemiring R\n\u22a2 1 = \u2191(single 0) 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : Semiring R\u271d\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\nR : Type u_3\ninst\u271d : CommSemiring R\nn : \u2115\nih : \u2191(ofPowerSeries \u0393 R) PowerSeries.X ^ n = \u2191(single \u2191n) 1\n\u22a2 \u2191(ofPowerSeries \u0393 R) PowerSeries.X ^ Nat.succ n = \u2191(single \u2191(Nat.succ n)) 1\n[PROOFSTEP]\nrw [pow_succ, pow_succ, ih, ofPowerSeries_X, mul_comm, single_mul_single, one_mul, Nat.cast_succ, add_comm]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\n\u22a2 (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) ((fun f => f.coeff) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nx\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) ((fun f => f.coeff) f)) x\u271d =\n coeff f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf : MvPowerSeries \u03c3 R\n\u22a2 (fun f => f.coeff) ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf : MvPowerSeries \u03c3 R\nn\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n\u271d)\n ((fun f => f.coeff) ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f)) =\n \u2191(MvPowerSeries.coeff R n\u271d) f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\n\u22a2 Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff) ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n (f * g) =\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n f *\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n g\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n)\n (Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n (f * g)) =\n \u2191(MvPowerSeries.coeff R n)\n (Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n f *\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n g)\n[PROOFSTEP]\nsimp only [MvPowerSeries.coeff_mul]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n) (f * g).coeff =\n \u2211 x in Finsupp.antidiagonal n, \u2191(MvPowerSeries.coeff R x.fst) f.coeff * \u2191(MvPowerSeries.coeff R x.snd) g.coeff\n[PROOFSTEP]\nclassical\nchange (f * g).coeff n = _\nsimp_rw [mul_coeff]\nrefine' sum_filter_ne_zero.symm.trans ((sum_congr _ fun _ _ => rfl).trans sum_filter_ne_zero)\next m\nsimp only [and_congr_left_iff, mem_addAntidiagonal, mem_filter, mem_support, Finsupp.mem_antidiagonal]\nrintro h\nrw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n) (f * g).coeff =\n \u2211 x in Finsupp.antidiagonal n, \u2191(MvPowerSeries.coeff R x.fst) f.coeff * \u2191(MvPowerSeries.coeff R x.snd) g.coeff\n[PROOFSTEP]\nchange (f * g).coeff n = _\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff (f * g) n =\n \u2211 x in Finsupp.antidiagonal n, \u2191(MvPowerSeries.coeff R x.fst) f.coeff * \u2191(MvPowerSeries.coeff R x.snd) g.coeff\n[PROOFSTEP]\nsimp_rw [mul_coeff]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n, coeff f ij.fst * coeff g ij.snd =\n \u2211 x in Finsupp.antidiagonal n, \u2191(MvPowerSeries.coeff R x.fst) f.coeff * \u2191(MvPowerSeries.coeff R x.snd) g.coeff\n[PROOFSTEP]\nrefine' sum_filter_ne_zero.symm.trans ((sum_congr _ fun _ _ => rfl).trans sum_filter_ne_zero)\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\n\u22a2 filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0)\n (addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n) =\n filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0) (Finsupp.antidiagonal n)\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\nm : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)\n\u22a2 m \u2208\n filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0)\n (addAntidiagonal (_ : Set.IsPwo (support f)) (_ : Set.IsPwo (support g)) n) \u2194\n m \u2208 filter (fun x => coeff f x.fst * coeff g x.snd \u2260 0) (Finsupp.antidiagonal n)\n[PROOFSTEP]\nsimp only [and_congr_left_iff, mem_addAntidiagonal, mem_filter, mem_support, Finsupp.mem_antidiagonal]\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\nm : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)\n\u22a2 coeff f m.fst * coeff g m.snd \u2260 0 \u2192 (coeff f m.fst \u2260 0 \u2227 coeff g m.snd \u2260 0 \u2227 m.fst + m.snd = n \u2194 m.fst + m.snd = n)\n[PROOFSTEP]\nrintro h\n[GOAL]\ncase h.a\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn : \u03c3 \u2192\u2080 \u2115\nm : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)\nh : coeff f m.fst * coeff g m.snd \u2260 0\n\u22a2 coeff f m.fst \u2260 0 \u2227 coeff g m.snd \u2260 0 \u2227 m.fst + m.snd = n \u2194 m.fst + m.snd = n\n[PROOFSTEP]\nrw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\n\u22a2 Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff) ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n (f + g) =\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n f +\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : StrictOrderedSemiring \u0393\n\u03c3 : Type u_3\ninst\u271d : Fintype \u03c3\nf g : HahnSeries (\u03c3 \u2192\u2080 \u2115) R\nn\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n\u271d)\n (Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n (f + g)) =\n \u2191(MvPowerSeries.coeff R n\u271d)\n (Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n f +\n Equiv.toFun\n { toFun := fun f => f.coeff,\n invFun := fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) },\n left_inv :=\n (_ :\n \u2200 (f : HahnSeries (\u03c3 \u2192\u2080 \u2115) R),\n (fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) })\n ((fun f => f.coeff) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : MvPowerSeries \u03c3 R),\n (fun f => f.coeff)\n ((fun f => { coeff := f, isPwo_support' := (_ : Set.IsPwo (Function.support f)) }) f) =\n f) }\n g)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\n\u22a2 Equiv.toFun src\u271d.toEquiv (\u2191(algebraMap R (HahnSeries \u2115 A)) r) = \u2191(algebraMap R (PowerSeries A)) r\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\nn : \u2115\n\u22a2 \u2191(PowerSeries.coeff A n) (Equiv.toFun src\u271d.toEquiv (\u2191(algebraMap R (HahnSeries \u2115 A)) r)) =\n \u2191(PowerSeries.coeff A n) (\u2191(algebraMap R (PowerSeries A)) r)\n[PROOFSTEP]\nsimp only [algebraMap_apply, PowerSeries.algebraMap_apply, C_apply, coeff_toPowerSeries]\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\nn : \u2115\n\u22a2 \u2191(PowerSeries.coeff A n) (Equiv.toFun toPowerSeries.toEquiv (\u2191(single 0) (\u2191(algebraMap R A) r))) =\n \u2191(PowerSeries.coeff A n) (\u2191(PowerSeries.C A) (\u2191(algebraMap R A) r))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase h.zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\n\u22a2 \u2191(PowerSeries.coeff A Nat.zero) (Equiv.toFun toPowerSeries.toEquiv (\u2191(single 0) (\u2191(algebraMap R A) r))) =\n \u2191(PowerSeries.coeff A Nat.zero) (\u2191(PowerSeries.C A) (\u2191(algebraMap R A) r))\n[PROOFSTEP]\nsimp [PowerSeries.coeff_zero_eq_constantCoeff, single_coeff_same]\n[GOAL]\ncase h.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\nn : \u2115\n\u22a2 \u2191(PowerSeries.coeff A (Nat.succ n)) (Equiv.toFun toPowerSeries.toEquiv (\u2191(single 0) (\u2191(algebraMap R A) r))) =\n \u2191(PowerSeries.coeff A (Nat.succ n)) (\u2191(PowerSeries.C A) (\u2191(algebraMap R A) r))\n[PROOFSTEP]\nsimp [n.succ_ne_zero, Ne.def, not_false_iff, single_coeff_of_ne]\n[GOAL]\ncase h.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : CommSemiring R\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : HahnSeries \u2115 A \u2243+* PowerSeries A := toPowerSeries\nr : R\nn : \u2115\n\u22a2 0 = \u2191(PowerSeries.coeff A (Nat.succ n)) (\u2191(PowerSeries.C A) (\u2191(algebraMap R A) r))\n[PROOFSTEP]\nrw [PowerSeries.coeff_C, if_neg n.succ_ne_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\n\u22a2 \u2191(order 1) = 0\n[PROOFSTEP]\nsimp [order_of_ne]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : x = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : x = 0\nhy : y = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : x = 0\nhy : \u00acy = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min ((fun x => if x = 0 then \u22a4 else \u2191(order x)) x) ((fun x => if x = 0 then \u22a4 else \u2191(order x)) y) \u2264\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x + y)\n[PROOFSTEP]\nsimp only [hx, hy, support_nonempty_iff, if_neg, not_false_iff, isWf_support]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min \u2191(order x) \u2191(order y) \u2264 if x + y = 0 then \u22a4 else \u2191(order (x + y))\n[PROOFSTEP]\nby_cases hxy : x + y = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : x + y = 0\n\u22a2 min \u2191(order x) \u2191(order y) \u2264 if x + y = 0 then \u22a4 else \u2191(order (x + y))\n[PROOFSTEP]\nsimp [hxy]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : \u00acx + y = 0\n\u22a2 min \u2191(order x) \u2191(order y) \u2264 if x + y = 0 then \u22a4 else \u2191(order (x + y))\n[PROOFSTEP]\nrw [if_neg hxy, \u2190 WithTop.coe_min, WithTop.coe_le_coe]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\nhxy : \u00acx + y = 0\n\u22a2 min (order x) (order y) \u2264 order (x + y)\n[PROOFSTEP]\nexact min_order_le_order_add hxy\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\n\u22a2 (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x * y) =\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) x + (fun x => if x = 0 then \u22a4 else \u2191(order x)) y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : x = 0\n\u22a2 (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x * y) =\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) x + (fun x => if x = 0 then \u22a4 else \u2191(order x)) y\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\n\u22a2 (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x * y) =\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) x + (fun x => if x = 0 then \u22a4 else \u2191(order x)) y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x * y) =\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) x + (fun x => if x = 0 then \u22a4 else \u2191(order x)) y\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (fun x => if x = 0 then \u22a4 else \u2191(order x)) (x * y) =\n (fun x => if x = 0 then \u22a4 else \u2191(order x)) x + (fun x => if x = 0 then \u22a4 else \u2191(order x)) y\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx y : HahnSeries \u0393 R\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(order (x * y))) = (if x = 0 then \u22a4 else \u2191(order x)) + if y = 0 then \u22a4 else \u2191(order y)\n[PROOFSTEP]\nrw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), \u2190 WithTop.coe_add, WithTop.coe_eq_coe, order_mul hx hy]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\ng : \u0393\nh : coeff x g \u2260 0\n\u22a2 \u2191(addVal \u0393 R) x \u2264 \u2191g\n[PROOFSTEP]\nrw [addVal_apply_of_ne (ne_zero_of_coeff_ne_zero h), WithTop.coe_le_coe]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\ng : \u0393\nh : coeff x g \u2260 0\n\u22a2 order x \u2264 g\n[PROOFSTEP]\nexact order_le_of_coeff_ne_zero h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\n[PROOFSTEP]\napply (x.isWf_support.isPwo.addSubmonoid_closure _).mono _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u2200 (x_1 : \u0393), x_1 \u2208 support x \u2192 0 \u2264 x_1\n[PROOFSTEP]\nexact fun g hg => WithTop.coe_le_coe.1 (le_trans (le_of_lt hx) (addVal_le_of_coeff_ne_zero hg))\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u22c3 (n : \u2115), support (x ^ n) \u2286 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nrefine' Set.iUnion_subset fun n => _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\n\u22a2 support (x ^ n) \u2286 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 support (x ^ Nat.zero) \u2286 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nintro g hn\n[GOAL]\ncase succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\nih : support (x ^ n) \u2286 \u2191(AddSubmonoid.closure (support x))\n\u22a2 support (x ^ Nat.succ n) \u2286 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nintro g hn\n[GOAL]\ncase zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhn : g \u2208 support (x ^ Nat.zero)\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nsimp only [Nat.zero_eq, pow_zero, support_one, Set.mem_singleton_iff] at hn \n[GOAL]\ncase zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nrw [hn, SetLike.mem_coe]\n[GOAL]\ncase zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 0 \u2208 AddSubmonoid.closure (support x)\n[PROOFSTEP]\nexact AddSubmonoid.zero_mem _\n[GOAL]\ncase succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\nih : support (x ^ n) \u2286 \u2191(AddSubmonoid.closure (support x))\ng : \u0393\nhn : g \u2208 support (x ^ Nat.succ n)\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nobtain \u27e8i, j, hi, hj, rfl\u27e9 := support_mul_subset_add_support hn\n[GOAL]\ncase succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\nih : support (x ^ n) \u2286 \u2191(AddSubmonoid.closure (support x))\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (npowRec n x)\nhn : (fun x x_1 => x + x_1) i j \u2208 support (x ^ Nat.succ n)\n\u22a2 (fun x x_1 => x + x_1) i j \u2208 \u2191(AddSubmonoid.closure (support x))\n[PROOFSTEP]\nexact SetLike.mem_coe.2 (AddSubmonoid.add_mem _ (AddSubmonoid.subset_closure hi) (ih hj))\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\n\u22a2 \u22c3 (a : \u03b1), support ((\u2191x + \u2191y) a) \u2286 (\u22c3 (a : \u03b1), support (\u2191x a)) \u222a \u22c3 (a : \u03b1), support (\u2191y a)\n[PROOFSTEP]\nrw [\u2190 Set.iUnion_union_distrib]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\n\u22a2 \u22c3 (a : \u03b1), support ((\u2191x + \u2191y) a) \u2286 \u22c3 (i : \u03b1), support (\u2191x i) \u222a support (\u2191y i)\n[PROOFSTEP]\nexact Set.iUnion_mono fun a => support_add_subset\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 {a | coeff ((\u2191x + \u2191y) a) g \u2260 0} \u2286\n (Function.support fun a => coeff (\u2191x a) g) \u222a Function.support fun a => coeff (\u2191y a) g\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\nha : a \u2208 {a | coeff ((\u2191x + \u2191y) a) g \u2260 0}\n\u22a2 a \u2208 (Function.support fun a => coeff (\u2191x a) g) \u222a Function.support fun a => coeff (\u2191y a) g\n[PROOFSTEP]\nchange (x a).coeff g + (y a).coeff g \u2260 0 at ha \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\nha : coeff (\u2191x a) g + coeff (\u2191y a) g \u2260 0\n\u22a2 a \u2208 (Function.support fun a => coeff (\u2191x a) g) \u222a Function.support fun a => coeff (\u2191y a) g\n[PROOFSTEP]\nrw [Set.mem_union, Function.mem_support, Function.mem_support]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\nha : coeff (\u2191x a) g + coeff (\u2191y a) g \u2260 0\n\u22a2 coeff (\u2191x a) g \u2260 0 \u2228 coeff (\u2191y a) g \u2260 0\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nx y : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\nha : coeff (\u2191x a) g = 0 \u2227 coeff (\u2191y a) g = 0\n\u22a2 coeff (\u2191x a) g + coeff (\u2191y a) g = 0\n[PROOFSTEP]\nrw [ha.1, ha.2, add_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b1), support (OfNat.ofNat 0 a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u22a2 \u2200 (g : \u0393), Set.Finite {a | coeff (OfNat.ofNat 0 a) g \u2260 0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nr s t : SummableFamily \u0393 R \u03b1\n\u22a2 r + s + t = r + (s + t)\n[PROOFSTEP]\next\n[GOAL]\ncase h.coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nr s t : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nx\u271d : \u0393\n\u22a2 coeff (\u2191(r + s + t) a\u271d) x\u271d = coeff (\u2191(r + (s + t)) a\u271d) x\u271d\n[PROOFSTEP]\napply add_assoc\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\n\u22a2 0 + s = s\n[PROOFSTEP]\next\n[GOAL]\ncase h.coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nx\u271d : \u0393\n\u22a2 coeff (\u2191(0 + s) a\u271d) x\u271d = coeff (\u2191s a\u271d) x\u271d\n[PROOFSTEP]\napply zero_add\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\n\u22a2 s + 0 = s\n[PROOFSTEP]\next\n[GOAL]\ncase h.coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nx\u271d : \u0393\n\u22a2 coeff (\u2191(s + 0) a\u271d) x\u271d = coeff (\u2191s a\u271d) x\u271d\n[PROOFSTEP]\napply add_zero\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\n\u22a2 s + t = t + s\n[PROOFSTEP]\next\n[GOAL]\ncase h.coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nx\u271d : \u0393\n\u22a2 coeff (\u2191(s + t) a\u271d) x\u271d = coeff (\u2191(t + s) a\u271d) x\u271d\n[PROOFSTEP]\napply add_comm\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 (g \u2208 Function.support fun g => \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g) \u2192 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\ncontrapose\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u00acg \u2208 \u22c3 (a : \u03b1), support (\u2191s a) \u2192 \u00acg \u2208 Function.support fun g => \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g\n[PROOFSTEP]\nrw [Set.mem_iUnion, not_exists, Function.mem_support, Classical.not_not]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 (\u2200 (x : \u03b1), \u00acg \u2208 support (\u2191s x)) \u2192 \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g = 0\n[PROOFSTEP]\nsimp_rw [mem_support, Classical.not_not]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 (\u2200 (x : \u03b1), coeff (\u2191s x) g = 0) \u2192 \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g = 0\n[PROOFSTEP]\nintro h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nh : \u2200 (x : \u03b1), coeff (\u2191s x) g = 0\n\u22a2 \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g = 0\n[PROOFSTEP]\nrw [finsum_congr h, finsum_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nhg : g \u2208 support (hsum s)\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nrw [mem_support, hsum_coeff, finsum_eq_sum _ (s.finite_co_support _)] at hg \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nhg : \u2211 i in Set.Finite.toFinset (_ : Set.Finite (Function.support fun a => coeff (\u2191s a) g)), coeff (\u2191s i) g \u2260 0\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nobtain \u27e8a, _, h2\u27e9 := exists_ne_zero_of_sum_ne_zero hg\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nhg : \u2211 i in Set.Finite.toFinset (_ : Set.Finite (Function.support fun a => coeff (\u2191s a) g)), coeff (\u2191s i) g \u2260 0\na : \u03b1\nleft\u271d : a \u2208 Set.Finite.toFinset (_ : Set.Finite (Function.support fun a => coeff (\u2191s a) g))\nh2 : coeff (\u2191s a) g \u2260 0\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nrw [Set.mem_iUnion]\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nhg : \u2211 i in Set.Finite.toFinset (_ : Set.Finite (Function.support fun a => coeff (\u2191s a) g)), coeff (\u2191s i) g \u2260 0\na : \u03b1\nleft\u271d : a \u2208 Set.Finite.toFinset (_ : Set.Finite (Function.support fun a => coeff (\u2191s a) g))\nh2 : coeff (\u2191s a) g \u2260 0\n\u22a2 \u2203 i, g \u2208 support (\u2191s i)\n[PROOFSTEP]\nexact \u27e8a, h2\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\n\u22a2 hsum (s + t) = hsum s + hsum t\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 coeff (hsum (s + t)) g = coeff (hsum s + hsum t) g\n[PROOFSTEP]\nsimp only [hsum_coeff, add_coeff, add_apply]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2211\u1da0 (i : \u03b1), (coeff (\u2191s i) g + coeff (\u2191t i) g) = \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g + \u2211\u1da0 (i : \u03b1), coeff (\u2191t i) g\n[PROOFSTEP]\nexact finsum_add_distrib (s.finite_co_support _) (t.finite_co_support _)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns\u271d t : SummableFamily \u0393 R \u03b1\na : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\ns : SummableFamily \u0393 R \u03b1\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b1), support ((fun a => -\u2191s a) a))\n[PROOFSTEP]\nsimp_rw [support_neg]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns\u271d t : SummableFamily \u0393 R \u03b1\na : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\ns : SummableFamily \u0393 R \u03b1\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))\n[PROOFSTEP]\nexact s.isPwo_iUnion_support'\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns\u271d t : SummableFamily \u0393 R \u03b1\na : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 Set.Finite {a | coeff ((fun a => -\u2191s a) a) g \u2260 0}\n[PROOFSTEP]\nsimp only [neg_coeff', Pi.neg_apply, Ne.def, neg_eq_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns\u271d t : SummableFamily \u0393 R \u03b1\na : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 Set.Finite {a | \u00accoeff (\u2191s a) g = 0}\n[PROOFSTEP]\nexact s.finite_co_support g\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\na : SummableFamily \u0393 R \u03b1\n\u22a2 -a + a = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h.coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommGroup R\n\u03b1 : Type u_3\ns t : SummableFamily \u0393 R \u03b1\na\u271d\u00b9 : \u03b1\nsrc\u271d : AddCommMonoid (SummableFamily \u0393 R \u03b1) := inferInstanceAs (AddCommMonoid (SummableFamily \u0393 R \u03b1))\na : SummableFamily \u0393 R \u03b1\na\u271d : \u03b1\nx\u271d : \u0393\n\u22a2 coeff (\u2191(-a + a) a\u271d) x\u271d = coeff (\u21910 a\u271d) x\u271d\n[PROOFSTEP]\napply add_left_neg\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b1), support ((fun a => x * \u2191s a) a))\n[PROOFSTEP]\napply (x.isPwo_support.add s.isPwo_iUnion_support).mono\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\n\u22a2 \u22c3 (a : \u03b1), support ((fun a => x * \u2191s a) a) \u2286 support x + \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nrefine' Set.Subset.trans (Set.iUnion_mono fun a => support_mul_subset_add_support) _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\n\u22a2 \u22c3 (i : \u03b1), support x + support (\u2191s i) \u2286 support x + \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nintro g\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 g \u2208 \u22c3 (i : \u03b1), support x + support (\u2191s i) \u2192 g \u2208 support x + \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, exists_imp]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2200 (x_1 : \u03b1), g \u2208 support x + support (\u2191s x_1) \u2192 g \u2208 support x + \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nexact fun a ha => (Set.add_subset_add (Set.Subset.refl _) (Set.subset_iUnion _ a)) ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 Set.Finite {a | coeff ((fun a => x * \u2191s a) a) g \u2260 0}\n[PROOFSTEP]\nrefine'\n ((addAntidiagonal x.isPwo_support s.isPwo_iUnion_support g).finite_toSet.biUnion' fun ij _ => _).subset fun a ha => _\n[GOAL]\ncase refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 (i : \u0393 \u00d7 \u0393) \u2192 i \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g) \u2192 Set \u03b1\n[PROOFSTEP]\nexact fun ij _ => Function.support fun a => (s a).coeff ij.2\n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\nij : \u0393 \u00d7 \u0393\nx\u271d : ij \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g)\n\u22a2 Set.Finite (Function.support fun a => coeff (\u2191s a) ij.snd)\n[PROOFSTEP]\napply s.finite_co_support\n[GOAL]\ncase refine'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\nha : a \u2208 {a | coeff ((fun a => x * \u2191s a) a) g \u2260 0}\n\u22a2 a \u2208\n \u22c3 (i : \u0393 \u00d7 \u0393) (_ :\n i \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g)),\n Function.support fun a => coeff (\u2191s a) i.snd\n[PROOFSTEP]\nobtain \u27e8i, j, hi, hj, rfl\u27e9 := support_mul_subset_add_support ha\n[GOAL]\ncase refine'_3.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\na : \u03b1\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (\u2191s a)\nha : a \u2208 {a | coeff ((fun a => x * \u2191s a) a) ((fun x x_1 => x + x_1) i j) \u2260 0}\n\u22a2 a \u2208\n \u22c3 (i_1 : \u0393 \u00d7 \u0393) (_ :\n i_1 \u2208\n \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a)))\n ((fun x x_1 => x + x_1) i j))),\n Function.support fun a => coeff (\u2191s a) i_1.snd\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion, mem_addAntidiagonal, mul_coeff, mem_support, isPwo_support, Prod.exists]\n[GOAL]\ncase refine'_3.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\na : \u03b1\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (\u2191s a)\nha : a \u2208 {a | coeff ((fun a => x * \u2191s a) a) ((fun x x_1 => x + x_1) i j) \u2260 0}\n\u22a2 \u2203 a_1 b,\n (a_1, b) \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) (i + j)) \u2227\n a \u2208 Function.support fun a => coeff (\u2191s a) b\n[PROOFSTEP]\nexact \u27e8i, j, mem_coe.2 (mem_addAntidiagonal.2 \u27e8hi, Set.mem_iUnion.2 \u27e8a, hj\u27e9, rfl\u27e9), hj\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\n\u22a2 hsum (x \u2022 s) = x * hsum s\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 coeff (hsum (x \u2022 s)) g = coeff (x * hsum s) g\n[PROOFSTEP]\nsimp only [mul_coeff, hsum_coeff, smul_apply]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2211\u1da0 (i : \u03b1),\n \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (\u2191s i))) g,\n coeff x ij.fst * coeff (\u2191s i) ij.snd =\n \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (hsum s))) g,\n coeff x x_1.fst * \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) x_1.snd\n[PROOFSTEP]\nrefine'\n (Eq.trans (finsum_congr fun a => _)\n (finsum_sum_comm (addAntidiagonal x.isPwo_support s.isPwo_iUnion_support g)\n (fun i ij => x.coeff (Prod.fst ij) * (s i).coeff ij.snd) _)).trans\n _\n[GOAL]\ncase coeff.h.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\n\u22a2 \u2211 ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (\u2191s a))) g,\n coeff x ij.fst * coeff (\u2191s a) ij.snd =\n \u2211 b in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g,\n coeff x b.fst * coeff (\u2191s a) b.snd\n[PROOFSTEP]\nrefine' sum_subset (addAntidiagonal_mono_right (Set.subset_iUnion (fun j => support (toFun s j)) a)) _\n[GOAL]\ncase coeff.h.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g \u2192\n \u00acx_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (\u2191s a))) g \u2192\n coeff x x_1.fst * coeff (\u2191s a) x_1.snd = 0\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hU ha\n[GOAL]\ncase coeff.h.refine'_1.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\ni j : \u0393\nhU : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\nha : \u00ac(i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (\u2191s a))) g\n\u22a2 coeff x (i, j).fst * coeff (\u2191s a) (i, j).snd = 0\n[PROOFSTEP]\nrw [mem_addAntidiagonal] at *\n[GOAL]\ncase coeff.h.refine'_1.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\na : \u03b1\ni j : \u0393\nhU : (i, j).fst \u2208 support x \u2227 (i, j).snd \u2208 \u22c3 (a : \u03b1), support (\u2191s a) \u2227 (i, j).fst + (i, j).snd = g\nha : \u00ac((i, j).fst \u2208 support x \u2227 (i, j).snd \u2208 support (\u2191s a) \u2227 (i, j).fst + (i, j).snd = g)\n\u22a2 coeff x (i, j).fst * coeff (\u2191s a) (i, j).snd = 0\n[PROOFSTEP]\nrw [Classical.not_not.1 fun con => ha \u27e8hU.1, con, hU.2.2\u27e9, mul_zero]\n[GOAL]\ncase coeff.h.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2200 (b : \u0393 \u00d7 \u0393),\n b \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g \u2192\n Set.Finite (Function.support fun a => (fun i ij => coeff x ij.fst * coeff (\u2191s i) ij.snd) a b)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 _\n[GOAL]\ncase coeff.h.refine'_2.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\n\u22a2 Set.Finite (Function.support fun a => (fun i ij => coeff x ij.fst * coeff (\u2191s i) ij.snd) a (i, j))\n[PROOFSTEP]\nrefine' (s.finite_co_support j).subset _\n[GOAL]\ncase coeff.h.refine'_2.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\n\u22a2 (Function.support fun a => (fun i ij => coeff x ij.fst * coeff (\u2191s i) ij.snd) a (i, j)) \u2286\n Function.support fun a => coeff (\u2191s a) j\n[PROOFSTEP]\nsimp_rw [Function.support_subset_iff', Function.mem_support, Classical.not_not]\n[GOAL]\ncase coeff.h.refine'_2.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\n\u22a2 \u2200 (x_1 : \u03b1), coeff (\u2191s x_1) j = 0 \u2192 coeff x i * coeff (\u2191s x_1) j = 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase coeff.h.refine'_2.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\na : \u03b1\nha : coeff (\u2191s a) j = 0\n\u22a2 coeff x i * coeff (\u2191s a) j = 0\n[PROOFSTEP]\nrw [ha, mul_zero]\n[GOAL]\ncase coeff.h.refine'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2211 b in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g,\n \u2211\u1da0 (a : \u03b1), coeff x b.fst * coeff (\u2191s a) b.snd =\n \u2211 x_1 in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (hsum s))) g,\n coeff x x_1.fst * \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) x_1.snd\n[PROOFSTEP]\nrefine' (sum_congr rfl _).trans (sum_subset (addAntidiagonal_mono_right _) _).symm\n[GOAL]\ncase coeff.h.refine'_3.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g \u2192\n \u2211\u1da0 (a : \u03b1), coeff x x_1.fst * coeff (\u2191s a) x_1.snd = coeff x x_1.fst * \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) x_1.snd\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 _\n[GOAL]\ncase coeff.h.refine'_3.refine'_1.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\n\u22a2 \u2211\u1da0 (a : \u03b1), coeff x (i, j).fst * coeff (\u2191s a) (i, j).snd =\n coeff x (i, j).fst * \u2211\u1da0 (i_1 : \u03b1), coeff (\u2191s i_1) (i, j).snd\n[PROOFSTEP]\nrw [mul_finsum]\n[GOAL]\ncase coeff.h.refine'_3.refine'_1.mk.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\na\u271d : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\n\u22a2 Set.Finite (Function.support fun i_1 => coeff (\u2191s i_1) (i, j).snd)\n[PROOFSTEP]\napply s.finite_co_support\n[GOAL]\ncase coeff.h.refine'_3.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 support (hsum s) \u2286 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase coeff.h.refine'_3.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx\u271d : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng x : \u0393\nhx : x \u2208 support (hsum s)\n\u22a2 x \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Ne.def, mem_support]\n[GOAL]\ncase coeff.h.refine'_3.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx\u271d : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng x : \u0393\nhx : x \u2208 support (hsum s)\n\u22a2 \u2203 i, \u00accoeff (\u2191s i) x = 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase coeff.h.refine'_3.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx\u271d : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng x : \u0393\nhx : \u2200 (i : \u03b1), coeff (\u2191s i) x = 0\n\u22a2 \u00acx \u2208 support (hsum s)\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase coeff.h.refine'_3.refine'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng : \u0393\n\u22a2 \u2200 (x_1 : \u0393 \u00d7 \u0393),\n x_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g \u2192\n \u00acx_1 \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (hsum s))) g \u2192\n coeff x x_1.fst * \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) x_1.snd = 0\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hU ha\n[GOAL]\ncase coeff.h.refine'_3.refine'_3.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\nhU : (i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191s a))) g\nha : \u00ac(i, j) \u2208 addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support (hsum s))) g\n\u22a2 coeff x (i, j).fst * \u2211\u1da0 (i_1 : \u03b1), coeff (\u2191s i_1) (i, j).snd = 0\n[PROOFSTEP]\nrw [mem_addAntidiagonal] at *\n[GOAL]\ncase coeff.h.refine'_3.refine'_3.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : Semiring R\n\u03b1 : Type u_3\nx : HahnSeries \u0393 R\ns : SummableFamily \u0393 R \u03b1\ng i j : \u0393\nhU : (i, j).fst \u2208 support x \u2227 (i, j).snd \u2208 \u22c3 (a : \u03b1), support (\u2191s a) \u2227 (i, j).fst + (i, j).snd = g\nha : \u00ac((i, j).fst \u2208 support x \u2227 (i, j).snd \u2208 support (hsum s) \u2227 (i, j).fst + (i, j).snd = g)\n\u22a2 coeff x (i, j).fst * \u2211\u1da0 (i_1 : \u03b1), coeff (\u2191s i_1) (i, j).snd = 0\n[PROOFSTEP]\nrw [\u2190 hsum_coeff, Classical.not_not.1 fun con => ha \u27e8hU.1, con, hU.2.2\u27e9, mul_zero]\n[GOAL]\n\u0393 : Type u_1\nR\u271d : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Semiring R\u271d\n\u03b1 : Type u_3\nR : Type u_4\ninst\u271d : Ring R\ns t : SummableFamily \u0393 R \u03b1\n\u22a2 hsum (s - t) = hsum s - hsum t\n[PROOFSTEP]\nrw [\u2190 lsum_apply, LinearMap.map_sub, lsum_apply, lsum_apply]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b1), support (\u2191f a))\n[PROOFSTEP]\napply (f.support.isPwo_bUnion.2 fun a _ => (f a).isPwo_support).mono\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\n\u22a2 \u22c3 (a : \u03b1), support (\u2191f a) \u2286 \u22c3 (i : \u03b1) (_ : i \u2208 f.support), support (\u2191f i)\n[PROOFSTEP]\nrefine' Set.iUnion_subset_iff.2 fun a g hg => _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\na : \u03b1\ng : \u0393\nhg : g \u2208 support (\u2191f a)\n\u22a2 g \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 f.support), support (\u2191f i)\n[PROOFSTEP]\nhave haf : a \u2208 f.support := by\n rw [Finsupp.mem_support_iff, \u2190 support_nonempty_iff]\n exact \u27e8g, hg\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\na : \u03b1\ng : \u0393\nhg : g \u2208 support (\u2191f a)\n\u22a2 a \u2208 f.support\n[PROOFSTEP]\nrw [Finsupp.mem_support_iff, \u2190 support_nonempty_iff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\na : \u03b1\ng : \u0393\nhg : g \u2208 support (\u2191f a)\n\u22a2 Set.Nonempty (support (\u2191f a))\n[PROOFSTEP]\nexact \u27e8g, hg\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\na : \u03b1\ng : \u0393\nhg : g \u2208 support (\u2191f a)\nhaf : a \u2208 f.support\n\u22a2 g \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 f.support), support (\u2191f i)\n[PROOFSTEP]\nexact Set.mem_biUnion haf hg\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\n\u22a2 Set.Finite {a | coeff (\u2191f a) g \u2260 0}\n[PROOFSTEP]\nrefine' f.support.finite_toSet.subset fun a ha => _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\na : \u03b1\nha : a \u2208 {a | coeff (\u2191f a) g \u2260 0}\n\u22a2 a \u2208 \u2191f.support\n[PROOFSTEP]\nsimp only [coeff.addMonoidHom_apply, mem_coe, Finsupp.mem_support_iff, Ne.def, Function.mem_support]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\na : \u03b1\nha : a \u2208 {a | coeff (\u2191f a) g \u2260 0}\n\u22a2 \u00ac\u2191f a = 0\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\na : \u03b1\nha : \u2191f a = 0\n\u22a2 \u00aca \u2208 {a | coeff (\u2191f a) g \u2260 0}\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\n\u22a2 hsum (ofFinsupp f) = Finsupp.sum f fun x => id\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\n\u22a2 coeff (hsum (ofFinsupp f)) g = coeff (Finsupp.sum f fun x => id) g\n[PROOFSTEP]\nsimp only [hsum_coeff, coe_ofFinsupp, Finsupp.sum, Ne.def]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2211\u1da0 (i : \u03b1), coeff (\u2191f i) g = coeff (\u2211 x in f.support, id (\u2191f x)) g\n[PROOFSTEP]\nsimp_rw [\u2190 coeff.addMonoidHom_apply, id.def]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\n\u22a2 \u2211\u1da0 (i : \u03b1), \u2191(coeff.addMonoidHom g) (\u2191f i) = \u2191(coeff.addMonoidHom g) (\u2211 x in f.support, \u2191f x)\n[PROOFSTEP]\nrw [map_sum, finsum_eq_sum_of_support_subset]\n[GOAL]\ncase coeff.h.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\n\u22a2 (Function.support fun i => \u2191(coeff.addMonoidHom g) (\u2191f i)) \u2286 \u2191f.support\n[PROOFSTEP]\nintro x h\n[GOAL]\ncase coeff.h.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\nx : \u03b1\nh : x \u2208 Function.support fun i => \u2191(coeff.addMonoidHom g) (\u2191f i)\n\u22a2 x \u2208 \u2191f.support\n[PROOFSTEP]\nsimp only [coeff.addMonoidHom_apply, mem_coe, Finsupp.mem_support_iff, Ne.def]\n[GOAL]\ncase coeff.h.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\nx : \u03b1\nh : x \u2208 Function.support fun i => \u2191(coeff.addMonoidHom g) (\u2191f i)\n\u22a2 \u00ac\u2191f x = 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase coeff.h.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\nf : \u03b1 \u2192\u2080 HahnSeries \u0393 R\ng : \u0393\nx : \u03b1\nh : \u2191f x = 0\n\u22a2 \u00acx \u2208 Function.support fun i => \u2191(coeff.addMonoidHom g) (\u2191f i)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\n\u22a2 Set.IsPwo (\u22c3 (a : \u03b2), support ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) a))\n[PROOFSTEP]\nrefine' s.isPwo_iUnion_support.mono (Set.iUnion_subset fun b g h => _)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nh : g \u2208 support ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) b)\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nby_cases hb : b \u2208 Set.range f\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nh : g \u2208 support ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) b)\nhb : b \u2208 Set.range \u2191f\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\ndsimp only at h \n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nh : g \u2208 support (if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0)\nhb : b \u2208 Set.range \u2191f\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nrw [dif_pos hb] at h \n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nhb : b \u2208 Set.range \u2191f\nh : g \u2208 support (\u2191s (Classical.choose hb))\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\nexact Set.mem_iUnion.2 \u27e8Classical.choose hb, h\u27e9\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nh : g \u2208 support ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) b)\nhb : \u00acb \u2208 Set.range \u2191f\n\u22a2 g \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nhb : \u00acb \u2208 Set.range \u2191f\nh : \u00acg \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n\u22a2 \u00acg \u2208 support (if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose (_ : b \u2208 Set.range \u2191f)) else 0)\n[PROOFSTEP]\nrw [dif_neg hb]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\nb : \u03b2\ng : \u0393\nhb : \u00acb \u2208 Set.range \u2191f\nh : \u00acg \u2208 \u22c3 (a : \u03b1), support (\u2191s a)\n\u22a2 \u00acg \u2208 support 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\n\u22a2 {a | coeff ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) a) g \u2260 0} \u2286\n \u2191f '' Function.support fun a => coeff (\u2191s a) g\n[PROOFSTEP]\nintro b h\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\nb : \u03b2\nh : b \u2208 {a | coeff ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) a) g \u2260 0}\n\u22a2 b \u2208 \u2191f '' Function.support fun a => coeff (\u2191s a) g\n[PROOFSTEP]\nby_cases hb : b \u2208 Set.range f\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\nb : \u03b2\nh : b \u2208 {a | coeff ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) a) g \u2260 0}\nhb : b \u2208 Set.range \u2191f\n\u22a2 b \u2208 \u2191f '' Function.support fun a => coeff (\u2191s a) g\n[PROOFSTEP]\nsimp only [Ne.def, Set.mem_setOf_eq, dif_pos hb] at h \n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\nb : \u03b2\nhb : b \u2208 Set.range \u2191f\nh : \u00accoeff (\u2191s (Classical.choose (_ : b \u2208 Set.range \u2191f))) g = 0\n\u22a2 b \u2208 \u2191f '' Function.support fun a => coeff (\u2191s a) g\n[PROOFSTEP]\nexact \u27e8Classical.choose hb, h, Classical.choose_spec hb\u27e9\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\nb : \u03b2\nh : b \u2208 {a | coeff ((fun b => if h : b \u2208 Set.range \u2191f then \u2191s (Classical.choose h) else 0) a) g \u2260 0}\nhb : \u00acb \u2208 Set.range \u2191f\n\u22a2 b \u2208 \u2191f '' Function.support fun a => coeff (\u2191s a) g\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\ng : \u0393\nb : \u03b2\nhb : \u00acb \u2208 Set.range \u2191f\nh : \u00acb \u2208 (fun a => \u2191f a) '' Function.support fun a => coeff (\u2191s a) g\n\u22a2 \u00acb \u2208 {a | coeff (if h : a \u2208 Set.range \u2191f then \u2191s (Classical.choose (_ : a \u2208 Set.range \u2191f)) else 0) g \u2260 0}\n[PROOFSTEP]\nsimp only [Ne.def, Set.mem_setOf_eq, dif_neg hb, Classical.not_not, zero_coeff]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2191(embDomain s f) (\u2191f a) = \u2191s a\n[PROOFSTEP]\nrw [embDomain_apply, dif_pos (Set.mem_range_self a)]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2191s (Classical.choose (_ : \u2191f a \u2208 Set.range \u2191f)) = \u2191s a\n[PROOFSTEP]\nexact congr rfl (f.injective (Classical.choose_spec (Set.mem_range_self a)))\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\nh : \u00acb \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain s f) b = 0\n[PROOFSTEP]\nrw [embDomain_apply, dif_neg h]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 hsum (embDomain s f) = hsum s\n[PROOFSTEP]\next g\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ng : \u0393\n\u22a2 coeff (hsum (embDomain s f)) g = coeff (hsum s) g\n[PROOFSTEP]\nsimp only [hsum_coeff, embDomain_apply, apply_dite HahnSeries.coeff, dite_apply, zero_coeff]\n[GOAL]\ncase coeff.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : AddCommMonoid R\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ns : SummableFamily \u0393 R \u03b1\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ng : \u0393\n\u22a2 (\u2211\u1da0 (i : \u03b2), if h : i \u2208 Set.range \u2191f then coeff (\u2191s (Classical.choose (_ : i \u2208 Set.range \u2191f))) g else 0) =\n \u2211\u1da0 (i : \u03b1), coeff (\u2191s i) g\n[PROOFSTEP]\nexact finsum_emb_domain f fun a => (s a).coeff g\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\n[PROOFSTEP]\nhave hpwo := isPwo_iUnion_support_powers hx\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\n[PROOFSTEP]\nby_cases hg : g \u2208 \u22c3 n : \u2115, {g | (x ^ n).coeff g \u2260 0}\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : \u00acg \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : \u00acg \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\n[PROOFSTEP]\nexact Set.finite_empty.subset fun n hn => hg (Set.mem_iUnion.2 \u27e8n, hn\u27e9)\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) g \u2260 0}\n[PROOFSTEP]\napply hpwo.isWf.induction hg\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\n\u22a2 \u2200 (y : \u0393),\n y \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192\n (\u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}) \u2192\n Set.Finite {a | coeff ((fun n => x ^ n) a) y \u2260 0}\n[PROOFSTEP]\nintro y ys hy\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\n\u22a2 Set.Finite {a | coeff ((fun n => x ^ n) a) y \u2260 0}\n[PROOFSTEP]\nrefine'\n ((((addAntidiagonal x.isPwo_support hpwo y).finite_toSet.biUnion fun ij hij => hy ij.snd _ _).image Nat.succ).union\n (Set.finite_singleton 0)).subset\n _\n[GOAL]\ncase pos.refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nij : \u0393 \u00d7 \u0393\nhij : ij \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo y)\n\u22a2 ij.snd \u2208 \u22c3 (n : \u2115), support (x ^ n)\n[PROOFSTEP]\nexact (mem_addAntidiagonal.1 (mem_coe.1 hij)).2.1\n[GOAL]\ncase pos.refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nij : \u0393 \u00d7 \u0393\nhij : ij \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo y)\n\u22a2 ij.snd < y\n[PROOFSTEP]\nobtain \u27e8hi, _, rfl\u27e9 := mem_addAntidiagonal.1 (mem_coe.1 hij)\n[GOAL]\ncase pos.refine'_2.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\nij : \u0393 \u00d7 \u0393\nhi : ij.fst \u2208 support x\nleft\u271d : ij.snd \u2208 \u22c3 (n : \u2115), support (x ^ n)\nys : ij.fst + ij.snd \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < ij.fst + ij.snd \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhij : ij \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo (ij.fst + ij.snd))\n\u22a2 ij.snd < ij.fst + ij.snd\n[PROOFSTEP]\nrw [\u2190 zero_add ij.snd, \u2190 add_assoc, add_zero]\n[GOAL]\ncase pos.refine'_2.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\nij : \u0393 \u00d7 \u0393\nhi : ij.fst \u2208 support x\nleft\u271d : ij.snd \u2208 \u22c3 (n : \u2115), support (x ^ n)\nys : ij.fst + ij.snd \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < ij.fst + ij.snd \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhij : ij \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo (ij.fst + ij.snd))\n\u22a2 0 + ij.snd < ij.fst + ij.snd\n[PROOFSTEP]\nexact add_lt_add_right (WithTop.coe_lt_coe.1 (lt_of_lt_of_le hx (addVal_le_of_coeff_ne_zero hi))) _\n[GOAL]\ncase pos.refine'_3\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\n\u22a2 {a | coeff ((fun n => x ^ n) a) y \u2260 0} \u2286\n (Nat.succ ''\n \u22c3 (i : \u0393 \u00d7 \u0393) (_ : i \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo y)),\n {a | coeff ((fun n => x ^ n) a) i.snd \u2260 0}) \u222a\n {0}\n[PROOFSTEP]\nrintro (_ | n) hn\n[GOAL]\ncase pos.refine'_3.zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhn : Nat.zero \u2208 {a | coeff ((fun n => x ^ n) a) y \u2260 0}\n\u22a2 Nat.zero \u2208\n (Nat.succ ''\n \u22c3 (i : \u0393 \u00d7 \u0393) (_ : i \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo y)),\n {a | coeff ((fun n => x ^ n) a) i.snd \u2260 0}) \u222a\n {0}\n[PROOFSTEP]\nexact Set.mem_union_right _ (Set.mem_singleton 0)\n[GOAL]\ncase pos.refine'_3.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\ny : \u0393\nys : y \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy : \u2200 (z : \u0393), z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < y \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nn : \u2115\nhn : Nat.succ n \u2208 {a | coeff ((fun n => x ^ n) a) y \u2260 0}\n\u22a2 Nat.succ n \u2208\n (Nat.succ ''\n \u22c3 (i : \u0393 \u00d7 \u0393) (_ : i \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo y)),\n {a | coeff ((fun n => x ^ n) a) i.snd \u2260 0}) \u222a\n {0}\n[PROOFSTEP]\nobtain \u27e8i, j, hi, hj, rfl\u27e9 := support_mul_subset_add_support hn\n[GOAL]\ncase pos.refine'_3.succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\nn : \u2115\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (npowRec n x)\nys : (fun x x_1 => x + x_1) i j \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy :\n \u2200 (z : \u0393),\n z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < (fun x x_1 => x + x_1) i j \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhn : Nat.succ n \u2208 {a | coeff ((fun n => x ^ n) a) ((fun x x_1 => x + x_1) i j) \u2260 0}\n\u22a2 Nat.succ n \u2208\n (Nat.succ ''\n \u22c3 (i_1 : \u0393 \u00d7 \u0393) (_ : i_1 \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo ((fun x x_1 => x + x_1) i j))),\n {a | coeff ((fun n => x ^ n) a) i_1.snd \u2260 0}) \u222a\n {0}\n[PROOFSTEP]\nrefine' Set.mem_union_left _ \u27e8n, Set.mem_iUnion.2 \u27e8\u27e8i, j\u27e9, Set.mem_iUnion.2 \u27e8_, hj\u27e9\u27e9, rfl\u27e9\n[GOAL]\ncase pos.refine'_3.succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\nn : \u2115\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (npowRec n x)\nys : (fun x x_1 => x + x_1) i j \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy :\n \u2200 (z : \u0393),\n z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < (fun x x_1 => x + x_1) i j \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhn : Nat.succ n \u2208 {a | coeff ((fun n => x ^ n) a) ((fun x x_1 => x + x_1) i j) \u2260 0}\n\u22a2 (i, j) \u2208 \u2191(addAntidiagonal (_ : Set.IsPwo (support x)) hpwo ((fun x x_1 => x + x_1) i j))\n[PROOFSTEP]\nsimp only [and_true_iff, Set.mem_iUnion, mem_addAntidiagonal, mem_coe, eq_self_iff_true, Ne.def, mem_support,\n Set.mem_setOf_eq]\n[GOAL]\ncase pos.refine'_3.succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\ng : \u0393\nhpwo : Set.IsPwo (\u22c3 (n : \u2115), support (x ^ n))\nhg : g \u2208 \u22c3 (n : \u2115), {g | coeff (x ^ n) g \u2260 0}\nn : \u2115\ni j : \u0393\nhi : i \u2208 support x\nhj : j \u2208 support (npowRec n x)\nys : (fun x x_1 => x + x_1) i j \u2208 \u22c3 (n : \u2115), support (x ^ n)\nhy :\n \u2200 (z : \u0393),\n z \u2208 \u22c3 (n : \u2115), support (x ^ n) \u2192 z < (fun x x_1 => x + x_1) i j \u2192 Set.Finite {a | coeff ((fun n => x ^ n) a) z \u2260 0}\nhn : Nat.succ n \u2208 {a | coeff ((fun n => x ^ n) a) ((fun x x_1 => x + x_1) i j) \u2260 0}\n\u22a2 \u00accoeff x i = 0 \u2227 \u2203 i, \u00accoeff (x ^ i) j = 0\n[PROOFSTEP]\nexact \u27e8hi, n, hj\u27e9\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 embDomain (x \u2022 powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective } =\n powers x hx - ofFinsupp (Finsupp.single 0 1)\n[PROOFSTEP]\napply SummableFamily.ext\n[GOAL]\ncase h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u2200 (a : \u2115),\n \u2191(embDomain (x \u2022 powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) a =\n \u2191(powers x hx - ofFinsupp (Finsupp.single 0 1)) a\n[PROOFSTEP]\nrintro (_ | n)\n[GOAL]\ncase h.zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u2191(embDomain (x \u2022 powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) Nat.zero =\n \u2191(powers x hx - ofFinsupp (Finsupp.single 0 1)) Nat.zero\n[PROOFSTEP]\nrw [embDomain_notin_range, sub_apply, coe_powers, pow_zero, coe_ofFinsupp, Finsupp.single_eq_same, sub_self]\n[GOAL]\ncase h.zero.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u00acNat.zero \u2208 Set.range \u2191{ toFun := Nat.succ, inj' := Nat.succ_injective }\n[PROOFSTEP]\nrw [Set.mem_range, not_exists]\n[GOAL]\ncase h.zero.h\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 \u2200 (x : \u2115), \u00ac\u2191{ toFun := Nat.succ, inj' := Nat.succ_injective } x = Nat.zero\n[PROOFSTEP]\nexact Nat.succ_ne_zero\n[GOAL]\ncase h.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\n\u22a2 \u2191(embDomain (x \u2022 powers x hx) { toFun := Nat.succ, inj' := Nat.succ_injective }) (Nat.succ n) =\n \u2191(powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)\n[PROOFSTEP]\nrefine' Eq.trans (embDomain_image _ \u27e8Nat.succ, Nat.succ_injective\u27e9) _\n[GOAL]\ncase h.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\n\u22a2 \u2191(x \u2022 powers x hx) n = \u2191(powers x hx - ofFinsupp (Finsupp.single 0 1)) (Nat.succ n)\n[PROOFSTEP]\nsimp only [pow_succ, coe_powers, coe_sub, smul_apply, coe_ofFinsupp, Pi.sub_apply]\n[GOAL]\ncase h.succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\nn : \u2115\n\u22a2 x * x ^ n = x * x ^ n - \u2191(Finsupp.single 0 1) (Nat.succ n)\n[PROOFSTEP]\nrw [Finsupp.single_eq_of_ne n.succ_ne_zero.symm, sub_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 (1 - x) * hsum (powers x hx) = 1\n[PROOFSTEP]\nrw [\u2190 hsum_smul, sub_smul 1 x (powers x hx), one_smul, hsum_sub, \u2190\n hsum_embDomain (x \u2022 powers x hx) \u27e8Nat.succ, Nat.succ_injective\u27e9, embDomain_succ_smul_powers]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < \u2191(addVal \u0393 R) x\n\u22a2 hsum (powers x hx) - hsum (powers x hx - ofFinsupp (Finsupp.single 0 1)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\n\u22a2 0 < \u2191(addVal \u0393 R) (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nhave h10 : (1 : R) \u2260 0 := one_ne_zero\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\n\u22a2 0 < \u2191(addVal \u0393 R) (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nhave x0 : x \u2260 0 := ne_zero_of_coeff_ne_zero (right_ne_zero_of_mul_eq_one hr)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 0 < \u2191(addVal \u0393 R) (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nrefine' lt_of_le_of_ne ((addVal \u0393 R).map_le_sub (ge_of_eq (addVal \u0393 R).map_one) _) _\n[GOAL]\ncase refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 0 \u2264 \u2191(addVal \u0393 R) (\u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nsimp only [AddValuation.map_mul]\n[GOAL]\ncase refine'_1\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 0 \u2264 \u2191(addVal \u0393 R) (\u2191C r) + \u2191(addVal \u0393 R) (\u2191(single (-order x)) 1) + \u2191(addVal \u0393 R) x\n[PROOFSTEP]\nrw [addVal_apply_of_ne x0, addVal_apply_of_ne (single_ne_zero h10), addVal_apply_of_ne _, order_C, order_single h10,\n WithTop.coe_zero, zero_add, \u2190 WithTop.coe_add, neg_add_self, WithTop.coe_zero]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 \u2191C r \u2260 0\n[PROOFSTEP]\nexact C_ne_zero (left_ne_zero_of_mul_eq_one hr)\n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 0 \u2260 \u2191(addVal \u0393 R) (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nrw [addVal_apply, \u2190 WithTop.coe_zero]\n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\n\u22a2 \u21910 \u2260 if 1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0 then \u22a4 else \u2191(order (1 - \u2191C r * \u2191(single (-order x)) 1 * x))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\nh : 1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0\n\u22a2 \u21910 \u2260 \u22a4\n[PROOFSTEP]\napply WithTop.coe_ne_top\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\nh : \u00ac1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0\n\u22a2 \u21910 \u2260 \u2191(order (1 - \u2191C r * \u2191(single (-order x)) 1 * x))\n[PROOFSTEP]\nrw [Ne.def, WithTop.coe_eq_coe]\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\nh : \u00ac1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0\n\u22a2 \u00ac0 = order (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n[PROOFSTEP]\nintro con\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\nh : \u00ac1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0\ncon : 0 = order (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n\u22a2 False\n[PROOFSTEP]\napply coeff_order_ne_zero h\n[GOAL]\ncase neg\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nr : R\nhr : r * coeff x (order x) = 1\nh10 : 1 \u2260 0\nx0 : x \u2260 0\nh : \u00ac1 - \u2191C r * \u2191(single (-order x)) 1 * x = 0\ncon : 0 = order (1 - \u2191C r * \u2191(single (-order x)) 1 * x)\n\u22a2 coeff (1 - \u2191C r * \u2191(single (-order x)) 1 * x) (order (1 - \u2191C r * \u2191(single (-order x)) 1 * x)) = 0\n[PROOFSTEP]\nrw [\u2190 con, mul_assoc, sub_coeff, one_coeff, if_pos rfl, C_mul_eq_smul, smul_coeff, smul_eq_mul, \u2190 add_neg_self x.order,\n single_mul_coeff_add, one_mul, hr, sub_self]\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\n\u22a2 IsUnit x \u2194 IsUnit (coeff x (order x))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\n\u22a2 IsUnit x \u2192 IsUnit (coeff x (order x))\n[PROOFSTEP]\nrintro \u27e8\u27e8u, i, ui, iu\u27e9, rfl\u27e9\n[GOAL]\ncase mp.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nu i : HahnSeries \u0393 R\nui : u * i = 1\niu : i * u = 1\n\u22a2 IsUnit\n (coeff (\u2191{ val := u, inv := i, val_inv := ui, inv_val := iu })\n (order \u2191{ val := u, inv := i, val_inv := ui, inv_val := iu }))\n[PROOFSTEP]\nrefine' isUnit_of_mul_eq_one (u.coeff u.order) (i.coeff i.order) ((mul_coeff_order_add_order u i).symm.trans _)\n[GOAL]\ncase mp.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nu i : HahnSeries \u0393 R\nui : u * i = 1\niu : i * u = 1\n\u22a2 coeff (u * i) (order u + order i) = 1\n[PROOFSTEP]\nrw [ui, one_coeff, if_pos]\n[GOAL]\ncase mp.intro.mk.hc\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nu i : HahnSeries \u0393 R\nui : u * i = 1\niu : i * u = 1\n\u22a2 order u + order i = 0\n[PROOFSTEP]\nrw [\u2190 order_mul (left_ne_zero_of_mul_eq_one ui) (right_ne_zero_of_mul_eq_one ui), ui, order_one]\n[GOAL]\ncase mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\n\u22a2 IsUnit (coeff x (order x)) \u2192 IsUnit x\n[PROOFSTEP]\nrintro \u27e8\u27e8u, i, ui, iu\u27e9, h\u27e9\n[GOAL]\ncase mpr.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nu i : R\nui : u * i = 1\niu : i * u = 1\nh : \u2191{ val := u, inv := i, val_inv := ui, inv_val := iu } = coeff x (order x)\n\u22a2 IsUnit x\n[PROOFSTEP]\nrw [Units.val_mk] at h \n[GOAL]\ncase mpr.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nu i : R\nui : u * i = 1\niu : i * u = 1\nh : u = coeff x (order x)\n\u22a2 IsUnit x\n[PROOFSTEP]\nrw [h] at iu \n[GOAL]\ncase mpr.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nu i : R\nui : u * i = 1\niu : i * coeff x (order x) = 1\nh : u = coeff x (order x)\n\u22a2 IsUnit x\n[PROOFSTEP]\nhave h := SummableFamily.one_sub_self_mul_hsum_powers (unit_aux x iu)\n[GOAL]\ncase mpr.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nu i : R\nui : u * i = 1\niu : i * coeff x (order x) = 1\nh\u271d : u = coeff x (order x)\nh :\n (1 - (1 - \u2191C i * \u2191(single (-order x)) 1 * x)) *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C i * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C i * \u2191(single (-order x)) 1 * x))) =\n 1\n\u22a2 IsUnit x\n[PROOFSTEP]\nrw [sub_sub_cancel] at h \n[GOAL]\ncase mpr.intro.mk\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u0393\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nu i : R\nui : u * i = 1\niu : i * coeff x (order x) = 1\nh\u271d : u = coeff x (order x)\nh :\n \u2191C i * \u2191(single (-order x)) 1 * x *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C i * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C i * \u2191(single (-order x)) 1 * x))) =\n 1\n\u22a2 IsUnit x\n[PROOFSTEP]\nexact isUnit_of_mul_isUnit_right (isUnit_of_mul_eq_one _ _ h)\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u0393\ninst\u271d : Field R\nsrc\u271d\u00b9 : IsDomain (HahnSeries \u0393 R) := inferInstanceAs (IsDomain (HahnSeries \u0393 R))\nsrc\u271d : CommRing (HahnSeries \u0393 R) := inferInstanceAs (CommRing (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx0 : x \u2260 0\n\u22a2 x * x\u207b\u00b9 = 1\n[PROOFSTEP]\nrefine' (congr rfl (dif_neg x0)).trans _\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u0393\ninst\u271d : Field R\nsrc\u271d\u00b9 : IsDomain (HahnSeries \u0393 R) := inferInstanceAs (IsDomain (HahnSeries \u0393 R))\nsrc\u271d : CommRing (HahnSeries \u0393 R) := inferInstanceAs (CommRing (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx0 : x \u2260 0\n\u22a2 x *\n (\u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)))) =\n 1\n[PROOFSTEP]\nhave h := SummableFamily.one_sub_self_mul_hsum_powers (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))\n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u0393\ninst\u271d : Field R\nsrc\u271d\u00b9 : IsDomain (HahnSeries \u0393 R) := inferInstanceAs (IsDomain (HahnSeries \u0393 R))\nsrc\u271d : CommRing (HahnSeries \u0393 R) := inferInstanceAs (CommRing (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx0 : x \u2260 0\nh :\n (1 - (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)) *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x))) =\n 1\n\u22a2 x *\n (\u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)))) =\n 1\n[PROOFSTEP]\nrw [sub_sub_cancel] at h \n[GOAL]\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u0393\ninst\u271d : Field R\nsrc\u271d\u00b9 : IsDomain (HahnSeries \u0393 R) := inferInstanceAs (IsDomain (HahnSeries \u0393 R))\nsrc\u271d : CommRing (HahnSeries \u0393 R) := inferInstanceAs (CommRing (HahnSeries \u0393 R))\nx : HahnSeries \u0393 R\nx0 : x \u2260 0\nh :\n \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x))) =\n 1\n\u22a2 x *\n (\u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 *\n SummableFamily.hsum\n (SummableFamily.powers (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)\n (_ : 0 < \u2191(addVal \u0393 R) (1 - \u2191C (coeff x (order x))\u207b\u00b9 * \u2191(single (-order x)) 1 * x)))) =\n 1\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm x, h]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.HahnSeries", "llama_tokens": 116267, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.29401824317072395}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\n\u22a2 card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n[PROOFSTEP]\nlet i : DecidableRel ((\u00b7 \u2286 \u00b7) : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop) := fun _ _ => Classical.dec _\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\n\u22a2 card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n[PROOFSTEP]\nrefine' card_mul_le_card_mul' (\u00b7 \u2286 \u00b7) (fun s hs => _) (fun s hs => _)\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \ud835\udc9c\n\u22a2 r \u2264 card (bipartiteBelow (fun x x_1 => x \u2286 x_1) (\u2202 \ud835\udc9c) s)\n[PROOFSTEP]\nrw [\u2190 h\ud835\udc9c hs, \u2190 card_image_of_injOn s.erase_injOn]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \ud835\udc9c\n\u22a2 card (image (erase s) s) \u2264 card (bipartiteBelow (fun x x_1 => x \u2286 x_1) (\u2202 \ud835\udc9c) s)\n[PROOFSTEP]\nrefine' card_le_of_subset _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \ud835\udc9c\n\u22a2 image (erase s) s \u2286 bipartiteBelow (fun x x_1 => x \u2286 x_1) (\u2202 \ud835\udc9c) s\n[PROOFSTEP]\nsimp_rw [image_subset_iff, mem_bipartiteBelow]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \ud835\udc9c\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 erase s x \u2208 \u2202 \ud835\udc9c \u2227 erase s x \u2286 s\n[PROOFSTEP]\nexact fun a ha => \u27e8erase_mem_shadow hs ha, erase_subset _ _\u27e9\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\n\u22a2 card (bipartiteAbove (fun x x_1 => x \u2286 x_1) \ud835\udc9c s) \u2264 Fintype.card \u03b1 - r + 1\n[PROOFSTEP]\nrefine' le_trans _ tsub_tsub_le_tsub_add\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\n\u22a2 card (bipartiteAbove (fun x x_1 => x \u2286 x_1) \ud835\udc9c s) \u2264 Fintype.card \u03b1 - (r - 1)\n[PROOFSTEP]\nrw [\u2190 (Set.Sized.shadow h\ud835\udc9c) hs, \u2190 card_compl, \u2190 card_image_of_injOn (insert_inj_on' _)]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\n\u22a2 card (bipartiteAbove (fun x x_1 => x \u2286 x_1) \ud835\udc9c s) \u2264 card (image (fun a => insert a s) s\u1d9c)\n[PROOFSTEP]\nrefine'\n card_le_of_subset fun t ht =>\n _\n -- porting note: commented out the following line\n -- infer_instance\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 bipartiteAbove (fun x x_1 => x \u2286 x_1) \ud835\udc9c s\n\u22a2 t \u2208 image (fun a => insert a s) s\u1d9c\n[PROOFSTEP]\nrw [mem_bipartiteAbove] at ht \n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\n\u22a2 t \u2208 image (fun a => insert a s) s\u1d9c\n[PROOFSTEP]\nhave : \u2205 \u2209 \ud835\udc9c := by\n rw [\u2190 mem_coe, h\ud835\udc9c.empty_mem_iff, coe_eq_singleton]\n rintro rfl\n rw [shadow_singleton_empty] at hs \n exact not_mem_empty s hs\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\n\u22a2 \u00ac\u2205 \u2208 \ud835\udc9c\n[PROOFSTEP]\nrw [\u2190 mem_coe, h\ud835\udc9c.empty_mem_iff, coe_eq_singleton]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\n\u22a2 \u00ac\ud835\udc9c = {\u2205}\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nr : \u2115\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns t : Finset \u03b1\nh\ud835\udc9c : Set.Sized r \u2191{\u2205}\nhs : s \u2208 \u2202 {\u2205}\nht : t \u2208 {\u2205} \u2227 s \u2286 t\n\u22a2 False\n[PROOFSTEP]\nrw [shadow_singleton_empty] at hs \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nr : \u2115\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns t : Finset \u03b1\nh\ud835\udc9c : Set.Sized r \u2191{\u2205}\nhs : s \u2208 \u2205\nht : t \u2208 {\u2205} \u2227 s \u2286 t\n\u22a2 False\n[PROOFSTEP]\nexact not_mem_empty s hs\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\nthis : \u00ac\u2205 \u2208 \ud835\udc9c\n\u22a2 t \u2208 image (fun a => insert a s) s\u1d9c\n[PROOFSTEP]\nhave h :=\n exists_eq_insert_iff.2 \u27e8ht.2, by rw [(sized_shadow_iff this).1 (Set.Sized.shadow h\ud835\udc9c) ht.1, (Set.Sized.shadow h\ud835\udc9c) hs]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\nthis : \u00ac\u2205 \u2208 \ud835\udc9c\n\u22a2 card s + 1 = card t\n[PROOFSTEP]\nrw [(sized_shadow_iff this).1 (Set.Sized.shadow h\ud835\udc9c) ht.1, (Set.Sized.shadow h\ud835\udc9c) hs]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c \u2227 s \u2286 t\nthis : \u00ac\u2205 \u2208 \ud835\udc9c\nh : \u2203 a x, insert a s = t\n\u22a2 t \u2208 image (fun a => insert a s) s\u1d9c\n[PROOFSTEP]\nrcases h with \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\ni : DecidableRel fun x x_1 => x \u2286 x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x \u2286 x_2) x x_1)\ns : Finset \u03b1\nhs : s \u2208 \u2202 \ud835\udc9c\nthis : \u00ac\u2205 \u2208 \ud835\udc9c\na : \u03b1\nha : \u00aca \u2208 s\nht : insert a s \u2208 \ud835\udc9c \u2227 s \u2286 insert a s\n\u22a2 insert a s \u2208 image (fun a => insert a s) s\u1d9c\n[PROOFSTEP]\nexact mem_image_of_mem _ (mem_compl.2 ha)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\n\u22a2 \u2191(card \ud835\udc9c) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 \u2191(card (\u2202 \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nobtain hr' | hr' := lt_or_le (Fintype.card \u03b1) r\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\nhr' : Fintype.card \u03b1 < r\n\u22a2 \u2191(card \ud835\udc9c) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 \u2191(card (\u2202 \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt hr', cast_zero, div_zero]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\nhr' : Fintype.card \u03b1 < r\n\u22a2 0 \u2264 \u2191(card (\u2202 \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nexact div_nonneg (cast_nonneg _) (cast_nonneg _)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\nhr' : r \u2264 Fintype.card \u03b1\n\u22a2 \u2191(card \ud835\udc9c) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 \u2191(card (\u2202 \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nreplace h\ud835\udc9c := card_mul_le_card_shadow_mul h\ud835\udc9c\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 \u2191(card \ud835\udc9c) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 \u2191(card (\u2202 \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nrw [div_le_div_iff]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 \u2191(card \ud835\udc9c) * \u2191(Nat.choose (Fintype.card \u03b1) (r - 1)) \u2264 \u2191(card (\u2202 \ud835\udc9c)) * \u2191(Nat.choose (Fintype.card \u03b1) r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.b0\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.d0\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) (r - 1))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (r - 1) \u2264 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\ncases' r with r\n[GOAL]\ncase inr.zero\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nhr : zero \u2260 0\nhr' : zero \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * zero \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - zero + 1)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (zero - 1) \u2264 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) zero\n[PROOFSTEP]\nexact (hr rfl).elim\n[GOAL]\ncase inr.succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : succ r \u2260 0\nhr' : succ r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * succ r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - succ r + 1)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (succ r - 1) \u2264 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) (succ r)\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one] at *\n[GOAL]\ncase inr.succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - (r + 1) + 1)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (r + 1 - 1) \u2264 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) (r + 1)\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub hr', add_tsub_add_eq_tsub_right] at h\ud835\udc9c \n[GOAL]\ncase inr.succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (r + 1 - 1) \u2264 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) (r + 1)\n[PROOFSTEP]\napply le_of_mul_le_mul_right _ (pos_iff_ne_zero.2 hr)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (r + 1 - 1) * (r + 1) \u2264\n card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) (r + 1) * (r + 1)\n[PROOFSTEP]\nconvert Nat.mul_le_mul_right ((Fintype.card \u03b1).choose r) h\ud835\udc9c using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 card \ud835\udc9c * Nat.choose (Fintype.card \u03b1) (r + 1 - 1) * (r + 1) = card \ud835\udc9c * (r + 1) * Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\nsimp [mul_assoc, Nat.choose_succ_right_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 Nat.choose (Fintype.card \u03b1) r * (r + 1) = (r + 1) * Nat.choose (Fintype.card \u03b1) r \u2228 \ud835\udc9c = \u2205\n[PROOFSTEP]\nexact Or.inl (mul_comm _ _)\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 card (\u2202 \ud835\udc9c) * Nat.choose (Fintype.card \u03b1) (r + 1) * (r + 1) =\n card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r) * Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\nsimp only [mul_assoc, choose_succ_right_eq, mul_eq_mul_left_iff]\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r + 1 \u2260 0\nhr' : r + 1 \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * (r + 1) \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r)\n\u22a2 Nat.choose (Fintype.card \u03b1) r * (Fintype.card \u03b1 - r) = (Fintype.card \u03b1 - r) * Nat.choose (Fintype.card \u03b1) r \u2228\n card (\u2202 \ud835\udc9c) = 0\n[PROOFSTEP]\nexact Or.inl (mul_comm _ _)\n[GOAL]\ncase inr.b0\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 0 < Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\nexact Nat.choose_pos hr'\n[GOAL]\ncase inr.d0\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nr : \u2115\nhr : r \u2260 0\nhr' : r \u2264 Fintype.card \u03b1\nh\ud835\udc9c : card \ud835\udc9c * r \u2264 card (\u2202 \ud835\udc9c) * (Fintype.card \u03b1 - r + 1)\n\u22a2 0 < Nat.choose (Fintype.card \u03b1) (r - 1)\n[PROOFSTEP]\nexact Nat.choose_pos (r.pred_le.trans hr')\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\n\u22a2 s \u2208 falling k \ud835\udc9c \u2194 (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k\n[PROOFSTEP]\nsimp_rw [falling, mem_sup, mem_powersetLen]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\n\u22a2 (\u2203 v, v \u2208 \ud835\udc9c \u2227 s \u2286 v \u2227 card s = k) \u2194 (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k\n[PROOFSTEP]\naesop\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\n\u22a2 \ud835\udc9c # k \u222a \u2202 (falling (k + 1) \ud835\udc9c) = falling k \ud835\udc9c\n[PROOFSTEP]\next s\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\n\u22a2 s \u2208 \ud835\udc9c # k \u222a \u2202 (falling (k + 1) \ud835\udc9c) \u2194 s \u2208 falling k \ud835\udc9c\n[PROOFSTEP]\nsimp_rw [mem_union, mem_slice, mem_shadow_iff, mem_falling]\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\n\u22a2 (s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s) \u2194\n (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\n\u22a2 (s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s) \u2192\n (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k\n[PROOFSTEP]\nrintro (h | \u27e8s, \u27e8\u27e8t, ht, hst\u27e9, hs\u27e9, a, ha, rfl\u27e9)\n[GOAL]\ncase a.mp.inl\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nh : s \u2208 \ud835\udc9c \u2227 card s = k\n\u22a2 (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k\n[PROOFSTEP]\nexact \u27e8\u27e8s, h.1, Subset.refl _\u27e9, h.2\u27e9\n[GOAL]\ncase a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k + 1\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\na : \u03b1\nha : a \u2208 s\n\u22a2 (\u2203 t, t \u2208 \ud835\udc9c \u2227 erase s a \u2286 t) \u2227 card (erase s a) = k\n[PROOFSTEP]\nrefine' \u27e8\u27e8t, ht, (erase_subset _ _).trans hst\u27e9, _\u27e9\n[GOAL]\ncase a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k + 1\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\na : \u03b1\nha : a \u2208 s\n\u22a2 card (erase s a) = k\n[PROOFSTEP]\nrw [card_erase_of_mem ha, hs]\n[GOAL]\ncase a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k + 1\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\na : \u03b1\nha : a \u2208 s\n\u22a2 k + 1 - 1 = k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.mpr\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\n\u22a2 (\u2203 t, t \u2208 \ud835\udc9c \u2227 s \u2286 t) \u2227 card s = k \u2192\n s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n[PROOFSTEP]\nrintro \u27e8\u27e8t, ht, hst\u27e9, hs\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\n\u22a2 s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n[PROOFSTEP]\nby_cases h : s \u2208 \ud835\udc9c\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\nh : s \u2208 \ud835\udc9c\n\u22a2 s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n[PROOFSTEP]\nexact Or.inl \u27e8h, hs\u27e9\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\nh : \u00acs \u2208 \ud835\udc9c\n\u22a2 s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n[PROOFSTEP]\nobtain \u27e8a, ha, hst\u27e9 := ssubset_iff.1 (ssubset_of_subset_of_ne hst (ht.ne_of_not_mem h).symm)\n[GOAL]\ncase neg.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst\u271d : s \u2286 t\nh : \u00acs \u2208 \ud835\udc9c\na : \u03b1\nha : \u00aca \u2208 s\nhst : insert a s \u2286 t\n\u22a2 s \u2208 \ud835\udc9c \u2227 card s = k \u2228 \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = k + 1) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n[PROOFSTEP]\nrefine' Or.inr \u27e8insert a s, \u27e8\u27e8t, ht, hst\u27e9, _\u27e9, a, mem_insert_self _ _, erase_insert ha\u27e9\n[GOAL]\ncase neg.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d s : Finset \u03b1\nhs : card s = k\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst\u271d : s \u2286 t\nh : \u00acs \u2208 \ud835\udc9c\na : \u03b1\nha : \u00aca \u2208 s\nhst : insert a s \u2286 t\n\u22a2 card (insert a s) = k + 1\n[PROOFSTEP]\nrw [card_insert_of_not_mem ha, hs]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\nm n : \u2115\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\ns : Finset \u03b1\nh\u2081 : s \u2208 \u2202 (falling n \ud835\udc9c)\nh\u2082 : s \u2208 \ud835\udc9c # m\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [mem_shadow_iff, mem_falling] at h\u2081 \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\nm n : \u2115\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\ns : Finset \u03b1\nh\u2082 : s \u2208 \ud835\udc9c # m\nh\u2081 : \u2203 t, ((\u2203 t_1, t_1 \u2208 \ud835\udc9c \u2227 t \u2286 t_1) \u2227 card t = n) \u2227 \u2203 a, a \u2208 t \u2227 erase t a = s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8s, \u27e8\u27e8t, ht, hst\u27e9, _\u27e9, a, ha, rfl\u27e9 := h\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\nm n : \u2115\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\ns : Finset \u03b1\nright\u271d : card s = n\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\na : \u03b1\nha : a \u2208 s\nh\u2082 : erase s a \u2208 \ud835\udc9c # m\n\u22a2 False\n[PROOFSTEP]\nrefine' h\ud835\udc9c (slice_subset h\u2082) ht _ ((erase_subset _ _).trans hst)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\nm n : \u2115\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\ns : Finset \u03b1\nright\u271d : card s = n\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9c\nhst : s \u2286 t\na : \u03b1\nha : a \u2208 s\nh\u2082 : erase s a \u2208 \ud835\udc9c # m\n\u22a2 erase s a \u2260 t\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\nm n : \u2115\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\ns : Finset \u03b1\nright\u271d : card s = n\na : \u03b1\nha : a \u2208 s\nh\u2082 : erase s a \u2208 \ud835\udc9c # m\nht : erase s a \u2208 \ud835\udc9c\nhst : s \u2286 erase s a\n\u22a2 False\n[PROOFSTEP]\nexact not_mem_erase _ _ (hst ha)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk : k \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k))\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nhk : zero \u2264 Fintype.card \u03b1\n\u22a2 \u2211 r in range (zero + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - zero) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - zero))\n[PROOFSTEP]\nsimp only [tsub_zero, cast_one, cast_le, sum_singleton, div_one, choose_self, range_one, zero_eq, zero_add, range_one,\n ge_iff_le, sum_singleton, nonpos_iff_eq_zero, tsub_zero, choose_self, cast_one, div_one, cast_le]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nhk : zero \u2264 Fintype.card \u03b1\n\u22a2 card (\ud835\udc9c # Fintype.card \u03b1) \u2264 card (falling (Fintype.card \u03b1) \ud835\udc9c)\n[PROOFSTEP]\nexact card_le_of_subset (slice_subset_falling _ _)\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk\u271d : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k\u271d \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nk : \u2115\nih :\n k \u2264 Fintype.card \u03b1 \u2192\n \u2211 r in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k))\nhk : succ k \u2264 Fintype.card \u03b1\n\u22a2 \u2211 r in range (succ k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - succ k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - succ k))\n[PROOFSTEP]\nrw [succ_eq_add_one] at *\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk\u271d : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k\u271d \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nk : \u2115\nih :\n k \u2264 Fintype.card \u03b1 \u2192\n \u2211 r in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k))\nhk : k + 1 \u2264 Fintype.card \u03b1\n\u22a2 \u2211 r in range (k + 1 + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - (k + 1)) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - (k + 1)))\n[PROOFSTEP]\nrw [sum_range_succ, \u2190 slice_union_shadow_falling_succ,\n card_disjoint_union (IsAntichain.disjoint_slice_shadow_falling h\ud835\udc9c), cast_add, _root_.add_div, add_comm]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk\u271d : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k\u271d \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nk : \u2115\nih :\n k \u2264 Fintype.card \u03b1 \u2192\n \u2211 r in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k))\nhk : k + 1 \u2264 Fintype.card \u03b1\n\u22a2 \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - (k + 1)))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - (k + 1))) +\n \u2211 x in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - x))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - x)) \u2264\n \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - (k + 1)))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - (k + 1))) +\n \u2191(card (\u2202 (falling (Fintype.card \u03b1 - (k + 1) + 1) \ud835\udc9c))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - (k + 1)))\n[PROOFSTEP]\nrw [\u2190 tsub_tsub, tsub_add_cancel_of_le (le_tsub_of_add_le_left hk)]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\nk\u271d : \u2115\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhk\u271d : k\u271d \u2264 Fintype.card \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nk : \u2115\nih :\n k \u2264 Fintype.card \u03b1 \u2192\n \u2211 r in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n \u2191(card (falling (Fintype.card \u03b1 - k) \ud835\udc9c)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k))\nhk : k + 1 \u2264 Fintype.card \u03b1\n\u22a2 \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - k - 1))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k - 1)) +\n \u2211 x in range (k + 1), \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - x))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - x)) \u2264\n \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - k - 1))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k - 1)) +\n \u2191(card (\u2202 (falling (Fintype.card \u03b1 - k) \ud835\udc9c))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - k - 1))\n[PROOFSTEP]\nexact\n add_le_add_left\n ((ih <| le_of_succ_le hk).trans <|\n card_div_choose_le_card_shadow_div_choose (tsub_pos_iff_lt.2 <| Nat.succ_le_iff.1 hk).ne' <| sized_falling _ _)\n _\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in range (Fintype.card \u03b1 + 1), \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 1\n[PROOFSTEP]\nclassical\nrw [\u2190 sum_flip]\nrefine' (le_card_falling_div_choose le_rfl h\ud835\udc9c).trans _\nrw [div_le_iff] <;> norm_cast\n\u00b7 simpa only [Nat.sub_self, one_mul, Nat.choose_zero_right, falling] using Set.Sized.card_le (sized_falling 0 \ud835\udc9c)\n\u00b7 rw [tsub_self, choose_zero_right]\n exact zero_lt_one\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in range (Fintype.card \u03b1 + 1), \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 1\n[PROOFSTEP]\nrw [\u2190 sum_flip]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in range (Fintype.card \u03b1 + 1),\n \u2191(card (\ud835\udc9c # (Fintype.card \u03b1 - r))) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - r)) \u2264\n 1\n[PROOFSTEP]\nrefine' (le_card_falling_div_choose le_rfl h\ud835\udc9c).trans _\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2191(card (falling (Fintype.card \u03b1 - Fintype.card \u03b1) \ud835\udc9c)) /\n \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - Fintype.card \u03b1)) \u2264\n 1\n[PROOFSTEP]\nrw [div_le_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2191(card (falling (Fintype.card \u03b1 - Fintype.card \u03b1) \ud835\udc9c)) \u2264\n 1 * \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - Fintype.card \u03b1))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - Fintype.card \u03b1))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 card (falling (Fintype.card \u03b1 - Fintype.card \u03b1) \ud835\udc9c) \u2264 1 * Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - Fintype.card \u03b1)\n[PROOFSTEP]\nsimpa only [Nat.sub_self, one_mul, Nat.choose_zero_right, falling] using Set.Sized.card_le (sized_falling 0 \ud835\udc9c)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 0 < Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 - Fintype.card \u03b1)\n[PROOFSTEP]\nrw [tsub_self, choose_zero_right]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nk : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 0 < 1\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n[PROOFSTEP]\nclassical\nsuffices (\u2211 r in Iic (Fintype.card \u03b1), ((\ud835\udc9c # r).card : \u211a) / (Fintype.card \u03b1).choose (Fintype.card \u03b1 / 2)) \u2264 1\n by\n rw [\u2190 sum_div, \u2190 Nat.cast_sum, div_le_one] at this \n simp only [cast_le] at this \n rwa [sum_card_slice] at this \n simp only [cast_pos]\n exact choose_pos (Nat.div_le_self _ _)\nrw [Iic_eq_Icc, \u2190 Ico_succ_right, bot_eq_zero, Ico_zero_eq_range]\nrefine' (sum_le_sum fun r hr => _).trans (sum_card_slice_div_choose_le_one h\ud835\udc9c)\nrw [mem_range] at hr \nrefine' div_le_div_of_le_left _ _ _ <;> norm_cast\n\u00b7 exact Nat.zero_le _\n\u00b7 exact choose_pos (lt_succ_iff.1 hr)\n\u00b7 exact choose_le_middle _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n[PROOFSTEP]\nsuffices (\u2211 r in Iic (Fintype.card \u03b1), ((\ud835\udc9c # r).card : \u211a) / (Fintype.card \u03b1).choose (Fintype.card \u03b1 / 2)) \u2264 1\n by\n rw [\u2190 sum_div, \u2190 Nat.cast_sum, div_le_one] at this \n simp only [cast_le] at this \n rwa [sum_card_slice] at this \n simp only [cast_pos]\n exact choose_pos (Nat.div_le_self _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2211 r in Iic (Fintype.card \u03b1), \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n[PROOFSTEP]\nrw [\u2190 sum_div, \u2190 Nat.cast_sum, div_le_one] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2191(\u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x)) \u2264 \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2))\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2191(\u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2))\n[PROOFSTEP]\nsimp only [cast_le] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x) \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2191(\u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2))\n[PROOFSTEP]\nrwa [sum_card_slice] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2191(\u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2))\n[PROOFSTEP]\nsimp only [cast_pos]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nthis : \u2191(\u2211 x in Iic (Fintype.card \u03b1), card (\ud835\udc9c # x)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n\u22a2 0 < Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n[PROOFSTEP]\nexact choose_pos (Nat.div_le_self _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in Iic (Fintype.card \u03b1), \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n[PROOFSTEP]\nrw [Iic_eq_Icc, \u2190 Ico_succ_right, bot_eq_zero, Ico_zero_eq_range]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\n\u22a2 \u2211 r in range (succ (Fintype.card \u03b1)), \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264 1\n[PROOFSTEP]\nrefine' (sum_le_sum fun r hr => _).trans (sum_card_slice_div_choose_le_one h\ud835\udc9c)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r \u2208 range (succ (Fintype.card \u03b1))\n\u22a2 \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264\n \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) r)\n[PROOFSTEP]\nrw [mem_range] at hr \n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)) \u2264\n \u2191(card (\ud835\udc9c # r)) / \u2191(Nat.choose (Fintype.card \u03b1) r)\n[PROOFSTEP]\nrefine' div_le_div_of_le_left _ _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 0 \u2264 \u2191(card (\ud835\udc9c # r))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 0 < \u2191(Nat.choose (Fintype.card \u03b1) r)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 \u2191(Nat.choose (Fintype.card \u03b1) r) \u2264 \u2191(Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 0 \u2264 card (\ud835\udc9c # r)\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 0 < Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\nexact choose_pos (lt_succ_iff.1 hr)\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsAntichain (fun x x_1 => x \u2286 x_1) \u2191\ud835\udc9c\nr : \u2115\nhr : r < succ (Fintype.card \u03b1)\n\u22a2 Nat.choose (Fintype.card \u03b1) r \u2264 Nat.choose (Fintype.card \u03b1) (Fintype.card \u03b1 / 2)\n[PROOFSTEP]\nexact choose_le_middle _ _\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SetFamily.LYM", "llama_tokens": 20903, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.29401824317072395}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u22a2 LeftInverse toList ofList\n[PROOFSTEP]\nintro xs\n[GOAL]\n\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 toList (ofList xs) = xs\n[PROOFSTEP]\ninduction xs\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u22a2 toList (ofList []) = []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : toList (ofList tail\u271d) = tail\u271d\n\u22a2 toList (ofList (head\u271d :: tail\u271d)) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimpa [ofList, toList]\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 Function.RightInverse toList ofList\n[PROOFSTEP]\nintro xs\n[GOAL]\n\u03b1 : Type u_1\nxs : LazyList \u03b1\n\u22a2 ofList (toList xs) = xs\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u22a2 ofList (toList nil) = nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\ntl_ih\u271d : ?m.1024 tl\u271d\n\u22a2 ofList (toList (cons hd\u271d tl\u271d)) = cons hd\u271d tl\u271d\n[PROOFSTEP]\nsimpa only [toList, ofList, cons.injEq, true_and]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), ofList (toList (fn\u271d a)) = fn\u271d a\n\u22a2 { fn := fun x => ofList (toList (Thunk.get { fn := fn\u271d })) } = { fn := fn\u271d }\n[PROOFSTEP]\nrw [Thunk.get, ih]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : \u00acThunk.get xs = Thunk.get ys\n\u22a2 Decidable (cons x xs = cons y ys)\n[PROOFSTEP]\napply isFalse\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : \u00acThunk.get xs = Thunk.get ys\n\u22a2 \u00accons x xs = cons y ys\n[PROOFSTEP]\nsimp only [cons.injEq, not_and]\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : \u00acThunk.get xs = Thunk.get ys\n\u22a2 x = y \u2192 \u00acxs = ys\n[PROOFSTEP]\nintro _ xs_ys\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : \u00acThunk.get xs = Thunk.get ys\na\u271d : x = y\nxs_ys : xs = ys\n\u22a2 False\n[PROOFSTEP]\napply h2\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : \u00acThunk.get xs = Thunk.get ys\na\u271d : x = y\nxs_ys : xs = ys\n\u22a2 Thunk.get xs = Thunk.get ys\n[PROOFSTEP]\nrw [xs_ys]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : Thunk.get xs = Thunk.get ys\n\u22a2 Decidable (cons x xs = cons y ys)\n[PROOFSTEP]\napply isTrue\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : Thunk.get xs = Thunk.get ys\n\u22a2 cons x xs = cons y ys\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_tl\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : Thunk.get xs = Thunk.get ys\n\u22a2 xs = ys\n[PROOFSTEP]\next\n[GOAL]\ncase h.e_tl.eq\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\nh2 : Thunk.get xs = Thunk.get ys\n\u22a2 Thunk.get xs = Thunk.get ys\n[PROOFSTEP]\nexact h2\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\n\u22a2 Decidable (cons x xs = cons y ys)\n[PROOFSTEP]\napply isFalse\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\n\u22a2 \u00accons x xs = cons y ys\n[PROOFSTEP]\nsimp only [cons.injEq, not_and]\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\n\u22a2 x = y \u2192 \u00acxs = ys\n[PROOFSTEP]\nintro\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\ny : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\na\u271d : x = y\n\u22a2 \u00acxs = ys\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\n\u22a2 Decidable (nil = cons hd\u271d tl\u271d)\n[PROOFSTEP]\napply isFalse\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\n\u22a2 \u00acnil = cons hd\u271d tl\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\n\u22a2 Decidable (cons hd\u271d tl\u271d = nil)\n[PROOFSTEP]\napply isFalse\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\n\u22a2 \u00accons hd\u271d tl\u271d = nil\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 LawfulTraversable LazyList\n[PROOFSTEP]\napply Equiv.isLawfulTraversable' listEquivLazyList\n[GOAL]\ncase h\u2080\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.19468} (f : \u03b1 \u2192 \u03b2), Functor.map f = Equiv.map listEquivLazyList f\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.19468} (f : \u03b2), Functor.mapConst f = (Equiv.map listEquivLazyList \u2218 const \u03b1) f\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2082\n\u22a2 \u2200 {F : Type ?u.19468 \u2192 Type ?u.19468} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type ?u.19468}\n (f : \u03b1 \u2192 F \u03b2), traverse f = Equiv.traverse listEquivLazyList f\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2080\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\n\u22a2 Functor.map f\u271d = Equiv.map listEquivLazyList f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b2\u271d\n\u22a2 Functor.mapConst f\u271d = (Equiv.map listEquivLazyList \u2218 const \u03b1\u271d) f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2082\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\n\u22a2 traverse f\u271d = Equiv.traverse listEquivLazyList f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080.h\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : LazyList \u03b1\u271d\n\u22a2 f\u271d <$> x\u271d = Equiv.map listEquivLazyList f\u271d x\u271d\n[PROOFSTEP]\nrename_i f xs\n[GOAL]\ncase h\u2081.h\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b2\u271d\nx\u271d : LazyList \u03b1\u271d\n\u22a2 Functor.mapConst f\u271d x\u271d = (Equiv.map listEquivLazyList \u2218 const \u03b1\u271d) f\u271d x\u271d\n[PROOFSTEP]\nrename_i f xs\n[GOAL]\ncase h\u2082.h\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf\u271d : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nx\u271d : LazyList \u03b1\u271d\n\u22a2 traverse f\u271d x\u271d = Equiv.traverse listEquivLazyList f\u271d x\u271d\n[PROOFSTEP]\nrename_i f xs\n[GOAL]\ncase h\u2080.h\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 f <$> xs = Equiv.map listEquivLazyList f xs\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase h\u2080.h.nil\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 \u03b2\u271d\n\u22a2 f <$> nil = Equiv.map listEquivLazyList f nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2080.h.cons\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.19696 tl\u271d\n\u22a2 f <$> cons hd\u271d tl\u271d = Equiv.map listEquivLazyList f (cons hd\u271d tl\u271d)\n[PROOFSTEP]\nsimpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, LazyList.traverse,\n Seq.seq, toList, ofList, cons.injEq, true_and]\n[GOAL]\ncase h\u2080.h.mk\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), f <$> fn\u271d a = Equiv.map listEquivLazyList f (fn\u271d a)\n\u22a2 Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn\u271d })) =\n { fn := fun x => ofList (List.map f (toList (Thunk.get { fn := fn\u271d }))) }\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080.h.mk.eq\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), f <$> fn\u271d a = Equiv.map listEquivLazyList f (fn\u271d a)\n\u22a2 Thunk.get (Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn\u271d }))) =\n Thunk.get { fn := fun x => ofList (List.map f (toList (Thunk.get { fn := fn\u271d }))) }\n[PROOFSTEP]\napply ih\n[GOAL]\ncase h\u2081.h\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 Functor.mapConst f xs = (Equiv.map listEquivLazyList \u2218 const \u03b1\u271d) f xs\n[PROOFSTEP]\nsimp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp, Functor.mapConst]\n[GOAL]\ncase h\u2081.h\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 LazyList.traverse (const \u03b1\u271d f) xs = ofList (const \u03b1\u271d f <$> toList xs)\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase h\u2081.h.nil\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\n\u22a2 LazyList.traverse (const \u03b1\u271d f) nil = ofList (const \u03b1\u271d f <$> toList nil)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2081.h.cons\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.21260 tl\u271d\n\u22a2 LazyList.traverse (const \u03b1\u271d f) (cons hd\u271d tl\u271d) = ofList (const \u03b1\u271d f <$> toList (cons hd\u271d tl\u271d))\n[PROOFSTEP]\nsimpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]\n[GOAL]\ncase h\u2081.h.mk\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), LazyList.traverse (const \u03b1\u271d f) (fn\u271d a) = ofList (const \u03b1\u271d f <$> toList (fn\u271d a))\n\u22a2 Thunk.pure (LazyList.traverse (const \u03b1\u271d f) (Thunk.get { fn := fn\u271d })) =\n { fn := fun x => ofList (List.map (const \u03b1\u271d f) (toList (Thunk.get { fn := fn\u271d }))) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h\u2081.h.mk.e_a\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), LazyList.traverse (const \u03b1\u271d f) (fn\u271d a) = ofList (const \u03b1\u271d f <$> toList (fn\u271d a))\n\u22a2 LazyList.traverse (const \u03b1\u271d f) (Thunk.get { fn := fn\u271d }) =\n ofList (List.map (const \u03b1\u271d f) (toList (Thunk.get { fn := fn\u271d })))\n[PROOFSTEP]\napply ih\n[GOAL]\ncase h\u2082.h\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 traverse f xs = Equiv.traverse listEquivLazyList f xs\n[PROOFSTEP]\nsimp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]\n[GOAL]\ncase h\u2082.h\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 LazyList.traverse f xs = ofList <$> List.traverse f (toList xs)\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ tl ih _ ih\n[GOAL]\ncase h\u2082.h.nil\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\n\u22a2 LazyList.traverse f nil = ofList <$> List.traverse f (toList nil)\n[PROOFSTEP]\nsimp only [List.traverse, map_pure]\n[GOAL]\ncase h\u2082.h.nil\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\n\u22a2 LazyList.traverse f nil = pure (ofList [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.h.cons\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl : Thunk (LazyList \u03b1\u271d)\nih : ?m.22684 tl\n\u22a2 LazyList.traverse f (cons hd\u271d tl) = ofList <$> List.traverse f (toList (cons hd\u271d tl))\n[PROOFSTEP]\nhave : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih\n[GOAL]\ncase h\u2082.h.cons\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl : Thunk (LazyList \u03b1\u271d)\nih this : LazyList.traverse f (Thunk.get tl) = ofList <$> List.traverse f (toList (Thunk.get tl))\n\u22a2 LazyList.traverse f (cons hd\u271d tl) = ofList <$> List.traverse f (toList (cons hd\u271d tl))\n[PROOFSTEP]\nsimp only [traverse._eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq]\n[GOAL]\ncase h\u2082.h.cons\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl : Thunk (LazyList \u03b1\u271d)\nih this : LazyList.traverse f (Thunk.get tl) = ofList <$> List.traverse f (toList (Thunk.get tl))\n\u22a2 (Seq.seq (((fun x => x \u2218 Thunk.pure \u2218 ofList) \u2218 cons) <$> f hd\u271d) fun x => List.traverse f (toList (Thunk.get tl))) =\n Seq.seq (((fun x => ofList \u2218 x) \u2218 List.cons) <$> f hd\u271d) fun x => List.traverse f (toList (Thunk.get tl))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.h.mk\nF\u271d : Type ?u.19468 \u2192 Type ?u.19468\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d : Type ?u.19468\nf : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), LazyList.traverse f (fn\u271d a) = ofList <$> List.traverse f (toList (fn\u271d a))\n\u22a2 LazyList.traverse f (Thunk.get { fn := fn\u271d }) = ofList <$> List.traverse f (toList (Thunk.get { fn := fn\u271d }))\n[PROOFSTEP]\napply ih\n[GOAL]\n\u03b1 : Type u_1\nxs : LazyList \u03b1\n\u22a2 append xs (Thunk.pure nil) = xs\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u22a2 append nil (Thunk.pure nil) = nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\ntl_ih\u271d : ?m.62141 tl\u271d\n\u22a2 append (cons hd\u271d tl\u271d) (Thunk.pure nil) = cons hd\u271d tl\u271d\n[PROOFSTEP]\nsimpa only [append, cons.injEq, true_and]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = { fn := fn\u271d }\n[PROOFSTEP]\next\n[GOAL]\ncase mk.eq\n\u03b1 : Type u_1\nxs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih : \u2200 (a : Unit), append (fn\u271d a) (Thunk.pure nil) = fn\u271d a\n\u22a2 Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) (Thunk.pure nil) } = Thunk.get { fn := fn\u271d }\n[PROOFSTEP]\napply ih\n[GOAL]\n\u03b1 : Type u_1\nxs ys zs : LazyList \u03b1\n\u22a2 append (append xs { fn := fun x => ys }) { fn := fun x => zs } =\n append xs { fn := fun x => append ys { fn := fun x => zs } }\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nys zs : LazyList \u03b1\n\u22a2 append (append nil { fn := fun x => ys }) { fn := fun x => zs } =\n append nil { fn := fun x => append ys { fn := fun x => zs } }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nxs ys zs : LazyList \u03b1\nhd\u271d : \u03b1\ntl\u271d : Thunk (LazyList \u03b1)\ntl_ih\u271d : ?m.65384 tl\u271d\n\u22a2 append (append (cons hd\u271d tl\u271d) { fn := fun x => ys }) { fn := fun x => zs } =\n append (cons hd\u271d tl\u271d) { fn := fun x => append ys { fn := fun x => zs } }\n[PROOFSTEP]\nsimpa only [append, cons.injEq, true_and]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nxs ys zs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih :\n \u2200 (a : Unit),\n append (append (fn\u271d a) { fn := fun x => ys }) { fn := fun x => zs } =\n append (fn\u271d a) { fn := fun x => append ys { fn := fun x => zs } }\n\u22a2 {\n fn := fun x =>\n append (Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) { fn := fun x => ys } })\n { fn := fun x => zs } } =\n { fn := fun x => append (Thunk.get { fn := fn\u271d }) { fn := fun x => append ys { fn := fun x => zs } } }\n[PROOFSTEP]\next\n[GOAL]\ncase mk.eq\n\u03b1 : Type u_1\nxs ys zs : LazyList \u03b1\nfn\u271d : Unit \u2192 LazyList \u03b1\nih :\n \u2200 (a : Unit),\n append (append (fn\u271d a) { fn := fun x => ys }) { fn := fun x => zs } =\n append (fn\u271d a) { fn := fun x => append ys { fn := fun x => zs } }\n\u22a2 Thunk.get\n {\n fn := fun x =>\n append (Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) { fn := fun x => ys } })\n { fn := fun x => zs } } =\n Thunk.get { fn := fun x => append (Thunk.get { fn := fn\u271d }) { fn := fun x => append ys { fn := fun x => zs } } }\n[PROOFSTEP]\napply ih\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\n\u22a2 LazyList.bind (append xs ys) f = append (LazyList.bind xs f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n[PROOFSTEP]\nmatch xs with\n| LazyList.nil => rfl\n| LazyList.cons x xs =>\n simp only [append, Thunk.get, LazyList.bind]\n have := append_bind xs.get ys f\n simp only [Thunk.get] at this \n rw [this, append_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\n\u22a2 LazyList.bind (append nil ys) f = append (LazyList.bind nil f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\n\u22a2 LazyList.bind (append (cons x xs) ys) f =\n append (LazyList.bind (cons x xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n[PROOFSTEP]\nsimp only [append, Thunk.get, LazyList.bind]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n[PROOFSTEP]\nhave := append_bind xs.get ys f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n LazyList.bind (append (Thunk.get xs) ys) f =\n append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n[PROOFSTEP]\nsimp only [Thunk.get] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nxs\u271d : LazyList \u03b1\nys : Thunk (LazyList \u03b1)\nf : \u03b1 \u2192 LazyList \u03b2\nx : \u03b1\nxs : Thunk (LazyList \u03b1)\nthis :\n LazyList.bind (append (Thunk.fn xs ()) ys) f =\n append (LazyList.bind (Thunk.fn xs ()) f) { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n\u22a2 append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =\n append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })\n { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }\n[PROOFSTEP]\nrw [this, append_assoc]\n[GOAL]\n\u22a2 \u2200 {\u03b1 : Type ?u.75696} (x : LazyList \u03b1), id <$> x = x\n[PROOFSTEP]\nintro _ xs\n[GOAL]\n\u03b1\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\n\u22a2 id <$> xs = xs\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1\u271d : Type ?u.75696\n\u22a2 id <$> nil = nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.75815 tl\u271d\n\u22a2 id <$> cons hd\u271d tl\u271d = cons hd\u271d tl\u271d\n[PROOFSTEP]\nsimpa only [Functor.map, traverse._eq_2, id_eq, Id.map_eq, Seq.seq, cons.injEq, true_and]\n[GOAL]\ncase mk\n\u03b1\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), id <$> fn\u271d a = fn\u271d a\n\u22a2 Thunk.pure (LazyList.traverse id (Thunk.get { fn := fn\u271d })) = { fn := fn\u271d }\n[PROOFSTEP]\next\n[GOAL]\ncase mk.eq\n\u03b1\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), id <$> fn\u271d a = fn\u271d a\n\u22a2 Thunk.get (Thunk.pure (LazyList.traverse id (Thunk.get { fn := fn\u271d }))) = Thunk.get { fn := fn\u271d }\n[PROOFSTEP]\napply ih\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.75696} (x : \u03b1) (f : \u03b1 \u2192 LazyList \u03b2), pure x >>= f = f x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\n\u22a2 pure x\u271d >>= f\u271d = f\u271d x\u271d\n[PROOFSTEP]\nsimp only [bind, pure, singleton, LazyList.bind]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\n\u22a2 append (f\u271d x\u271d) { fn := fun x => LazyList.bind (Thunk.get (Thunk.pure nil)) f\u271d } = f\u271d x\u271d\n[PROOFSTEP]\napply append_nil\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.75696} (x : LazyList \u03b1) (f : \u03b1 \u2192 LazyList \u03b2) (g : \u03b2 \u2192 LazyList \u03b3),\n x >>= f >>= g = x >>= fun x => f x >>= g\n[PROOFSTEP]\nintro _ _ _ xs _ _\n[GOAL]\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\n\u22a2 xs >>= f\u271d >>= g\u271d = xs >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\n\u22a2 nil >>= f\u271d >>= g\u271d = nil >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.76333 tl\u271d\n\u22a2 cons hd\u271d tl\u271d >>= f\u271d >>= g\u271d = cons hd\u271d tl\u271d >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nsimp only [bind, LazyList.bind, append_bind]\n[GOAL]\ncase cons\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.76333 tl\u271d\n\u22a2 append (LazyList.bind (f\u271d hd\u271d) g\u271d)\n { fn := fun x => LazyList.bind (Thunk.get { fn := fun x => LazyList.bind (Thunk.get tl\u271d) f\u271d }) g\u271d } =\n append (LazyList.bind (f\u271d hd\u271d) g\u271d) { fn := fun x => LazyList.bind (Thunk.get tl\u271d) fun x => LazyList.bind (f\u271d x) g\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), fn\u271d a >>= f\u271d >>= g\u271d = fn\u271d a >>= fun x => f\u271d x >>= g\u271d\n\u22a2 { fn := fun x => LazyList.bind (Thunk.get { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) f\u271d }) g\u271d } =\n { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) fun x => LazyList.bind (f\u271d x) g\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.e_fn\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), fn\u271d a >>= f\u271d >>= g\u271d = fn\u271d a >>= fun x => f\u271d x >>= g\u271d\n\u22a2 (fun x => LazyList.bind (Thunk.get { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) f\u271d }) g\u271d) = fun x =>\n LazyList.bind (Thunk.get { fn := fn\u271d }) fun x => LazyList.bind (f\u271d x) g\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.e_fn.h\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.75696\nxs : LazyList \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 LazyList \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 LazyList \u03b3\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), fn\u271d a >>= f\u271d >>= g\u271d = fn\u271d a >>= fun x => f\u271d x >>= g\u271d\nx\u271d : Unit\n\u22a2 LazyList.bind (Thunk.get { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) f\u271d }) g\u271d =\n LazyList.bind (Thunk.get { fn := fn\u271d }) fun x => LazyList.bind (f\u271d x) g\u271d\n[PROOFSTEP]\napply ih\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.75696} (f : \u03b1 \u2192 \u03b2) (x : LazyList \u03b1),\n (do\n let y \u2190 x\n pure (f y)) =\n f <$> x\n[PROOFSTEP]\nintro _ _ f xs\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 (do\n let y \u2190 xs\n pure (f y)) =\n f <$> xs\n[PROOFSTEP]\nsimp only [bind, Functor.map, pure, singleton]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\n\u22a2 (LazyList.bind xs fun y => cons (f y) (Thunk.pure nil)) = LazyList.traverse f xs\n[PROOFSTEP]\ninduction' xs using LazyList.rec with _ _ _ _ ih\n[GOAL]\ncase nil\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\n\u22a2 (LazyList.bind nil fun y => cons (f y) (Thunk.pure nil)) = LazyList.traverse f nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.77095 tl\u271d\n\u22a2 (LazyList.bind (cons hd\u271d tl\u271d) fun y => cons (f y) (Thunk.pure nil)) = LazyList.traverse f (cons hd\u271d tl\u271d)\n[PROOFSTEP]\nsimp only [bind._eq_2, append, traverse._eq_2, Id.map_eq, cons.injEq, true_and]\n[GOAL]\ncase cons\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nhd\u271d : \u03b1\u271d\ntl\u271d : Thunk (LazyList \u03b1\u271d)\ntl_ih\u271d : ?m.77095 tl\u271d\n\u22a2 cons (f hd\u271d)\n {\n fn := fun x =>\n append (Thunk.get (Thunk.pure nil))\n { fn := fun x => LazyList.bind (Thunk.get tl\u271d) fun y => cons (f y) (Thunk.pure nil) } } =\n Seq.seq (cons (f hd\u271d)) fun x => Thunk.pure (LazyList.traverse f (Thunk.get tl\u271d))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), (LazyList.bind (fn\u271d a) fun y => cons (f y) (Thunk.pure nil)) = LazyList.traverse f (fn\u271d a)\n\u22a2 {\n fn := fun x =>\n append (Thunk.get (Thunk.pure nil))\n { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) fun y => cons (f y) (Thunk.pure nil) } } =\n (fun x => Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn\u271d }))) ()\n[PROOFSTEP]\next\n[GOAL]\ncase mk.eq\n\u03b1\u271d \u03b2\u271d : Type ?u.75696\nf : \u03b1\u271d \u2192 \u03b2\u271d\nxs : LazyList \u03b1\u271d\nfn\u271d : Unit \u2192 LazyList \u03b1\u271d\nih : \u2200 (a : Unit), (LazyList.bind (fn\u271d a) fun y => cons (f y) (Thunk.pure nil)) = LazyList.traverse f (fn\u271d a)\n\u22a2 Thunk.get\n {\n fn := fun x =>\n append (Thunk.get (Thunk.pure nil))\n { fn := fun x => LazyList.bind (Thunk.get { fn := fn\u271d }) fun y => cons (f y) (Thunk.pure nil) } } =\n Thunk.get ((fun x => Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn\u271d }))) ())\n[PROOFSTEP]\napply ih\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 Decidable (x \u2208 nil)\n[PROOFSTEP]\napply Decidable.isFalse\n[GOAL]\ncase h\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 nil\n[PROOFSTEP]\nsimp [Membership.mem, LazyList.Mem]\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\n\u22a2 Decidable (x \u2208 cons y ys)\n[PROOFSTEP]\napply Decidable.isTrue\n[GOAL]\ncase h\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\n\u22a2 x \u2208 cons y ys\n[PROOFSTEP]\nsimp only [Membership.mem, LazyList.Mem]\n[GOAL]\ncase h\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : x = y\n\u22a2 x = y \u2228 LazyList.Mem x (Thunk.get ys)\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\n\u22a2 Decidable (x \u2208 cons y ys)\n[PROOFSTEP]\nhave := Mem.decidable x ys.get\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\nthis : Decidable (x \u2208 Thunk.get ys)\n\u22a2 Decidable (x \u2208 cons y ys)\n[PROOFSTEP]\nhave : (x \u2208 ys.get) \u2194 (x \u2208 cons y ys) := by simp [(\u00b7 \u2208 \u00b7), LazyList.Mem, h]\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\nthis : Decidable (x \u2208 Thunk.get ys)\n\u22a2 x \u2208 Thunk.get ys \u2194 x \u2208 cons y ys\n[PROOFSTEP]\nsimp [(\u00b7 \u2208 \u00b7), LazyList.Mem, h]\n[GOAL]\n\u03b1 : Type ?u.93639\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\nh : \u00acx = y\nthis\u271d : Decidable (x \u2208 Thunk.get ys)\nthis : x \u2208 Thunk.get ys \u2194 x \u2208 cons y ys\n\u22a2 Decidable (x \u2208 cons y ys)\n[PROOFSTEP]\nexact decidable_of_decidable_of_iff this\n[GOAL]\n\u03b1 : Type u_1\nx y : \u03b1\nys : Thunk (LazyList \u03b1)\n\u22a2 x \u2208 cons y ys \u2194 x = y \u2228 x \u2208 Thunk.get ys\n[PROOFSTEP]\nsimp [Membership.mem, LazyList.Mem]\n[GOAL]\n\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\na : \u03b1\nl : Thunk (LazyList \u03b1)\n\u22a2 (\u2200 (x : \u03b1), x \u2208 cons a l \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 Thunk.get l \u2192 p x\n[PROOFSTEP]\nsimp only [Membership.mem, LazyList.Mem, or_imp, forall_and, forall_eq]\n", "meta": {"mathlib_filename": "Mathlib.Data.LazyList.Basic", "llama_tokens": 13525, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.2935607743132893}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\n\u22a2 IsTotalPreorder \u03b1 fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n -- porting note: added\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.lt \u2194 a < b\n[PROOFSTEP]\nrw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.lt \u2194 a < b\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a < b\n\u22a2 Ordering.lt = Ordering.lt \u2194 a < b\n[PROOFSTEP]\nsimp only [*, lt_irrefl]\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False \u2194 a < b\n[PROOFSTEP]\nsimp only [*, lt_irrefl]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False \u2194 a < b\n[PROOFSTEP]\nsimp only [*, lt_irrefl]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.gt \u2194 a > b\n[PROOFSTEP]\nrw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.gt \u2194 a > b\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a < b\n\u22a2 False \u2194 a > b\n[PROOFSTEP]\nsimp only [*, lt_irrefl, not_lt_of_gt]\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False \u2194 a > b\n[PROOFSTEP]\nsimp only [*, lt_irrefl, not_lt_of_gt]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 Ordering.gt = Ordering.gt \u2194 a > b\n[PROOFSTEP]\nsimp only [*, lt_irrefl, not_lt_of_gt]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 True \u2194 a > b\n[PROOFSTEP]\ncase _ h\u2081 h\u2082 =>\n have h : b < a := lt_trichotomy a b |>.resolve_left h\u2081 |>.resolve_left h\u2082\n exact true_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u2081 : \u00aca < b\nh\u2082 : \u00aca = b\n\u22a2 True \u2194 a > b\n[PROOFSTEP]\ncase _ h\u2081 h\u2082 =>\n have h : b < a := lt_trichotomy a b |>.resolve_left h\u2081 |>.resolve_left h\u2082\n exact true_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u2081 : \u00aca < b\nh\u2082 : \u00aca = b\n\u22a2 True \u2194 a > b\n[PROOFSTEP]\nhave h : b < a := lt_trichotomy a b |>.resolve_left h\u2081 |>.resolve_left h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u2081 : \u00aca < b\nh\u2082 : \u00aca = b\nh : b < a\n\u22a2 True \u2194 a > b\n[PROOFSTEP]\nexact true_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.eq \u2194 a = b\n[PROOFSTEP]\nrw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.eq \u2194 a = b\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a < b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\ntry simp only []\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a < b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 Ordering.eq = Ordering.eq \u2194 a = b\n[PROOFSTEP]\ntry simp only []\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 Ordering.eq = Ordering.eq \u2194 a = b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\ntry simp only []\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : a < b\n\u22a2 False \u2194 a = b\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 True \u2194 a = b\ncase neg \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : \u00aca = b \u22a2 False \u2194 a = b\n[PROOFSTEP]\ncase _ h => exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\ncase _ h => exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\nexact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 True \u2194 a = b\ncase neg \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : \u00aca = b \u22a2 False \u2194 a = b\n[PROOFSTEP]\ncase _ _ h => exact true_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : \u00aca < b\nh : a = b\n\u22a2 True \u2194 a = b\n[PROOFSTEP]\ncase _ _ h => exact true_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : \u00aca < b\nh : a = b\n\u22a2 True \u2194 a = b\n[PROOFSTEP]\nexact true_iff_iff.2 h\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\ncase _ _ h => exact false_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : \u00aca < b\nh : \u00aca = b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\ncase _ _ h => exact false_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh\u271d : \u00aca < b\nh : \u00aca = b\n\u22a2 False \u2194 a = b\n[PROOFSTEP]\nexact false_iff_iff.2 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b \u2260 Ordering.gt \u2194 a \u2264 b\n[PROOFSTEP]\ncases h : compare a b\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 Ordering.lt \u2260 Ordering.gt \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 Ordering.eq \u2260 Ordering.gt \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 Ordering.gt \u2260 Ordering.gt \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 True \u2194 a \u2264 b\n[PROOFSTEP]\nexact true_iff_iff.2 <| le_of_lt <| compare_lt_iff_lt.1 h\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 True \u2194 a \u2264 b\n[PROOFSTEP]\nexact true_iff_iff.2 <| le_of_eq <| compare_eq_iff_eq.1 h\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 False \u2194 a \u2264 b\n[PROOFSTEP]\nexact false_iff_iff.2 <| not_le_of_gt <| compare_gt_iff_gt.1 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b \u2260 Ordering.lt \u2194 a \u2265 b\n[PROOFSTEP]\ncases h : compare a b\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 Ordering.lt \u2260 Ordering.lt \u2194 a \u2265 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 Ordering.eq \u2260 Ordering.lt \u2194 a \u2265 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 Ordering.gt \u2260 Ordering.lt \u2194 a \u2265 b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 False \u2194 a \u2265 b\n[PROOFSTEP]\nexact false_iff_iff.2 <| (lt_iff_not_ge a b).1 <| compare_lt_iff_lt.1 h\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 True \u2194 a \u2265 b\n[PROOFSTEP]\nexact true_iff_iff.2 <| le_of_eq <| (\u00b7.symm) <| compare_eq_iff_eq.1 h\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 True \u2194 a \u2265 b\n[PROOFSTEP]\nexact true_iff_iff.2 <| le_of_lt <| compare_gt_iff_gt.1 h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\no : Ordering\n\u22a2 compare a b = o \u2194 Ordering.toRel o a b\n[PROOFSTEP]\ncases o\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.lt \u2194 Ordering.toRel Ordering.lt a b\n[PROOFSTEP]\nsimp only [Ordering.toRel]\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.eq \u2194 Ordering.toRel Ordering.eq a b\n[PROOFSTEP]\nsimp only [Ordering.toRel]\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.gt \u2194 Ordering.toRel Ordering.gt a b\n[PROOFSTEP]\nsimp only [Ordering.toRel]\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.lt \u2194 a < b\n[PROOFSTEP]\nexact compare_lt_iff_lt\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.eq \u2194 a = b\n[PROOFSTEP]\nexact compare_eq_iff_eq\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.gt \u2194 a > b\n[PROOFSTEP]\nexact compare_gt_iff_gt\n[GOAL]\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 Ordering.swap (compare a b) = compare b a\n[PROOFSTEP]\ncases h : compare a b\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 Ordering.swap Ordering.lt = compare b a\n[PROOFSTEP]\nsimp only [Ordering.swap]\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 Ordering.swap Ordering.eq = compare b a\n[PROOFSTEP]\nsimp only [Ordering.swap]\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 Ordering.swap Ordering.gt = compare b a\n[PROOFSTEP]\nsimp only [Ordering.swap]\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 Ordering.gt = compare b a\n[PROOFSTEP]\nsymm\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 Ordering.eq = compare b a\n[PROOFSTEP]\nsymm\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 Ordering.lt = compare b a\n[PROOFSTEP]\nsymm\n[GOAL]\ncase lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.lt\n\u22a2 compare b a = Ordering.gt\n[PROOFSTEP]\nexact compare_gt_iff_gt.2 <| compare_lt_iff_lt.1 h\n[GOAL]\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.eq\n\u22a2 compare b a = Ordering.eq\n[PROOFSTEP]\nexact compare_eq_iff_eq.2 <| compare_eq_iff_eq.1 h |>.symm\n[GOAL]\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : compare a b = Ordering.gt\n\u22a2 compare b a = Ordering.lt\n[PROOFSTEP]\nexact compare_lt_iff_lt.2 <| compare_gt_iff_gt.1 h\n", "meta": {"mathlib_filename": "Mathlib.Init.Algebra.Order", "llama_tokens": 4714, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.293560766853691}} {"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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA B : StructuredArrow S T\nf : A \u27f6 B\n\u22a2 A.hom \u226b T.map f.right = B.hom\n[PROOFSTEP]\nhave := f.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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA B : StructuredArrow S T\nf : A \u27f6 B\nthis : (Functor.fromPUnit S).map f.left \u226b B.hom = A.hom \u226b T.map f.right\n\u22a2 A.hom \u226b T.map f.right = B.hom\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf f' : StructuredArrow S T\ng : f.right \u27f6 f'.right\nw : autoParam (f.hom \u226b T.map g = f'.hom) _auto\u271d\n\u22a2 f.left = f'.left\n[PROOFSTEP]\next\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf f' : StructuredArrow S T\ng : f.right \u27f6 f'.right\nw : autoParam (f.hom \u226b T.map g = f'.hom) _auto\u271d\n\u22a2 (Functor.fromPUnit S).map (eqToHom (_ : f.left = f'.left)) \u226b f'.hom = f.hom \u226b T.map 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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf f' : StructuredArrow S T\ng : f.right \u27f6 f'.right\nw : autoParam (f.hom \u226b T.map g = f'.hom) _auto\u271d\n\u22a2 \ud835\udfd9 S \u226b f'.hom = f.hom \u226b T.map g\n[PROOFSTEP]\nsimpa using w.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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf : StructuredArrow S T\ng : f.right \u27f6 Y'\n\u22a2 f.left = (mk (f.hom \u226b T.map g)).left\n[PROOFSTEP]\next\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf f' : StructuredArrow S T\ng : f.right \u2245 f'.right\nw : autoParam (f.hom \u226b T.map g.hom = f'.hom) _auto\u271d\n\u22a2 f.left = f'.left\n[PROOFSTEP]\next\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf f' : StructuredArrow S T\ng : f.right \u2245 f'.right\nw : autoParam (f.hom \u226b T.map g.hom = f'.hom) _auto\u271d\n\u22a2 (Functor.fromPUnit S).map (eqToIso (_ : f.left = f'.left)).hom \u226b f'.hom = f.hom \u226b T.map g.hom\n[PROOFSTEP]\nsimpa [eqToHom_map] using w.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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf : StructuredArrow S T\n\u22a2 (map (\ud835\udfd9 S)).obj f = f\n[PROOFSTEP]\nrw [eq_mk 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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf : StructuredArrow S T\n\u22a2 (map (\ud835\udfd9 S)).obj (mk f.hom) = mk f.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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf : S \u27f6 S'\nf' : S' \u27f6 S''\nh : StructuredArrow S'' T\n\u22a2 (map (f \u226b f')).obj h = (map f).obj ((map f').obj h)\n[PROOFSTEP]\nrw [eq_mk 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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nf : S \u27f6 S'\nf' : S' \u27f6 S''\nh : StructuredArrow S'' T\n\u22a2 (map (f \u226b f')).obj (mk h.hom) = (map f).obj ((map f').obj (mk h.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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 Z.hom \u226b T.map (inv ((proj S T).map f)) = Y.hom\n[PROOFSTEP]\nrw [Functor.map_inv, IsIso.comp_inv_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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 Z.hom = Y.hom \u226b T.map ((proj S T).map 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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 f \u226b homMk (inv ((proj S T).map f)) = \ud835\udfd9 Y \u2227 homMk (inv ((proj S T).map f)) \u226b f = \ud835\udfd9 Z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 f \u226b homMk (inv ((proj S T).map f)) = \ud835\udfd9 Y\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase right\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 homMk (inv ((proj S T).map f)) \u226b f = \ud835\udfd9 Z\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left.left\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (f \u226b homMk (inv ((proj S T).map f))).left = (\ud835\udfd9 Y).left\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase left.right\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (f \u226b homMk (inv ((proj S T).map f))).right = (\ud835\udfd9 Y).right\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase right.left\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (homMk (inv ((proj S T).map f)) \u226b f).left = (\ud835\udfd9 Z).left\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase right.right\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (homMk (inv ((proj S T).map f)) \u226b f).right = (\ud835\udfd9 Z).right\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase left.right\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso f.right\n\u22a2 f.right \u226b inv f.right = \ud835\udfd9 Y.right\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.right\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\nS S' S'' : D\nY\u271d Y' : C\nT T' : C \u2964 D\nY Z : StructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso f.right\n\u22a2 inv f.right \u226b f.right = \ud835\udfd9 Z.right\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\ninst\u271d\u00b9 : Full T\ninst\u271d : Faithful T\nc : Cocone (Functor.empty (StructuredArrow (T.obj Y) T))\nm : (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 m = (fun c => homMk (T.preimage c.pt.hom)) c\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\ninst\u271d\u00b9 : Full T\ninst\u271d : Faithful T\nc : Cocone (Functor.empty (StructuredArrow (T.obj Y) T))\nm : (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 m.left = ((fun c => homMk (T.preimage c.pt.hom)) c).left\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase right\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\ninst\u271d\u00b9 : Full T\ninst\u271d : Faithful T\nc : Cocone (Functor.empty (StructuredArrow (T.obj Y) T))\nm : (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 m.right = ((fun c => homMk (T.preimage c.pt.hom)) c).right\n[PROOFSTEP]\napply T.map_injective\n[GOAL]\ncase right.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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\ninst\u271d\u00b9 : Full T\ninst\u271d : Faithful T\nc : Cocone (Functor.empty (StructuredArrow (T.obj Y) T))\nm : (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app (asEmptyCocone (mk (\ud835\udfd9 (T.obj Y)))).\u03b9 j \u226b m = NatTrans.app c.\u03b9 j\n\u22a2 T.map m.right = T.map ((fun c => homMk (T.preimage c.pt.hom)) c).right\n[PROOFSTEP]\nsimpa only [homMk_right, T.image_preimage, \u2190 w m] using (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\nS\u271d S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} B\nS : C\nF : B \u2964 C\nG : C \u2964 D\nX\u271d Y\u271d : StructuredArrow S F\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun X => mk (G.map X.hom)) X\u271d).hom \u226b (F \u22d9 G).map f.right = ((fun X => mk (G.map X.hom)) Y\u271d).hom\n[PROOFSTEP]\nsimp [Functor.comp_map, \u2190 G.map_comp, \u2190 f.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\nS\u271d S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} B\nS : C\nF : B \u2964 C\nG : C \u2964 D\nx\u271d\u00b3 x\u271d\u00b2 : StructuredArrow S F\nx\u271d\u00b9 x\u271d : x\u271d\u00b3 \u27f6 x\u271d\u00b2\nh : (post S F G).map x\u271d\u00b9 = (post S F G).map x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nsimpa [ext_iff] using h\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\nS\u271d S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} B\nS : C\nF : B \u2964 C\nG : C \u2964 D\ninst\u271d : Faithful G\nx\u271d\u00b9 x\u271d : StructuredArrow S F\nf : (post S F G).obj x\u271d\u00b9 \u27f6 (post S F G).obj x\u271d\n\u22a2 G.map (x\u271d\u00b9.hom \u226b F.map f.right) = G.map x\u271d.hom\n[PROOFSTEP]\nsimpa using f.w.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\nS\u271d S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} B\nS : C\nF : B \u2964 C\nG : C \u2964 D\ninst\u271d : Full G\nh : StructuredArrow (G.obj S) (F \u22d9 G)\n\u22a2 ((post S F G).obj (mk (G.preimage h.hom))).hom \u226b\n (F \u22d9 G).map (Iso.refl ((post S F G).obj (mk (G.preimage h.hom))).right).hom =\n h.hom\n[PROOFSTEP]\nsimp\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191((proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2)\n[PROOFSTEP]\nsuffices (proj S T).obj \u207b\u00b9' \ud835\udca2 = Set.range fun f : \u03a3 G : \ud835\udca2, S \u27f6 T.obj G => mk f.2\n by\n rw [this]\n infer_instance\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nthis : (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191((proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2)\n[PROOFSTEP]\nrw [this]\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nthis : (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191(Set.range fun f => mk f.snd)\n[PROOFSTEP]\ninfer_instance\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\n\u22a2 (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n[PROOFSTEP]\nexact Set.ext fun X => \u27e8fun h => \u27e8\u27e8\u27e8_, h\u27e9, X.hom\u27e9, (eq_mk _).symm\u27e9, by aesop_cat\u27e9\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\nS S' S'' : D\nY Y' : C\nT T' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nX : StructuredArrow S T\n\u22a2 (X \u2208 Set.range fun f => mk f.snd) \u2192 X \u2208 (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA B : CostructuredArrow S T\nf : A \u27f6 B\n\u22a2 S.map f.left \u226b B.hom = A.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nX Y : CostructuredArrow S T\nh : X = Y\n\u22a2 X.left = Y.left\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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nX Y : CostructuredArrow S T\nh : X = Y\n\u22a2 (eqToHom h).left = eqToHom (_ : X.left = Y.left)\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nX : CostructuredArrow S T\n\u22a2 (eqToHom (_ : X = X)).left = eqToHom (_ : X.left = X.left)\n[PROOFSTEP]\nsimp only [eqToHom_refl, id_left]\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf f' : CostructuredArrow S T\ng : f.left \u27f6 f'.left\nw : autoParam (S.map g \u226b f'.hom = f.hom) _auto\u271d\n\u22a2 f.right = f'.right\n[PROOFSTEP]\next\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf f' : CostructuredArrow S T\ng : f.left \u27f6 f'.left\nw : autoParam (S.map g \u226b f'.hom = f.hom) _auto\u271d\n\u22a2 S.map g \u226b f'.hom = f.hom \u226b (Functor.fromPUnit T).map (eqToHom (_ : f.right = f'.right))\n[PROOFSTEP]\nsimpa [eqToHom_map] using 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf : CostructuredArrow S T\ng : Y' \u27f6 f.left\n\u22a2 (mk (S.map g \u226b f.hom)).right = f.right\n[PROOFSTEP]\next\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf f' : CostructuredArrow S T\ng : f.left \u2245 f'.left\nw : autoParam (S.map g.hom \u226b f'.hom = f.hom) _auto\u271d\n\u22a2 f.right = f'.right\n[PROOFSTEP]\next\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf f' : CostructuredArrow S T\ng : f.left \u2245 f'.left\nw : autoParam (S.map g.hom \u226b f'.hom = f.hom) _auto\u271d\n\u22a2 S.map g.hom \u226b f'.hom = f.hom \u226b (Functor.fromPUnit T).map (eqToIso (_ : f.right = f'.right)).hom\n[PROOFSTEP]\nsimpa [eqToHom_map] using 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf : CostructuredArrow S T\n\u22a2 (map (\ud835\udfd9 T)).obj f = f\n[PROOFSTEP]\nrw [eq_mk 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf : CostructuredArrow S T\n\u22a2 (map (\ud835\udfd9 T)).obj (mk f.hom) = mk f.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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf : T \u27f6 T'\nf' : T' \u27f6 T''\nh : CostructuredArrow S T\n\u22a2 (map (f \u226b f')).obj h = (map f').obj ((map f).obj h)\n[PROOFSTEP]\nrw [eq_mk 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nf : T \u27f6 T'\nf' : T' \u27f6 T''\nh : CostructuredArrow S T\n\u22a2 (map (f \u226b f')).obj (mk h.hom) = (map f').obj ((map f).obj (mk h.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 S.map (inv ((proj S T).map f)) \u226b Y.hom = Z.hom\n[PROOFSTEP]\nrw [Functor.map_inv, IsIso.inv_comp_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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 Y.hom = S.map ((proj S T).map f) \u226b Z.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 f \u226b homMk (inv ((proj S T).map f)) = \ud835\udfd9 Y \u2227 homMk (inv ((proj S T).map f)) \u226b f = \ud835\udfd9 Z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 f \u226b homMk (inv ((proj S T).map f)) = \ud835\udfd9 Y\n[PROOFSTEP]\next\n[GOAL]\ncase right\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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 homMk (inv ((proj S T).map f)) \u226b f = \ud835\udfd9 Z\n[PROOFSTEP]\next\n[GOAL]\ncase left.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (f \u226b homMk (inv ((proj S T).map f))).left = (\ud835\udfd9 Y).left\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase right.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso ((proj S T).map f)\n\u22a2 (homMk (inv ((proj S T).map f)) \u226b f).left = (\ud835\udfd9 Z).left\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\ncase left.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso f.left\n\u22a2 f.left \u226b inv f.left = \ud835\udfd9 Y.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.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\nT T' T'' : D\nY\u271d Y' : C\nS S' : C \u2964 D\nY Z : CostructuredArrow S T\nf : Y \u27f6 Z\nt : IsIso f.left\n\u22a2 inv f.left \u226b f.left = \ud835\udfd9 Z.left\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\ninst\u271d\u00b9 : Full S\ninst\u271d : Faithful S\n\u22a2 \u2200 (s : Cone (Functor.empty (CostructuredArrow S (S.obj Y)))) (m : s.pt \u27f6 (asEmptyCone (mk (\ud835\udfd9 (S.obj Y)))).pt),\n (\u2200 (j : Discrete PEmpty), m \u226b NatTrans.app (asEmptyCone (mk (\ud835\udfd9 (S.obj Y)))).\u03c0 j = NatTrans.app s.\u03c0 j) \u2192\n m = (fun c => homMk (S.preimage c.pt.hom)) s\n[PROOFSTEP]\nrintro c m -\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\ninst\u271d\u00b9 : Full S\ninst\u271d : Faithful S\nc : Cone (Functor.empty (CostructuredArrow S (S.obj Y)))\nm : c.pt \u27f6 (asEmptyCone (mk (\ud835\udfd9 (S.obj Y)))).pt\n\u22a2 m = (fun c => homMk (S.preimage c.pt.hom)) c\n[PROOFSTEP]\next\n[GOAL]\ncase 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\ninst\u271d\u00b9 : Full S\ninst\u271d : Faithful S\nc : Cone (Functor.empty (CostructuredArrow S (S.obj Y)))\nm : c.pt \u27f6 (asEmptyCone (mk (\ud835\udfd9 (S.obj Y)))).pt\n\u22a2 m.left = ((fun c => homMk (S.preimage c.pt.hom)) c).left\n[PROOFSTEP]\napply S.map_injective\n[GOAL]\ncase 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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\ninst\u271d\u00b9 : Full S\ninst\u271d : Faithful S\nc : Cone (Functor.empty (CostructuredArrow S (S.obj Y)))\nm : c.pt \u27f6 (asEmptyCone (mk (\ud835\udfd9 (S.obj Y)))).pt\n\u22a2 S.map m.left = S.map ((fun c => homMk (S.preimage c.pt.hom)) c).left\n[PROOFSTEP]\nsimpa only [homMk_left, S.image_preimage, \u2190 w m] using (Category.comp_id _).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\nT T' T'' : D\nY Y' : C\nS\u271d S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} B\nF : B \u2964 C\nG : C \u2964 D\nS : C\nX\u271d Y\u271d : CostructuredArrow F S\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (F \u22d9 G).map f.left \u226b ((fun X => mk (G.map X.hom)) Y\u271d).hom = ((fun X => mk (G.map X.hom)) X\u271d).hom\n[PROOFSTEP]\nsimp [Functor.comp_map, \u2190 G.map_comp, \u2190 f.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\nT T' T'' : D\nY Y' : C\nS\u271d S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d : Category.{v\u2084, u\u2084} B\nF : B \u2964 C\nG : C \u2964 D\nS : C\nx\u271d\u00b3 x\u271d\u00b2 : CostructuredArrow F S\nx\u271d\u00b9 x\u271d : x\u271d\u00b3 \u27f6 x\u271d\u00b2\nh : (post F G S).map x\u271d\u00b9 = (post F G S).map x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nsimpa [ext_iff] using h\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\nT T' T'' : D\nY Y' : C\nS\u271d S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} B\nF : B \u2964 C\nG : C \u2964 D\nS : C\ninst\u271d : Faithful G\nx\u271d\u00b9 x\u271d : CostructuredArrow F S\nf : (post F G S).obj x\u271d\u00b9 \u27f6 (post F G S).obj x\u271d\n\u22a2 G.map (F.map f.left \u226b x\u271d.hom) = G.map x\u271d\u00b9.hom\n[PROOFSTEP]\nsimpa using f.w\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\nT T' T'' : D\nY Y' : C\nS\u271d S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b9 : Category.{v\u2084, u\u2084} B\nF : B \u2964 C\nG : C \u2964 D\nS : C\ninst\u271d : Full G\nh : CostructuredArrow (F \u22d9 G) (G.obj S)\n\u22a2 (F \u22d9 G).map (Iso.refl ((post F G S).obj (mk (G.preimage h.hom))).left).hom \u226b h.hom =\n ((post F G S).obj (mk (G.preimage h.hom))).hom\n[PROOFSTEP]\nsimp\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191((proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2)\n[PROOFSTEP]\nsuffices (proj S T).obj \u207b\u00b9' \ud835\udca2 = Set.range fun f : \u03a3 G : \ud835\udca2, S.obj G \u27f6 T => mk f.2\n by\n rw [this]\n infer_instance\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nthis : (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191((proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2)\n[PROOFSTEP]\nrw [this]\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nthis : (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n\u22a2 Small.{v\u2081, max u\u2081 v\u2082} \u2191(Set.range fun f => mk f.snd)\n[PROOFSTEP]\ninfer_instance\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\n\u22a2 (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2 = Set.range fun f => mk f.snd\n[PROOFSTEP]\nexact Set.ext fun X => \u27e8fun h => \u27e8\u27e8\u27e8_, h\u27e9, X.hom\u27e9, (eq_mk _).symm\u27e9, by aesop_cat\u27e9\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\nT T' T'' : D\nY Y' : C\nS S' : C \u2964 D\nA : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} A\nB : Type u\u2084\ninst\u271d\u00b2 : Category.{v\u2084, u\u2084} B\n\ud835\udca2 : Set C\ninst\u271d\u00b9 : Small.{v\u2081, u\u2081} \u2191\ud835\udca2\ninst\u271d : LocallySmall D\nX : CostructuredArrow S T\n\u22a2 (X \u2208 Set.range fun f => mk f.snd) \u2192 X \u2208 (proj S T).toPrefunctor.obj \u207b\u00b9' \ud835\udca2\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 : C \u2964 D\nd : D\nX\u271d Y\u271d : (StructuredArrow d F)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 F.op.map f.unop.right.op \u226b ((fun X => CostructuredArrow.mk X.unop.hom.op) Y\u271d).hom =\n ((fun X => CostructuredArrow.mk X.unop.hom.op) X\u271d).hom\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\nd : D\nX\u271d Y\u271d : (StructuredArrow d F)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (F.map f.unop.right).op \u226b Y\u271d.unop.hom.op = X\u271d.unop.hom.op\n[PROOFSTEP]\nrw [\u2190 op_comp, \u2190 f.unop.w, Functor.const_obj_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (StructuredArrow d F)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfd9 d \u226b X\u271d.unop.hom).op = X\u271d.unop.hom.op\n[PROOFSTEP]\nerw [Category.id_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (StructuredArrow (op d) F.op)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f.unop.right.unop \u226b ((fun X => CostructuredArrow.mk X.unop.hom.unop) Y\u271d).hom =\n ((fun X => CostructuredArrow.mk X.unop.hom.unop) X\u271d).hom\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\nd : D\nX\u271d Y\u271d : (StructuredArrow (op d) F.op)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f.unop.right.unop \u226b Y\u271d.unop.hom.unop = X\u271d.unop.hom.unop\n[PROOFSTEP]\nrw [\u2190 Quiver.Hom.unop_op (F.map (Quiver.Hom.unop f.unop.right)), \u2190 unop_comp, \u2190 F.op_map, \u2190 f.unop.w,\n Functor.const_obj_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (StructuredArrow (op d) F.op)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfd9 (op d) \u226b X\u271d.unop.hom).unop = X\u271d.unop.hom.unop\n[PROOFSTEP]\nerw [Category.id_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (CostructuredArrow F d)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun X => StructuredArrow.mk X.unop.hom.op) X\u271d).hom \u226b F.op.map f.unop.left.op =\n ((fun X => StructuredArrow.mk X.unop.hom.op) Y\u271d).hom\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\nd : D\nX\u271d Y\u271d : (CostructuredArrow F d)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 X\u271d.unop.hom.op \u226b (F.map f.unop.left).op = Y\u271d.unop.hom.op\n[PROOFSTEP]\nrw [\u2190 op_comp, f.unop.w, Functor.const_obj_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (CostructuredArrow F d)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (Y\u271d.unop.hom \u226b \ud835\udfd9 d).op = Y\u271d.unop.hom.op\n[PROOFSTEP]\nerw [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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (CostructuredArrow F.op (op d))\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun X => StructuredArrow.mk X.unop.hom.unop) X\u271d).hom \u226b F.map f.unop.left.unop =\n ((fun X => StructuredArrow.mk X.unop.hom.unop) Y\u271d).hom\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\nd : D\nX\u271d Y\u271d : (CostructuredArrow F.op (op d))\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 X\u271d.unop.hom.unop \u226b F.map f.unop.left.unop = Y\u271d.unop.hom.unop\n[PROOFSTEP]\nrw [\u2190 Quiver.Hom.unop_op (F.map f.unop.left.unop), \u2190 unop_comp, \u2190 F.op_map, f.unop.w, Functor.const_obj_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\nF : C \u2964 D\nd : D\nX\u271d Y\u271d : (CostructuredArrow F.op (op d))\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (Y\u271d.unop.hom \u226b \ud835\udfd9 (op d)).unop = Y\u271d.unop.hom.unop\n[PROOFSTEP]\nerw [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\nF : C \u2964 D\nd : D\nX Y : (StructuredArrow d F)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (StructuredArrow d F)\u1d52\u1d56).map f \u226b ((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) Y).hom).unop =\n (((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) X).hom \u226b\n (StructuredArrow.toCostructuredArrow F d \u22d9 (CostructuredArrow.toStructuredArrow' F d).rightOp).map f).unop\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left\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\nd : D\nX Y : (StructuredArrow d F)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (StructuredArrow d F)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) Y).hom).unop.left =\n (((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) X).hom \u226b\n (StructuredArrow.toCostructuredArrow F d \u22d9 (CostructuredArrow.toStructuredArrow' F d).rightOp).map\n f).unop.left\n[PROOFSTEP]\ndsimp [StructuredArrow.isoMk, Comma.isoMk, StructuredArrow.homMk]\n[GOAL]\ncase right\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\nd : D\nX Y : (StructuredArrow d F)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (StructuredArrow d F)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) Y).hom).unop.right =\n (((fun X => Iso.op (StructuredArrow.isoMk (Iso.refl X.unop.right))) X).hom \u226b\n (StructuredArrow.toCostructuredArrow F d \u22d9 (CostructuredArrow.toStructuredArrow' F d).rightOp).map\n f).unop.right\n[PROOFSTEP]\ndsimp [StructuredArrow.isoMk, Comma.isoMk, StructuredArrow.homMk]\n[GOAL]\ncase right\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\nd : D\nX Y : (StructuredArrow d F)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 Y.unop.right \u226b f.unop.right = f.unop.right \u226b \ud835\udfd9 X.unop.right\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\nd : D\nX Y : CostructuredArrow F.op (op d)\nf : X \u27f6 Y\n\u22a2 ((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).map f \u226b\n ((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n Y).hom =\n ((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n X).hom \u226b\n (\ud835\udfed (CostructuredArrow F.op (op d))).map f\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left\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\nd : D\nX Y : CostructuredArrow F.op (op d)\nf : X \u27f6 Y\n\u22a2 (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).map f \u226b\n ((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n Y).hom).left =\n (((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n X).hom \u226b\n (\ud835\udfed (CostructuredArrow F.op (op d))).map f).left\n[PROOFSTEP]\ndsimp [CostructuredArrow.isoMk, Comma.isoMk, CostructuredArrow.homMk]\n[GOAL]\ncase right\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\nd : D\nX Y : CostructuredArrow F.op (op d)\nf : X \u27f6 Y\n\u22a2 (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).map f \u226b\n ((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n Y).hom).right =\n (((fun X =>\n CostructuredArrow.isoMk\n (Iso.refl\n (((CostructuredArrow.toStructuredArrow' F d).rightOp \u22d9 StructuredArrow.toCostructuredArrow F d).obj\n X).left))\n X).hom \u226b\n (\ud835\udfed (CostructuredArrow F.op (op d))).map f).right\n[PROOFSTEP]\ndsimp [CostructuredArrow.isoMk, Comma.isoMk, CostructuredArrow.homMk]\n[GOAL]\ncase left\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\nd : D\nX Y : CostructuredArrow F.op (op d)\nf : X \u27f6 Y\n\u22a2 f.left \u226b \ud835\udfd9 Y.left = \ud835\udfd9 X.left \u226b f.left\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\nd : D\nX Y : (CostructuredArrow F d)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (CostructuredArrow F d)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) Y).hom).unop =\n (((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) X).hom \u226b\n (CostructuredArrow.toStructuredArrow F d \u22d9 (StructuredArrow.toCostructuredArrow' F d).rightOp).map f).unop\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left\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\nd : D\nX Y : (CostructuredArrow F d)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (CostructuredArrow F d)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) Y).hom).unop.left =\n (((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) X).hom \u226b\n (CostructuredArrow.toStructuredArrow F d \u22d9 (StructuredArrow.toCostructuredArrow' F d).rightOp).map\n f).unop.left\n[PROOFSTEP]\ndsimp [CostructuredArrow.isoMk, CostructuredArrow.homMk, Comma.isoMk]\n[GOAL]\ncase right\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\nd : D\nX Y : (CostructuredArrow F d)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((\ud835\udfed (CostructuredArrow F d)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) Y).hom).unop.right =\n (((fun X => Iso.op (CostructuredArrow.isoMk (Iso.refl X.unop.left))) X).hom \u226b\n (CostructuredArrow.toStructuredArrow F d \u22d9 (StructuredArrow.toCostructuredArrow' F d).rightOp).map\n f).unop.right\n[PROOFSTEP]\ndsimp [CostructuredArrow.isoMk, CostructuredArrow.homMk, Comma.isoMk]\n[GOAL]\ncase left\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\nd : D\nX Y : (CostructuredArrow F d)\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 \ud835\udfd9 Y.unop.left \u226b f.unop.left = f.unop.left \u226b \ud835\udfd9 X.unop.left\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\nd : D\nX Y : StructuredArrow (op d) F.op\nf : X \u27f6 Y\n\u22a2 ((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).map f \u226b\n ((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n Y).hom =\n ((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n X).hom \u226b\n (\ud835\udfed (StructuredArrow (op d) F.op)).map f\n[PROOFSTEP]\napply CommaMorphism.ext\n[GOAL]\ncase left\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\nd : D\nX Y : StructuredArrow (op d) F.op\nf : X \u27f6 Y\n\u22a2 (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).map f \u226b\n ((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n Y).hom).left =\n (((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n X).hom \u226b\n (\ud835\udfed (StructuredArrow (op d) F.op)).map f).left\n[PROOFSTEP]\ndsimp [StructuredArrow.isoMk, StructuredArrow.homMk, Comma.isoMk]\n[GOAL]\ncase right\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\nd : D\nX Y : StructuredArrow (op d) F.op\nf : X \u27f6 Y\n\u22a2 (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).map f \u226b\n ((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n Y).hom).right =\n (((fun X =>\n StructuredArrow.isoMk\n (Iso.refl\n (((StructuredArrow.toCostructuredArrow' F d).rightOp \u22d9 CostructuredArrow.toStructuredArrow F d).obj\n X).right))\n X).hom \u226b\n (\ud835\udfed (StructuredArrow (op d) F.op)).map f).right\n[PROOFSTEP]\ndsimp [StructuredArrow.isoMk, StructuredArrow.homMk, Comma.isoMk]\n[GOAL]\ncase right\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\nd : D\nX Y : StructuredArrow (op d) F.op\nf : X \u27f6 Y\n\u22a2 f.right \u226b \ud835\udfd9 Y.right = \ud835\udfd9 X.right \u226b f.right\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.StructuredArrow", "llama_tokens": 19041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.293560766853691}} {"text": "[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 NeBot (cocompact \u211a \u2293 \ud835\udcdd p)\n[PROOFSTEP]\nrefine' (hasBasis_cocompact.inf (nhds_basis_opens _)).neBot_iff.2 _\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 \u2200 {i : Set \u211a \u00d7 Set \u211a}, IsCompact i.fst \u2227 p \u2208 i.snd \u2227 IsOpen i.snd \u2192 Set.Nonempty (i.fst\u1d9c \u2229 i.snd)\n[PROOFSTEP]\nrintro \u27e8s, o\u27e9 \u27e8hs, hpo, ho\u27e9\n[GOAL]\ncase mk.intro.intro\np q : \u211a\ns\u271d t s o : Set \u211a\nhs : IsCompact (s, o).fst\nhpo : p \u2208 (s, o).snd\nho : IsOpen (s, o).snd\n\u22a2 Set.Nonempty ((s, o).fst\u1d9c \u2229 (s, o).snd)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\ncase mk.intro.intro\np q : \u211a\ns\u271d t s o : Set \u211a\nhs : IsCompact (s, o).fst\nhpo : p \u2208 (s, o).snd\nho : IsOpen (s, o).snd\n\u22a2 Set.Nonempty ((s, o).snd \u2229 (s, o).fst\u1d9c)\n[PROOFSTEP]\nexact (dense_compl_compact hs).inter_open_nonempty _ ho \u27e8p, hpo\u27e9\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 \u00acIsCountablyGenerated (cocompact \u211a)\n[PROOFSTEP]\nintro H\n[GOAL]\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\n\u22a2 False\n[PROOFSTEP]\nrcases exists_seq_tendsto (cocompact \u211a \u2293 \ud835\udcdd 0) with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhx : Tendsto x atTop (cocompact \u211a \u2293 \ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nrw [tendsto_inf] at hx \n[GOAL]\ncase intro\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhx : Tendsto x atTop (cocompact \u211a) \u2227 Tendsto x atTop (\ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nrcases hx with \u27e8hxc, hx0\u27e9\n[GOAL]\ncase intro.intro\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhxc : Tendsto x atTop (cocompact \u211a)\nhx0 : Tendsto x atTop (\ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, x n \u2209 insert (0 : \u211a) (range x)\n[GOAL]\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhxc : Tendsto x atTop (cocompact \u211a)\nhx0 : Tendsto x atTop (\ud835\udcdd 0)\n\u22a2 \u2203 n, \u00acx n \u2208 insert 0 (range x)\ncase intro.intro.intro\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhxc : Tendsto x atTop (cocompact \u211a)\nhx0 : Tendsto x atTop (\ud835\udcdd 0)\nn : \u2115\nhn : \u00acx n \u2208 insert 0 (range x)\n\u22a2 False\n[PROOFSTEP]\nexact (hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists\n[GOAL]\ncase intro.intro.intro\np q : \u211a\ns t : Set \u211a\nH : IsCountablyGenerated (cocompact \u211a)\nx : \u2115 \u2192 \u211a\nhxc : Tendsto x atTop (cocompact \u211a)\nhx0 : Tendsto x atTop (\ud835\udcdd 0)\nn : \u2115\nhn : \u00acx n \u2208 insert 0 (range x)\n\u22a2 False\n[PROOFSTEP]\nexact hn (Or.inr \u27e8n, rfl\u27e9)\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 \u00acIsCountablyGenerated (\ud835\udcdd \u221e)\n[PROOFSTEP]\nintro\n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : IsCountablyGenerated (\ud835\udcdd \u221e)\n\u22a2 False\n[PROOFSTEP]\nhave : IsCountablyGenerated (comap (OnePoint.some : \u211a \u2192 \u211a\u221e) (\ud835\udcdd \u221e)) := by infer_instance\n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : IsCountablyGenerated (\ud835\udcdd \u221e)\n\u22a2 IsCountablyGenerated (comap OnePoint.some (\ud835\udcdd \u221e))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : IsCountablyGenerated (\ud835\udcdd \u221e)\nthis : IsCountablyGenerated (comap OnePoint.some (\ud835\udcdd \u221e))\n\u22a2 False\n[PROOFSTEP]\nrw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this \n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : IsCountablyGenerated (\ud835\udcdd \u221e)\nthis : IsCountablyGenerated (cocompact \u211a)\n\u22a2 False\n[PROOFSTEP]\nexact not_countably_generated_cocompact this\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 \u00acFirstCountableTopology \u211a\u221e\n[PROOFSTEP]\nintro\n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : FirstCountableTopology \u211a\u221e\n\u22a2 False\n[PROOFSTEP]\nexact not_countably_generated_nhds_infty_opc inferInstance\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 \u00acSecondCountableTopology \u211a\u221e\n[PROOFSTEP]\nintro\n[GOAL]\np q : \u211a\ns t : Set \u211a\na\u271d : SecondCountableTopology \u211a\u221e\n\u22a2 False\n[PROOFSTEP]\nexact not_firstCountableTopology_opc inferInstance\n[GOAL]\np q : \u211a\ns t : Set \u211a\n\u22a2 TotallyDisconnectedSpace \u211a\n[PROOFSTEP]\nrefine' \u27e8fun s hsu hs x hx y hy => _\u27e9\n[GOAL]\np q : \u211a\ns\u271d t s : Set \u211a\nhsu : s \u2286 univ\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\n\u22a2 x = y\n[PROOFSTEP]\nclear hsu\n[GOAL]\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\n\u22a2 x = y\n[PROOFSTEP]\nby_contra' H : x \u2260 y\n[GOAL]\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\n\u22a2 False\n[PROOFSTEP]\nwlog hlt : x < y\n[GOAL]\ncase inr\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nthis :\n \u2200 {p q : \u211a} {s t : Set \u211a} (s : Set \u211a), IsPreconnected s \u2192 \u2200 (x : \u211a), x \u2208 s \u2192 \u2200 (y : \u211a), y \u2208 s \u2192 x \u2260 y \u2192 x < y \u2192 False\nhlt : \u00acx < y\n\u22a2 False\n[PROOFSTEP]\nrefine' this s hs y hy x hx H.symm <| H.lt_or_lt.resolve_left hlt\n[GOAL]\ncase inr.refine'_1\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nthis :\n \u2200 {p q : \u211a} {s t : Set \u211a} (s : Set \u211a), IsPreconnected s \u2192 \u2200 (x : \u211a), x \u2208 s \u2192 \u2200 (y : \u211a), y \u2208 s \u2192 x \u2260 y \u2192 x < y \u2192 False\nhlt : \u00acx < y\n\u22a2 \u211a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.refine'_2\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nthis :\n \u2200 {p q : \u211a} {s t : Set \u211a} (s : Set \u211a), IsPreconnected s \u2192 \u2200 (x : \u211a), x \u2208 s \u2192 \u2200 (y : \u211a), y \u2208 s \u2192 x \u2260 y \u2192 x < y \u2192 False\nhlt : \u00acx < y\n\u22a2 \u211a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.refine'_3\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nthis :\n \u2200 {p q : \u211a} {s t : Set \u211a} (s : Set \u211a), IsPreconnected s \u2192 \u2200 (x : \u211a), x \u2208 s \u2192 \u2200 (y : \u211a), y \u2208 s \u2192 x \u2260 y \u2192 x < y \u2192 False\nhlt : \u00acx < y\n\u22a2 Set \u211a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.refine'_4\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nthis :\n \u2200 {p q : \u211a} {s t : Set \u211a} (s : Set \u211a), IsPreconnected s \u2192 \u2200 (x : \u211a), x \u2208 s \u2192 \u2200 (y : \u211a), y \u2208 s \u2192 x \u2260 y \u2192 x < y \u2192 False\nhlt : \u00acx < y\n\u22a2 Set \u211a\n[PROOFSTEP]\nassumption\n[GOAL]\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nhlt : x < y\n\u22a2 False\n[PROOFSTEP]\nrcases exists_irrational_btwn (Rat.cast_lt.2 hlt) with \u27e8z, hz, hxz, hzy\u27e9\n[GOAL]\ncase intro.intro.intro\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nhlt : x < y\nz : \u211d\nhz : Irrational z\nhxz : \u2191x < z\nhzy : z < \u2191y\n\u22a2 False\n[PROOFSTEP]\nhave := hs.image _ continuous_coe_real.continuousOn\n[GOAL]\ncase intro.intro.intro\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nhlt : x < y\nz : \u211d\nhz : Irrational z\nhxz : \u2191x < z\nhzy : z < \u2191y\nthis : IsPreconnected (Rat.cast '' s)\n\u22a2 False\n[PROOFSTEP]\nrw [isPreconnected_iff_ordConnected] at this \n[GOAL]\ncase intro.intro.intro\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nhlt : x < y\nz : \u211d\nhz : Irrational z\nhxz : \u2191x < z\nhzy : z < \u2191y\nthis\u271d : IsPreconnected (Rat.cast '' s)\nthis : OrdConnected (Rat.cast '' s)\n\u22a2 False\n[PROOFSTEP]\nhave : z \u2208 Rat.cast '' s := this.out (mem_image_of_mem _ hx) (mem_image_of_mem _ hy) \u27e8hxz.le, hzy.le\u27e9\n[GOAL]\ncase intro.intro.intro\np q : \u211a\ns\u271d t s : Set \u211a\nhs : IsPreconnected s\nx : \u211a\nhx : x \u2208 s\ny : \u211a\nhy : y \u2208 s\nH : x \u2260 y\nhlt : x < y\nz : \u211d\nhz : Irrational z\nhxz : \u2191x < z\nhzy : z < \u2191y\nthis\u271d\u00b9 : IsPreconnected (Rat.cast '' s)\nthis\u271d : OrdConnected (Rat.cast '' s)\nthis : z \u2208 Rat.cast '' s\n\u22a2 False\n[PROOFSTEP]\nexact hz (image_subset_range _ _ this)\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.RatLemmas", "llama_tokens": 3649, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5964331319177488, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.2935573112807121}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring R\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring S\ninst\u271d : NonUnitalNonAssocSemiring T\nf\u271d : R \u2192\u2099+* S\ng : R \u2192\u2099+* T\nf : R \u2192\u2099+* S \u00d7 T\nx : R\n\u22a2 \u2191(NonUnitalRingHom.prod (comp (fst S T) f) (comp (snd S T) f)) x = \u2191f x\n[PROOFSTEP]\nsimp only [prod_apply, coe_fst, coe_snd, comp_apply, Prod.mk.eta]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u00b2 : NonAssocSemiring R\ninst\u271d\u00b9 : NonAssocSemiring S\ninst\u271d : NonAssocSemiring T\nf\u271d : R \u2192+* S\ng : R \u2192+* T\nf : R \u2192+* S \u00d7 T\nx : R\n\u22a2 \u2191(RingHom.prod (comp (fst S T) f) (comp (snd S T) f)) x = \u2191f x\n[PROOFSTEP]\nsimp only [prod_apply, coe_fst, coe_snd, comp_apply, Prod.mk.eta]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\nx : R \u00d7 S\n\u22a2 (fun x => (x, 0)) x.fst = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\nfst\u271d : R\nsnd\u271d : S\n\u22a2 (fun x => (x, 0)) (fst\u271d, snd\u271d).fst = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n (x * y) =\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n x *\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n (x + y) =\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n x +\n Equiv.toFun\n { toFun := fun x => (x, 0), invFun := Prod.fst,\n left_inv := (_ : \u2200 (x : R), ((fun x => (x, 0)) x).fst = ((fun x => (x, 0)) x).fst),\n right_inv := (_ : \u2200 (x : R \u00d7 S), (fun x => (x, 0)) x.fst = x) }\n y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\nx : S \u00d7 R\n\u22a2 (fun x => (0, x)) x.snd = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\nfst\u271d : S\nsnd\u271d : R\n\u22a2 (fun x => (0, x)) (fst\u271d, snd\u271d).snd = (fst\u271d, snd\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n (x * y) =\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n x *\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\nR' : Type u_4\nS : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\ninst\u271d\u2074 : NonAssocSemiring R\ninst\u271d\u00b3 : NonAssocSemiring S\ninst\u271d\u00b2 : NonAssocSemiring R'\ninst\u271d\u00b9 : NonAssocSemiring S'\ninst\u271d : Subsingleton S\n\u22a2 \u2200 (x y : R),\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n (x + y) =\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n x +\n Equiv.toFun\n { toFun := fun x => (0, x), invFun := Prod.snd,\n left_inv := (_ : \u2200 (x : R), ((fun x => (0, x)) x).snd = ((fun x => (0, x)) x).snd),\n right_inv := (_ : \u2200 (x : S \u00d7 R), (fun x => (0, x)) x.snd = x) }\n y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\n\u22a2 False\n[PROOFSTEP]\nhave := NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero (show ((0 : R), (1 : S)) * (1, 0) = 0 by simp)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\n\u22a2 (0, 1) * (1, 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\nthis : (0, 1) = 0 \u2228 (1, 0) = 0\n\u22a2 False\n[PROOFSTEP]\nrw [Prod.mk_eq_zero, Prod.mk_eq_zero] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\nthis : 0 = 0 \u2227 1 = 0 \u2228 1 = 0 \u2227 0 = 0\n\u22a2 False\n[PROOFSTEP]\nrcases this with (\u27e8_, h\u27e9 | \u27e8h, _\u27e9)\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\nleft\u271d : 0 = 0\nh : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one h.symm\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d : Type u_3\nR' : Type u_4\nS\u271d : Type u_5\nS' : Type u_6\nT : Type u_7\nT' : Type u_8\nR : Type u_9\nS : Type u_10\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : IsDomain (R \u00d7 S)\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : Nontrivial S\nh : 1 = 0\nright\u271d : 0 = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one h.symm\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Prod", "llama_tokens": 4235, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.29352114538869767}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\n\u22a2 Measurable fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s)\n[PROOFSTEP]\nsuffices Measurable fun y => \u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z : G \u00d7 G => ((1 : G), z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) by\n convert this using 1; ext1 x; congr 1 with y : 1; simp\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\nthis : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\n\u22a2 Measurable fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s)\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\nthis : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\n\u22a2 (fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s)) = fun y =>\n \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_5.h\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\nthis : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\nx : G\n\u22a2 \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s) = \u2191\u2191\u03bc ((fun x_1 => (x_1, x)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\n[PROOFSTEP]\ncongr 1 with y : 1\n[GOAL]\ncase h.e'_5.h.e_a.h\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\nthis : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\nx y : G\n\u22a2 y \u2208 (fun y => y * x) \u207b\u00b9' s \u2194 y \u2208 (fun x_1 => (x_1, x)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s)\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\n\u22a2 Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s))\n[PROOFSTEP]\napply measurable_measure_prod_mk_right\n[GOAL]\ncase hs\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\n\u22a2 MeasurableSet ((fun z => (1, z.fst * z.snd)) \u207b\u00b9' univ \u00d7\u02e2 s)\n[PROOFSTEP]\napply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs)\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b9 : SigmaFinite \u03bd\ninst\u271d : SigmaFinite \u03bc\ns : Set G\nhs : MeasurableSet s\n\u22a2 MeasurableSpace G\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\n\u22a2 MeasurePreserving fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n[PROOFSTEP]\nconvert (measurePreserving_prod_inv_mul_swap \u03bd \u03bc).comp (measurePreserving_prod_mul_swap \u03bc \u03bd) using 1\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\n\u22a2 (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) = (fun z => (z.snd, z.snd\u207b\u00b9 * z.fst)) \u2218 fun z => (z.snd, z.snd * z.fst)\n[PROOFSTEP]\next1 \u27e8x, y\u27e9\n[GOAL]\ncase h.e'_5.h.mk\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nx y : G\n\u22a2 ((x, y).snd * (x, y).fst, (x, y).fst\u207b\u00b9) =\n ((fun z => (z.snd, z.snd\u207b\u00b9 * z.fst)) \u2218 fun z => (z.snd, z.snd * z.fst)) (x, y)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, mul_inv_rev, inv_mul_cancel_right]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\n\u22a2 QuasiMeasurePreserving Inv.inv\n[PROOFSTEP]\nrefine' \u27e8measurable_inv, AbsolutelyContinuous.mk fun s hsm h\u03bcs => _\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191(map Inv.inv \u03bc) s = 0\n[PROOFSTEP]\nrw [map_apply measurable_inv hsm, inv_preimage]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0\n[PROOFSTEP]\nhave hf : Measurable fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0\n[PROOFSTEP]\nsuffices map (fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9)) (\u03bc.prod \u03bc) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0 by\n simpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e), or_self_iff] using this\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nthis : \u2191\u2191(map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0\n[PROOFSTEP]\nsimpa only [(measurePreserving_mul_prod_inv \u03bc \u03bc).map_eq, prod_prod, mul_eq_zero (M\u2080 := \u211d\u22650\u221e), or_self_iff] using this\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0\n[PROOFSTEP]\nhave hsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) := hsm.inv.prod hsm.inv\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nhsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s = 0\nhf : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nhsm' : MeasurableSet (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9)\n\u22a2 \u2191\u2191(map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bc)) (s\u207b\u00b9 \u00d7\u02e2 s\u207b\u00b9) = 0\n[PROOFSTEP]\nsimp_rw [map_apply hf hsm', prod_apply_symm (\u03bc := \u03bc) (\u03bd := \u03bc) (hf hsm'), preimage_preimage, mk_preimage_prod,\n inv_preimage, inv_inv, measure_mono_null (inter_subset_right _ _) h\u03bcs, lintegral_zero]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9 = 0 \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrefine' \u27e8fun hs => _, (quasiMeasurePreserving_inv \u03bc).preimage_null\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\nhs : \u2191\u2191\u03bc s\u207b\u00b9 = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [\u2190 inv_inv s]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\nhs : \u2191\u2191\u03bc s\u207b\u00b9 = 0\n\u22a2 \u2191\u2191\u03bc s\u207b\u00b9\u207b\u00b9 = 0\n[PROOFSTEP]\nexact (quasiMeasurePreserving_inv \u03bc).preimage_null hs\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\n\u22a2 \u03bc \u226a Measure.inv \u03bc\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s _ => _\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ns : Set G\nx\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.inv \u03bc) s = 0 \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsimp_rw [inv_apply \u03bc s, measure_inv_null, imp_self]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave h : Measurable fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nhave h2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9) (\u03bc.prod \u03bd) :=\n hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv \u03bc \u03bd).quasiMeasurePreserving\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [lintegral_lintegral h2f, lintegral_lintegral hf]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 \u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd = \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nconv_rhs => rw [\u2190 (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n| \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n| \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n| \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nrw [\u2190 (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 \u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd =\n \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nsymm\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd) =\n \u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd\n[PROOFSTEP]\nexact lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq.absolutelyContinuous) h.aemeasurable\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ny : G\n\u22a2 \u2191\u2191\u03bc ((fun x => x * y) \u207b\u00b9' s) = 0 \u2194 \u2191\u2191\u03bc ((fun x => y\u207b\u00b9 * x) \u207b\u00b9' s\u207b\u00b9)\u207b\u00b9 = 0\n[PROOFSTEP]\nsimp_rw [\u2190 inv_preimage, preimage_preimage, mul_inv_rev, inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ny : G\n\u22a2 \u2191\u2191\u03bc ((fun x => y\u207b\u00b9 * x) \u207b\u00b9' s\u207b\u00b9)\u207b\u00b9 = 0 \u2194 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsimp only [measure_inv_null \u03bc, measure_preimage_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 \u03bc \u226a map (fun x => x * g) \u03bc\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s hs => _\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map (fun x => x * g) \u03bc) s = 0 \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nrw [map_apply (measurable_mul_const g) hs, measure_mul_right_null]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nexact id\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 \u03bc \u226a map (fun h => g / h) \u03bc\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 \u03bc \u226a map (fun h => g * h\u207b\u00b9) \u03bc\n[PROOFSTEP]\nerw [\u2190 map_map (measurable_const_mul g) measurable_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 \u03bc \u226a map (fun x => g * x) (map Inv.inv \u03bc)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 map_mul_left_eq_self \u03bc g]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n| \u03bc\n[PROOFSTEP]\nrw [\u2190 map_mul_left_eq_self \u03bc g]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n| \u03bc\n[PROOFSTEP]\nrw [\u2190 map_mul_left_eq_self \u03bc g]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n| \u03bc\n[PROOFSTEP]\nrw [\u2190 map_mul_left_eq_self \u03bc g]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 map (fun x => g * x) \u03bc \u226a map (fun x => g * x) (map Inv.inv \u03bc)\n[PROOFSTEP]\nexact (absolutelyContinuous_inv \u03bc).map (measurable_const_mul g)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 set_lintegral_one, \u2190 lintegral_indicator _ sm, \u2190\n lintegral_lintegral_mul (measurable_const.indicator sm).aemeasurable hf.aemeasurable, \u2190\n lintegral_lintegral_mul_inv \u03bc \u03bd]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\ncase hf\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)\n[PROOFSTEP]\nswap\n[GOAL]\ncase hf\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 AEMeasurable (uncurry fun x y => indicator s (fun x => 1) x * f y)\n[PROOFSTEP]\nexact (((measurable_const.indicator sm).comp measurable_fst).mul (hf.comp measurable_snd)).aemeasurable\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n[PROOFSTEP]\nhave ms : \u2200 x : G, Measurable fun y => ((fun z => z * x) \u207b\u00b9' s).indicator (fun _ => (1 : \u211d\u22650\u221e)) y := fun x =>\n measurable_const.indicator (measurable_mul_const _ sm)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 x y, s.indicator (fun _ : G => (1 : \u211d\u22650\u221e)) (y * x) = ((fun z => z * x) \u207b\u00b9' s).indicator (fun b : G => 1) y :=\n by intro x y; symm; convert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2; ext1; rfl\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\n\u22a2 \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y\n[PROOFSTEP]\nintro x y\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y\n[PROOFSTEP]\nsymm\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y = indicator s (fun x => 1) (y * x)\n[PROOFSTEP]\nconvert indicator_comp_right (M := \u211d\u22650\u221e) fun y => y * x using 2\n[GOAL]\ncase h.e'_2.h.e'_5\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y : G\n\u22a2 (fun b => 1) = (fun x => 1) \u2218 fun y => y * x\n[PROOFSTEP]\next1\n[GOAL]\ncase h.e'_2.h.e'_5.h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nx y x\u271d : G\n\u22a2 1 = ((fun x => 1) \u2218 fun y => y * x) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\nms : \u2200 (x : G), Measurable fun y => indicator ((fun z => z * x) \u207b\u00b9' s) (fun x => 1) y\nthis : \u2200 (x y : G), indicator s (fun x => 1) (y * x) = indicator ((fun z => z * x) \u207b\u00b9' s) (fun b => 1) y\n\u22a2 \u222b\u207b (x : G), \u222b\u207b (y : G), indicator s (fun x => 1) (y * x) * f x\u207b\u00b9 \u2202\u03bd \u2202\u03bc =\n \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * f x\u207b\u00b9 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [this, lintegral_mul_const _ (ms _), lintegral_indicator _ (measurable_mul_const _ sm), set_lintegral_one]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nh\u03bd : \u03bd \u2260 0\n\u22a2 \u03bc \u226a \u03bd\n[PROOFSTEP]\nrefine' AbsolutelyContinuous.mk fun s sm h\u03bds => _\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nh\u03bd : \u03bd \u2260 0\ns : Set G\nsm : MeasurableSet s\nh\u03bds : \u2191\u2191\u03bd s = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nhave h1 := measure_mul_lintegral_eq \u03bc \u03bd sm 1 measurable_one\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nh\u03bd : \u03bd \u2260 0\ns : Set G\nsm : MeasurableSet s\nh\u03bds : \u2191\u2191\u03bd s = 0\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), OfNat.ofNat 1 y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * OfNat.ofNat 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsimp_rw [Pi.one_apply, lintegral_one, mul_one, (measure_mul_right_null \u03bd _).mpr h\u03bds, lintegral_zero,\n mul_eq_zero (M\u2080 := \u211d\u22650\u221e), measure_univ_eq_zero.not.mpr h\u03bd, or_false_iff] at h1 \n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nh\u03bd : \u03bd \u2260 0\ns : Set G\nsm : MeasurableSet s\nh\u03bds : \u2191\u2191\u03bd s = 0\nh1 : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nexact h1\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nrefine' ae_of_forall_measure_lt_top_ae_restrict' \u03bd.inv _ _\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (s_1 : Set G),\n MeasurableSet s_1 \u2192\n \u2191\u2191\u03bc s_1 < \u22a4 \u2192 \u2191\u2191(Measure.inv \u03bd) s_1 < \u22a4 \u2192 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc s_1, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nintro A hA _ h3A\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191(Measure.inv \u03bd) A < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nsimp only [\u03bd.inv_apply] at h3A \n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202Measure.restrict \u03bc A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\napply ae_lt_top (measurable_measure_mul_right \u03bd sm)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nhave h1 := measure_mul_lintegral_eq \u03bc \u03bd sm (A\u207b\u00b9.indicator 1) (measurable_one.indicator hA.inv)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator A\u207b\u00b9 1 y \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrw [lintegral_indicator _ hA.inv] at h1 \n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u222b\u207b (a : G) in A\u207b\u00b9, OfNat.ofNat 1 a \u2202\u03bd = \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * indicator A\u207b\u00b9 1 x\u207b\u00b9 \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nsimp_rw [Pi.one_apply, set_lintegral_one, \u2190 image_inv, indicator_image inv_injective, image_inv, \u2190\n indicator_mul_right _ fun x => \u03bd ((fun y => y * x) \u207b\u00b9' s), Function.comp, Pi.one_apply, mul_one] at h1 \n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u222b\u207b (x : G) in A, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) \u2202\u03bc \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 lintegral_indicator _ hA, \u2190 h1]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nA : Set G\nhA : MeasurableSet A\na\u271d : \u2191\u2191\u03bc A < \u22a4\nh3A : \u2191\u2191\u03bd A\u207b\u00b9 < \u22a4\nh1 : \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 = \u222b\u207b (x : G), indicator A (fun a => \u2191\u2191\u03bd ((fun y => y * a) \u207b\u00b9' s)) x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd A\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_ne_top h\u03bcs h3A.ne\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : G) \u2202\u03bc, \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n[PROOFSTEP]\nrefine' (ae_measure_preimage_mul_right_lt_top \u03bd \u03bd sm h3s).filter_mono _\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 ae \u03bc \u2264 ae \u03bd\n[PROOFSTEP]\nrefine' (absolutelyContinuous_of_isMulLeftInvariant \u03bc \u03bd _).ae_le\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd \u2260 0\n[PROOFSTEP]\nrefine' mt _ h2s\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bd = 0 \u2192 \u2191\u2191\u03bd s = 0\n[PROOFSTEP]\nintro h\u03bd\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh\u03bd : \u03bd = 0\n\u22a2 \u2191\u2191\u03bd s = 0\n[PROOFSTEP]\nrw [h\u03bd, Measure.coe_zero, Pi.zero_apply]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 \u2191\u2191\u03bc s * \u222b\u207b (y : G), f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd = \u222b\u207b (x : G), f x \u2202\u03bc\n[PROOFSTEP]\nset g := fun y => f y\u207b\u00b9 / \u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc\n[PROOFSTEP]\nhave hg : Measurable g := (hf.comp measurable_inv).div ((measurable_measure_mul_right \u03bd sm).comp measurable_inv)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [measure_mul_lintegral_eq \u03bc \u03bd sm g hg, inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc = \u222b\u207b (x : G), f x \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr_ae _\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x\n[PROOFSTEP]\nrefine' (ae_measure_preimage_mul_right_lt_top_of_ne_zero \u03bc \u03bd sm h2s h3s).mono fun x hx => _\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\nx : G\nhx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x\n[PROOFSTEP]\nsimp_rw [ENNReal.mul_div_cancel' (measure_mul_right_ne_zero \u03bd h2s _) hx.ne]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave h1 := measure_lintegral_div_measure \u03bd \u03bd hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh1 :\n \u2191\u2191\u03bd s * \u222b\u207b (y : G), indicator t (fun x => 1) y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd =\n \u222b\u207b (x : G), indicator t (fun x => 1) x \u2202\u03bd\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave h2 := measure_lintegral_div_measure \u03bc \u03bd hs h2s h3s (t.indicator fun _ => 1) (measurable_const.indicator ht)\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh1 :\n \u2191\u2191\u03bd s * \u222b\u207b (y : G), indicator t (fun x => 1) y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd =\n \u222b\u207b (x : G), indicator t (fun x => 1) x \u2202\u03bd\nh2 :\n \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator t (fun x => 1) y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd =\n \u222b\u207b (x : G), indicator t (fun x => 1) x \u2202\u03bc\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [lintegral_indicator _ ht, set_lintegral_one] at h1 h2 \n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nh1 : \u2191\u2191\u03bd s * \u222b\u207b (y : G), indicator t (fun x => 1) y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd = \u2191\u2191\u03bd t\nh2 : \u2191\u2191\u03bc s * \u222b\u207b (y : G), indicator t (fun x => 1) y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd = \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc s * \u2191\u2191\u03bd t = \u2191\u2191\u03bd s * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [\u2190 h1, mul_left_comm, h2]\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\n\u22a2 \u03bc = (\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd\n[PROOFSTEP]\next1 t ht\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nhs : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nt : Set G\nht : MeasurableSet t\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191((\u2191\u2191\u03bc s / \u2191\u2191\u03bd s) \u2022 \u03bd) t\n[PROOFSTEP]\nrw [smul_apply, smul_eq_mul, mul_comm, \u2190 mul_div_assoc, mul_comm, measure_mul_measure_eq \u03bc \u03bd hs ht h2s h3s,\n mul_div_assoc, ENNReal.mul_div_cancel' h2s h3s]\n[GOAL]\nG : Type u_1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\ns : Set G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 MeasurePreserving ?m.111352\n[PROOFSTEP]\napply measurePreserving_prod_mul_swap_right \u03bc \u03bd\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 MeasurePreserving ?m.118254\n[PROOFSTEP]\napply measurePreserving_prod_div_swap \u03bc \u03bd\n[GOAL]\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulRightInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bd\n\u22a2 MeasurePreserving fun z => (z.fst * z.snd, z.fst\u207b\u00b9)\n[PROOFSTEP]\nconvert (measurePreserving_prod_div_swap \u03bd \u03bc).comp (measurePreserving_prod_mul_swap_right \u03bc \u03bd) using 1\n[GOAL]\ncase h.e'_5\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulRightInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bd\n\u22a2 (fun z => (z.fst * z.snd, z.fst\u207b\u00b9)) = (fun z => (z.snd, z.fst / z.snd)) \u2218 fun z => (z.snd, z.fst * z.snd)\n[PROOFSTEP]\next1 \u27e8x, y\u27e9\n[GOAL]\ncase h.e'_5.h.mk\nG : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulRightInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bd\nx y : G\n\u22a2 ((x, y).fst * (x, y).snd, (x, y).fst\u207b\u00b9) = ((fun z => (z.snd, z.fst / z.snd)) \u2218 fun z => (z.snd, z.fst * z.snd)) (x, y)\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, div_mul_eq_div_div_swap, div_self', one_div]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 QuasiMeasurePreserving Inv.inv\n[PROOFSTEP]\nrw [\u2190 \u03bc.inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 QuasiMeasurePreserving Inv.inv\n[PROOFSTEP]\nexact (quasiMeasurePreserving_inv \u03bc.inv).mono (inv_absolutelyContinuous \u03bc.inv) (absolutelyContinuous_inv \u03bc.inv)\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g / h\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g * h\u207b\u00b9\n[PROOFSTEP]\nexact (measurePreserving_mul_left \u03bc g).quasiMeasurePreserving.comp (quasiMeasurePreserving_inv \u03bc)\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g / h\n[PROOFSTEP]\nrw [\u2190 \u03bc.inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g / h\n[PROOFSTEP]\nexact (quasiMeasurePreserving_div_left \u03bc.inv g).mono (inv_absolutelyContinuous \u03bc.inv) (absolutelyContinuous_inv \u03bc.inv)\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\n\u22a2 QuasiMeasurePreserving fun p => p.fst / p.snd\n[PROOFSTEP]\nrefine' QuasiMeasurePreserving.prod_of_left measurable_div (eventually_of_forall fun y => _)\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ny : G\n\u22a2 QuasiMeasurePreserving fun x => (x, y).fst / (x, y).snd\n[PROOFSTEP]\nexact (measurePreserving_div_right \u03bc y).quasiMeasurePreserving\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => h * g\n[PROOFSTEP]\nrefine' \u27e8measurable_mul_const g, AbsolutelyContinuous.mk fun s hs => _\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191(map (fun h => h * g) \u03bc) s = 0\n[PROOFSTEP]\nrw [map_apply (measurable_mul_const g) hs, measure_mul_right_null]\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nexact id\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\n\u22a2 QuasiMeasurePreserving fun h => g * h\n[PROOFSTEP]\nhave :=\n (quasiMeasurePreserving_mul_right \u03bc.inv g\u207b\u00b9).mono (inv_absolutelyContinuous \u03bc.inv) (absolutelyContinuous_inv \u03bc.inv)\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\n\u22a2 QuasiMeasurePreserving fun h => g * h\n[PROOFSTEP]\nrw [\u03bc.inv_inv] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\n\u22a2 QuasiMeasurePreserving fun h => g * h\n[PROOFSTEP]\nhave :=\n (quasiMeasurePreserving_inv_of_right_invariant \u03bc).comp (this.comp (quasiMeasurePreserving_inv_of_right_invariant \u03bc))\n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving (Inv.inv \u2218 (fun h => h * g\u207b\u00b9) \u2218 Inv.inv)\n\u22a2 QuasiMeasurePreserving fun h => g * h\n[PROOFSTEP]\nsimp_rw [Function.comp, mul_inv_rev, inv_inv] at this \n[GOAL]\nG : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nthis\u271d : QuasiMeasurePreserving fun h => h * g\u207b\u00b9\nthis : QuasiMeasurePreserving fun x => g * x\n\u22a2 QuasiMeasurePreserving fun h => g * h\n[PROOFSTEP]\nexact this\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Group.Prod", "llama_tokens": 24810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.293258998007607}} {"text": "[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2081.toMulOneClass = m\u2082.toMulOneClass := MulOneClass.ext h_mul\n[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis : toMulOneClass = toMulOneClass\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave h\u2081 : m\u2081.one = m\u2082.one := congr_arg (\u00b7.one) (this)\n[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nlet f : @MonoidHom M M m\u2081.toMulOneClass m\u2082.toMulOneClass :=\n @MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h\u2081) (fun x y => congr_fun (congr_fun h_mul x) y)\n[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2081.npow = m\u2082.npow := by\n ext n x\n exact @MonoidHom.map_pow M M m\u2081 m\u2082 f x n\n[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\n\u22a2 npow = npow\n[PROOFSTEP]\next n x\n[GOAL]\ncase h.h\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nn : \u2115\nx : M\n\u22a2 npow n x = npow n x\n[PROOFSTEP]\nexact @MonoidHom.map_pow M M m\u2081 m\u2082 f x n\n[GOAL]\nM : Type u\nm\u2081 m\u2082 : Monoid M\nh_mul : Mul.mul = Mul.mul\nthis\u271d : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : npow = npow\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nrcases m\u2081 with @\u27e8@\u27e8\u27e8_\u27e9\u27e9, \u27e8_\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u\nm\u2082 : Monoid M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nmul\u271d : M \u2192 M \u2192 M\nmul_assoc\u271d : \u2200 (a b c : M), a * b * c = a * (b * c)\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\none\u271d : M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nh_mul : Mul.mul = Mul.mul\nthis\u271d : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : npow = npow\n\u22a2 mk one_mul\u271d mul_one\u271d npow\u271d = m\u2082\n[PROOFSTEP]\nrcases m\u2082 with @\u27e8@\u27e8\u27e8_\u27e9\u27e9, \u27e8_\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nM : Type u\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nmul\u271d\u00b9 : M \u2192 M \u2192 M\nmul_assoc\u271d\u00b9 : \u2200 (a b c : M), a * b * c = a * (b * c)\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\none\u271d\u00b9 : M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow\u271d : \u2115 \u2192 M \u2192 M\nmul\u271d : M \u2192 M \u2192 M\nmul_assoc\u271d : \u2200 (a b c : M), a * b * c = a * (b * c)\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\none\u271d : M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nh_mul : Mul.mul = Mul.mul\nthis\u271d : toMulOneClass = toMulOneClass\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : npow = npow\n\u22a2 mk one_mul\u271d\u00b9 mul_one\u271d\u00b9 npow\u271d\u00b9 = mk one_mul\u271d mul_one\u271d npow\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u\n\u22a2 Function.Injective (@toMonoid M)\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 h\n[GOAL]\ncase mk.mk\nM : Type u\ntoMonoid\u271d\u00b9 : Monoid M\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoMonoid\u271d : Monoid M\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toMonoid = toMonoid\n\u22a2 mk mul_comm\u271d\u00b9 = mk mul_comm\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u\n\u22a2 Function.Injective (@toMonoid M)\n[PROOFSTEP]\nrintro @\u27e8@\u27e8\u27e9\u27e9 @\u27e8@\u27e8\u27e9\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 mk one_mul\u271d\u00b9 mul_one\u271d\u00b9 npow\u271d\u00b9 = mk one_mul\u271d mul_one\u271d npow\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.e_toLeftCancelSemigroup.e_toSemigroup\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 toSemigroup\u271d\u00b9 = toSemigroup\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase mk.mk.mk.mk.e_toOne\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 toOne\u271d\u00b9 = toOne\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase mk.mk.mk.mk.e_npow\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 npow\u271d\u00b9 = npow\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\nM : Type u\n\u22a2 Function.Injective (@toMonoid M)\n[PROOFSTEP]\nrintro @\u27e8@\u27e8\u27e9\u27e9 @\u27e8@\u27e8\u27e9\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_right_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_right_cancel\u271d : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 mk one_mul\u271d\u00b9 mul_one\u271d\u00b9 npow\u271d\u00b9 = mk one_mul\u271d mul_one\u271d npow\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.e_toRightCancelSemigroup.e_toSemigroup\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_right_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_right_cancel\u271d : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 toSemigroup\u271d\u00b9 = toSemigroup\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase mk.mk.mk.mk.e_toOne\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_right_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_right_cancel\u271d : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 toOne\u271d\u00b9 = toOne\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\ncase mk.mk.mk.mk.e_npow\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_right_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_right_cancel\u271d : \u2200 (a b c : M), a * b = c * b \u2192 a = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nh : toMonoid = toMonoid\n\u22a2 npow\u271d\u00b9 = npow\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\nM : Type u\n\u22a2 Function.Injective (@toLeftCancelMonoid M)\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 h\n[GOAL]\ncase mk.mk\nM : Type u\ntoLeftCancelMonoid\u271d\u00b9 : LeftCancelMonoid M\nmul_right_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = c * b \u2192 a = c\ntoLeftCancelMonoid\u271d : LeftCancelMonoid M\nmul_right_cancel\u271d : \u2200 (a b c : M), a * b = c * b \u2192 a = c\nh : toLeftCancelMonoid = toLeftCancelMonoid\n\u22a2 mk mul_right_cancel\u271d\u00b9 = mk mul_right_cancel\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u\n\u22a2 Function.Injective (@toCommMonoid M)\n[PROOFSTEP]\nrintro @\u27e8@\u27e8@\u27e8\u27e9\u27e9\u27e9 @\u27e8@\u27e8@\u27e8\u27e9\u27e9\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 mk mul_comm\u271d\u00b9 = mk mul_comm\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toLeftCancelSemigroup.e_toSemigroup\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 toSemigroup\u271d\u00b9 = toSemigroup\u271d\n[PROOFSTEP]\n{ injection h with h'\n injection h'\n}\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toLeftCancelSemigroup.e_toSemigroup\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 toSemigroup\u271d\u00b9 = toSemigroup\u271d\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toLeftCancelSemigroup.e_toSemigroup\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh' :\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow =\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow\n\u22a2 toSemigroup\u271d\u00b9 = toSemigroup\u271d\n[PROOFSTEP]\ninjection h'\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toOne\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 toOne\u271d\u00b9 = toOne\u271d\n[PROOFSTEP]\n{ injection h with h'\n injection h'\n}\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toOne\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 toOne\u271d\u00b9 = toOne\u271d\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_toOne\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh' :\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow =\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow\n\u22a2 toOne\u271d\u00b9 = toOne\u271d\n[PROOFSTEP]\ninjection h'\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_npow\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 npow\u271d\u00b9 = npow\u271d\n[PROOFSTEP]\n{ injection h with h'\n injection h'\n}\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_npow\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh : toCommMonoid = toCommMonoid\n\u22a2 npow\u271d\u00b9 = npow\u271d\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.e_toLeftCancelMonoid.e_npow\nM : Type u\ntoOne\u271d\u00b9 : One M\nnpow\u271d\u00b9 : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d\u00b9 : \u2200 (x : M), npow\u271d\u00b9 0 x = 1\ntoSemigroup\u271d\u00b9 : Semigroup M\nmul_left_cancel\u271d\u00b9 : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d\u00b9 : \u2200 (a : M), 1 * a = a\nmul_one\u271d\u00b9 : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d\u00b9 : \u2200 (n : \u2115) (x : M), npow\u271d\u00b9 (n + 1) x = x * npow\u271d\u00b9 n x\nmul_comm\u271d\u00b9 : \u2200 (a b : M), a * b = b * a\ntoOne\u271d : One M\nnpow\u271d : \u2115 \u2192 M \u2192 M\nnpow_zero\u271d : \u2200 (x : M), npow\u271d 0 x = 1\ntoSemigroup\u271d : Semigroup M\nmul_left_cancel\u271d : \u2200 (a b c : M), a * b = a * c \u2192 b = c\none_mul\u271d : \u2200 (a : M), 1 * a = a\nmul_one\u271d : \u2200 (a : M), a * 1 = a\nnpow_succ\u271d : \u2200 (n : \u2115) (x : M), npow\u271d (n + 1) x = x * npow\u271d n x\nmul_comm\u271d : \u2200 (a b : M), a * b = b * a\nh' :\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow =\n Monoid.mk (_ : \u2200 (a : M), 1 * a = a) (_ : \u2200 (a : M), a * 1 = a) LeftCancelMonoid.npow\n\u22a2 npow\u271d\u00b9 = npow\u271d\n[PROOFSTEP]\ninjection h'\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave h_mon := Monoid.ext h_mul\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave h\u2081 : m\u2081.one = m\u2082.one := congr_arg (\u00b7.one) h_mon\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nlet f : @MonoidHom M M m\u2081.toMulOneClass m\u2082.toMulOneClass :=\n @MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h\u2081) (fun x y => congr_fun (congr_fun h_mul x) y)\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2081.npow = m\u2082.npow := congr_arg (\u00b7.npow) h_mon\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : Monoid.npow = Monoid.npow\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2081.zpow = m\u2082.zpow := by\n ext m x\n exact @MonoidHom.map_zpow' M M m\u2081 m\u2082 f (congr_fun h_inv) x m\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : Monoid.npow = Monoid.npow\n\u22a2 zpow = zpow\n[PROOFSTEP]\next m x\n[GOAL]\ncase h.h\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis : Monoid.npow = Monoid.npow\nm : \u2124\nx : M\n\u22a2 zpow m x = zpow m x\n[PROOFSTEP]\nexact @MonoidHom.map_zpow' M M m\u2081 m\u2082 f (congr_fun h_inv) x m\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d : Monoid.npow = Monoid.npow\nthis : zpow = zpow\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2081.div = m\u2082.div := by\n ext a b\n exact @map_div' _ _ (@MonoidHom _ _ (_) _) (id _) _ (@MonoidHom.monoidHomClass _ _ (_) _) f (congr_fun h_inv) a b\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d : Monoid.npow = Monoid.npow\nthis : zpow = zpow\n\u22a2 Div.div = Div.div\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.h\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d : Monoid.npow = Monoid.npow\nthis : zpow = zpow\na b : M\n\u22a2 Div.div a b = Div.div a b\n[PROOFSTEP]\nexact @map_div' _ _ (@MonoidHom _ _ (_) _) (id _) _ (@MonoidHom.monoidHomClass _ _ (_) _) f (congr_fun h_inv) a b\n[GOAL]\nM : Type u_1\nm\u2081 m\u2082 : DivInvMonoid M\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d\u00b9 : Monoid.npow = Monoid.npow\nthis\u271d : zpow = zpow\nthis : Div.div = Div.div\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nrcases m\u2081 with @\u27e8_, \u27e8_\u27e9, \u27e8_\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk\nM : Type u_1\nm\u2082 : DivInvMonoid M\ntoMonoid\u271d : Monoid M\nzpow\u271d : \u2124 \u2192 M \u2192 M\nzpow_zero'\u271d : \u2200 (a : M), zpow\u271d 0 a = 1\nzpow_succ'\u271d : \u2200 (n : \u2115) (a : M), zpow\u271d (Int.ofNat (Nat.succ n)) a = a * zpow\u271d (Int.ofNat n) a\ninv\u271d : M \u2192 M\nzpow_neg'\u271d : \u2200 (n : \u2115) (a : M), zpow\u271d (Int.negSucc n) a = (zpow\u271d (\u2191(Nat.succ n)) a)\u207b\u00b9\ndiv\u271d : M \u2192 M \u2192 M\ndiv_eq_mul_inv\u271d : \u2200 (a b : M), a / b = a * b\u207b\u00b9\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d\u00b9 : Monoid.npow = Monoid.npow\nthis\u271d : zpow = zpow\nthis : Div.div = Div.div\n\u22a2 mk zpow\u271d = m\u2082\n[PROOFSTEP]\nrcases m\u2082 with @\u27e8_, \u27e8_\u27e9, \u27e8_\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nM : Type u_1\ntoMonoid\u271d\u00b9 : Monoid M\nzpow\u271d\u00b9 : \u2124 \u2192 M \u2192 M\nzpow_zero'\u271d\u00b9 : \u2200 (a : M), zpow\u271d\u00b9 0 a = 1\nzpow_succ'\u271d\u00b9 : \u2200 (n : \u2115) (a : M), zpow\u271d\u00b9 (Int.ofNat (Nat.succ n)) a = a * zpow\u271d\u00b9 (Int.ofNat n) a\ninv\u271d\u00b9 : M \u2192 M\nzpow_neg'\u271d\u00b9 : \u2200 (n : \u2115) (a : M), zpow\u271d\u00b9 (Int.negSucc n) a = (zpow\u271d\u00b9 (\u2191(Nat.succ n)) a)\u207b\u00b9\ndiv\u271d\u00b9 : M \u2192 M \u2192 M\ndiv_eq_mul_inv\u271d\u00b9 : \u2200 (a b : M), a / b = a * b\u207b\u00b9\ntoMonoid\u271d : Monoid M\nzpow\u271d : \u2124 \u2192 M \u2192 M\nzpow_zero'\u271d : \u2200 (a : M), zpow\u271d 0 a = 1\nzpow_succ'\u271d : \u2200 (n : \u2115) (a : M), zpow\u271d (Int.ofNat (Nat.succ n)) a = a * zpow\u271d (Int.ofNat n) a\ninv\u271d : M \u2192 M\nzpow_neg'\u271d : \u2200 (n : \u2115) (a : M), zpow\u271d (Int.negSucc n) a = (zpow\u271d (\u2191(Nat.succ n)) a)\u207b\u00b9\ndiv\u271d : M \u2192 M \u2192 M\ndiv_eq_mul_inv\u271d : \u2200 (a b : M), a / b = a * b\u207b\u00b9\nh_mul : Mul.mul = Mul.mul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh\u2081 : One.one = One.one\nf : M \u2192* M := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : M), Mul.mul x y = Mul.mul x y) }\nthis\u271d\u00b9 : Monoid.npow = Monoid.npow\nthis\u271d : zpow = zpow\nthis : Div.div = Div.div\n\u22a2 mk zpow\u271d\u00b9 = mk zpow\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nG : Type u_1\ng\u2081 g\u2082 : Group G\nh_mul : Mul.mul = Mul.mul\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nhave h\u2081 : g\u2081.one = g\u2082.one := congr_arg (\u00b7.one) (Monoid.ext h_mul)\n[GOAL]\nG : Type u_1\ng\u2081 g\u2082 : Group G\nh_mul : Mul.mul = Mul.mul\nh\u2081 : One.one = One.one\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nlet f : @MonoidHom G G g\u2081.toMulOneClass g\u2082.toMulOneClass :=\n @MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h\u2081) (fun x y => congr_fun (congr_fun h_mul x) y)\n[GOAL]\nG : Type u_1\ng\u2081 g\u2082 : Group G\nh_mul : Mul.mul = Mul.mul\nh\u2081 : One.one = One.one\nf : G \u2192* G := { toOneHom := { toFun := id, map_one' := h\u2081 }, map_mul' := (_ : \u2200 (x y : G), Mul.mul x y = Mul.mul x y) }\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nexact\n Group.toDivInvMonoid_injective (DivInvMonoid.ext h_mul (funext <| @MonoidHom.map_inv G G g\u2081 g\u2082.toDivisionMonoid f))\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Ext", "llama_tokens": 14486, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.29313835555535167}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\n\u22a2 \u2200 (x : \u2191(toTopCat X)),\n \u2203 U R, Nonempty (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245 Spec.toLocallyRingedSpace.obj (op R))\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\n\u22a2 \u2203 U R, Nonempty (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245 Spec.toLocallyRingedSpace.obj (op R))\n[PROOFSTEP]\nobtain \u27e8R, f, h\u2081, h\u2082\u27e9 := h x\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 \u2203 U R, Nonempty (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245 Spec.toLocallyRingedSpace.obj (op R))\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8_, h\u2082.base_open.open_range\u27e9, h\u2081\u27e9, R, \u27e8_\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n { obj := { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) },\n property := h\u2081 }.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)\n[PROOFSTEP]\napply LocallyRingedSpace.isoOfSheafedSpaceIso\n[GOAL]\ncase intro.intro.intro.f\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 (restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n { obj := { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) },\n property := h\u2081 }.obj))).toSheafedSpace \u2245\n (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace\n[PROOFSTEP]\nrefine' SheafedSpace.forgetToPresheafedSpace.preimageIso _\n[GOAL]\ncase intro.intro.intro.f\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.obj\n (restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n { obj := { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) },\n property := h\u2081 }.obj))).toSheafedSpace \u2245\n SheafedSpace.forgetToPresheafedSpace.obj (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro.intro.intro.f\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.obj\n (restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n { obj := { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) },\n property := h\u2081 }.obj))).toSheafedSpace \u2245\n SheafedSpace.forgetToPresheafedSpace.obj (Spec.toLocallyRingedSpace.obj (op R)).toSheafedSpace\n[PROOFSTEP]\napply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1\n[GOAL]\ncase intro.intro.intro.f\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 Set.range \u2191(PresheafedSpace.ofRestrict X.toPresheafedSpace ?m.3415).base = Set.range \u2191f.val.base\n[PROOFSTEP]\nexact Subtype.range_coe_subtype\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : LocallyRingedSpace\nh : \u2200 (x : \u2191(toTopCat X)), \u2203 R f, x \u2208 Set.range \u2191f.val.base \u2227 IsOpenImmersion f\nx : \u2191(toTopCat X)\nR : CommRingCat\nf : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X\nh\u2081 : x \u2208 Set.range \u2191f.val.base\nh\u2082 : IsOpenImmersion f\n\u22a2 OpenEmbedding\n \u2191(Opens.inclusion\n { obj := { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) },\n property := h\u2081 }.obj)\n[PROOFSTEP]\nexact\n Opens.openEmbedding\n _\n -- Porting note : was `infer_instance`\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace),\n x \u2208\n Set.range\n \u2191((fun x =>\n (Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv \u226b\n LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)))\n ((fun x => x) x)).val.base\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n \u2191((fun x =>\n (Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv \u226b\n LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)))\n ((fun x => x) x)).val.base\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n (\u2191(LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj))).val.base \u2218\n \u2191(Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base)\n[PROOFSTEP]\nrw [Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n \u2191(LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj))).val.base\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base\n[PROOFSTEP]\nerw [Subtype.range_coe_subtype]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n {x_1 |\n x_1 \u2208\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj}\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base\n[PROOFSTEP]\nexact (X.local_affine x).choose.2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Epi\n (Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv.val.base\n[PROOFSTEP]\nchange Epi ((SheafedSpace.forget _).map (LocallyRingedSpace.forgetToSheafedSpace.map _))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Epi\n ((SheafedSpace.forget CommRingCat).map\n (LocallyRingedSpace.forgetToSheafedSpace.map\n (Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsOpenImmersion\n ((fun x =>\n (Nonempty.some\n (_ :\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj\n (op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))))).inv \u226b\n LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)))\n x)\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) PresheafedSpace.IsOpenImmersion.comp\n[GOAL]\ncase hg\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 PresheafedSpace.IsOpenImmersion\n (LocallyRingedSpace.ofRestrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj))).val\n[PROOFSTEP]\napply PresheafedSpace.IsOpenImmersion.ofRestrict\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n \u2191((fun x => map (f x.fst) x.snd \u226b map \ud835\udcb0 x.fst)\n ((fun x =>\n { fst := AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x,\n snd :=\n AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (Exists.choose\n (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)) })\n x)).val.base\n[PROOFSTEP]\nlet y := (\ud835\udcb0.Covers x).choose\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\n\u22a2 x \u2208\n Set.range\n \u2191((fun x => map (f x.fst) x.snd \u226b map \ud835\udcb0 x.fst)\n ((fun x =>\n { fst := AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x,\n snd :=\n AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (Exists.choose\n (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)) })\n x)).val.base\n[PROOFSTEP]\nhave hy : (\ud835\udcb0.map (\ud835\udcb0.f x)).val.base y = x := (\ud835\udcb0.Covers x).choose_spec\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\nhy : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base y = x\n\u22a2 x \u2208\n Set.range\n \u2191((fun x => map (f x.fst) x.snd \u226b map \ud835\udcb0 x.fst)\n ((fun x =>\n { fst := AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x,\n snd :=\n AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (Exists.choose\n (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)) })\n x)).val.base\n[PROOFSTEP]\nrcases(f (\ud835\udcb0.f x)).Covers y with \u27e8z, hz\u27e9\n[GOAL]\ncase intro\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\nhy : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base y = x\nz :\n (forget TopCat).obj\n \u2191(obj (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).toPresheafedSpace\nhz :\n \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).val.base\n z =\n y\n\u22a2 x \u2208\n Set.range\n \u2191((fun x => map (f x.fst) x.snd \u226b map \ud835\udcb0 x.fst)\n ((fun x =>\n { fst := AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x,\n snd :=\n AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (Exists.choose\n (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)) })\n x)).val.base\n[PROOFSTEP]\nchange x \u2208 Set.range ((f (\ud835\udcb0.f x)).map ((f (\ud835\udcb0.f x)).f y) \u226b \ud835\udcb0.map (\ud835\udcb0.f x)).1.base\n[GOAL]\ncase intro\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\nhy : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base y = x\nz :\n (forget TopCat).obj\n \u2191(obj (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).toPresheafedSpace\nhz :\n \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).val.base\n z =\n y\n\u22a2 x \u2208\n Set.range\n \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y) \u226b\n map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base\n[PROOFSTEP]\nuse z\n[GOAL]\ncase h\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\nhy : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base y = x\nz :\n (forget TopCat).obj\n \u2191(obj (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).toPresheafedSpace\nhz :\n \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).val.base\n z =\n y\n\u22a2 \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y) \u226b\n map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base\n z =\n x\n[PROOFSTEP]\nerw [comp_apply]\n[GOAL]\ncase h\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (map \ud835\udcb0 x \u226b f\u271d) g\nf : (x : \ud835\udcb0.J) \u2192 OpenCover (obj \ud835\udcb0 x)\nx : \u2191\u2191X.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).toPresheafedSpace :=\n Exists.choose (_ : x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base)\nhy : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base y = x\nz :\n (forget TopCat).obj\n \u2191(obj (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).toPresheafedSpace\nhz :\n \u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).val.base\n z =\n y\n\u22a2 \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base\n (\u2191(map (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x))\n (AlgebraicGeometry.Scheme.OpenCover.f (f (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) y)).val.base\n z) =\n x\n[PROOFSTEP]\nrw [hz, hy]\n -- Porting note : weirdly, even though no input is needed, `inferInstance` does not work\n -- `PresheafedSpace.IsOpenImmersion.comp` is marked as `instance`\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 x \u2208 Set.range \u2191((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\nrw [Set.range_iff_surjective.mpr]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 x \u2208 Set.univ\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 Function.Surjective \u2191((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\nall_goals try trivial\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 x \u2208 Set.univ\n[PROOFSTEP]\ntry trivial\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 x \u2208 Set.univ\n[PROOFSTEP]\ntrivial\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 Function.Surjective \u2191((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\ntry trivial\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 Function.Surjective \u2191((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\ntrivial\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 Function.Surjective \u2191((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 Epi ((fun x => f) ((fun x => PUnit.unit) x)).val.base\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 Set.range \u2191(map ((fun x => \u2191e\u2081.symm (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) x)).val.base\n[PROOFSTEP]\nrw [e\u2082, Scheme.comp_val_base, coe_comp, Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ,\n e\u2081.rightInverse_symm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 Set.range \u2191(AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base\n[PROOFSTEP]\nexact \ud835\udcb0.Covers x\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Surjective \u2191(e\u2082\u271d ((fun x => \u2191e\u2081.symm (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) x)).hom.val.base\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Epi (e\u2082\u271d ((fun x => \u2191e\u2081.symm (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)) x)).hom.val.base\n[PROOFSTEP]\ninfer_instance\n -- Porting note : weirdly, even though no input is needed, `inferInstance` does not work\n -- `PresheafedSpace.IsOpenImmersion.comp` is marked as `instance`\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\ni : J\n\u22a2 IsOpenImmersion (map i)\n[PROOFSTEP]\nrw [e\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nJ : Type u_1\nobj : J \u2192 Scheme\nmap : (i : J) \u2192 obj i \u27f6 X\ne\u2081 : J \u2243 \ud835\udcb0.J\ne\u2082\u271d : (i : J) \u2192 obj i \u2245 AlgebraicGeometry.Scheme.OpenCover.obj \ud835\udcb0 (\u2191e\u2081 i)\ne\u2082 : \u2200 (i : J), map i = (e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i)\ni : J\n\u22a2 IsOpenImmersion ((e\u2082\u271d i).hom \u226b AlgebraicGeometry.Scheme.OpenCover.map \ud835\udcb0 (\u2191e\u2081 i))\n[PROOFSTEP]\nexact PresheafedSpace.IsOpenImmersion.comp _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f\u271d) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : Y \u27f6 X\ninst\u271d : IsOpenImmersion f\n\u22a2 \u2200 (x : Option \ud835\udcb0.J), IsOpenImmersion ((fun i => Option.rec f \ud835\udcb0.map i) x)\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f\u271d) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : Y \u27f6 X\ninst\u271d : IsOpenImmersion f\n\u22a2 IsOpenImmersion ((fun i => Option.rec f \ud835\udcb0.map i) none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f\u271d) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : Y \u27f6 X\ninst\u271d : IsOpenImmersion f\nval\u271d : \ud835\udcb0.J\n\u22a2 IsOpenImmersion ((fun i => Option.rec f \ud835\udcb0.map i) (some val\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f\u271d) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : Y \u27f6 X\ninst\u271d : IsOpenImmersion f\n\u22a2 IsOpenImmersion f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase some\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\ninst\u271d\u00b9 : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f\u271d) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nY : Scheme\nf : Y \u27f6 X\ninst\u271d : IsOpenImmersion f\nval\u271d : \ud835\udcb0.J\n\u22a2 IsOpenImmersion (map \ud835\udcb0 val\u271d)\n[PROOFSTEP]\ninfer_instance\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nR : CommRingCat\nf : \u2191R\n\u22a2 IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op)\n[PROOFSTEP]\napply SheafedSpace.IsOpenImmersion.of_stalk_iso (H := ?_)\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nR : CommRingCat\nf : \u2191R\n\u22a2 OpenEmbedding \u2191(Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op).val.base\n[PROOFSTEP]\nexact (PrimeSpectrum.localization_away_openEmbedding (Localization.Away f) f : _)\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nR : CommRingCat\nf : \u2191R\nhf : OpenEmbedding \u2191(Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op).val.base\n\u22a2 \u2200 (x : \u2191\u2191(Spec.obj (op (CommRingCat.of (Localization.Away f)))).toPresheafedSpace),\n IsIso (PresheafedSpace.stalkMap (Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op).val x)\n[PROOFSTEP]\nintro x\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 : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f\u271d) g\nR : CommRingCat\nf : \u2191R\nhf : OpenEmbedding \u2191(Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op).val.base\nx : \u2191\u2191(Spec.obj (op (CommRingCat.of (Localization.Away f)))).toPresheafedSpace\n\u22a2 IsIso (PresheafedSpace.stalkMap (Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away f))).op).val x)\n[PROOFSTEP]\nexact Spec_map_localization_isIso R (Submonoid.powers f) x\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nR : CommRingCat\nr : \u2191\u2191(Spec.obj (op R)).toPresheafedSpace\n\u22a2 r \u2208 Set.range \u2191((fun r => Spec.map (algebraMap (\u2191R) (Localization.Away r)).op) ((fun x => 1) r)).val.base\n[PROOFSTEP]\nrw [Set.range_iff_surjective.mpr ((TopCat.epi_iff_surjective _).mp _)]\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nR : CommRingCat\nr : \u2191\u2191(Spec.obj (op R)).toPresheafedSpace\n\u22a2 r \u2208 Set.univ\n[PROOFSTEP]\nexact trivial\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nR : CommRingCat\nr : \u2191\u2191(Spec.obj (op R)).toPresheafedSpace\n\u22a2 Epi ((fun r => Spec.map (algebraMap (\u2191R) (Localization.Away r)).op) ((fun x => 1) r)).val.base\n[PROOFSTEP]\nchange Epi (Spec.map (CommRingCat.ofHom (algebraMap _ _)).op).1.base\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nR : CommRingCat\nr : \u2191\u2191(Spec.obj (op R)).toPresheafedSpace\n\u22a2 Epi (Spec.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization.Away ((fun x => 1) r)))).op).val.base\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\nr :\n \u2191(Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))\n\u22a2 Set.range \u2191(OpenCover.map (affineBasisCover X) { fst := x, snd := r }).val.base =\n \u2191(OpenCover.map (affineCover X) x).val.base '' (PrimeSpectrum.basicOpen r).carrier\n[PROOFSTEP]\nerw [coe_comp, Set.range_comp]\n -- Porting note : `congr` fails to see the goal is comparing image of the same function\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\nr :\n \u2191(Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))\n\u22a2 \u2191(OpenCover.map (affineCover X) { fst := x, snd := r }.fst).val.base ''\n Set.range\n \u2191(OpenCover.map\n ((fun x =>\n affineBasisCoverOfAffine\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))\n { fst := x, snd := r }.fst)\n { fst := x, snd := r }.snd).val.base =\n \u2191(OpenCover.map (affineCover X) x).val.base '' (PrimeSpectrum.basicOpen r).carrier\n[PROOFSTEP]\nrefine congr_arg (_ '' \u00b7) ?_\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\nx : \u2191\u2191X.toPresheafedSpace\nr :\n \u2191(Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))\n\u22a2 Set.range\n \u2191(OpenCover.map\n ((fun x =>\n affineBasisCoverOfAffine\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))))\n { fst := x, snd := r }.fst)\n { fst := x, snd := r }.snd).val.base =\n (PrimeSpectrum.basicOpen r).carrier\n[PROOFSTEP]\nexact (PrimeSpectrum.localization_away_comap_range (Localization.Away r) r : _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\n\u22a2 IsTopologicalBasis {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base}\n[PROOFSTEP]\napply TopologicalSpace.isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\n\u22a2 \u2200 (u : Set \u2191\u2191X.toPresheafedSpace),\n u \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8a, rfl\u27e9\n[GOAL]\ncase h_open.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : (affineBasisCover X).J\n\u22a2 IsOpen (Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base)\n[PROOFSTEP]\nexact IsOpenImmersion.open_range (X.affineBasisCover.map a)\n[GOAL]\ncase h_nhds\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\n\u22a2 \u2200 (a : \u2191\u2191X.toPresheafedSpace) (u : Set \u2191\u2191X.toPresheafedSpace),\n a \u2208 u \u2192\n IsOpen u \u2192 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro a U haU hU\n[GOAL]\ncase h_nhds\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\n\u22a2 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrcases X.affineCover.Covers a with \u27e8x, e\u27e9\n[GOAL]\ncase h_nhds.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\n\u22a2 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nlet U' := (X.affineCover.map (X.affineCover.f a)).1.base \u207b\u00b9' U\n[GOAL]\ncase h_nhds.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\n\u22a2 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nhave hxU' : x \u2208 U' := by rw [\u2190 e] at haU ; exact haU\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\n\u22a2 x \u2208 U'\n[PROOFSTEP]\nrw [\u2190 e] at haU \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\nhaU : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x \u2208 U\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\n\u22a2 x \u2208 U'\n[PROOFSTEP]\nexact haU\n[GOAL]\ncase h_nhds.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\n\u22a2 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrcases PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hxU'\n ((X.affineCover.map (X.affineCover.f a)).1.base.continuous_toFun.isOpen_preimage _ hU) with\n \u27e8_, \u27e8_, \u27e8s, rfl\u27e9, rfl\u27e9, hxV, hVU\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 \u2203 v, v \u2208 {x | \u2203 a, x = Set.range \u2191(OpenCover.map (affineBasisCover X) a).val.base} \u2227 a \u2208 v \u2227 v \u2286 U\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8\u27e8_, s\u27e9, rfl\u27e9, _, _\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 a \u2208 Set.range \u2191(OpenCover.map (affineBasisCover X) { fst := OpenCover.f (affineCover X) a, snd := s }).val.base\n[PROOFSTEP]\nerw [affineBasisCover_map_range]\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 Set.range \u2191(OpenCover.map (affineBasisCover X) { fst := OpenCover.f (affineCover X) a, snd := s }).val.base \u2286 U\n[PROOFSTEP]\nerw [affineBasisCover_map_range]\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 a \u2208 \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base '' (PrimeSpectrum.basicOpen s).carrier\n[PROOFSTEP]\nexact \u27e8x, hxV, e\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base '' (PrimeSpectrum.basicOpen s).carrier \u2286 U\n[PROOFSTEP]\nrw [Set.image_subset_iff]\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0 : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nX : Scheme\na : \u2191\u2191X.toPresheafedSpace\nU : Set \u2191\u2191X.toPresheafedSpace\nhaU : a \u2208 U\nhU : IsOpen U\nx : (forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace\ne : \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base x = a\nU' : Set ((forget TopCat).obj \u2191(OpenCover.obj (affineCover X) (OpenCover.f (affineCover X) a)).toPresheafedSpace) :=\n \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\nhxU' : x \u2208 U'\ns :\n \u2191(op\n (Exists.choose\n (_ :\n \u2203 R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n (Exists.choose\n (_ :\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict X.toLocallyRingedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R)))).obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))))).unop\nhxV : x \u2208 \u2191(PrimeSpectrum.basicOpen s)\nhVU : \u2191(PrimeSpectrum.basicOpen s) \u2286 U'\n\u22a2 (PrimeSpectrum.basicOpen s).carrier \u2286 \u2191(OpenCover.map (affineCover X) (OpenCover.f (affineCover X) a)).val.base \u207b\u00b9' U\n[PROOFSTEP]\nexact hVU\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 OpenCover X\n[PROOFSTEP]\nhave :=\n @CompactSpace.elim_nhds_subcover _ _ H (fun x : X => Set.range (\ud835\udcb0.map (\ud835\udcb0.f x)).1.base) fun x =>\n (IsOpenImmersion.open_range (\ud835\udcb0.map (\ud835\udcb0.f x))).mem_nhds (\ud835\udcb0.Covers x)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\n\u22a2 OpenCover X\n[PROOFSTEP]\nlet t := this.choose\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\n\u22a2 OpenCover X\n[PROOFSTEP]\nhave h : \u2200 x : X, \u2203 y : t, x \u2208 Set.range (\ud835\udcb0.map (\ud835\udcb0.f y)).1.base :=\n by\n intro x\n have h' : x \u2208 (\u22a4 : Set X) := trivial\n rw [\u2190 Classical.choose_spec this, Set.mem_iUnion] at h' \n rcases h' with \u27e8y, _, \u27e8hy, rfl\u27e9, hy'\u27e9\n exact \u27e8\u27e8y, hy\u27e9, hy'\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace), \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n[PROOFSTEP]\nhave h' : x \u2208 (\u22a4 : Set X) := trivial\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\nx : \u2191\u2191X.toPresheafedSpace\nh' : x \u2208 \u22a4\n\u22a2 \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n[PROOFSTEP]\nrw [\u2190 Classical.choose_spec this, Set.mem_iUnion] at h' \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\nx : \u2191\u2191X.toPresheafedSpace\nh' :\n \u2203 i,\n x \u2208\n \u22c3 (_ : i \u2208 Classical.choose this),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) i\n\u22a2 \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n[PROOFSTEP]\nrcases h' with \u27e8y, _, \u27e8hy, rfl\u27e9, hy'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\nx y : \u2191\u2191X.toPresheafedSpace\nhy : y \u2208 Classical.choose this\nhy' : x \u2208 (fun h => (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) y) hy\n\u22a2 \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n[PROOFSTEP]\nexact \u27e8\u27e8y, hy\u27e9, hy'\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y Z : Scheme\n\ud835\udcb0\u271d : OpenCover X\u271d\nf : X\u271d \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (x : \ud835\udcb0\u271d.J), HasPullback (map \ud835\udcb0\u271d x \u226b f) g\nX : Scheme\n\ud835\udcb0 : OpenCover X\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\nthis :\n \u2203 t,\n \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t),\n (fun x => Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 x)).val.base) x =\n \u22a4\nt : Finset \u2191\u2191X.toPresheafedSpace := Exists.choose this\nh : \u2200 (x : \u2191\u2191X.toPresheafedSpace), \u2203 y, x \u2208 Set.range \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 \u2191y)).val.base\n\u22a2 OpenCover X\n[PROOFSTEP]\nexact\n { J := t\n obj := fun x => \ud835\udcb0.obj (\ud835\udcb0.f x.1)\n map := fun x => \ud835\udcb0.map (\ud835\udcb0.f x.1)\n f := fun x => (h x).choose\n Covers := fun x => (h x).choose_spec }\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 Fintype (OpenCover.finiteSubcover \ud835\udcb0).J\n[PROOFSTEP]\ndelta OpenCover.finiteSubcover\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 (x : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 x \u226b f) g\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 Fintype\n (let_fun this :=\n (_ :\n \u2203 t, \u22c3 (x : \u2191\u2191X.toPresheafedSpace) (_ : x \u2208 t), Set.range \u2191(OpenCover.map \ud835\udcb0 (OpenCover.f \ud835\udcb0 x)).val.base = \u22a4);\n let t := Exists.choose this;\n let_fun h :=\n (_ : \u2200 (x : \u2191\u2191X.toPresheafedSpace), \u2203 y, x \u2208 Set.range \u2191(OpenCover.map \ud835\udcb0 (OpenCover.f \ud835\udcb0 \u2191y)).val.base);\n OpenCover.mk { x // x \u2208 t } (fun x => OpenCover.obj \ud835\udcb0 (OpenCover.f \ud835\udcb0 \u2191x))\n (fun x => OpenCover.map \ud835\udcb0 (OpenCover.f \ud835\udcb0 \u2191x))\n (fun x => Exists.choose (_ : \u2203 y, x \u2208 Set.range \u2191(OpenCover.map \ud835\udcb0 (OpenCover.f \ud835\udcb0 \u2191y)).val.base))\n (_ :\n \u2200 (x : \u2191\u2191X.toPresheafedSpace),\n x \u2208\n Set.range\n \u2191(OpenCover.map \ud835\udcb0\n (OpenCover.f \ud835\udcb0\n \u2191(Exists.choose\n (_ : \u2203 y, x \u2208 Set.range \u2191(OpenCover.map \ud835\udcb0 (OpenCover.f \ud835\udcb0 \u2191y)).val.base)))).val.base)).J\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\n\u22a2 Scheme\n[PROOFSTEP]\napply LocallyRingedSpace.IsOpenImmersion.scheme (toLocallyRingedSpace _ f)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\n\u22a2 \u2200 (x : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))),\n \u2203 R f_1, x \u2208 Set.range \u2191f_1.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f_1\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\n\u22a2 \u2203 R f_1, x \u2208 Set.range \u2191f_1.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f_1\n[PROOFSTEP]\nobtain \u27e8_, \u27e8i, rfl\u27e9, hx, hi\u27e9 :=\n Y.affineBasisCover_is_basis.exists_subset_of_mem_open (Set.mem_range_self x) H.base_open.open_range\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 \u2203 R f_1, x \u2208 Set.range \u2191f_1.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f_1\n[PROOFSTEP]\nuse Y.affineBasisCoverRing i\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 \u2203 f_1, x \u2208 Set.range \u2191f_1.val.base \u2227 LocallyRingedSpace.IsOpenImmersion f_1\n[PROOFSTEP]\nuse LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom _ f) _ hi\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 x \u2208\n Set.range\n \u2191(LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom Y.toLocallyRingedSpace f)\n (Scheme.OpenCover.map (Scheme.affineBasisCover Y) i) hi).val.base \u2227\n LocallyRingedSpace.IsOpenImmersion\n (LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom Y.toLocallyRingedSpace f)\n (Scheme.OpenCover.map (Scheme.affineBasisCover Y) i) hi)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 x \u2208\n Set.range\n \u2191(LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom Y.toLocallyRingedSpace f)\n (Scheme.OpenCover.map (Scheme.affineBasisCover Y) i) hi).val.base\n[PROOFSTEP]\nrw [LocallyRingedSpace.IsOpenImmersion.lift_range]\n[GOAL]\ncase h.left\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 x \u2208\n \u2191(toLocallyRingedSpaceHom Y.toLocallyRingedSpace f).val.base \u207b\u00b9'\n Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase h.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 LocallyRingedSpace.IsOpenImmersion\n (LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom Y.toLocallyRingedSpace f)\n (Scheme.OpenCover.map (Scheme.affineBasisCover Y) i) hi)\n[PROOFSTEP]\ndelta LocallyRingedSpace.IsOpenImmersion.lift\n[GOAL]\ncase h.right\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH : IsOpenImmersion f\nx : \u2191(LocallyRingedSpace.toTopCat (toLocallyRingedSpace Y.toLocallyRingedSpace f))\ni : (Scheme.affineBasisCover Y).J\nhx : \u2191f.base x \u2208 Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base\nhi : Set.range \u2191(Scheme.OpenCover.map (Scheme.affineBasisCover Y) i).val.base \u2286 Set.range \u2191f.base\n\u22a2 LocallyRingedSpace.IsOpenImmersion\n (let_fun this := (_ : IsIso pullback.snd);\n inv pullback.snd \u226b pullback.fst)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d : PresheafedSpace CommRingCat\nY\u271d : Scheme\nf : X\u271d \u27f6 Y\u271d.toPresheafedSpace\nH\u271d : IsOpenImmersion f\nX Y : Scheme\nH : X.toLocallyRingedSpace = Y.toLocallyRingedSpace\n\u22a2 X = Y\n[PROOFSTEP]\ncases X\n[GOAL]\ncase mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY\u271d : Scheme\nf : X \u27f6 Y\u271d.toPresheafedSpace\nH\u271d : IsOpenImmersion f\nY : Scheme\ntoLocallyRingedSpace\u271d : LocallyRingedSpace\nlocal_affine\u271d :\n \u2200 (x : \u2191(LocallyRingedSpace.toTopCat toLocallyRingedSpace\u271d)),\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict toLocallyRingedSpace\u271d (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))\nH :\n { toLocallyRingedSpace := toLocallyRingedSpace\u271d, local_affine := local_affine\u271d }.toLocallyRingedSpace =\n Y.toLocallyRingedSpace\n\u22a2 { toLocallyRingedSpace := toLocallyRingedSpace\u271d, local_affine := local_affine\u271d } = Y\n[PROOFSTEP]\ncases Y\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : PresheafedSpace CommRingCat\nY : Scheme\nf : X \u27f6 Y.toPresheafedSpace\nH\u271d : IsOpenImmersion f\ntoLocallyRingedSpace\u271d\u00b9 : LocallyRingedSpace\nlocal_affine\u271d\u00b9 :\n \u2200 (x : \u2191(LocallyRingedSpace.toTopCat toLocallyRingedSpace\u271d\u00b9)),\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict toLocallyRingedSpace\u271d\u00b9 (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))\ntoLocallyRingedSpace\u271d : LocallyRingedSpace\nlocal_affine\u271d :\n \u2200 (x : \u2191(LocallyRingedSpace.toTopCat toLocallyRingedSpace\u271d)),\n \u2203 U R,\n Nonempty\n (LocallyRingedSpace.restrict toLocallyRingedSpace\u271d (_ : OpenEmbedding \u2191(Opens.inclusion U.obj)) \u2245\n Spec.toLocallyRingedSpace.obj (op R))\nH :\n { toLocallyRingedSpace := toLocallyRingedSpace\u271d\u00b9, local_affine := local_affine\u271d\u00b9 }.toLocallyRingedSpace =\n { toLocallyRingedSpace := toLocallyRingedSpace\u271d, local_affine := local_affine\u271d }.toLocallyRingedSpace\n\u22a2 { toLocallyRingedSpace := toLocallyRingedSpace\u271d\u00b9, local_affine := local_affine\u271d\u00b9 } =\n { toLocallyRingedSpace := toLocallyRingedSpace\u271d, local_affine := local_affine\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d : PresheafedSpace CommRingCat\nY\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d.toPresheafedSpace\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : AlgebraicGeometry.IsOpenImmersion f\n\u22a2 toScheme Y f.val = X\n[PROOFSTEP]\napply scheme_eq_of_locallyRingedSpace_eq\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d : PresheafedSpace CommRingCat\nY\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d.toPresheafedSpace\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : AlgebraicGeometry.IsOpenImmersion f\n\u22a2 (toScheme Y f.val).toLocallyRingedSpace = X.toLocallyRingedSpace\n[PROOFSTEP]\nexact locallyRingedSpace_toLocallyRingedSpace f\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nU : TopCat\nX : Scheme\nf : U \u27f6 TopCat.of \u2191\u2191X.toPresheafedSpace\nh : OpenEmbedding \u2191f\n\u22a2 PresheafedSpace.IsOpenImmersion (PresheafedSpace.ofRestrict X.toPresheafedSpace h)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsIso g\n\u22a2 IsIso ((inducedFunctor Scheme.toLocallyRingedSpace).map g)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 IsIso f \u2194 IsIso f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n[PROOFSTEP]\nrw [isIso_iff_isOpenImmersion, IsOpenImmersion.iff_stalk_iso, and_comm, \u2190 and_assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 ((Epi f.val.base \u2227 OpenEmbedding \u2191f.val.base) \u2227\n \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)) \u2194\n IsIso f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n[PROOFSTEP]\nrefine' and_congr \u27e8_, _\u27e9 Iff.rfl\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 Epi f.val.base \u2227 OpenEmbedding \u2191f.val.base \u2192 IsIso f.val.base\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase refine'_1.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\nh\u2081 : Epi f.val.base\nh\u2082 : OpenEmbedding \u2191f.val.base\n\u22a2 IsIso f.val.base\n[PROOFSTEP]\nconvert_to\n IsIso\n (TopCat.isoOfHomeo\n (Homeomorph.homeomorphOfContinuousOpen (Equiv.ofBijective _ \u27e8h\u2082.inj, (TopCat.epi_iff_surjective _).mp h\u2081\u27e9)\n h\u2082.continuous h\u2082.isOpenMap)).hom\n[GOAL]\ncase refine'_1.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\nh\u2081 : Epi f.val.base\nh\u2082 : OpenEmbedding \u2191f.val.base\n\u22a2 IsIso\n (TopCat.isoOfHomeo\n (Homeomorph.homeomorphOfContinuousOpen\n (Equiv.ofBijective \u2191f.val.base (_ : Function.Injective \u2191f.val.base \u2227 Function.Surjective \u2191f.val.base))\n (_ : Continuous \u2191f.val.base) (_ : IsOpenMap \u2191f.val.base))).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 IsIso f.val.base \u2192 Epi f.val.base \u2227 OpenEmbedding \u2191f.val.base\n[PROOFSTEP]\nintro H\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng : Y\u271d \u27f6 Z\nH\u271d : IsOpenImmersion f\u271d\nX Y : Scheme\nf : X \u27f6 Y\nH : IsIso f.val.base\n\u22a2 Epi f.val.base \u2227 OpenEmbedding \u2191f.val.base\n[PROOFSTEP]\nexact \u27e8inferInstance, (TopCat.homeoOfIso (asIso f.1.base)).openEmbedding\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Mono f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan f g \u22d9 forget)\n[PROOFSTEP]\napply @hasLimitOfIso _ _ _ _ _ _ ?_ (diagramIsoCospan.{u} _).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan ((cospan f g \u22d9 forget).map WalkingCospan.Hom.inl) ((cospan f g \u22d9 forget).map WalkingCospan.Hom.inr))\n[PROOFSTEP]\nchange HasLimit (cospan ((forget).map f) ((forget).map g))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan (forget.map f) (forget.map g))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan g f \u22d9 forget)\n[PROOFSTEP]\napply @hasLimitOfIso _ _ _ _ _ _ ?_ (diagramIsoCospan.{u} _).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan ((cospan g f \u22d9 forget).map Hom.inl) ((cospan g f \u22d9 forget).map Hom.inr))\n[PROOFSTEP]\nchange HasLimit (cospan ((forget).map g) ((forget).map f))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 HasLimit (cospan (forget.map g) (forget.map f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 forget.obj (PresheafedSpace.IsOpenImmersion.toScheme Y pullback.snd.val) =\n limit (cospan ((cospan f g \u22d9 forget).map Hom.inl) ((cospan f g \u22d9 forget).map Hom.inr))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 forget.obj (PresheafedSpace.IsOpenImmersion.toScheme Y pullback.fst.val) =\n limit (cospan ((cospan g f \u22d9 forget).map Hom.inl) ((cospan g f \u22d9 forget).map Hom.inr))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 IsOpenImmersion pullback.snd\n[PROOFSTEP]\nhave := PreservesPullback.iso_hom_snd forget f g\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\nthis : (PreservesPullback.iso forget f g).hom \u226b pullback.snd = forget.map pullback.snd\n\u22a2 IsOpenImmersion pullback.snd\n[PROOFSTEP]\ndsimp only [Scheme.forgetToLocallyRingedSpace, inducedFunctor_map] at this \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\nthis : (PreservesPullback.iso (inducedFunctor Scheme.toLocallyRingedSpace) f g).hom \u226b pullback.snd = pullback.snd\n\u22a2 IsOpenImmersion pullback.snd\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\nthis : (PreservesPullback.iso (inducedFunctor Scheme.toLocallyRingedSpace) f g).hom \u226b pullback.snd = pullback.snd\n\u22a2 IsOpenImmersion ((PreservesPullback.iso (inducedFunctor Scheme.toLocallyRingedSpace) f g).hom \u226b pullback.snd)\n[PROOFSTEP]\nchange LocallyRingedSpace.IsOpenImmersion _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\nthis : (PreservesPullback.iso (inducedFunctor Scheme.toLocallyRingedSpace) f g).hom \u226b pullback.snd = pullback.snd\n\u22a2 LocallyRingedSpace.IsOpenImmersion\n ((PreservesPullback.iso (inducedFunctor Scheme.toLocallyRingedSpace) f g).hom \u226b pullback.snd)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 IsOpenImmersion pullback.fst\n[PROOFSTEP]\nrw [\u2190 pullbackSymmetry_hom_comp_snd]\n -- Porting note : was just `infer_instance`, it is a bit weird that no explicit class instance is\n -- provided but still class inference fail to find this\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 IsOpenImmersion ((pullbackSymmetry g f).hom \u226b pullback.snd)\n[PROOFSTEP]\nexact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\n\u22a2 IsOpenImmersion (limit.\u03c0 (cospan f g) one)\n[PROOFSTEP]\nrw [\u2190 limit.w (cospan f g) WalkingCospan.Hom.inl]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\n\u22a2 IsOpenImmersion (limit.\u03c0 (cospan f g) left \u226b (cospan f g).map Hom.inl)\n[PROOFSTEP]\nchange\n IsOpenImmersion\n (_ \u226b f)\n -- Porting note : was just `infer_instance`, it is a bit weird that no explicit class instance is\n -- provided but still class inference fail to find this\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\n\u22a2 IsOpenImmersion (limit.\u03c0 (cospan f g) left \u226b f)\n[PROOFSTEP]\nexact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan f g) Scheme.forgetToTop\n[PROOFSTEP]\ndelta Scheme.forgetToTop\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan f g) (forget \u22d9 LocallyRingedSpace.forgetToTop)\n[PROOFSTEP]\napply @Limits.compPreservesLimit (K := cospan f g) (F := forget) (G := LocallyRingedSpace.forgetToTop) ?_ ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan f g) forget\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan f g \u22d9 forget) LocallyRingedSpace.forgetToTop\n[PROOFSTEP]\napply @preservesLimitOfIsoDiagram (F := _) _ _ _ _ _ _ (diagramIsoCospan.{u} _).symm ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan ((cospan f g \u22d9 forget).map Hom.inl) ((cospan f g \u22d9 forget).map Hom.inr))\n LocallyRingedSpace.forgetToTop\n[PROOFSTEP]\ndsimp [LocallyRingedSpace.forgetToTop]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 PreservesLimit (cospan f g) (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forget CommRingCat)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191pullback.snd.val.base =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nrw [\u2190 show _ = (pullback.snd : pullback f g \u27f6 _).1.base from PreservesPullback.iso_hom_snd Scheme.forgetToTop f g]\n -- Porting note : was `rw`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191((PreservesPullback.iso Scheme.forgetToTop f g).hom \u226b pullback.snd) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range (\u2191pullback.snd \u2218 \u2191(PreservesPullback.iso Scheme.forgetToTop f g).hom) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nrw [Set.range_comp, Set.range_iff_surjective.mpr, \u2190\n @Set.preimage_univ _ _ (pullback.fst : pullback f.1.base g.1.base \u27f6 _)]\n -- Porting note : was `rw`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191pullback.snd '' (\u2191pullback.fst \u207b\u00b9' Set.univ) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop f g).hom\n[PROOFSTEP]\nerw [TopCat.pullback_snd_image_fst_preimage]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191(Scheme.forgetToTop.map g) \u207b\u00b9' (\u2191(Scheme.forgetToTop.map f) '' Set.univ) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop f g).hom\n[PROOFSTEP]\nrw [Set.image_univ]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191(Scheme.forgetToTop.map g) \u207b\u00b9' Set.range \u2191(Scheme.forgetToTop.map f) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop f g).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop f g).hom\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Epi (PreservesPullback.iso Scheme.forgetToTop f g).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191pullback.fst.val.base =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nrw [\u2190 show _ = (pullback.fst : pullback g f \u27f6 _).1.base from PreservesPullback.iso_hom_fst Scheme.forgetToTop g f]\n -- Porting note : was `rw`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191((PreservesPullback.iso Scheme.forgetToTop g f).hom \u226b pullback.fst) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range (\u2191pullback.fst \u2218 \u2191(PreservesPullback.iso Scheme.forgetToTop g f).hom) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\n[PROOFSTEP]\nrw [Set.range_comp, Set.range_iff_surjective.mpr, \u2190\n @Set.preimage_univ _ _ (pullback.snd : pullback g.1.base f.1.base \u27f6 _)]\n -- Porting note : was `rw`\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191pullback.fst '' (\u2191pullback.snd \u207b\u00b9' Set.univ) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop g f).hom\n[PROOFSTEP]\nerw [TopCat.pullback_fst_image_snd_preimage]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191(Scheme.forgetToTop.map g) \u207b\u00b9' (\u2191(Scheme.forgetToTop.map f) '' Set.univ) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop g f).hom\n[PROOFSTEP]\nrw [Set.image_univ]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 \u2191(Scheme.forgetToTop.map g) \u207b\u00b9' Set.range \u2191(Scheme.forgetToTop.map f) =\n ((Opens.map g.val.base).obj\n { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }).carrier\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop g f).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Function.Surjective \u2191(PreservesPullback.iso Scheme.forgetToTop g f).hom\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Epi (PreservesPullback.iso Scheme.forgetToTop g f).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191(pullback.fst \u226b f).val.base = Set.range \u2191f.val.base \u2229 Set.range \u2191g.val.base\n[PROOFSTEP]\nrw [pullback.condition, Scheme.comp_val_base, coe_comp, Set.range_comp, range_pullback_snd_of_left,\n Opens.carrier_eq_coe, Opens.map_obj, Opens.coe_mk, Set.image_preimage_eq_inter_range, Set.inter_comm]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\n\u22a2 Set.range \u2191(pullback.fst \u226b g).val.base = Set.range \u2191g.val.base \u2229 Set.range \u2191f.val.base\n[PROOFSTEP]\nrw [Scheme.comp_val_base, coe_comp, Set.range_comp, range_pullback_fst_of_right, Opens.map_obj, Opens.carrier_eq_coe,\n Opens.coe_mk, Set.image_preimage_eq_inter_range, Set.inter_comm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\ne : Set.range \u2191f.val.base = Set.range \u2191g.val.base\n\u22a2 lift g f (_ : Set.range \u2191f.val.base \u2264 Set.range \u2191g.val.base) \u226b\n lift f g (_ : Set.range \u2191g.val.base \u2264 Set.range \u2191f.val.base) =\n \ud835\udfd9 X\n[PROOFSTEP]\nrw [\u2190 cancel_mono f]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\ne : Set.range \u2191f.val.base = Set.range \u2191g.val.base\n\u22a2 (lift g f (_ : Set.range \u2191f.val.base \u2264 Set.range \u2191g.val.base) \u226b\n lift f g (_ : Set.range \u2191g.val.base \u2264 Set.range \u2191f.val.base)) \u226b\n f =\n \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\ne : Set.range \u2191f.val.base = Set.range \u2191g.val.base\n\u22a2 lift f g (_ : Set.range \u2191g.val.base \u2264 Set.range \u2191f.val.base) \u226b\n lift g f (_ : Set.range \u2191f.val.base \u2264 Set.range \u2191g.val.base) =\n \ud835\udfd9 Y\n[PROOFSTEP]\nrw [\u2190 cancel_mono g]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Z\ng : Y \u27f6 Z\nH : IsOpenImmersion f\ninst\u271d : IsOpenImmersion g\ne : Set.range \u2191f.val.base = Set.range \u2191g.val.base\n\u22a2 (lift f g (_ : Set.range \u2191g.val.base \u2264 Set.range \u2191f.val.base) \u226b\n lift g f (_ : Set.range \u2191f.val.base \u2264 Set.range \u2191g.val.base)) \u226b\n g =\n \ud835\udfd9 Y \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH\u271d : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nfg : Y \u27f6 X\nH : fg = f \u226b g\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 (Opens.map f.val.base).obj V = (Opens.map fg.val.base).obj ((Scheme.Hom.opensFunctor g).obj V)\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 (Opens.map f.val.base).obj V = (Opens.map (f \u226b g).val.base).obj ((Scheme.Hom.opensFunctor g).obj V)\n[PROOFSTEP]\nrw [Scheme.comp_val_base, Opens.map_comp_obj]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 (Opens.map f.val.base).obj V =\n (Opens.map f.val.base).obj ((Opens.map g.val.base).obj ((Scheme.Hom.opensFunctor g).obj V))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 V = (Opens.map g.val.base).obj ((Scheme.Hom.opensFunctor g).obj V)\n[PROOFSTEP]\next1\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 \u2191V = \u2191((Opens.map g.val.base).obj ((Scheme.Hom.opensFunctor g).obj V))\n[PROOFSTEP]\nexact (Set.preimage_image_eq _ h.base_open.inj).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH\u271d : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nfg : Y \u27f6 X\nH : fg = f \u226b g\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 NatTrans.app f.val.c (op V) =\n Scheme.Hom.invApp g V \u226b\n NatTrans.app fg.val.c (op ((Scheme.Hom.opensFunctor g).obj V)) \u226b\n Y.presheaf.map\n (eqToHom\n (_ : (Opens.map f.val.base).obj V = (Opens.map fg.val.base).obj ((Scheme.Hom.opensFunctor g).obj V))).op\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 NatTrans.app f.val.c (op V) =\n Scheme.Hom.invApp g V \u226b\n NatTrans.app (f \u226b g).val.c (op ((Scheme.Hom.opensFunctor g).obj V)) \u226b\n Y.presheaf.map\n (eqToHom\n (_ :\n (Opens.map f.val.base).obj V = (Opens.map (f \u226b g).val.base).obj ((Scheme.Hom.opensFunctor g).obj V))).op\n[PROOFSTEP]\nrw [Scheme.comp_val_c_app, Category.assoc, Scheme.Hom.invApp, PresheafedSpace.IsOpenImmersion.invApp_app_assoc,\n f.val.c.naturality_assoc, TopCat.Presheaf.pushforwardObj_map, \u2190 Functor.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\n\u22a2 NatTrans.app f.val.c (op V) =\n NatTrans.app f.val.c (op V) \u226b\n Y.presheaf.map\n ((Opens.map f.val.base).op.map\n (eqToHom\n (_ :\n op V =\n op ((Opens.map g.val.base).obj (op ((PresheafedSpace.IsOpenImmersion.openFunctor h).obj V)).unop))) \u226b\n (eqToHom\n (_ :\n (Opens.map f.val.base).obj V =\n (Opens.map (f \u226b g).val.base).obj ((Scheme.Hom.opensFunctor g).obj V))).op)\n[PROOFSTEP]\nconvert (Category.comp_id <| f.1.c.app (op V)).symm\n[GOAL]\ncase h.e'_3.h.e'_7.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z : Scheme\nf\u271d : X\u271d \u27f6 Z\ng\u271d : Y\u271d \u27f6 Z\nH : IsOpenImmersion f\u271d\nX Y U : Scheme\nf : Y \u27f6 U\ng : U \u27f6 X\nh : IsOpenImmersion g\nV : Opens \u2191\u2191U.toPresheafedSpace\ne_5\u271d : Y.presheaf.obj (op ((Opens.map f.val.base).obj V)) = (f.val.base _* Y.presheaf).obj (op V)\n\u22a2 Y.presheaf.map\n ((Opens.map f.val.base).op.map\n (eqToHom\n (_ :\n op V =\n op ((Opens.map g.val.base).obj (op ((PresheafedSpace.IsOpenImmersion.openFunctor h).obj V)).unop))) \u226b\n (eqToHom\n (_ :\n (Opens.map f.val.base).obj V =\n (Opens.map (f \u226b g).val.base).obj ((Scheme.Hom.opensFunctor g).obj V))).op) =\n \ud835\udfd9 ((f.val.base _* Y.presheaf).obj (op V))\n[PROOFSTEP]\nconvert Y.presheaf.map_id _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\n\u22a2 (Hom.opensFunctor f).obj (basicOpen X r) = basicOpen Y (\u2191(Hom.invApp f U) r)\n[PROOFSTEP]\nhave e := Scheme.preimage_basicOpen f (Scheme.Hom.invApp f U r)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(Hom.invApp f U) r)) =\n basicOpen X (\u2191(NatTrans.app f.val.c (op ((Hom.opensFunctor f).obj U))) (\u2191(Hom.invApp f U) r))\n\u22a2 (Hom.opensFunctor f).obj (basicOpen X r) = basicOpen Y (\u2191(Hom.invApp f U) r)\n[PROOFSTEP]\nrw [Scheme.Hom.invApp] at e \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n basicOpen X\n (\u2191(NatTrans.app f.val.c (op ((Hom.opensFunctor f).obj U))) (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r))\n\u22a2 (Hom.opensFunctor f).obj (basicOpen X r) = basicOpen Y (\u2191(Hom.invApp f U) r)\n[PROOFSTEP]\nerw [PresheafedSpace.IsOpenImmersion.invApp_app_apply] at e \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n basicOpen X\n (\u2191(X.presheaf.map\n (eqToHom\n (_ :\n op U =\n op ((Opens.map f.val.base).obj (op ((PresheafedSpace.IsOpenImmersion.openFunctor H).obj U)).unop))))\n r)\n\u22a2 (Hom.opensFunctor f).obj (basicOpen X r) = basicOpen Y (\u2191(Hom.invApp f U) r)\n[PROOFSTEP]\nrw [Scheme.basicOpen_res, inf_eq_right.mpr _] at e \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 (Hom.opensFunctor f).obj (basicOpen X r) = basicOpen Y (\u2191(Hom.invApp f U) r)\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 (Hom.opensFunctor f).obj\n ((Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r))) =\n basicOpen Y (\u2191(Hom.invApp f U) r)\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\next1\n -- Porting note : this `dsimp` was not necessary\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 \u2191((Hom.opensFunctor f).obj\n ((Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)))) =\n \u2191(basicOpen Y (\u2191(Hom.invApp f U) r))\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\ndsimp [Opens.map]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 \u2191f.val.base '' (\u2191f.val.base \u207b\u00b9' \u2191(basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r))) =\n \u2191(basicOpen Y (\u2191(Hom.invApp f U) r))\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\nrefine' Set.image_preimage_eq_inter_range.trans _\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 \u2191(basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) \u2229 Set.range \u2191f.val.base =\n \u2191(basicOpen Y (\u2191(Hom.invApp f U) r))\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\nerw [Set.inter_eq_left_iff_subset]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne : (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) = basicOpen X r\n\u22a2 \u2191(basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) \u2286 Set.range \u2191f.val.base\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\nrefine' Set.Subset.trans (Scheme.basicOpen_le _ _) (Set.image_subset_range _ _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 basicOpen X r \u2264 (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\nrefine' le_trans (Scheme.basicOpen_le _ _) (le_of_eq _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 U = (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nU : Opens \u2191\u2191X.toPresheafedSpace\nr : \u2191(X.presheaf.obj (op U))\ne :\n (Opens.map f.val.base).obj (basicOpen Y (\u2191(PresheafedSpace.IsOpenImmersion.invApp H U) r)) =\n (Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U) \u2293 basicOpen X r\n\u22a2 \u2191U = \u2191((Opens.map f.val.base).obj ((Hom.opensFunctor f).obj U))\n[PROOFSTEP]\nexact (Set.preimage_image_eq _ H.base_open.inj).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base\n[PROOFSTEP]\ndsimp [ofRestrict, LocallyRingedSpace.ofRestrict, Opens.inclusion]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ContinuousMap.mk Subtype.val) \u2286 Set.range \u2191(ContinuousMap.mk Subtype.val)\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Subtype.range_val, Subtype.range_val]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 \u2191U \u2286 \u2191V\n[PROOFSTEP]\nexact i.le\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 { obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n (\ud835\udfd9 U) =\n \ud835\udfd9\n ({ obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.obj\n U)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 ({ obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n (\ud835\udfd9 U)).left =\n (\ud835\udfd9\n ({ obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.obj\n U)).left\n[PROOFSTEP]\ndsimp only [Over.homMk_left, Over.id_left]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base) =\n \ud835\udfd9 (Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))).left\n[PROOFSTEP]\nrw [\u2190 cancel_mono (X.ofRestrict U.openEmbedding), Category.id_comp, IsOpenImmersion.lift_fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 { obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n (i \u226b j) =\n { obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n i \u226b\n { obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n j\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 ({ obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n (i \u226b j)).left =\n ({ obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n i \u226b\n { obj := fun U => Over.mk (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))),\n map := fun {U V} i =>\n Over.homMk\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base)) }.map\n j).left\n[PROOFSTEP]\ndsimp only [Over.homMk_left, Over.comp_left]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) =\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base) \u226b\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base)\n[PROOFSTEP]\nrw [\u2190 cancel_mono (X.ofRestrict W.openEmbedding), Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)) =\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base) \u226b\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))\n[PROOFSTEP]\niterate 3 rw [IsOpenImmersion.lift_fac]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)) =\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base) \u226b\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))\n[PROOFSTEP]\nrw [IsOpenImmersion.lift_fac]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)) =\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base) \u226b\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion W))\n[PROOFSTEP]\nrw [IsOpenImmersion.lift_fac]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V W : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)) =\n IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ :\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base) \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))\n[PROOFSTEP]\nrw [IsOpenImmersion.lift_fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base\n[PROOFSTEP]\ndsimp [ofRestrict, LocallyRingedSpace.ofRestrict, Opens.inclusion]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ContinuousMap.mk Subtype.val) \u2286 Set.range \u2191(ContinuousMap.mk Subtype.val)\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Subtype.range_val, Subtype.range_val]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 \u2191U \u2286 \u2191V\n[PROOFSTEP]\nexact i.le\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base \u2286\n Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base\n[PROOFSTEP]\ndsimp [ofRestrict, LocallyRingedSpace.ofRestrict, Opens.inclusion]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 Set.range \u2191(ContinuousMap.mk Subtype.val) \u2286 Set.range \u2191(ContinuousMap.mk Subtype.val)\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Subtype.range_val, Subtype.range_val]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 \u2191U \u2286 \u2191V\n[PROOFSTEP]\nexact i.le\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\n\u22a2 ((restrictFunctor X).map i).left.val.base = (Opens.toTopCat \u2191X.toPresheafedSpace).map i\n[PROOFSTEP]\next a\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\na : (forget TopCat).obj \u2191((restrictFunctor X).obj U).left.toPresheafedSpace\n\u22a2 \u2191((restrictFunctor X).map i).left.val.base a = \u2191((Opens.toTopCat \u2191X.toPresheafedSpace).map i) a\n[PROOFSTEP]\nrefine\n Subtype.ext\n ?_\n -- Porting note : `ext` did not pick up `Subtype.ext`\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\na : (forget TopCat).obj \u2191((restrictFunctor X).obj U).left.toPresheafedSpace\n\u22a2 \u2191(\u2191((restrictFunctor X).map i).left.val.base a) = \u2191(\u2191((Opens.toTopCat \u2191X.toPresheafedSpace).map i) a)\n[PROOFSTEP]\nexact (congr_arg (fun f : X.restrict U.openEmbedding \u27f6 X => f.1.base a) (X.restrictFunctor_map_ofRestrict i))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W\n[PROOFSTEP]\nsimp only [\u2190 SetLike.coe_subset_coe, IsOpenMap.functor_obj_coe, Set.image_subset_iff, Scheme.restrictFunctor_map_base,\n Opens.map_coe, Opens.inclusion_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\n\u22a2 \u2191((Opens.toTopCat \u2191X.toPresheafedSpace).map i) \u207b\u00b9' \u2191W \u2286\n (fun a => \u2191(Opens.inclusion U) a) \u207b\u00b9' ((fun a => \u2191(Opens.inclusion V) a) '' \u2191W)\n[PROOFSTEP]\nrintro _ h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\na\u271d : (forget TopCat).obj ((Opens.toTopCat \u2191X.toPresheafedSpace).obj U)\nh : a\u271d \u2208 \u2191((Opens.toTopCat \u2191X.toPresheafedSpace).map i) \u207b\u00b9' \u2191W\n\u22a2 a\u271d \u2208 (fun a => \u2191(Opens.inclusion U) a) \u207b\u00b9' ((fun a => \u2191(Opens.inclusion V) a) '' \u2191W)\n[PROOFSTEP]\nexact \u27e8_, h, rfl\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\n\u22a2 NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nhave e\u2081 := Scheme.congr_app (X.restrictFunctor_map_ofRestrict i) (op <| V.openEmbedding.isOpenMap.functor.obj W)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n NatTrans.app (((restrictFunctor X).map i).left \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n ((restrictFunctor X).obj U).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (Opens.map\n (((restrictFunctor X).map i).left \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))\n\u22a2 NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nrw [Scheme.comp_val_c_app] at e\u2081 \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n NatTrans.app ((restrictFunctor X).map i).left.val.c\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n ((restrictFunctor X).obj U).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (Opens.map\n (((restrictFunctor X).map i).left \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))\n\u22a2 NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nhave e\u2082 := (X.restrictFunctor.map i).1.val.c.naturality (eqToHom <| W.map_functor_eq (U := V)).op\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n NatTrans.app ((restrictFunctor X).map i).left.val.c\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n ((restrictFunctor X).obj U).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (Opens.map\n (((restrictFunctor X).map i).left \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))\ne\u2082 :\n ((restrictFunctor X).obj V).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)).op \u226b\n NatTrans.app ((restrictFunctor X).map i).left.val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) \u226b\n (((restrictFunctor X).map i).left.val.base _* ((restrictFunctor X).obj U).left.presheaf).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)).op\n\u22a2 NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_inv_comp] at e\u2082 \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n NatTrans.app ((restrictFunctor X).map i).left.val.c\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) \u226b\n ((restrictFunctor X).obj U).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (Opens.map\n (((restrictFunctor X).map i).left \u226b\n ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))\ne\u2082 :\n NatTrans.app ((restrictFunctor X).map i).left.val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n inv\n (((restrictFunctor X).obj V).left.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)).op) \u226b\n NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) \u226b\n (((restrictFunctor X).map i).left.val.base _* ((restrictFunctor X).obj U).left.presheaf).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)).op\n\u22a2 NatTrans.app ((restrictFunctor X).map i).left.val.c (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \u22a2\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n X.presheaf.map\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion V))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n X.presheaf.map\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n (eqToHom\n (_ :\n op\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n op\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))).unop).op\ne\u2082 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n inv\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W))).op) \u226b\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)))).op\n\u22a2 NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nrw [e\u2082, W.adjunction_counit_map_functor (U := V), \u2190 IsIso.eq_inv_comp, IsIso.inv_comp_eq, \u2190 IsIso.eq_comp_inv] at e\u2081 \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2081 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W))).op \u226b\n inv\n (X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n (\ud835\udfed (Opens \u2191\u2191X.toPresheafedSpace)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))).op) \u226b\n X.presheaf.map\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n (eqToHom\n (_ :\n op\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n op\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))).unop).op) \u226b\n inv\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)))).op)\ne\u2082 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n inv\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W))).op) \u226b\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)))).op\n\u22a2 NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nsimp_rw [eqToHom_map (Opens.map _), eqToHom_map (IsOpenMap.functor _), \u2190 Functor.map_inv, \u2190 Functor.map_comp] at e\u2081 \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2082 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n inv\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W))).op) \u226b\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)))).op\ne\u2081 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n X.presheaf.map\n (((eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n inv\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n (\ud835\udfed (Opens \u2191\u2191X.toPresheafedSpace)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))).op \u226b\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n (eqToHom\n (_ :\n op\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n op\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))).unop).op) \u226b\n inv\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n W))).op)\n\u22a2 NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\nrw [e\u2081]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : Opens \u2191\u2191X.toPresheafedSpace\ni : U \u27f6 V\nW : Opens { x // x \u2208 V }\ne\u2082 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op ((Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n inv\n (X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W))).op) \u226b\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).map\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n W)))).op\ne\u2081 :\n NatTrans.app\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.c\n (op W) =\n X.presheaf.map\n (((eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n inv\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n (\ud835\udfed (Opens \u2191\u2191X.toPresheafedSpace)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))).op \u226b\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n (eqToHom\n (_ :\n op\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n op\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))).unop).op) \u226b\n inv\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n W))).op)\n\u22a2 X.presheaf.map\n (((eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n inv\n (eqToHom\n (_ :\n (Opens.map (Opens.inclusion V) \u22d9 IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W) =\n (\ud835\udfed (Opens \u2191\u2191X.toPresheafedSpace)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))).op \u226b\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op \u226b\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).map\n (eqToHom\n (_ :\n op\n ((Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)) =\n op\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))))).unop).op) \u226b\n inv\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n ((Opens.map (Opens.inclusion V)).obj\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W))) =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map\n (IsOpenImmersion.lift (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : Set.range Subtype.val \u2286 Set.range Subtype.val)).val.base).obj\n W))).op) =\n X.presheaf.map\n (homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map ((restrictFunctor X).map i).left.val.base).obj W) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V))).obj W)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\u22a2 \u2200 {X_1 Y : (Opens \u2191\u2191X.toPresheafedSpace)\u1d52\u1d56} (f : X_1 \u27f6 Y),\n ((restrictFunctor X).op \u22d9 (Over.forget X).op \u22d9 \u0393).map f \u226b\n ((fun U =>\n X.presheaf.mapIso\n (Iso.op\n (eqToIso (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4 = U.unop)).symm))\n Y).hom =\n ((fun U =>\n X.presheaf.mapIso\n (Iso.op\n (eqToIso (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4 = U.unop)).symm))\n X_1).hom \u226b\n X.presheaf.map f\n[PROOFSTEP]\nintro U V i\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : (Opens \u2191\u2191X.toPresheafedSpace)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 ((restrictFunctor X).op \u22d9 (Over.forget X).op \u22d9 \u0393).map i \u226b\n ((fun U =>\n X.presheaf.mapIso\n (Iso.op\n (eqToIso (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4 = U.unop)).symm))\n V).hom =\n ((fun U =>\n X.presheaf.mapIso\n (Iso.op\n (eqToIso (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4 = U.unop)).symm))\n U).hom \u226b\n X.presheaf.map i\n[PROOFSTEP]\ndsimp [-Scheme.restrictFunctor_map_left]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : (Opens \u2191\u2191X.toPresheafedSpace)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 NatTrans.app ((restrictFunctor X).map i.unop).left.val.c (op \u22a4) \u226b\n X.presheaf.map (eqToHom (_ : V.unop = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V.unop))).obj \u22a4)).op =\n X.presheaf.map (eqToHom (_ : U.unop = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4)).op \u226b\n X.presheaf.map i\n[PROOFSTEP]\nrw [X.restrictFunctor_map_app, \u2190 Functor.map_comp, \u2190 Functor.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\nU V : (Opens \u2191\u2191X.toPresheafedSpace)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 X.presheaf.map\n ((homOfLE\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V.unop))).obj\n ((Opens.map ((restrictFunctor X).map i.unop).left.val.base).obj \u22a4) \u2264\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4)).op \u226b\n (eqToHom (_ : V.unop = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion V.unop))).obj \u22a4)).op) =\n X.presheaf.map ((eqToHom (_ : U.unop = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U.unop))).obj \u22a4)).op \u226b i)\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u2245\n restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\n[PROOFSTEP]\napply\n IsOpenImmersion.isoOfRangeEq (f := X.ofRestrict _ \u226b f) (H :=\n PresheafedSpace.IsOpenImmersion.comp (hf := inferInstance) (hg := inferInstance)) (Y.ofRestrict _) _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Set.range \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b f).val.base =\n Set.range \u2191(ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base\n[PROOFSTEP]\ndsimp [Opens.inclusion]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Set.range \u2191(ContinuousMap.mk Subtype.val \u226b f.val.base) = Set.range \u2191(ContinuousMap.mk Subtype.val)\n[PROOFSTEP]\nrw [coe_comp, Set.range_comp, ContinuousMap.coe_mk, ContinuousMap.coe_mk]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base '' Set.range Subtype.val = Set.range Subtype.val\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base '' Set.range Subtype.val = Set.range Subtype.val\n[PROOFSTEP]\nrw [Subtype.range_val, Subtype.range_coe]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base '' \u2191((Opens.map f.val.base).obj U) = \u2191U\n[PROOFSTEP]\nrefine' @Set.image_preimage_eq _ _ f.1.base U.1 _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Function.Surjective \u2191f.val.base\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Epi f.val.base\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 x \u2208\n Set.range \u2191((fun x => pullback.fst) ((fun x => AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)) x)).val.base\n[PROOFSTEP]\nrw [\u2190\n show _ = (pullback.fst : pullback f (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x))) \u27f6 _).1.base from\n PreservesPullback.iso_hom_fst Scheme.forgetToTop f (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x)))]\n -- Porting note : `rw` to `erw` on this single lemma\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n \u2191((PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom \u226b\n pullback.fst)\n[PROOFSTEP]\nerw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n (\u2191pullback.fst \u2218\n \u2191(PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom)\n[PROOFSTEP]\nrw [Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ, TopCat.pullback_fst_range]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 x \u2208\n {x_1 |\n \u2203 y,\n \u2191(forgetToTop.map f) x_1 =\n \u2191(forgetToTop.map (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))) y}\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom\n[PROOFSTEP]\nobtain \u27e8y, h\u27e9 := \ud835\udcb0.Covers (f.1.base x)\n[GOAL]\ncase intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\ny : (forget TopCat).obj \u2191(obj \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x))).toPresheafedSpace\nh : \u2191(map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x))).val.base y = \u2191f.val.base x\n\u22a2 x \u2208\n {x_1 |\n \u2203 y,\n \u2191(forgetToTop.map f) x_1 =\n \u2191(forgetToTop.map (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))) y}\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom\n[PROOFSTEP]\nexact \u27e8y, h.symm\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 Function.Surjective\n \u2191(PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom\n[PROOFSTEP]\nrw [\u2190 TopCat.epi_iff_surjective]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nW : Scheme\nf : W \u27f6 X\nx : \u2191\u2191W.toPresheafedSpace\n\u22a2 Epi (PreservesPullback.iso forgetToTop f (map \ud835\udcb0 (AlgebraicGeometry.Scheme.OpenCover.f \ud835\udcb0 (\u2191f.val.base x)))).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u22c3 (i : \ud835\udcb0.J), Set.range \u2191(map \ud835\udcb0 i).val.base = Set.univ\n[PROOFSTEP]\nrw [Set.eq_univ_iff_forall]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u2200 (x : (forget TopCat).obj \u2191X.toPresheafedSpace), x \u2208 \u22c3 (i : \ud835\udcb0.J), Set.range \u2191(map \ud835\udcb0 i).val.base\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u22c3 (i : \ud835\udcb0.J), Set.range \u2191(map \ud835\udcb0 i).val.base\n[PROOFSTEP]\nrw [Set.mem_iUnion]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\nx : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 \u2203 i, x \u2208 Set.range \u2191(map \ud835\udcb0 i).val.base\n[PROOFSTEP]\nexact \u27e8\ud835\udcb0.f x, \ud835\udcb0.Covers x\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u2191(\u2a06 (i : \ud835\udcb0.J), Hom.opensRange (map \ud835\udcb0 i)) = \u2191\u22a4\n[PROOFSTEP]\nrw [Opens.coe_iSup]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\n\u22a2 \u22c3 (i : \ud835\udcb0.J), \u2191(Hom.opensRange (map \ud835\udcb0 i)) = \u2191\u22a4\n[PROOFSTEP]\nexact \ud835\udcb0.iUnion_range\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\n\u22a2 CompactSpace \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\ncases nonempty_fintype \ud835\udcb0.J\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\nval\u271d : Fintype \ud835\udcb0.J\n\u22a2 CompactSpace \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\nrw [\u2190 isCompact_univ_iff, \u2190 \ud835\udcb0.iUnion_range]\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\nval\u271d : Fintype \ud835\udcb0.J\n\u22a2 IsCompact (\u22c3 (i : \ud835\udcb0.J), Set.range \u2191(map \ud835\udcb0 i).val.base)\n[PROOFSTEP]\napply isCompact_iUnion\n[GOAL]\ncase intro.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\nval\u271d : Fintype \ud835\udcb0.J\n\u22a2 \u2200 (i : \ud835\udcb0.J), IsCompact (Set.range \u2191(map \ud835\udcb0 i).val.base)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase intro.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\nval\u271d : Fintype \ud835\udcb0.J\ni : \ud835\udcb0.J\n\u22a2 IsCompact (Set.range \u2191(map \ud835\udcb0 i).val.base)\n[PROOFSTEP]\nrw [isCompact_iff_compactSpace]\n[GOAL]\ncase intro.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : Scheme\n\ud835\udcb0 : OpenCover X\ninst\u271d : Finite \ud835\udcb0.J\nH : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(obj \ud835\udcb0 i).toPresheafedSpace\nval\u271d : Fintype \ud835\udcb0.J\ni : \ud835\udcb0.J\n\u22a2 CompactSpace \u2191(Set.range \u2191(map \ud835\udcb0 i).val.base)\n[PROOFSTEP]\nexact\n @Homeomorph.compactSpace _ _ _ _ (H i)\n (TopCat.homeoOfIso\n (asIso\n (IsOpenImmersion.isoOfRangeEq (\ud835\udcb0.map i)\n (X.ofRestrict (Opens.openEmbedding \u27e8_, (\ud835\udcb0.IsOpen i).base_open.open_range\u27e9))\n Subtype.range_coe.symm).hom.1.base))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0\u2081 : OpenCover X\n\ud835\udcb0\u2082 : OpenCover X\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 Set.range \u2191((fun ij => pullback.fst \u226b map \ud835\udcb0\u2081 ij.fst) ((fun x => (f \ud835\udcb0\u2081 x, f \ud835\udcb0\u2082 x)) x)).val.base\n[PROOFSTEP]\nrw [IsOpenImmersion.range_pullback_to_base_of_left]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Scheme\n\ud835\udcb0\u2081 : OpenCover X\n\ud835\udcb0\u2082 : OpenCover X\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range \u2191(map \ud835\udcb0\u2081 ((fun x => (f \ud835\udcb0\u2081 x, f \ud835\udcb0\u2082 x)) x).fst).val.base \u2229\n Set.range \u2191(map \ud835\udcb0\u2082 ((fun x => (f \ud835\udcb0\u2081 x, f \ud835\udcb0\u2082 x)) x).snd).val.base\n[PROOFSTEP]\nexact\n \u27e8\ud835\udcb0\u2081.Covers x, \ud835\udcb0\u2082.Covers x\u27e9\n -- Porting note : was automatic\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\ns : Type u_1\nX : Scheme\nU : s \u2192 Opens \u2191\u2191X.toPresheafedSpace\nhU : \u2a06 (i : s), U i = \u22a4\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntriv\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\ns : Type u_1\nX : Scheme\nU : s \u2192 Opens \u2191\u2191X.toPresheafedSpace\nhU : \u2a06 (i : s), U i = \u22a4\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n Set.range\n \u2191((fun i => ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion (U i))))\n ((fun x => Exists.choose (_ : \u2203 i, x \u2208 U i)) x)).val.base\n[PROOFSTEP]\nerw [Subtype.range_coe]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\ns : Type u_1\nX : Scheme\nU : s \u2192 Opens \u2191\u2191X.toPresheafedSpace\nhU : \u2a06 (i : s), U i = \u22a4\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191(U ((fun x => Exists.choose (_ : \u2203 i, x \u2208 U i)) x))\n[PROOFSTEP]\nhave : x \u2208 \u2a06 i, U i := hU.symm \u25b8 show x \u2208 (\u22a4 : Opens X) by triv\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\ns : Type u_1\nX : Scheme\nU : s \u2192 Opens \u2191\u2191X.toPresheafedSpace\nhU : \u2a06 (i : s), U i = \u22a4\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntriv\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\ns : Type u_1\nX : Scheme\nU : s \u2192 Opens \u2191\u2191X.toPresheafedSpace\nhU : \u2a06 (i : s), U i = \u22a4\nx : \u2191\u2191X.toPresheafedSpace\nthis : x \u2208 \u2a06 (i : s), U i\n\u22a2 x \u2208 \u2191(U ((fun x => Exists.choose (_ : \u2203 i, x \u2208 U i)) x))\n[PROOFSTEP]\nexact (Opens.mem_iSup.mp this).choose_spec\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 pullback f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) \u2245\n Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))\n[PROOFSTEP]\nrefine' IsOpenImmersion.isoOfRangeEq pullback.fst (X.ofRestrict _) _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Set.range \u2191pullback.fst.val.base =\n Set.range \u2191(Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.base\n[PROOFSTEP]\nrw [IsOpenImmersion.range_pullback_fst_of_right]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 ((Opens.map f.val.base).obj\n { carrier := Set.range \u2191(Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base,\n is_open' :=\n (_ :\n IsOpen (Set.range \u2191(Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base)) }).carrier =\n Set.range \u2191(Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.base\n[PROOFSTEP]\ndsimp [Opens.inclusion]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base \u207b\u00b9' Set.range \u2191(ContinuousMap.mk Subtype.val) = Set.range \u2191(ContinuousMap.mk Subtype.val)\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Subtype.range_val, Subtype.range_coe]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base \u207b\u00b9' \u2191U = \u2191((Opens.map f.val.base).obj U)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict f U).inv \u226b pullback.fst =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))\n[PROOFSTEP]\ndelta pullbackRestrictIsoRestrict\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (IsOpenImmersion.isoOfRangeEq pullback.fst\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ :\n Set.range \u2191pullback.fst.val.base =\n Set.range\n \u2191(Scheme.ofRestrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.base)).inv \u226b\n pullback.fst =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict f U).hom \u226b\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) =\n pullback.fst\n[PROOFSTEP]\ndelta pullbackRestrictIsoRestrict\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (IsOpenImmersion.isoOfRangeEq pullback.fst\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ :\n Set.range \u2191pullback.fst.val.base =\n Set.range\n \u2191(Scheme.ofRestrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.base)).hom \u226b\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) =\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)) =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b f\n[PROOFSTEP]\ndelta morphismRestrict\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 ((pullbackRestrictIsoRestrict f U).inv \u226b pullback.snd) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)) =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b f\n[PROOFSTEP]\nrw [Category.assoc, pullback.condition.symm, pullbackRestrictIsoRestrict_inv_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsPullback (f \u2223_ U) (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n[PROOFSTEP]\ndelta morphismRestrict\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsPullback ((pullbackRestrictIsoRestrict f U).inv \u226b pullback.snd)\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n[PROOFSTEP]\nrw [\u2190 Category.id_comp f]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsPullback ((pullbackRestrictIsoRestrict (\ud835\udfd9 X \u226b f) U).inv \u226b pullback.snd)\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (\ud835\udfd9 X \u226b f).val.base).obj U))))\n (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) (\ud835\udfd9 X \u226b f)\n[PROOFSTEP]\nrefine' (IsPullback.of_horiz_isIso \u27e8_\u27e9).paste_horiz (IsPullback.of_hasPullback f (Y.ofRestrict U.openEmbedding)).flip\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict (\ud835\udfd9 X \u226b f) U).inv \u226b pullback.fst =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (\ud835\udfd9 X \u226b f).val.base).obj U))) \u226b \ud835\udfd9 X\n[PROOFSTEP]\nerw [pullbackRestrictIsoRestrict_inv_fst]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (\ud835\udfd9 X \u226b f).val.base).obj U))) =\n Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (\ud835\udfd9 X \u226b f).val.base).obj U))) \u226b \ud835\udfd9 X\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (f \u226b g) \u2223_ U = (f \u2223_ (Opens.map g.val.base).obj U) \u226b g \u2223_ U\n[PROOFSTEP]\ndelta morphismRestrict\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b pullback.snd =\n ((pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b pullback.snd) \u226b\n (pullbackRestrictIsoRestrict g U).inv \u226b pullback.snd\n[PROOFSTEP]\nrw [\u2190 pullbackRightPullbackFstIso_inv_snd_snd]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b\n (pullbackRightPullbackFstIso g (Scheme.ofRestrict Z (_ : OpenEmbedding \u2191(Opens.inclusion U))) f).inv \u226b\n pullback.snd \u226b pullback.snd =\n ((pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b pullback.snd) \u226b\n (pullbackRestrictIsoRestrict g U).inv \u226b pullback.snd\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (((pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b\n (pullbackRightPullbackFstIso g (Scheme.ofRestrict Z (_ : OpenEmbedding \u2191(Opens.inclusion U))) f).inv) \u226b\n pullback.snd) \u226b\n pullback.snd =\n (((pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b pullback.snd) \u226b\n (pullbackRestrictIsoRestrict g U).inv) \u226b\n pullback.snd\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 ((pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b\n (pullbackRightPullbackFstIso g (Scheme.ofRestrict Z (_ : OpenEmbedding \u2191(Opens.inclusion U))) f).inv) \u226b\n pullback.snd =\n ((pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b pullback.snd) \u226b\n (pullbackRestrictIsoRestrict g U).inv\n[PROOFSTEP]\nrw [\u2190 cancel_mono pullback.fst]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (((pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b\n (pullbackRightPullbackFstIso g (Scheme.ofRestrict Z (_ : OpenEmbedding \u2191(Opens.inclusion U))) f).inv) \u226b\n pullback.snd) \u226b\n pullback.fst =\n (((pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b pullback.snd) \u226b\n (pullbackRestrictIsoRestrict g U).inv) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp_rw [Category.assoc]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : Opens \u2191\u2191Z.toPresheafedSpace\n\u22a2 (pullbackRestrictIsoRestrict (f \u226b g) U).inv \u226b\n (pullbackRightPullbackFstIso g (Scheme.ofRestrict Z (_ : OpenEmbedding \u2191(Opens.inclusion U))) f).inv \u226b\n pullback.snd \u226b pullback.fst =\n (pullbackRestrictIsoRestrict f ((Opens.map g.val.base).obj U)).inv \u226b\n pullback.snd \u226b (pullbackRestrictIsoRestrict g U).inv \u226b pullback.fst\n[PROOFSTEP]\nrw [pullbackRestrictIsoRestrict_inv_fst, pullbackRightPullbackFstIso_inv_snd_fst, \u2190 pullback.condition,\n pullbackRestrictIsoRestrict_inv_fst_assoc, pullbackRestrictIsoRestrict_inv_fst_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsIso (f \u2223_ U)\n[PROOFSTEP]\ndelta morphismRestrict\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsIso ((pullbackRestrictIsoRestrict f U).inv \u226b pullback.snd)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V)) =\n \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V)) \u2194\n x \u2208 \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208\n \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V)) \u2192\n x \u2208 \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))\n[PROOFSTEP]\nrintro \u27e8\u27e8x, hx\u27e9, hx' : (f \u2223_ U).1.base _ \u2208 V, rfl\u27e9\n[GOAL]\ncase h.h.mp.intro.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : x \u2208 (Opens.map f.val.base).obj U\nhx' : \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V\n\u22a2 \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)) { val := x, property := hx } \u2208\n \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))\n[PROOFSTEP]\nrefine'\n \u27e8\u27e8_, hx\u27e9, _, rfl\u27e9\n -- Porting note : this rewrite was not necessary\n[GOAL]\ncase h.h.mp.intro.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : x \u2208 (Opens.map f.val.base).obj U\nhx' : \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V\n\u22a2 { val := \u2191f.val.base x, property := hx } \u2208 \u2191V\n[PROOFSTEP]\nrw [SetLike.mem_coe]\n[GOAL]\ncase h.h.mp.intro.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : x \u2208 (Opens.map f.val.base).obj U\nhx' : \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V\n\u22a2 { val := \u2191f.val.base x, property := hx } \u2208 V\n[PROOFSTEP]\nconvert hx'\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : x \u2208 (Opens.map f.val.base).obj U\nhx' : \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V\ne_1\u271d :\n (forget TopCat).obj ((Opens.toTopCat \u2191Y.toPresheafedSpace).obj U) =\n (fun x =>\n (forget TopCat).obj\n \u2191(Scheme.restrict Y\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n { val := x, property := hx }\n\u22a2 { val := \u2191f.val.base x, property := hx } = \u2191(f \u2223_ U).val.base { val := x, property := hx }\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : x \u2208 (Opens.map f.val.base).obj U\nhx' : \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V\ne_1\u271d :\n (forget TopCat).obj ((Opens.toTopCat \u2191Y.toPresheafedSpace).obj U) =\n (fun x =>\n (forget TopCat).obj\n \u2191(Scheme.restrict Y\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n { val := x, property := hx }\n\u22a2 \u2191{ val := \u2191f.val.base x, property := hx } = \u2191(\u2191(f \u2223_ U).val.base { val := x, property := hx })\n[PROOFSTEP]\nexact (morphismRestrict_base_coe f U \u27e8x, hx\u27e9).symm\n[GOAL]\ncase h.h.mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u2192\n x \u2208\n \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V))\n[PROOFSTEP]\nrintro \u27e8\u27e8x, hx\u27e9, hx' : _ \u2208 V.1, rfl : x = _\u27e9\n[GOAL]\ncase h.h.mpr.intro.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : \u2191f.val.base x \u2208 U\nhx' : { val := \u2191f.val.base x, property := hx } \u2208 V.carrier\n\u22a2 x \u2208\n \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V))\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, hx\u27e9, (_ : (f \u2223_ U).1.base \u27e8x, hx\u27e9 \u2208 V.1), rfl\u27e9\n[GOAL]\ncase h.h.mpr.intro.mk.intro\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : \u2191f.val.base x \u2208 U\nhx' : { val := \u2191f.val.base x, property := hx } \u2208 V.carrier\n\u22a2 \u2191(f \u2223_ U).val.base { val := x, property := hx } \u2208 V.carrier\n[PROOFSTEP]\nconvert hx'\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : \u2191f.val.base x \u2208 U\nhx' : { val := \u2191f.val.base x, property := hx } \u2208 V.carrier\ne_1\u271d :\n (fun x =>\n (forget TopCat).obj\n \u2191(Scheme.restrict Y\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n { val := x, property := hx } =\n { x // x \u2208 U }\n\u22a2 \u2191(f \u2223_ U).val.base { val := x, property := hx } = { val := \u2191f.val.base x, property := hx }\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase h.e'_4.h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nx : \u2191\u2191X.toPresheafedSpace\nhx : \u2191f.val.base x \u2208 U\nhx' : { val := \u2191f.val.base x, property := hx } \u2208 V.carrier\ne_1\u271d :\n (fun x =>\n (forget TopCat).obj\n \u2191(Scheme.restrict Y\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n { val := x, property := hx } =\n { x // x \u2208 U }\n\u22a2 \u2191(\u2191(f \u2223_ U).val.base { val := x, property := hx }) = \u2191{ val := \u2191f.val.base x, property := hx }\n[PROOFSTEP]\nexact morphismRestrict_base_coe f U \u27e8x, hx\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nhave := Scheme.congr_app (morphismRestrict_\u03b9 f U) (op (U.openEmbedding.isOpenMap.functor.obj V))\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b f).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nrw [Scheme.comp_val_c_app, Scheme.comp_val_c_app_assoc] at this \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nhave e : (Opens.map U.inclusion).obj (U.openEmbedding.isOpenMap.functor.obj V) = V := by ext1;\n exact Set.preimage_image_eq _ Subtype.coe_injective\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\n\u22a2 (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\n\u22a2 \u2191((Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) = \u2191V\n[PROOFSTEP]\nexact Set.preimage_image_eq _ Subtype.coe_injective\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nhave : _ \u226b X.presheaf.map _ = _ := (((f \u2223_ U).1.c.naturality (eqToHom e).op).symm.trans ?_).trans this\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis\u271d :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\nthis :\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n X.presheaf.map\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op)) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nrw [\u2190 IsIso.eq_comp_inv, \u2190 Functor.map_inv, Category.assoc] at this \n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis\u271d :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\nthis :\n NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op)))\n\u22a2 NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase refine_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis\u271d :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\nthis :\n NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op)))\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine_2.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis\u271d :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\nthis :\n NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op)))\n\u22a2 (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op))) =\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\nerw [\u2190 X.presheaf.map_comp, \u2190 X.presheaf.map_comp]\n[GOAL]\ncase refine_2.e_a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis\u271d :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\nthis :\n NatTrans.app (f \u2223_ U).val.c (op V) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n (NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n X.presheaf.map\n (inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op)))\n\u22a2 X.presheaf.map\n (((NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).counit\n ((Opens.map f.val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).unop).op \u226b\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))) \u226b\n inv\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).op.map\n ((Opens.map (f \u2223_ U).val.base).op.map (eqToHom e).op))) =\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\n\u22a2 (Scheme.restrict Y\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom e).op \u226b\n NatTrans.app (f \u2223_ U).val.c\n (op ((Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)))\n[PROOFSTEP]\nchange Y.presheaf.map _ \u226b _ = Y.presheaf.map _ \u226b _\n[GOAL]\ncase refine_1\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\nthis :\n NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n NatTrans.app (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).val.c\n ((Opens.map f.val.base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) \u226b\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (Scheme.ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) \u226b\n f).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) =\n (Opens.map ((f \u2223_ U) \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))))\ne : (Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V) = V\n\u22a2 Y.presheaf.map ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).op.map (eqToHom e).op) \u226b\n NatTrans.app (f \u2223_ U).val.c\n (op ((Opens.map (Opens.inclusion U)).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) =\n Y.presheaf.map\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion U))).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)).unop).op \u226b\n NatTrans.app (f \u2223_ U).val.c\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).op.obj\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Scheme.\u0393.map (f \u2223_ U).op =\n Y.presheaf.map (eqToHom (_ : U = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)).op \u226b\n NatTrans.app f.val.c (op U) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj \u22a4 =\n (Opens.map f.val.base).obj U)).op\n[PROOFSTEP]\nrw [Scheme.\u0393_map_op, morphismRestrict_c_app f U \u22a4, f.val.c.naturality_assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj \u22a4) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4))).op =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)) \u226b\n (f.val.base _* X.presheaf).map\n (eqToHom (_ : U = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)).op \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj \u22a4 =\n (Opens.map f.val.base).obj U)).op\n[PROOFSTEP]\nerw [\u2190 X.presheaf.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj \u22a4) =\n (Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4))).op =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)) \u226b\n X.presheaf.map\n ((Opens.map f.val.base).op.map\n (eqToHom (_ : U = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)).op \u226b\n (eqToHom\n (_ :\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj \u22a4 =\n (Opens.map f.val.base).obj U)).op)\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\n\u22a2 Arrow.mk (f \u2223_ Scheme.Hom.opensRange g) \u2245 Arrow.mk pullback.snd\n[PROOFSTEP]\nlet V : Opens Y := Scheme.Hom.opensRange g\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\n\u22a2 Arrow.mk (f \u2223_ Scheme.Hom.opensRange g) \u2245 Arrow.mk pullback.snd\n[PROOFSTEP]\nlet e := IsOpenImmersion.isoOfRangeEq g (Y.ofRestrict V.openEmbedding) Subtype.range_coe.symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\n\u22a2 Arrow.mk (f \u2223_ Scheme.Hom.opensRange g) \u2245 Arrow.mk pullback.snd\n[PROOFSTEP]\nlet t : pullback f g \u27f6 pullback f (Y.ofRestrict V.openEmbedding) :=\n pullback.map _ _ _ _ (\ud835\udfd9 _) e.hom (\ud835\udfd9 _) (by rw [Category.comp_id, Category.id_comp])\n (by rw [Category.comp_id, IsOpenImmersion.isoOfRangeEq_hom, IsOpenImmersion.lift_fac])\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\n\u22a2 f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nrw [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\n\u22a2 g \u226b \ud835\udfd9 Y = e.hom \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))\n[PROOFSTEP]\nrw [Category.comp_id, IsOpenImmersion.isoOfRangeEq_hom, IsOpenImmersion.lift_fac]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\nt : pullback f g \u27f6 pullback f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) :=\n pullback.map f g f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) (\ud835\udfd9 X) e.hom (\ud835\udfd9 Y)\n (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f) (_ : g \u226b \ud835\udfd9 Y = e.hom \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n\u22a2 Arrow.mk (f \u2223_ Scheme.Hom.opensRange g) \u2245 Arrow.mk pullback.snd\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\nt : pullback f g \u27f6 pullback f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) :=\n pullback.map f g f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) (\ud835\udfd9 X) e.hom (\ud835\udfd9 Y)\n (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f) (_ : g \u226b \ud835\udfd9 Y = e.hom \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n\u22a2 Arrow.mk pullback.snd \u2245 Arrow.mk (f \u2223_ Scheme.Hom.opensRange g)\n[PROOFSTEP]\nrefine' Arrow.isoMk (asIso t \u226a\u226b pullbackRestrictIsoRestrict f V) e _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y U : Scheme\nf : X \u27f6 Y\ng : U \u27f6 Y\nhg : IsOpenImmersion g\nV : Opens \u2191\u2191Y.toPresheafedSpace := Scheme.Hom.opensRange g\ne : U \u2245 Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)) :=\n IsOpenImmersion.isoOfRangeEq g (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ : Set.range \u2191g.val.base = Set.range Subtype.val)\nt : pullback f g \u27f6 pullback f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) :=\n pullback.map f g f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V))) (\ud835\udfd9 X) e.hom (\ud835\udfd9 Y)\n (_ : f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f) (_ : g \u226b \ud835\udfd9 Y = e.hom \u226b Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n\u22a2 (asIso t \u226a\u226b pullbackRestrictIsoRestrict f V).hom \u226b (Arrow.mk (f \u2223_ Scheme.Hom.opensRange g)).hom =\n (Arrow.mk pullback.snd).hom \u226b e.hom\n[PROOFSTEP]\nrw [Iso.trans_hom, asIso_hom, \u2190 Iso.comp_inv_eq, \u2190 cancel_mono g, Arrow.mk_hom, Arrow.mk_hom,\n IsOpenImmersion.isoOfRangeEq_inv, Category.assoc, Category.assoc, Category.assoc, IsOpenImmersion.lift_fac, \u2190\n pullback.condition, morphismRestrict_\u03b9, pullbackRestrictIsoRestrict_hom_restrict_assoc, pullback.lift_fst_assoc,\n Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU V : Opens \u2191\u2191Y.toPresheafedSpace\ne : U = V\n\u22a2 Arrow.mk (f \u2223_ U) = Arrow.mk (f \u2223_ V)\n[PROOFSTEP]\nsubst e\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\n\u22a2 Arrow.mk (f \u2223_ U) = Arrow.mk (f \u2223_ U)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nset g := ((Y.restrict U.openEmbedding).ofRestrict (V.openEmbedding (X := TopCat.of U)) \u226b Y.ofRestrict U.openEmbedding)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nhave i1 : IsOpenImmersion g := PresheafedSpace.IsOpenImmersion.comp _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nhave i2 : HasPullback f g := IsOpenImmersion.hasPullback_of_right g f\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nset h : _ \u27f6 pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry _ _).hom \u226b\n pullback.map _ _ _ _ (\ud835\udfd9 _) ((pullbackRestrictIsoRestrict f U).inv \u226b (pullbackSymmetry _ _).hom) (\ud835\udfd9 _)\n ((Category.comp_id _).trans (Category.id_comp _).symm) (by aesop_cat) \u226b\n (pullbackRightPullbackFstIso _ _ _).hom \u226b (pullbackSymmetry _ _).hom\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\n\u22a2 (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nhave i3 : IsIso h\n[GOAL]\ncase i3\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\n\u22a2 IsIso h\n[PROOFSTEP]\nrepeat apply (config := { allowSynthFailures := true }) IsIso.comp_isIso\n[GOAL]\ncase i3\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\n\u22a2 IsIso h\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsIso.comp_isIso\n[GOAL]\ncase inst\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\n\u22a2 IsIso\n ((pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom)\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsIso.comp_isIso\n[GOAL]\ncase inst\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\n\u22a2 IsIso\n (pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom)\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsIso.comp_isIso\n[GOAL]\n\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsIso.comp_isIso\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nhave : (f \u2223_ U \u2223_ V) \u226b (Iso.refl _).hom = (asIso h).hom \u226b pullback.snd (f := f) (g := g)\n[GOAL]\ncase this\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\n\u22a2 (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n[PROOFSTEP]\nsimp only [Category.comp_id, pullbackRightPullbackFstIso_hom_fst, Iso.refl_hom, Category.assoc,\n pullbackSymmetry_hom_comp_snd, asIso_hom, pullback.lift_fst, pullbackSymmetry_hom_comp_fst]\n[GOAL]\ncase this\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\n\u22a2 f \u2223_ U \u2223_ V = (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b pullback.snd\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\nrefine' Arrow.isoMk' _ _ _ _ this.symm \u226a\u226b (morphismRestrictOpensRange _ _).symm \u226a\u226b morphismRestrictEq _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 Scheme.Hom.opensRange g = (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 \u2191(Scheme.Hom.opensRange g) = \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 Set.range \u2191(Opens.inclusion V \u226b Opens.inclusion U) = \u2191(Opens.inclusion U) '' \u2191V\n[PROOFSTEP]\nrw [coe_comp, Set.range_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 \u2191(Opens.inclusion U) '' Set.range \u2191(Opens.inclusion V) = \u2191(Opens.inclusion U) '' \u2191V\n[PROOFSTEP]\napply congr_arg (U.inclusion '' \u00b7)\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nV : Opens { x // x \u2208 U }\ng : Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u27f6\n Y :=\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))\ni1 : IsOpenImmersion g\ni2 : HasPullback f g\nh : Scheme.restrict (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))))\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map (f \u2223_ U).val.base).obj V))) \u27f6\n pullback f g :=\n (pullbackRestrictIsoRestrict (f \u2223_ U) V).inv \u226b\n (pullbackSymmetry (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n pullback.map\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (f \u2223_ U)\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n pullback.fst\n (\ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))))\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom)\n (\ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))))\n (_ :\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n \ud835\udfd9\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V))) \u226b\n Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))\n (_ :\n (f \u2223_ U) \u226b \ud835\udfd9 (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) =\n ((pullbackRestrictIsoRestrict f U).inv \u226b\n (pullbackSymmetry f (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom) \u226b\n pullback.fst) \u226b\n (pullbackRightPullbackFstIso (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))) f\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom \u226b\n (pullbackSymmetry\n (Scheme.ofRestrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)) \u226b\n Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n f).hom\ni3 : IsIso h\nthis :\n (f \u2223_ U \u2223_ V) \u226b\n (Iso.refl\n (Scheme.restrict (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (_ : OpenEmbedding \u2191(Opens.inclusion V)))).hom =\n (asIso h).hom \u226b pullback.snd\n\u22a2 Set.range \u2191(Opens.inclusion V) = \u2191V\n[PROOFSTEP]\nexact Subtype.range_val\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\n\u22a2 Arrow.mk\n (f \u2223_ U \u2223_\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op) r)) \u2245\n Arrow.mk (f \u2223_ Scheme.basicOpen Y r)\n[PROOFSTEP]\nrefine' morphismRestrictRestrict _ _ _ \u226a\u226b morphismRestrictEq _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op) r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nhave e := Scheme.preimage_basicOpen (Y.ofRestrict U.openEmbedding) r\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(NatTrans.app (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.c (op U)) r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op) r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nerw [Scheme.ofRestrict_val_c_app, Opens.adjunction_counit_app_self, eqToHom_op] at e \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op) r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nrw [\u2190 (Y.restrict U.openEmbedding).basicOpen_res_eq _ (eqToHom U.inclusion_map_eq_top).op]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191((Scheme.restrict Y\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom (_ : (Opens.map (Opens.inclusion U)).obj U = \u22a4)).op)\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op)\n r))) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nerw [\u2190 comp_apply]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op \u226b\n (Scheme.restrict Y\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.map\n (eqToHom (_ : (Opens.map (Opens.inclusion U)).obj U = \u22a4)).op)\n r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nerw [\u2190 Y.presheaf.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n ((eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op \u226b\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).op.map\n (eqToHom (_ : (Opens.map (Opens.inclusion U)).obj U = \u22a4)).op))\n r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nrw [eqToHom_op, eqToHom_op, eqToHom_map, eqToHom_trans]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n (Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op U =\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).op.obj\n (op ((Opens.map (Opens.inclusion U)).obj U)))))\n r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\nerw [\u2190 e]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r)) =\n Scheme.basicOpen Y r\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj\n (Scheme.basicOpen Y r))) =\n \u2191(Scheme.basicOpen Y r)\n[PROOFSTEP]\ndsimp [Opens.map, Opens.inclusion]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 \u2191(ContinuousMap.mk Subtype.val) '' (\u2191(ContinuousMap.mk Subtype.val) \u207b\u00b9' \u2191(Scheme.basicOpen Y r)) =\n \u2191(Scheme.basicOpen Y r)\n[PROOFSTEP]\nrw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset, ContinuousMap.coe_mk, Subtype.range_val]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nr : \u2191(Y.presheaf.obj (op U))\ne :\n (Opens.map (Scheme.ofRestrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).val.base).obj (Scheme.basicOpen Y r) =\n Scheme.basicOpen (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n (\u2191(Y.presheaf.map\n (eqToHom\n (_ :\n op (op U).unop =\n op\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj\n ((Opens.map (Opens.inclusion U)).obj (op U).unop)))))\n r)\n\u22a2 \u2191(Scheme.basicOpen Y r) \u2286 \u2191U\n[PROOFSTEP]\nexact Y.basicOpen_le r\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 Arrow.mk (PresheafedSpace.stalkMap (f \u2223_ U).val x) \u2245 Arrow.mk (PresheafedSpace.stalkMap f.val \u2191x)\n[PROOFSTEP]\nfapply Arrow.isoMk'\n[GOAL]\ncase e\u2081\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 PresheafedSpace.stalk\n (Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n (\u2191(f \u2223_ U).val.base x) \u2245\n PresheafedSpace.stalk Y.toPresheafedSpace (\u2191f.val.base \u2191x)\n[PROOFSTEP]\nrefine' Y.restrictStalkIso U.openEmbedding ((f \u2223_ U).1.1 x) \u226a\u226b TopCat.Presheaf.stalkCongr _ _\n[GOAL]\ncase e\u2081\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 Inseparable (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) (\u2191f.val.base \u2191x)\n[PROOFSTEP]\napply Inseparable.of_eq\n[GOAL]\ncase e\u2081.e\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 \u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x) = \u2191f.val.base \u2191x\n[PROOFSTEP]\nexact morphismRestrict_base_coe f U x\n[GOAL]\ncase e\u2082\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 PresheafedSpace.stalk\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n x \u2245\n PresheafedSpace.stalk X.toPresheafedSpace \u2191x\n[PROOFSTEP]\nexact X.restrictStalkIso (Opens.openEmbedding _) _\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 autoParam\n ((PresheafedSpace.restrictStalkIso Y.toPresheafedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U))\n (\u2191(f \u2223_ U).val.base x) \u226a\u226b\n TopCat.Presheaf.stalkCongr Y.presheaf\n (_ : Inseparable (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) (\u2191f.val.base \u2191x))).hom \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n PresheafedSpace.stalkMap (f \u2223_ U).val x \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom)\n _auto\u271d\n[PROOFSTEP]\napply TopCat.Presheaf.stalk_hom_ext\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 \u2200\n (U_1 :\n Opens\n \u2191\u2191(Scheme.restrict Y\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n (hxU : \u2191(f \u2223_ U).val.base x \u2208 U_1),\n TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxU } \u226b\n (PresheafedSpace.restrictStalkIso Y.toPresheafedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U))\n (\u2191(f \u2223_ U).val.base x) \u226a\u226b\n TopCat.Presheaf.stalkCongr Y.presheaf\n (_ : Inseparable (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) (\u2191f.val.base \u2191x))).hom \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxU } \u226b\n PresheafedSpace.stalkMap (f \u2223_ U).val x \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nintro V hxV\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso Y.toPresheafedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U))\n (\u2191(f \u2223_ U).val.base x) \u226a\u226b\n TopCat.Presheaf.stalkCongr Y.presheaf\n (_ : Inseparable (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) (\u2191f.val.base \u2191x))).hom \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxV } \u226b\n PresheafedSpace.stalkMap (f \u2223_ U).val x \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nsimp only [TopCat.Presheaf.stalkCongr_hom, CategoryTheory.Category.assoc, CategoryTheory.Iso.trans_hom]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso Y.toPresheafedSpace (_ : OpenEmbedding \u2191(Opens.inclusion U))\n (\u2191(f \u2223_ U).val.base x)).hom \u226b\n TopCat.Presheaf.stalkSpecializes Y.presheaf\n (_ : nhds (\u2191f.val.base \u2191x) \u2264 nhds (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x))) \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxV } \u226b\n PresheafedSpace.stalkMap (f \u2223_ U).val x \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nerw [PresheafedSpace.restrictStalkIso_hom_eq_germ_assoc]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 TopCat.Presheaf.germ Y.presheaf\n { val := \u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x),\n property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) } \u226b\n TopCat.Presheaf.stalkSpecializes Y.presheaf\n (_ : nhds (\u2191f.val.base \u2191x) \u2264 nhds (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x))) \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n TopCat.Presheaf.germ\n (Scheme.restrict Y\n (_ :\n OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := \u2191(f \u2223_ U).val.base x, property := hxV } \u226b\n PresheafedSpace.stalkMap (f \u2223_ U).val x \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap_germ_assoc _ V \u27e8_, hxV\u27e9]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 TopCat.Presheaf.germ Y.presheaf\n { val := \u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x),\n property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x)) } \u226b\n TopCat.Presheaf.stalkSpecializes Y.presheaf\n (_ : nhds (\u2191f.val.base \u2191x) \u2264 nhds (\u2191(Opens.inclusion U) (\u2191(f \u2223_ U).val.base x))) \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n TopCat.Presheaf.germ\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nrw [TopCat.Presheaf.germ_stalk_specializes'_assoc]\n -- Porting note : explicit variables and proofs were not necessary\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 TopCat.Presheaf.germ Y.presheaf\n { val := \u2191f.val.base \u2191x,\n property := (_ : \u2191f.val.base \u2191x \u2208 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) } \u226b\n PresheafedSpace.stalkMap f.val \u2191x =\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n TopCat.Presheaf.germ\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap_germ _ (U.openEmbedding.isOpenMap.functor.obj V) \u27e8x.1, \u27e8\u27e8f.1.base x.1, x.2\u27e9, _, rfl\u27e9\u27e9]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n TopCat.Presheaf.germ X.presheaf\n { val := \u2191x, property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191f.val.base \u2191x) } =\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n TopCat.Presheaf.germ\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 { val := \u2191f.val.base \u2191x, property := (_ : \u2191x \u2208 (Opens.map f.val.base).obj U) } \u2208 \u2191V\n[PROOFSTEP]\nswap\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 { val := \u2191f.val.base \u2191x, property := (_ : \u2191x \u2208 (Opens.map f.val.base).obj U) } \u2208 \u2191V\n[PROOFSTEP]\nrw [morphismRestrict_val_base] at hxV \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : Set.restrictPreimage U.carrier (\u2191f.val.base) x \u2208 V\n\u22a2 { val := \u2191f.val.base \u2191x, property := (_ : \u2191x \u2208 (Opens.map f.val.base).obj U) } \u2208 \u2191V\n[PROOFSTEP]\nexact hxV\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n TopCat.Presheaf.germ X.presheaf\n { val := \u2191x, property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191f.val.base \u2191x) } =\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n TopCat.Presheaf.germ\n (Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj\n U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf\n { val := x, property := hxV } \u226b\n (PresheafedSpace.restrictStalkIso X.toPresheafedSpace\n (_ : OpenEmbedding \u2191(Opens.inclusion ((Opens.map f.val.base).obj U))) x).hom\n[PROOFSTEP]\nerw [PresheafedSpace.restrictStalkIso_hom_eq_germ]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n TopCat.Presheaf.germ X.presheaf\n { val := \u2191x, property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191f.val.base \u2191x) } =\n NatTrans.app (f \u2223_ U).val.c (op V) \u226b\n TopCat.Presheaf.germ X.presheaf\n { val := \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)) x,\n property :=\n (_ :\n \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)) x \u2208\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V)) }\n[PROOFSTEP]\nrw [morphismRestrict_c_app, Category.assoc, TopCat.Presheaf.germ_res]\n[GOAL]\ncase h.ih\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\nx :\n \u2191\u2191(Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion\n ((Opens.map f.val.base).obj U)))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nV :\n Opens\n \u2191\u2191(Scheme.restrict Y (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nhxV : \u2191(f \u2223_ U).val.base x \u2208 V\n\u22a2 NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n TopCat.Presheaf.germ X.presheaf\n { val := \u2191x, property := (_ : \u2203 a, a \u2208 \u2191V \u2227 \u2191(Opens.inclusion U) a = \u2191f.val.base \u2191x) } =\n NatTrans.app f.val.c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V)) \u226b\n TopCat.Presheaf.germ X.presheaf\n ((fun x =>\n { val := \u2191x,\n property :=\n (_ :\n \u2191x \u2208\n \u2191((Opens.map f.val.base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj V))) })\n { val := \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)) x,\n property :=\n (_ :\n \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)) x \u2208\n (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion ((Opens.map f.val.base).obj U)))).obj\n ((Opens.map (f \u2223_ U).val.base).obj V)) })\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\ninst\u271d : IsOpenImmersion f\n\u22a2 IsOpenImmersion (f \u2223_ U)\n[PROOFSTEP]\ndelta morphismRestrict\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y : Scheme\nf : X \u27f6 Y\nU : Opens \u2191\u2191Y.toPresheafedSpace\ninst\u271d : IsOpenImmersion f\n\u22a2 IsOpenImmersion ((pullbackRestrictIsoRestrict f U).inv \u226b pullback.snd)\n[PROOFSTEP]\nrefine PresheafedSpace.IsOpenImmersion.comp _ _\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.OpenImmersion.Scheme", "llama_tokens": 132619, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.29248779362094973}} {"text": "[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nx\u271d\u00b3 : \u03b1\nx\u271d\u00b2 x\u271d\u00b9 : List \u03b1\nx\u271d : x\u271d\u00b2 ~ x\u271d\u00b9\nhs : a \u2208 x\u271d\u00b2 \u2194 a \u2208 x\u271d\u00b9\n\u22a2 a \u2208 x\u271d\u00b3 :: x\u271d\u00b2 \u2194 a \u2208 x\u271d\u00b3 :: x\u271d\u00b9\n[PROOFSTEP]\nsimp only [mem_cons, hs]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nx\u271d\u00b2 x\u271d\u00b9 : \u03b1\nx\u271d : List \u03b1\n\u22a2 a \u2208 x\u271d\u00b9 :: x\u271d\u00b2 :: x\u271d \u2194 a \u2208 x\u271d\u00b2 :: x\u271d\u00b9 :: x\u271d\n[PROOFSTEP]\nsimp only [mem_cons, or_left_comm]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 a :: (l ++ []) ~ a :: l\n[PROOFSTEP]\nrw [append_nil]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d l\u2082 : List \u03b1\n\u22a2 [] ++ l\u2082 ~ l\u2082 ++ []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\na : \u03b1\n\u22a2 concat l a ~ a :: l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\n_x : \u03b1\nl\u2081 l\u2082 : List \u03b1\n_p : l\u2081 ~ l\u2082\nr : length l\u2081 = length l\u2082\n\u22a2 length (_x :: l\u2081) = length (_x :: l\u2082)\n[PROOFSTEP]\nsimp [r]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n_x _y : \u03b1\nl : List \u03b1\n\u22a2 length (_y :: _x :: l) = length (_x :: _y :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl : List \u03b1\nx\u271d : [] ~ x :: l\np : [] ~ x :: l := x\u271d\n\u22a2 False\n[PROOFSTEP]\ninjection p.symm.eq_nil\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 reverse (a :: l) ~ a :: l\n[PROOFSTEP]\nrw [reverse_cons]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 reverse l ++ [a] ~ a :: l\n[PROOFSTEP]\nexact (perm_append_singleton _ _).trans ((reverse_perm l).cons a)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\n\u22a2 [a] ~ [b] \u2194 a = b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 List.filterMap f l\u2081 ~ List.filterMap f l\u2082\n[PROOFSTEP]\ninduction p with\n| nil => simp\n| cons x _p IH => cases h : f x <;> simp [h, filterMap, IH, Perm.cons]\n| swap x y l\u2082 => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap, swap]\n| trans _p\u2081 _p\u2082 IH\u2081 IH\u2082 => exact IH\u2081.trans IH\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 List.filterMap f l\u2081 ~ List.filterMap f l\u2082\n[PROOFSTEP]\ninduction p with\n| nil => simp\n| cons x _p IH => cases h : f x <;> simp [h, filterMap, IH, Perm.cons]\n| swap x y l\u2082 => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap, swap]\n| trans _p\u2081 _p\u2082 IH\u2081 IH\u2082 => exact IH\u2081.trans IH\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\n\u22a2 List.filterMap f [] ~ List.filterMap f []\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\n\u22a2 List.filterMap f [] ~ List.filterMap f []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\n\u22a2 List.filterMap f (x :: l\u2081\u271d) ~ List.filterMap f (x :: l\u2082\u271d)\n[PROOFSTEP]\n\n| cons x _p IH => cases h : f x <;> simp [h, filterMap, IH, Perm.cons]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\n\u22a2 List.filterMap f (x :: l\u2081\u271d) ~ List.filterMap f (x :: l\u2082\u271d)\n[PROOFSTEP]\ncases h : f x\n[GOAL]\ncase cons.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\nh : f x = none\n\u22a2 List.filterMap f (x :: l\u2081\u271d) ~ List.filterMap f (x :: l\u2082\u271d)\n[PROOFSTEP]\nsimp [h, filterMap, IH, Perm.cons]\n[GOAL]\ncase cons.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\nval\u271d : \u03b2\nh : f x = some val\u271d\n\u22a2 List.filterMap f (x :: l\u2081\u271d) ~ List.filterMap f (x :: l\u2082\u271d)\n[PROOFSTEP]\nsimp [h, filterMap, IH, Perm.cons]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\n\n| swap x y l\u2082 => cases hx : f x <;> cases hy : f y <;> simp [hx, hy, filterMap, swap]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\ncases hx : f x\n[GOAL]\ncase swap.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nhx : f x = none\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\ncases hy : f y\n[GOAL]\ncase swap.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nval\u271d : \u03b2\nhx : f x = some val\u271d\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\ncases hy : f y\n[GOAL]\ncase swap.none.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nhx : f x = none\nhy : f y = none\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\nsimp [hx, hy, filterMap, swap]\n[GOAL]\ncase swap.none.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nhx : f x = none\nval\u271d : \u03b2\nhy : f y = some val\u271d\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\nsimp [hx, hy, filterMap, swap]\n[GOAL]\ncase swap.some.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nval\u271d : \u03b2\nhx : f x = some val\u271d\nhy : f y = none\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\nsimp [hx, hy, filterMap, swap]\n[GOAL]\ncase swap.some.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082\u271d : List \u03b1\nx y : \u03b1\nl\u2082 : List \u03b1\nval\u271d\u00b9 : \u03b2\nhx : f x = some val\u271d\u00b9\nval\u271d : \u03b2\nhy : f y = some val\u271d\n\u22a2 List.filterMap f (y :: x :: l\u2082) ~ List.filterMap f (x :: y :: l\u2082)\n[PROOFSTEP]\nsimp [hx, hy, filterMap, swap]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\n_p\u2081 : l\u2081\u271d ~ l\u2082\u271d\n_p\u2082 : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\nIH\u2082 : List.filterMap f l\u2082\u271d ~ List.filterMap f l\u2083\u271d\n\u22a2 List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2083\u271d\n[PROOFSTEP]\n\n| trans _p\u2081 _p\u2082 IH\u2081 IH\u2082 => exact IH\u2081.trans IH\u2082\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\n_p\u2081 : l\u2081\u271d ~ l\u2082\u271d\n_p\u2082 : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2082\u271d\nIH\u2082 : List.filterMap f l\u2082\u271d ~ List.filterMap f l\u2083\u271d\n\u22a2 List.filterMap f l\u2081\u271d ~ List.filterMap f l\u2083\u271d\n[PROOFSTEP]\nexact IH\u2081.trans IH\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p\u271d a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 p\u271d a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 p\u271d a\n\u22a2 List.pmap f l\u2081 H\u2081 ~ List.pmap f l\u2082 H\u2082\n[PROOFSTEP]\ninduction p with\n| nil => simp\n| cons x _p IH => simp [IH, Perm.cons]\n| swap x y => simp [swap]\n| trans _p\u2081 p\u2082 IH\u2081 IH\u2082 =>\n refine' IH\u2081.trans IH\u2082\n exact fun a m => H\u2082 a (p\u2082.subset m)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p\u271d a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 p\u271d a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 p\u271d a\n\u22a2 List.pmap f l\u2081 H\u2081 ~ List.pmap f l\u2082 H\u2082\n[PROOFSTEP]\ninduction p with\n| nil => simp\n| cons x _p IH => simp [IH, Perm.cons]\n| swap x y => simp [swap]\n| trans _p\u2081 p\u2082 IH\u2081 IH\u2082 =>\n refine' IH\u2081.trans IH\u2082\n exact fun a m => H\u2082 a (p\u2082.subset m)\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nH\u2081 H\u2082 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a\n\u22a2 List.pmap f [] H\u2081 ~ List.pmap f [] H\u2082\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nH\u2081 H\u2082 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a\n\u22a2 List.pmap f [] H\u2081 ~ List.pmap f [] H\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a}, List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2082\u271d H\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 x :: l\u2081\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 x :: l\u2082\u271d \u2192 p a\n\u22a2 List.pmap f (x :: l\u2081\u271d) H\u2081 ~ List.pmap f (x :: l\u2082\u271d) H\u2082\n[PROOFSTEP]\n\n| cons x _p IH => simp [IH, Perm.cons]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\n_p : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a}, List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2082\u271d H\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 x :: l\u2081\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 x :: l\u2082\u271d \u2192 p a\n\u22a2 List.pmap f (x :: l\u2081\u271d) H\u2081 ~ List.pmap f (x :: l\u2082\u271d) H\u2082\n[PROOFSTEP]\nsimp [IH, Perm.cons]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nx y : \u03b1\nl\u271d : List \u03b1\nH\u2081 : \u2200 (a : \u03b1), a \u2208 y :: x :: l\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 x :: y :: l\u271d \u2192 p a\n\u22a2 List.pmap f (y :: x :: l\u271d) H\u2081 ~ List.pmap f (x :: y :: l\u271d) H\u2082\n[PROOFSTEP]\n\n| swap x y => simp [swap]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nx y : \u03b1\nl\u271d : List \u03b1\nH\u2081 : \u2200 (a : \u03b1), a \u2208 y :: x :: l\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 x :: y :: l\u271d \u2192 p a\n\u22a2 List.pmap f (y :: x :: l\u271d) H\u2081 ~ List.pmap f (x :: y :: l\u271d) H\u2082\n[PROOFSTEP]\nsimp [swap]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\n_p\u2081 : l\u2081\u271d ~ l\u2082\u271d\np\u2082 : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a}, List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2082\u271d H\u2082\nIH\u2082 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a}, List.pmap f l\u2082\u271d H\u2081 ~ List.pmap f l\u2083\u271d H\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a\n\u22a2 List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2083\u271d H\u2082\n[PROOFSTEP]\n\n| trans _p\u2081 p\u2082 IH\u2081 IH\u2082 =>\n refine' IH\u2081.trans IH\u2082\n exact fun a m => H\u2082 a (p\u2082.subset m)\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\n_p\u2081 : l\u2081\u271d ~ l\u2082\u271d\np\u2082 : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a}, List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2082\u271d H\u2082\nIH\u2082 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a}, List.pmap f l\u2082\u271d H\u2081 ~ List.pmap f l\u2083\u271d H\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a\n\u22a2 List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2083\u271d H\u2082\n[PROOFSTEP]\nrefine' IH\u2081.trans IH\u2082\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\n_p\u2081 : l\u2081\u271d ~ l\u2082\u271d\np\u2082 : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a}, List.pmap f l\u2081\u271d H\u2081 ~ List.pmap f l\u2082\u271d H\u2082\nIH\u2082 : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a}, List.pmap f l\u2082\u271d H\u2081 ~ List.pmap f l\u2083\u271d H\u2082\nH\u2081 : \u2200 (a : \u03b1), a \u2208 l\u2081\u271d \u2192 p a\nH\u2082 : \u2200 (a : \u03b1), a \u2208 l\u2083\u271d \u2192 p a\n\u22a2 \u2200 (a : \u03b1), a \u2208 l\u2082\u271d \u2192 p a\n[PROOFSTEP]\nexact fun a m => H\u2082 a (p\u2082.subset m)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\n\u22a2 List.filter p l\u2081 ~ List.filter p l\u2082\n[PROOFSTEP]\nrw [\u2190 filterMap_eq_filter]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\n\u22a2 List.filterMap (Option.guard fun x => p x = true) l\u2081 ~ List.filterMap (Option.guard fun x => p x = true) l\u2082\n[PROOFSTEP]\napply s.filterMap _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\n\u22a2 filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\n[PROOFSTEP]\ninduction' l with x l ih\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 filter p [] ++ filter (fun x => decide \u00acp x = true) [] ~ []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\n\u22a2 filter p (x :: l) ++ filter (fun x => decide \u00acp x = true) (x :: l) ~ x :: l\n[PROOFSTEP]\nby_cases h : p x\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\nh : p x = true\n\u22a2 filter p (x :: l) ++ filter (fun x => decide \u00acp x = true) (x :: l) ~ x :: l\n[PROOFSTEP]\nsimp only [h, filter_cons_of_pos, filter_cons_of_neg, not_true, not_false_iff, cons_append]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\nh : p x = true\n\u22a2 x :: (filter p l ++ filter (fun x => decide \u00acp x = true) l) ~ x :: l\n[PROOFSTEP]\nexact ih.cons x\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\nh : \u00acp x = true\n\u22a2 filter p (x :: l) ++ filter (fun x => decide \u00acp x = true) (x :: l) ~ x :: l\n[PROOFSTEP]\nsimp only [h, filter_cons_of_neg, not_false_iff, filter_cons_of_pos]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\nh : \u00acp x = true\n\u22a2 filter p l ++ x :: filter (fun x => decide \u00acp x = true) l ~ x :: l\n[PROOFSTEP]\nrefine' Perm.trans _ (ih.cons x)\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : filter p l ++ filter (fun x => decide \u00acp x = true) l ~ l\nh : \u00acp x = true\n\u22a2 filter p l ++ x :: filter (fun x => decide \u00acp x = true) l ~\n x :: (filter p l ++ filter (fun x => decide \u00acp x = true) l)\n[PROOFSTEP]\nexact perm_append_comm.trans (perm_append_comm.cons _)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l\u2082' : List \u03b1\ns : l\u2081 <+ l\u2082\np : l\u2082 ~ l\u2082'\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ l\u2082'\n[PROOFSTEP]\ninduction p generalizing l\u2081 with\n| nil => exact \u27e8[], eq_nil_of_sublist_nil s \u25b8 Perm.refl _, nil_sublist _\u27e9\n| cons x _ IH =>\n cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8l\u2081', p', s'.cons _\u27e9\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8x :: l\u2081', p'.cons x, s'.cons\u2082 _\u27e9\n| swap x y _ =>\n cases' s with _ _ _ s l\u2081 _ _ s <;> cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7 exact \u27e8l\u2081, Perm.refl _, (s.cons _).cons _\u27e9\n \u00b7 exact \u27e8x :: l\u2081, Perm.refl _, (s.cons _).cons\u2082 _\u27e9\n \u00b7 exact \u27e8y :: l\u2081, Perm.refl _, (s.cons\u2082 _).cons _\u27e9\n \u00b7 exact \u27e8x :: y :: l\u2081, Perm.swap _ _ _, (s.cons\u2082 _).cons\u2082 _\u27e9\n| trans _ _ IH\u2081 IH\u2082 =>\n exact\n let \u27e8m\u2081, pm, sm\u27e9 := IH\u2081 s\n let \u27e8r\u2081, pr, sr\u27e9 := IH\u2082 sm\n \u27e8r\u2081, pr.trans pm, sr\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l\u2082' : List \u03b1\ns : l\u2081 <+ l\u2082\np : l\u2082 ~ l\u2082'\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ l\u2082'\n[PROOFSTEP]\ninduction p generalizing l\u2081 with\n| nil => exact \u27e8[], eq_nil_of_sublist_nil s \u25b8 Perm.refl _, nil_sublist _\u27e9\n| cons x _ IH =>\n cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8l\u2081', p', s'.cons _\u27e9\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8x :: l\u2081', p'.cons x, s'.cons\u2082 _\u27e9\n| swap x y _ =>\n cases' s with _ _ _ s l\u2081 _ _ s <;> cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7 exact \u27e8l\u2081, Perm.refl _, (s.cons _).cons _\u27e9\n \u00b7 exact \u27e8x :: l\u2081, Perm.refl _, (s.cons _).cons\u2082 _\u27e9\n \u00b7 exact \u27e8y :: l\u2081, Perm.refl _, (s.cons\u2082 _).cons _\u27e9\n \u00b7 exact \u27e8x :: y :: l\u2081, Perm.swap _ _ _, (s.cons\u2082 _).cons\u2082 _\u27e9\n| trans _ _ IH\u2081 IH\u2082 =>\n exact\n let \u27e8m\u2081, pm, sm\u27e9 := IH\u2081 s\n let \u27e8r\u2081, pr, sr\u27e9 := IH\u2082 sm\n \u27e8r\u2081, pr.trans pm, sr\u27e9\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' l\u2081 : List \u03b1\ns : l\u2081 <+ []\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ []\n[PROOFSTEP]\n\n| nil => exact \u27e8[], eq_nil_of_sublist_nil s \u25b8 Perm.refl _, nil_sublist _\u27e9\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' l\u2081 : List \u03b1\ns : l\u2081 <+ []\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ []\n[PROOFSTEP]\nexact \u27e8[], eq_nil_of_sublist_nil s \u25b8 Perm.refl _, nil_sublist _\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ x :: l\u2081\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: l\u2082\u271d\n[PROOFSTEP]\n\n| cons x _ IH =>\n cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8l\u2081', p', s'.cons _\u27e9\n \u00b7\n exact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8x :: l\u2081', p'.cons x, s'.cons\u2082 _\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ x :: l\u2081\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: l\u2082\u271d\n[PROOFSTEP]\ncases' s with _ _ _ s l\u2081 _ _ s\n[GOAL]\ncase cons.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ l\u2081\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: l\u2082\u271d\n[PROOFSTEP]\nexact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8l\u2081', p', s'.cons _\u27e9\n[GOAL]\ncase cons.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nIH : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ l\u2081\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: l\u2082\u271d\n[PROOFSTEP]\nexact\n let \u27e8l\u2081', p', s'\u27e9 := IH s\n \u27e8x :: l\u2081', p'.cons x, s'.cons\u2082 _\u27e9\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ y :: x :: l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\n\n| swap x y _ =>\n cases' s with _ _ _ s l\u2081 _ _ s <;> cases' s with _ _ _ s l\u2081 _ _ s\n \u00b7 exact \u27e8l\u2081, Perm.refl _, (s.cons _).cons _\u27e9\n \u00b7 exact \u27e8x :: l\u2081, Perm.refl _, (s.cons _).cons\u2082 _\u27e9\n \u00b7 exact \u27e8y :: l\u2081, Perm.refl _, (s.cons\u2082 _).cons _\u27e9\n \u00b7 exact \u27e8x :: y :: l\u2081, Perm.swap _ _ _, (s.cons\u2082 _).cons\u2082 _\u27e9\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ y :: x :: l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\ncases' s with _ _ _ s l\u2081 _ _ s\n[GOAL]\ncase swap.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ x :: l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\ncases' s with _ _ _ s l\u2081 _ _ s\n[GOAL]\ncase swap.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ x :: l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\ncases' s with _ _ _ s l\u2081 _ _ s\n[GOAL]\ncase swap.cons.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\nexact \u27e8l\u2081, Perm.refl _, (s.cons _).cons _\u27e9\n[GOAL]\ncase swap.cons.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\nexact \u27e8x :: l\u2081, Perm.refl _, (s.cons _).cons\u2082 _\u27e9\n[GOAL]\ncase swap.cons\u2082.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\nexact \u27e8y :: l\u2081, Perm.refl _, (s.cons\u2082 _).cons _\u27e9\n[GOAL]\ncase swap.cons\u2082.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2082 l\u2082' : List \u03b1\nx y : \u03b1\nl\u271d l\u2081 : List \u03b1\ns : l\u2081 <+ l\u271d\n\u22a2 \u2203 l\u2081' x_1, l\u2081' <+ x :: y :: l\u271d\n[PROOFSTEP]\nexact \u27e8x :: y :: l\u2081, Perm.swap _ _ _, (s.cons\u2082 _).cons\u2082 _\u27e9\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nIH\u2082 : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2082\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2083\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ l\u2081\u271d\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ l\u2083\u271d\n[PROOFSTEP]\n\n| trans _ _ IH\u2081 IH\u2082 =>\n exact\n let \u27e8m\u2081, pm, sm\u27e9 := IH\u2081 s\n let \u27e8r\u2081, pr, sr\u27e9 := IH\u2082 sm\n \u27e8r\u2081, pr.trans pm, sr\u27e9\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2082 l\u2082' l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nIH\u2081 : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2081\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2082\u271d\nIH\u2082 : \u2200 {l\u2081 : List \u03b1}, l\u2081 <+ l\u2082\u271d \u2192 \u2203 l\u2081' x, l\u2081' <+ l\u2083\u271d\nl\u2081 : List \u03b1\ns : l\u2081 <+ l\u2081\u271d\n\u22a2 \u2203 l\u2081' x, l\u2081' <+ l\u2083\u271d\n[PROOFSTEP]\nexact\n let \u27e8m\u2081, pm, sm\u27e9 := IH\u2081 s\n let \u27e8r\u2081, pr, sr\u27e9 := IH\u2082 sm\n \u27e8r\u2081, pr.trans pm, sr\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 sizeOf l\u2081 = sizeOf l\u2082\n[PROOFSTEP]\ninduction h with\n -- hd l\u2081 l\u2082 h\u2081\u2082 h_sz\u2081\u2082 a b l l\u2081 l\u2082 l\u2083 h\u2081\u2082 h\u2082\u2083 h_sz\u2081\u2082 h_sz\u2082\u2083\n| nil => rfl\n| cons _ _ h_sz\u2081\u2082 => simp [h_sz\u2081\u2082]\n| swap => simp [add_left_comm]\n| trans _ _ h_sz\u2081\u2082 h_sz\u2082\u2083 => simp [h_sz\u2081\u2082, h_sz\u2082\u2083]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 sizeOf l\u2081 = sizeOf l\u2082\n[PROOFSTEP]\ninduction h with\n -- hd l\u2081 l\u2082 h\u2081\u2082 h_sz\u2081\u2082 a b l l\u2081 l\u2082 l\u2083 h\u2081\u2082 h\u2082\u2083 h_sz\u2081\u2082 h_sz\u2082\u2083\n| nil => rfl\n| cons _ _ h_sz\u2081\u2082 => simp [h_sz\u2081\u2082]\n| swap => simp [add_left_comm]\n| trans _ _ h_sz\u2081\u2082 h_sz\u2082\u2083 => simp [h_sz\u2081\u2082, h_sz\u2082\u2083]\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 sizeOf [] = sizeOf []\n[PROOFSTEP]\n\n| nil => rfl\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 sizeOf [] = sizeOf []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nh_sz\u2081\u2082 : sizeOf l\u2081\u271d = sizeOf l\u2082\u271d\n\u22a2 sizeOf (x\u271d :: l\u2081\u271d) = sizeOf (x\u271d :: l\u2082\u271d)\n[PROOFSTEP]\n\n| cons _ _ h_sz\u2081\u2082 => simp [h_sz\u2081\u2082]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nh_sz\u2081\u2082 : sizeOf l\u2081\u271d = sizeOf l\u2082\u271d\n\u22a2 sizeOf (x\u271d :: l\u2081\u271d) = sizeOf (x\u271d :: l\u2082\u271d)\n[PROOFSTEP]\nsimp [h_sz\u2081\u2082]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 sizeOf (y\u271d :: x\u271d :: l\u271d) = sizeOf (x\u271d :: y\u271d :: l\u271d)\n[PROOFSTEP]\n\n| swap => simp [add_left_comm]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 sizeOf (y\u271d :: x\u271d :: l\u271d) = sizeOf (x\u271d :: y\u271d :: l\u271d)\n[PROOFSTEP]\nsimp [add_left_comm]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nh_sz\u2081\u2082 : sizeOf l\u2081\u271d = sizeOf l\u2082\u271d\nh_sz\u2082\u2083 : sizeOf l\u2082\u271d = sizeOf l\u2083\u271d\n\u22a2 sizeOf l\u2081\u271d = sizeOf l\u2083\u271d\n[PROOFSTEP]\n\n| trans _ _ h_sz\u2081\u2082 h_sz\u2082\u2083 => simp [h_sz\u2081\u2082, h_sz\u2082\u2083]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : SizeOf \u03b1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nh_sz\u2081\u2082 : sizeOf l\u2081\u271d = sizeOf l\u2082\u271d\nh_sz\u2082\u2083 : sizeOf l\u2082\u271d = sizeOf l\u2083\u271d\n\u22a2 sizeOf l\u2081\u271d = sizeOf l\u2083\u271d\n[PROOFSTEP]\nsimp [h_sz\u2081\u2082, h_sz\u2082\u2083]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\n\u22a2 Perm \u2218r Perm = Perm\n[PROOFSTEP]\nfunext a c\n[GOAL]\ncase h.h\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na c : List \u03b1\n\u22a2 (Perm \u2218r Perm) a c = (a ~ c)\n[PROOFSTEP]\napply propext\n[GOAL]\ncase h.h.a\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na c : List \u03b1\n\u22a2 (Perm \u2218r Perm) a c \u2194 a ~ c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.a.mp\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na c : List \u03b1\n\u22a2 (Perm \u2218r Perm) a c \u2192 a ~ c\n[PROOFSTEP]\nexact fun \u27e8b, hab, hba\u27e9 => Perm.trans hab hba\n[GOAL]\ncase h.h.a.mpr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na c : List \u03b1\n\u22a2 a ~ c \u2192 (Perm \u2218r Perm) a c\n[PROOFSTEP]\nexact fun h => \u27e8a, Perm.refl a, h\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nv : List \u03b2\nhlu : l ~ u\nhuv : Forall\u2082 r u v\n\u22a2 (Forall\u2082 r \u2218r Perm) l v\n[PROOFSTEP]\ninduction hlu generalizing v\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r [] v\n\u22a2 (Forall\u2082 r \u2218r Perm) [] v\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r (x\u271d :: l\u2082\u271d) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (x\u271d :: l\u2081\u271d) v\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r (x\u271d :: y\u271d :: l\u271d) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (y\u271d :: x\u271d :: l\u271d) v\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2083\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2082\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r l\u2083\u271d v\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\n[PROOFSTEP]\ncase nil => cases huv; exact \u27e8[], Forall\u2082.nil, Perm.nil\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r [] v\n\u22a2 (Forall\u2082 r \u2218r Perm) [] v\n[PROOFSTEP]\ncase nil => cases huv; exact \u27e8[], Forall\u2082.nil, Perm.nil\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r [] v\n\u22a2 (Forall\u2082 r \u2218r Perm) [] v\n[PROOFSTEP]\ncases huv\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\n\u22a2 (Forall\u2082 r \u2218r Perm) [] []\n[PROOFSTEP]\nexact \u27e8[], Forall\u2082.nil, Perm.nil\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r (x\u271d :: l\u2082\u271d) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (x\u271d :: l\u2081\u271d) v\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r (x\u271d :: y\u271d :: l\u271d) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (y\u271d :: x\u271d :: l\u271d) v\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2083\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2082\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r l\u2083\u271d v\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\n[PROOFSTEP]\ncase cons a l u _hlu ih =>\n cases' huv with _ b _ v hab huv'\n rcases ih huv' with \u27e8l\u2082, h\u2081\u2082, h\u2082\u2083\u27e9\n exact \u27e8b :: l\u2082, Forall\u2082.cons hab h\u2081\u2082, h\u2082\u2083.cons _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u271d u\u271d : List \u03b1\na : \u03b1\nl u : List \u03b1\n_hlu : l ~ u\nih : \u2200 {v : List \u03b2}, Forall\u2082 r u v \u2192 (Forall\u2082 r \u2218r Perm) l v\nv : List \u03b2\nhuv : Forall\u2082 r (a :: u) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a :: l) v\n[PROOFSTEP]\ncase cons a l u _hlu ih =>\n cases' huv with _ b _ v hab huv'\n rcases ih huv' with \u27e8l\u2082, h\u2081\u2082, h\u2082\u2083\u27e9\n exact \u27e8b :: l\u2082, Forall\u2082.cons hab h\u2081\u2082, h\u2082\u2083.cons _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u271d u\u271d : List \u03b1\na : \u03b1\nl u : List \u03b1\n_hlu : l ~ u\nih : \u2200 {v : List \u03b2}, Forall\u2082 r u v \u2192 (Forall\u2082 r \u2218r Perm) l v\nv : List \u03b2\nhuv : Forall\u2082 r (a :: u) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a :: l) v\n[PROOFSTEP]\ncases' huv with _ b _ v hab huv'\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u271d u\u271d : List \u03b1\na : \u03b1\nl u : List \u03b1\n_hlu : l ~ u\nih : \u2200 {v : List \u03b2}, Forall\u2082 r u v \u2192 (Forall\u2082 r \u2218r Perm) l v\nb : \u03b2\nv : List \u03b2\nhab : r a b\nhuv' : Forall\u2082 r u v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a :: l) (b :: v)\n[PROOFSTEP]\nrcases ih huv' with \u27e8l\u2082, h\u2081\u2082, h\u2082\u2083\u27e9\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u271d u\u271d : List \u03b1\na : \u03b1\nl u : List \u03b1\n_hlu : l ~ u\nih : \u2200 {v : List \u03b2}, Forall\u2082 r u v \u2192 (Forall\u2082 r \u2218r Perm) l v\nb : \u03b2\nv : List \u03b2\nhab : r a b\nhuv' : Forall\u2082 r u v\nl\u2082 : List \u03b2\nh\u2081\u2082 : Forall\u2082 r l l\u2082\nh\u2082\u2083 : l\u2082 ~ v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a :: l) (b :: v)\n[PROOFSTEP]\nexact \u27e8b :: l\u2082, Forall\u2082.cons hab h\u2081\u2082, h\u2082\u2083.cons _\u27e9\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r (x\u271d :: y\u271d :: l\u271d) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (y\u271d :: x\u271d :: l\u271d) v\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2083\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2082\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r l\u2083\u271d v\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\n[PROOFSTEP]\ncase swap a\u2081 a\u2082 h\u2082\u2083 =>\n cases' huv with _ b\u2081 _ l\u2082 h\u2081 hr\u2082\u2083\n cases' hr\u2082\u2083 with _ b\u2082 _ l\u2082 h\u2082 h\u2081\u2082\n exact \u27e8b\u2082 :: b\u2081 :: l\u2082, Forall\u2082.cons h\u2082 (Forall\u2082.cons h\u2081 h\u2081\u2082), Perm.swap _ _ _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\na\u2081 a\u2082 : \u03b1\nh\u2082\u2083 : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r (a\u2081 :: a\u2082 :: h\u2082\u2083) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a\u2082 :: a\u2081 :: h\u2082\u2083) v\n[PROOFSTEP]\ncase swap a\u2081 a\u2082 h\u2082\u2083 =>\n cases' huv with _ b\u2081 _ l\u2082 h\u2081 hr\u2082\u2083\n cases' hr\u2082\u2083 with _ b\u2082 _ l\u2082 h\u2082 h\u2081\u2082\n exact \u27e8b\u2082 :: b\u2081 :: l\u2082, Forall\u2082.cons h\u2082 (Forall\u2082.cons h\u2081 h\u2081\u2082), Perm.swap _ _ _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\na\u2081 a\u2082 : \u03b1\nh\u2082\u2083 : List \u03b1\nv : List \u03b2\nhuv : Forall\u2082 r (a\u2081 :: a\u2082 :: h\u2082\u2083) v\n\u22a2 (Forall\u2082 r \u2218r Perm) (a\u2082 :: a\u2081 :: h\u2082\u2083) v\n[PROOFSTEP]\ncases' huv with _ b\u2081 _ l\u2082 h\u2081 hr\u2082\u2083\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\na\u2081 a\u2082 : \u03b1\nh\u2082\u2083 : List \u03b1\nb\u2081 : \u03b2\nl\u2082 : List \u03b2\nh\u2081 : r a\u2081 b\u2081\nhr\u2082\u2083 : Forall\u2082 r (a\u2082 :: h\u2082\u2083) l\u2082\n\u22a2 (Forall\u2082 r \u2218r Perm) (a\u2082 :: a\u2081 :: h\u2082\u2083) (b\u2081 :: l\u2082)\n[PROOFSTEP]\ncases' hr\u2082\u2083 with _ b\u2082 _ l\u2082 h\u2082 h\u2081\u2082\n[GOAL]\ncase cons.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u : List \u03b1\na\u2081 a\u2082 : \u03b1\nh\u2082\u2083 : List \u03b1\nb\u2081 : \u03b2\nh\u2081 : r a\u2081 b\u2081\nb\u2082 : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : r a\u2082 b\u2082\nh\u2081\u2082 : Forall\u2082 r h\u2082\u2083 l\u2082\n\u22a2 (Forall\u2082 r \u2218r Perm) (a\u2082 :: a\u2081 :: h\u2082\u2083) (b\u2081 :: b\u2082 :: l\u2082)\n[PROOFSTEP]\nexact \u27e8b\u2082 :: b\u2081 :: l\u2082, Forall\u2082.cons h\u2082 (Forall\u2082.cons h\u2081 h\u2081\u2082), Perm.swap _ _ _\u27e9\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2082\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\na_ih\u271d : \u2200 {v : List \u03b2}, Forall\u2082 r l\u2083\u271d v \u2192 (Forall\u2082 r \u2218r Perm) l\u2082\u271d v\nv : List \u03b2\nhuv : Forall\u2082 r l\u2083\u271d v\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081\u271d v\n[PROOFSTEP]\ncase trans la\u2081 la\u2082 la\u2083 _ _ ih\u2081 ih\u2082 =>\n rcases ih\u2082 huv with \u27e8lb\u2082, hab\u2082, h\u2082\u2083\u27e9\n rcases ih\u2081 hab\u2082 with \u27e8lb\u2081, hab\u2081, h\u2081\u2082\u27e9\n exact \u27e8lb\u2081, hab\u2081, Perm.trans h\u2081\u2082 h\u2082\u2083\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u la\u2081 la\u2082 la\u2083 : List \u03b1\na\u271d\u00b9 : la\u2081 ~ la\u2082\na\u271d : la\u2082 ~ la\u2083\nih\u2081 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2082 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2081 v\nih\u2082 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2083 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2082 v\nv : List \u03b2\nhuv : Forall\u2082 r la\u2083 v\n\u22a2 (Forall\u2082 r \u2218r Perm) la\u2081 v\n[PROOFSTEP]\ncase trans la\u2081 la\u2082 la\u2083 _ _ ih\u2081 ih\u2082 =>\n rcases ih\u2082 huv with \u27e8lb\u2082, hab\u2082, h\u2082\u2083\u27e9\n rcases ih\u2081 hab\u2082 with \u27e8lb\u2081, hab\u2081, h\u2081\u2082\u27e9\n exact \u27e8lb\u2081, hab\u2081, Perm.trans h\u2081\u2082 h\u2082\u2083\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u la\u2081 la\u2082 la\u2083 : List \u03b1\na\u271d\u00b9 : la\u2081 ~ la\u2082\na\u271d : la\u2082 ~ la\u2083\nih\u2081 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2082 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2081 v\nih\u2082 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2083 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2082 v\nv : List \u03b2\nhuv : Forall\u2082 r la\u2083 v\n\u22a2 (Forall\u2082 r \u2218r Perm) la\u2081 v\n[PROOFSTEP]\nrcases ih\u2082 huv with \u27e8lb\u2082, hab\u2082, h\u2082\u2083\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u la\u2081 la\u2082 la\u2083 : List \u03b1\na\u271d\u00b9 : la\u2081 ~ la\u2082\na\u271d : la\u2082 ~ la\u2083\nih\u2081 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2082 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2081 v\nih\u2082 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2083 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2082 v\nv : List \u03b2\nhuv : Forall\u2082 r la\u2083 v\nlb\u2082 : List \u03b2\nhab\u2082 : Forall\u2082 r la\u2082 lb\u2082\nh\u2082\u2083 : lb\u2082 ~ v\n\u22a2 (Forall\u2082 r \u2218r Perm) la\u2081 v\n[PROOFSTEP]\nrcases ih\u2081 hab\u2082 with \u27e8lb\u2081, hab\u2081, h\u2081\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl u la\u2081 la\u2082 la\u2083 : List \u03b1\na\u271d\u00b9 : la\u2081 ~ la\u2082\na\u271d : la\u2082 ~ la\u2083\nih\u2081 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2082 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2081 v\nih\u2082 : \u2200 {v : List \u03b2}, Forall\u2082 r la\u2083 v \u2192 (Forall\u2082 r \u2218r Perm) la\u2082 v\nv : List \u03b2\nhuv : Forall\u2082 r la\u2083 v\nlb\u2082 : List \u03b2\nhab\u2082 : Forall\u2082 r la\u2082 lb\u2082\nh\u2082\u2083 : lb\u2082 ~ v\nlb\u2081 : List \u03b2\nhab\u2081 : Forall\u2082 r la\u2081 lb\u2081\nh\u2081\u2082 : lb\u2081 ~ lb\u2082\n\u22a2 (Forall\u2082 r \u2218r Perm) la\u2081 v\n[PROOFSTEP]\nexact \u27e8lb\u2081, hab\u2081, Perm.trans h\u2081\u2082 h\u2082\u2083\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\n\u22a2 Forall\u2082 r \u2218r Perm = Perm \u2218r Forall\u2082 r\n[PROOFSTEP]\nfunext l\u2081 l\u2083\n[GOAL]\ncase h.h\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 : List \u03b2\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081 l\u2083 = (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\napply propext\n[GOAL]\ncase h.h.a\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 : List \u03b2\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081 l\u2083 \u2194 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.a.mp\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 : List \u03b2\n\u22a2 (Forall\u2082 r \u2218r Perm) l\u2081 l\u2083 \u2192 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.h.a.mp\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 : List \u03b2\nh : (Forall\u2082 r \u2218r Perm) l\u2081 l\u2083\n\u22a2 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nrcases h with \u27e8l\u2082, h\u2081\u2082, h\u2082\u2083\u27e9\n[GOAL]\ncase h.h.a.mp.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 l\u2082 : List \u03b2\nh\u2081\u2082 : Forall\u2082 r l\u2081 l\u2082\nh\u2082\u2083 : l\u2082 ~ l\u2083\n\u22a2 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nhave : Forall\u2082 (flip r) l\u2082 l\u2081 := h\u2081\u2082.flip\n[GOAL]\ncase h.h.a.mp.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 l\u2082 : List \u03b2\nh\u2081\u2082 : Forall\u2082 r l\u2081 l\u2082\nh\u2082\u2083 : l\u2082 ~ l\u2083\nthis : Forall\u2082 (flip r) l\u2082 l\u2081\n\u22a2 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nrcases perm_comp_forall\u2082 h\u2082\u2083.symm this with \u27e8l', h\u2081, h\u2082\u27e9\n[GOAL]\ncase h.h.a.mp.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 l\u2082 : List \u03b2\nh\u2081\u2082 : Forall\u2082 r l\u2081 l\u2082\nh\u2082\u2083 : l\u2082 ~ l\u2083\nthis : Forall\u2082 (flip r) l\u2082 l\u2081\nl' : List \u03b1\nh\u2081 : Forall\u2082 (flip r) l\u2083 l'\nh\u2082 : l' ~ l\u2081\n\u22a2 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083\n[PROOFSTEP]\nexact \u27e8l', h\u2082.symm, h\u2081.flip\u27e9\n[GOAL]\ncase h.h.a.mpr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2083 : List \u03b2\n\u22a2 (Perm \u2218r Forall\u2082 r) l\u2081 l\u2083 \u2192 (Forall\u2082 r \u2218r Perm) l\u2081 l\u2083\n[PROOFSTEP]\nexact fun \u27e8l\u2082, h\u2081\u2082, h\u2082\u2083\u27e9 => perm_comp_forall\u2082 h\u2081\u2082 h\u2082\u2083\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nhr : RightUnique r\na : List \u03b1\nb : List \u03b2\nh\u2081 : Forall\u2082 r a b\nc : List \u03b1\nd : List \u03b2\nh\u2082 : Forall\u2082 r c d\nh : a ~ c\nthis : (flip (Forall\u2082 r) \u2218r Perm \u2218r Forall\u2082 r) b d\n\u22a2 ((flip (Forall\u2082 r) \u2218r Forall\u2082 r) \u2218r Perm) b d\n[PROOFSTEP]\nrwa [\u2190 forall\u2082_comp_perm_eq_perm_comp_forall\u2082, \u2190 Relation.comp_assoc] at this \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\n\u22a2 [] <+ l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl l' : List \u03b1\nh : l <+~ l'\n\u22a2 List.filter p l <+~ List.filter p l'\n[PROOFSTEP]\nobtain \u27e8xs, hp, h\u27e9 := h\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl l' xs : List \u03b1\nhp : xs ~ l\nh : xs <+ l'\n\u22a2 List.filter p l <+~ List.filter p l'\n[PROOFSTEP]\nexact \u27e8_, hp.filter p, h.filter p\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\n\u22a2 countp p l\u2081 = countp p l\u2082\n[PROOFSTEP]\nrw [countp_eq_length_filter, countp_eq_length_filter]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\n\u22a2 length (List.filter p l\u2081) = length (List.filter p l\u2082)\n[PROOFSTEP]\nexact (s.filter _).length_eq\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\np p' : \u03b1 \u2192 Bool\nhp : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\n\u22a2 countp p l\u2081 = countp p' l\u2082\n[PROOFSTEP]\nrw [\u2190 s.countp_eq p']\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 ~ l\u2082\np p' : \u03b1 \u2192 Bool\nhp : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\n\u22a2 countp p l\u2081 = countp p' l\u2081\n[PROOFSTEP]\nclear s\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np p' : \u03b1 \u2192 Bool\nhp : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\n\u22a2 countp p l\u2081 = countp p' l\u2081\n[PROOFSTEP]\ninduction' l\u2081 with y s hs\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np p' : \u03b1 \u2192 Bool\nhp\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\nhp : \u2200 (x : \u03b1), x \u2208 [] \u2192 p x = p' x\n\u22a2 countp p [] = countp p' []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np p' : \u03b1 \u2192 Bool\nhp\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\ny : \u03b1\ns : List \u03b1\nhs : (\u2200 (x : \u03b1), x \u2208 s \u2192 p x = p' x) \u2192 countp p s = countp p' s\nhp : \u2200 (x : \u03b1), x \u2208 y :: s \u2192 p x = p' x\n\u22a2 countp p (y :: s) = countp p' (y :: s)\n[PROOFSTEP]\nsimp only [mem_cons, forall_eq_or_imp] at hp \n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np p' : \u03b1 \u2192 Bool\nhp\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x = p' x\ny : \u03b1\ns : List \u03b1\nhs : (\u2200 (x : \u03b1), x \u2208 s \u2192 p x = p' x) \u2192 countp p s = countp p' s\nhp : p y = p' y \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 p a = p' a\n\u22a2 countp p (y :: s) = countp p' (y :: s)\n[PROOFSTEP]\nsimp only [countp_cons, hs hp.2, hp.1]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\np q : \u03b1 \u2192 Bool\n\u22a2 countp p l = countp p (filter q l) + countp p (filter (fun a => decide \u00acq a = true) l)\n[PROOFSTEP]\nrw [\u2190 countp_append]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\np q : \u03b1 \u2192 Bool\n\u22a2 countp p l = countp p (filter q l ++ filter (fun a => decide \u00acq a = true) l)\n[PROOFSTEP]\nexact Perm.countp_eq _ (filter_append_perm _ _).symm\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr :\n (\u2200 (x : \u03b1), x \u2208 t\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x) y = f (f z y) x) \u2192\n \u2200 (b : \u03b2), foldl f b t\u2081 = foldl f b t\u2082\nH : \u2200 (x_1 : \u03b1), x_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (y_1 : \u03b1), y_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x_1) y_1 = f (f z y_1) x_1\nb : \u03b2\n\u22a2 foldl f b (y :: x :: t\u2081) = foldl f b (x :: y :: t\u2082)\n[PROOFSTEP]\nsimp only [foldl]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr :\n (\u2200 (x : \u03b1), x \u2208 t\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x) y = f (f z y) x) \u2192\n \u2200 (b : \u03b2), foldl f b t\u2081 = foldl f b t\u2082\nH : \u2200 (x_1 : \u03b1), x_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (y_1 : \u03b1), y_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x_1) y_1 = f (f z y_1) x_1\nb : \u03b2\n\u22a2 foldl f (f (f b y) x) t\u2081 = foldl f (f (f b x) y) t\u2082\n[PROOFSTEP]\nrw [H x (.tail _ <| .head _) y (.head _)]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr :\n (\u2200 (x : \u03b1), x \u2208 t\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x) y = f (f z y) x) \u2192\n \u2200 (b : \u03b2), foldl f b t\u2081 = foldl f b t\u2082\nH : \u2200 (x_1 : \u03b1), x_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (y_1 : \u03b1), y_1 \u2208 y :: x :: t\u2081 \u2192 \u2200 (z : \u03b2), f (f z x_1) y_1 = f (f z y_1) x_1\nb : \u03b2\n\u22a2 foldl f (f (f b y) x) t\u2081 = foldl f (f (f b y) x) t\u2082\n[PROOFSTEP]\nexact r (fun x hx y hy => H _ (.tail _ <| .tail _ hx) _ (.tail _ <| .tail _ hy)) _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nlcomm : LeftCommutative f\np : l\u2081 ~ l\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr : \u2200 (b : \u03b2), foldr f b t\u2081 = foldr f b t\u2082\nb : \u03b2\n\u22a2 foldr f b (x :: t\u2081) = foldr f b (x :: t\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nlcomm : LeftCommutative f\np : l\u2081 ~ l\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr : \u2200 (b : \u03b2), foldr f b t\u2081 = foldr f b t\u2082\nb : \u03b2\n\u22a2 f x (foldr f b t\u2081) = f x (foldr f b t\u2082)\n[PROOFSTEP]\nrw [r b]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nlcomm : LeftCommutative f\np : l\u2081 ~ l\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr : \u2200 (b : \u03b2), foldr f b t\u2081 = foldr f b t\u2082\nb : \u03b2\n\u22a2 foldr f b (y :: x :: t\u2081) = foldr f b (x :: y :: t\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nl\u2081 l\u2082 : List \u03b1\nlcomm : LeftCommutative f\np : l\u2081 ~ l\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\n_p : t\u2081 ~ t\u2082\nr : \u2200 (b : \u03b2), foldr f b t\u2081 = foldr f b t\u2082\nb : \u03b2\n\u22a2 f y (f x (foldr f b t\u2081)) = f x (f y (foldr f b t\u2082))\n[PROOFSTEP]\nrw [lcomm, r b]\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nhl : l ~ l'\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\n\u22a2 HEq (List.rec b f l) (List.rec b f l')\n[PROOFSTEP]\ninduction hl\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\n\u22a2 HEq (List.rec b f []) (List.rec b f [])\ncase cons\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\n\u22a2 HEq (List.rec b f (x\u271d :: l\u2081\u271d)) (List.rec b f (x\u271d :: l\u2082\u271d))\ncase swap\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 HEq (List.rec b f (y\u271d :: x\u271d :: l\u271d)) (List.rec b f (x\u271d :: y\u271d :: l\u271d))\ncase trans\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\na_ih\u271d : HEq (List.rec b f l\u2082\u271d) (List.rec b f l\u2083\u271d)\n\u22a2 HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2083\u271d)\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\n\u22a2 HEq (List.rec b f []) (List.rec b f [])\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\n\u22a2 HEq (List.rec b f []) (List.rec b f [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\n\u22a2 HEq (List.rec b f (x\u271d :: l\u2081\u271d)) (List.rec b f (x\u271d :: l\u2082\u271d))\ncase swap\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 HEq (List.rec b f (y\u271d :: x\u271d :: l\u271d)) (List.rec b f (x\u271d :: y\u271d :: l\u271d))\ncase trans\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\na_ih\u271d : HEq (List.rec b f l\u2082\u271d) (List.rec b f l\u2083\u271d)\n\u22a2 HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2083\u271d)\n[PROOFSTEP]\ncase cons a l l' h ih => exact f_congr h ih\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl\u271d l'\u271d : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\na : \u03b1\nl l' : List \u03b1\nh : l ~ l'\nih : HEq (List.rec b f l) (List.rec b f l')\n\u22a2 HEq (List.rec b f (a :: l)) (List.rec b f (a :: l'))\n[PROOFSTEP]\ncase cons a l l' h ih => exact f_congr h ih\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl\u271d l'\u271d : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\na : \u03b1\nl l' : List \u03b1\nh : l ~ l'\nih : HEq (List.rec b f l) (List.rec b f l')\n\u22a2 HEq (List.rec b f (a :: l)) (List.rec b f (a :: l'))\n[PROOFSTEP]\nexact f_congr h ih\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 HEq (List.rec b f (y\u271d :: x\u271d :: l\u271d)) (List.rec b f (x\u271d :: y\u271d :: l\u271d))\ncase trans\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\na_ih\u271d : HEq (List.rec b f l\u2082\u271d) (List.rec b f l\u2083\u271d)\n\u22a2 HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2083\u271d)\n[PROOFSTEP]\ncase swap a a' l => exact f_swap\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl\u271d l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\na a' : \u03b1\nl : List \u03b1\n\u22a2 HEq (List.rec b f (a' :: a :: l)) (List.rec b f (a :: a' :: l))\n[PROOFSTEP]\ncase swap a a' l => exact f_swap\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl\u271d l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\na a' : \u03b1\nl : List \u03b1\n\u22a2 HEq (List.rec b f (a' :: a :: l)) (List.rec b f (a :: a' :: l))\n[PROOFSTEP]\nexact f_swap\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2082\u271d)\na_ih\u271d : HEq (List.rec b f l\u2082\u271d) (List.rec b f l\u2083\u271d)\n\u22a2 HEq (List.rec b f l\u2081\u271d) (List.rec b f l\u2083\u271d)\n[PROOFSTEP]\ncase trans l\u2081 l\u2082 l\u2083 _h\u2081 _h\u2082 ih\u2081 ih\u2082 => exact HEq.trans ih\u2081 ih\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081 l\u2082 l\u2083 : List \u03b1\n_h\u2081 : l\u2081 ~ l\u2082\n_h\u2082 : l\u2082 ~ l\u2083\nih\u2081 : HEq (List.rec b f l\u2081) (List.rec b f l\u2082)\nih\u2082 : HEq (List.rec b f l\u2082) (List.rec b f l\u2083)\n\u22a2 HEq (List.rec b f l\u2081) (List.rec b f l\u2083)\n[PROOFSTEP]\ncase trans l\u2081 l\u2082 l\u2083 _h\u2081 _h\u2082 ih\u2081 ih\u2082 => exact HEq.trans ih\u2081 ih\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2\u271d : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\n\u03b2 : List \u03b1 \u2192 Sort u_1\nf : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 \u03b2 l \u2192 \u03b2 (a :: l)\nb : \u03b2 []\nl l' : List \u03b1\nf_congr : \u2200 {a : \u03b1} {l l' : List \u03b1} {b : \u03b2 l} {b' : \u03b2 l'}, l ~ l' \u2192 HEq b b' \u2192 HEq (f a l b) (f a l' b')\nf_swap : \u2200 {a a' : \u03b1} {l : List \u03b1} {b : \u03b2 l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))\nl\u2081 l\u2082 l\u2083 : List \u03b1\n_h\u2081 : l\u2081 ~ l\u2082\n_h\u2082 : l\u2082 ~ l\u2083\nih\u2081 : HEq (List.rec b f l\u2081) (List.rec b f l\u2082)\nih\u2082 : HEq (List.rec b f l\u2082) (List.rec b f l\u2083)\n\u22a2 HEq (List.rec b f l\u2081) (List.rec b f l\u2083)\n[PROOFSTEP]\nexact HEq.trans ih\u2081 ih\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 prod l\u2081 = prod l\u2082\n[PROOFSTEP]\nrefine h.foldl_eq' ?_ _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2081 \u2192 \u2200 (z : \u03b1), z * x * y = z * y * x\n[PROOFSTEP]\napply Pairwise.forall_of_forall\n[GOAL]\ncase H\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Symmetric fun x y => \u2200 (z : \u03b1), z * x * y = z * y * x\n[PROOFSTEP]\nintro x y h z\n[GOAL]\ncase H\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx y : \u03b1\nh : \u2200 (z : \u03b1), z * x * y = z * y * x\nz : \u03b1\n\u22a2 z * y * x = z * x * y\n[PROOFSTEP]\nexact (h z).symm\n[GOAL]\ncase H\u2081\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (z : \u03b1), z * x * x = z * x * x\n[PROOFSTEP]\nintros\n[GOAL]\ncase H\u2081\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx\u271d : \u03b1\na\u271d : x\u271d \u2208 l\u2081\nz\u271d : \u03b1\n\u22a2 z\u271d * x\u271d * x\u271d = z\u271d * x\u271d * x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Pairwise (fun x y => \u2200 (z : \u03b1), z * x * y = z * y * x) l\u2081\n[PROOFSTEP]\napply hc.imp\n[GOAL]\ncase H\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 {a b : \u03b1}, Commute a b \u2192 \u2200 (z : \u03b1), z * a * b = z * b * a\n[PROOFSTEP]\nintro a b h z\n[GOAL]\ncase H\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nM : Monoid \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\na b : \u03b1\nh : Commute a b\nz : \u03b1\n\u22a2 z * a * b = z * b * a\n[PROOFSTEP]\nrw [mul_assoc z, mul_assoc z, h]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\n\u22a2 l\u2081 ++ a :: r\u2081 ~ l\u2082 ++ a :: r\u2082 \u2192 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\ngeneralize e\u2081 : l\u2081 ++ a :: r\u2081 = s\u2081\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 r\u2081 r\u2082 s\u2081 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = s\u2081\n\u22a2 s\u2081 ~ l\u2082 ++ a :: r\u2082 \u2192 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\ngeneralize e\u2082 : l\u2082 ++ a :: r\u2082 = s\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 r\u2081 r\u2082 s\u2081 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = s\u2081\ns\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = s\u2082\n\u22a2 s\u2081 ~ s\u2082 \u2192 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nintro p\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 r\u2081 r\u2082 s\u2081 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = s\u2081\ns\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = s\u2082\np : s\u2081 ~ s\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrevert l\u2081 l\u2082 r\u2081 r\u2082 e\u2081 e\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\n\u22a2 \u2200 {l\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1}, l\u2081 ++ a :: r\u2081 = s\u2081 \u2192 l\u2082 ++ a :: r\u2082 = s\u2082 \u2192 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nclear l\u2081 l\u2082 \u03b2\n[GOAL]\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\n\u22a2 \u2200 {l\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1}, l\u2081 ++ a :: r\u2081 = s\u2081 \u2192 l\u2082 ++ a :: r\u2082 = s\u2082 \u2192 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nshow \u2200 _ _ _ _, _\n[GOAL]\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\n\u22a2 \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = s\u2081 \u2192 x_1 ++ a :: x_3 = s\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n[PROOFSTEP]\nrefine perm_induction_on p ?_ (fun x t\u2081 t\u2082 p IH => ?_) (fun x y t\u2081 t\u2082 p IH => ?_) fun t\u2081 t\u2082 t\u2083 p\u2081 p\u2082 IH\u2081 IH\u2082 => ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\n\u22a2 \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = [] \u2192 x_1 ++ a :: x_3 = [] \u2192 x ++ x_2 ~ x_1 ++ x_3\n[PROOFSTEP]\nintro l\u2081 l\u2082 r\u2081 r\u2082 e\u2081 e\u2082\n[GOAL]\ncase refine_2\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 \u2200 (x_1 x_2 x_3 x_4 : List \u03b1), x_1 ++ a :: x_3 = x :: t\u2081 \u2192 x_2 ++ a :: x_4 = x :: t\u2082 \u2192 x_1 ++ x_3 ~ x_2 ++ x_4\n[PROOFSTEP]\nintro l\u2081 l\u2082 r\u2081 r\u2082 e\u2081 e\u2082\n[GOAL]\ncase refine_3\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 \u2200 (x_1 x_2 x_3 x_4 : List \u03b1),\n x_1 ++ a :: x_3 = y :: x :: t\u2081 \u2192 x_2 ++ a :: x_4 = x :: y :: t\u2082 \u2192 x_1 ++ x_3 ~ x_2 ++ x_4\n[PROOFSTEP]\nintro l\u2081 l\u2082 r\u2081 r\u2082 e\u2081 e\u2082\n[GOAL]\ncase refine_4\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2081 t\u2082 t\u2083 : List \u03b1\np\u2081 : t\u2081 ~ t\u2082\np\u2082 : t\u2082 ~ t\u2083\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = t\u2083 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2083 \u2192 x ++ x_2 ~ x_1 ++ x_3\n[PROOFSTEP]\nintro l\u2081 l\u2082 r\u2081 r\u2082 e\u2081 e\u2082\n[GOAL]\ncase refine_1\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = []\ne\u2082 : l\u2082 ++ a :: r\u2082 = []\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\napply (not_mem_nil a).elim\n[GOAL]\ncase refine_1\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = []\ne\u2082 : l\u2082 ++ a :: r\u2082 = []\n\u22a2 a \u2208 []\n[PROOFSTEP]\nrw [\u2190 e\u2081]\n[GOAL]\ncase refine_1\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = []\ne\u2082 : l\u2082 ++ a :: r\u2082 = []\n\u22a2 a \u2208 l\u2081 ++ a :: r\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = x :: t\u2081\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: t\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\ncases' l\u2081 with y l\u2081\n[GOAL]\ncase refine_2.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: t\u2082\ne\u2081 : [] ++ a :: r\u2081 = x :: t\u2081\n\u22a2 [] ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\ncases' l\u2082 with z l\u2082\n[GOAL]\ncase refine_2.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: t\u2082\ny : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: l\u2081 ++ a :: r\u2081 = x :: t\u2081\n\u22a2 y :: l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\ncases' l\u2082 with z l\u2082\n[GOAL]\ncase refine_2.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : [] ++ a :: r\u2081 = x :: t\u2081\ne\u2082 : [] ++ a :: r\u2082 = x :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_2.nil.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : [] ++ a :: r\u2081 = x :: t\u2081\nz : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : z :: l\u2082 ++ a :: r\u2082 = x :: t\u2082\n\u22a2 [] ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_2.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: l\u2081 ++ a :: r\u2081 = x :: t\u2081\ne\u2082 : [] ++ a :: r\u2082 = x :: t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_2.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: l\u2081 ++ a :: r\u2081 = x :: t\u2081\nz : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : z :: l\u2082 ++ a :: r\u2082 = x :: t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_2.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : a :: r\u2081 = x :: t\u2081\ne\u2082 : a :: r\u2082 = x :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_2.nil.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : a :: r\u2081 = x :: t\u2081\nz : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : z :: (l\u2082 ++ a :: r\u2082) = x :: t\u2082\n\u22a2 [] ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_2.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: (l\u2081 ++ a :: r\u2081) = x :: t\u2081\ne\u2082 : a :: r\u2082 = x :: t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_2.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: (l\u2081 ++ a :: r\u2081) = x :: t\u2081\nz : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : z :: (l\u2082 ++ a :: r\u2082) = x :: t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_2.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nhead_eq\u271d\u00b9 : a = x\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d : a = x\ntail_eq\u271d : r\u2082 = t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase refine_2.nil.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2082 : List \u03b1\nhead_eq\u271d\u00b9 : a = x\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d : z = x\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\n\u22a2 [] ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase refine_2.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\nhead_eq\u271d\u00b9 : y = x\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d : a = x\ntail_eq\u271d : r\u2082 = t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase refine_2.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\nz : \u03b1\nl\u2082 : List \u03b1\nhead_eq\u271d\u00b9 : y = x\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d : z = x\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\n\u22a2 y :: l\u2081 ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase refine_2.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\ntail_eq\u271d : r\u2082 = t\u2082\nhead_eq\u271d : a = a\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts t\u2081 t\u2082\n[GOAL]\ncase refine_2.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nr\u2081 r\u2082 : List \u03b1\nhead_eq\u271d : a = a\np : r\u2081 ~ r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = r\u2081 \u2192 x_1 ++ a :: x_3 = r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nexact p\n[GOAL]\ncase refine_2.nil.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\nhead_eq\u271d : z = a\n\u22a2 [] ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts z t\u2081 t\u2082\n[GOAL]\ncase refine_2.nil.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nr\u2081 r\u2082 l\u2082 : List \u03b1\np : r\u2081 ~ l\u2082 ++ a :: r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = r\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [] ++ r\u2081 ~ a :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nexact p.trans perm_middle\n[GOAL]\ncase refine_2.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\ntail_eq\u271d : r\u2082 = t\u2082\nhead_eq\u271d : a = y\n\u22a2 y :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts y t\u2081 t\u2082\n[GOAL]\ncase refine_2.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nr\u2081 r\u2082 l\u2081 : List \u03b1\np : l\u2081 ++ a :: r\u2081 ~ r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 a :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nexact perm_middle.symm.trans p\n[GOAL]\ncase refine_2.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 : List \u03b1\nz : \u03b1\nl\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\nhead_eq\u271d : z = y\n\u22a2 y :: l\u2081 ++ r\u2081 ~ z :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts z t\u2081 t\u2082\n[GOAL]\ncase refine_2.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ++ a :: r\u2081 ~ l\u2082 ++ a :: r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 y :: l\u2081 ++ r\u2081 ~ y :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nexact (IH _ _ _ _ rfl rfl).cons y\n[GOAL]\ncase refine_3\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = y :: x :: t\u2081\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: y :: t\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrcases l\u2081 with (_ | \u27e8y, _ | \u27e8z, l\u2081\u27e9\u27e9)\n[GOAL]\ncase refine_3.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: y :: t\u2082\ne\u2081 : [] ++ a :: r\u2081 = y :: x :: t\u2081\n\u22a2 [] ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrcases l\u2082 with (_ | \u27e8u, _ | \u27e8v, l\u2082\u27e9\u27e9)\n[GOAL]\ncase refine_3.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: y\u271d :: t\u2082\ny : \u03b1\ne\u2081 : [y] ++ a :: r\u2081 = y\u271d :: x :: t\u2081\n\u22a2 [y] ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrcases l\u2082 with (_ | \u27e8u, _ | \u27e8v, l\u2082\u27e9\u27e9)\n[GOAL]\ncase refine_3.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2082 : l\u2082 ++ a :: r\u2082 = x :: y\u271d :: t\u2082\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: l\u2081 ++ a :: r\u2081 = y\u271d :: x :: t\u2081\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrcases l\u2082 with (_ | \u27e8u, _ | \u27e8v, l\u2082\u27e9\u27e9)\n[GOAL]\ncase refine_3.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : [] ++ a :: r\u2081 = y :: x :: t\u2081\ne\u2082 : [] ++ a :: r\u2082 = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : [] ++ a :: r\u2081 = y :: x :: t\u2081\nu : \u03b1\ne\u2082 : [u] ++ a :: r\u2082 = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : [] ++ a :: r\u2081 = y :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: l\u2082 ++ a :: r\u2082 = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : [y] ++ a :: r\u2081 = y\u271d :: x :: t\u2081\ne\u2082 : [] ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : [y] ++ a :: r\u2081 = y\u271d :: x :: t\u2081\nu : \u03b1\ne\u2082 : [u] ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : [y] ++ a :: r\u2081 = y\u271d :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: l\u2082 ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: l\u2081 ++ a :: r\u2081 = y\u271d :: x :: t\u2081\ne\u2082 : [] ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: l\u2081 ++ a :: r\u2081 = y\u271d :: x :: t\u2081\nu : \u03b1\ne\u2082 : [u] ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.cons.cons.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: l\u2081 ++ a :: r\u2081 = y\u271d :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: l\u2082 ++ a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ndsimp at e\u2081 e\u2082 \n[GOAL]\ncase refine_3.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : a :: r\u2081 = y :: x :: t\u2081\ne\u2082 : a :: r\u2082 = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : a :: r\u2081 = y :: x :: t\u2081\nu : \u03b1\ne\u2082 : u :: a :: r\u2082 = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ne\u2081 : a :: r\u2081 = y :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: (l\u2082 ++ a :: r\u2082) = x :: y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : y :: a :: r\u2081 = y\u271d :: x :: t\u2081\ne\u2082 : a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : y :: a :: r\u2081 = y\u271d :: x :: t\u2081\nu : \u03b1\ne\u2082 : u :: a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\ne\u2081 : y :: a :: r\u2081 = y\u271d :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: (l\u2082 ++ a :: r\u2082) = x :: y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: (l\u2081 ++ a :: r\u2081) = y\u271d :: x :: t\u2081\ne\u2082 : a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: (l\u2081 ++ a :: r\u2081) = y\u271d :: x :: t\u2081\nu : \u03b1\ne\u2082 : u :: a :: r\u2082 = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.cons.cons.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\ne\u2081 : y :: z :: (l\u2081 ++ a :: r\u2081) = y\u271d :: x :: t\u2081\nu v : \u03b1\nl\u2082 : List \u03b1\ne\u2082 : u :: v :: (l\u2082 ++ a :: r\u2082) = x :: y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\ninjections\n[GOAL]\ncase refine_3.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nhead_eq\u271d\u00b9 : a = y\ntail_eq\u271d\u00b9 : r\u2081 = x :: t\u2081\nhead_eq\u271d : a = x\ntail_eq\u271d : r\u2082 = y :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu : \u03b1\nhead_eq\u271d\u00b2 : a = y\ntail_eq\u271d\u00b9 : r\u2081 = x :: t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : a = y\ntail_eq\u271d : r\u2082 = t\u2082\n\u22a2 [] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu v : \u03b1\nl\u2082 : List \u03b1\nhead_eq\u271d\u00b2 : a = y\ntail_eq\u271d\u00b9 : r\u2081 = x :: t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : v = y\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\n\u22a2 [] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny : \u03b1\nhead_eq\u271d\u00b2 : y = y\u271d\nhead_eq\u271d\u00b9 : a = x\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d : a = x\ntail_eq\u271d : r\u2082 = y\u271d :: t\u2082\n\u22a2 [y] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny u : \u03b1\nhead_eq\u271d\u00b3 : y = y\u271d\nhead_eq\u271d\u00b2 : a = x\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : a = y\u271d\ntail_eq\u271d : r\u2082 = t\u2082\n\u22a2 [y] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny u v : \u03b1\nl\u2082 : List \u03b1\nhead_eq\u271d\u00b3 : y = y\u271d\nhead_eq\u271d\u00b2 : a = x\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : v = y\u271d\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\n\u22a2 [y] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\nhead_eq\u271d\u00b2 : y = y\u271d\nhead_eq\u271d\u00b9 : z = x\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d : a = x\ntail_eq\u271d : r\u2082 = y\u271d :: t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\nu : \u03b1\nhead_eq\u271d\u00b3 : y = y\u271d\nhead_eq\u271d\u00b2 : z = x\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : a = y\u271d\ntail_eq\u271d : r\u2082 = t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.cons.cons.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nx y\u271d : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ny z : \u03b1\nl\u2081 : List \u03b1\nu v : \u03b1\nl\u2082 : List \u03b1\nhead_eq\u271d\u00b3 : y = y\u271d\nhead_eq\u271d\u00b2 : z = x\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : u = x\nhead_eq\u271d : v = y\u271d\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts x y\n[GOAL]\ncase refine_3.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = a :: t\u2081\ntail_eq\u271d : r\u2082 = a :: t\u2082\n\u22a2 [] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 r\u2082\n[GOAL]\ncase refine_3.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [] ++ a :: t\u2081 ~ [] ++ a :: t\u2082\n[PROOFSTEP]\nexact p.cons a\n[GOAL]\ncase refine_3.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu : \u03b1\ntail_eq\u271d\u00b9 : r\u2082 = t\u2082\ntail_eq\u271d : r\u2081 = u :: t\u2081\nhead_eq\u271d : a = a\n\u22a2 [] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 r\u2082\n[GOAL]\ncase refine_3.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nu : \u03b1\nhead_eq\u271d : a = a\n\u22a2 [] ++ u :: t\u2081 ~ [u] ++ t\u2082\n[PROOFSTEP]\nexact p.cons u\n[GOAL]\ncase refine_3.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu v : \u03b1\nl\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : l\u2082 ++ a :: r\u2082 = t\u2082\ntail_eq\u271d : r\u2081 = u :: t\u2081\nhead_eq\u271d : v = a\n\u22a2 [] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 v t\u2082\n[GOAL]\ncase refine_3.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 r\u2082 : List \u03b1\nu : \u03b1\nl\u2082 : List \u03b1\np : t\u2081 ~ l\u2082 ++ a :: r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [] ++ u :: t\u2081 ~ u :: a :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nexact (p.trans perm_middle).cons u\n[GOAL]\ncase refine_3.cons.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\ntail_eq\u271d : r\u2082 = y :: t\u2082\nhead_eq\u271d : a = a\n\u22a2 [y] ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 r\u2082\n[GOAL]\ncase refine_3.cons.nil.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nhead_eq\u271d : a = a\n\u22a2 [y] ++ t\u2081 ~ [] ++ y :: t\u2082\n[PROOFSTEP]\nexact p.cons y\n[GOAL]\ncase refine_3.cons.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu : \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : a = y\ntail_eq\u271d : r\u2082 = t\u2082\nhead_eq\u271d : u = a\n\u22a2 [y] ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 r\u2082 y u\n[GOAL]\ncase refine_3.cons.nil.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [a] ++ t\u2081 ~ [a] ++ t\u2082\n[PROOFSTEP]\nexact p.cons a\n[GOAL]\ncase refine_3.cons.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nu v : \u03b1\nl\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : v = y\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\nhead_eq\u271d : u = a\n\u22a2 [y] ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2081 u v t\u2082\n[GOAL]\ncase refine_3.cons.nil.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 r\u2082 l\u2082 : List \u03b1\np : t\u2081 ~ l\u2082 ++ a :: r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 [y] ++ t\u2081 ~ a :: y :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nexact ((p.trans perm_middle).cons y).trans (swap _ _ _)\n[GOAL]\ncase refine_3.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2081 : List \u03b1\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\ntail_eq\u271d : r\u2082 = y :: t\u2082\nhead_eq\u271d : a = z\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2082 z t\u2081\n[GOAL]\ncase refine_3.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2082 r\u2081 l\u2081 : List \u03b1\np : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 y :: a :: l\u2081 ++ r\u2081 ~ [] ++ y :: t\u2082\n[PROOFSTEP]\nexact (perm_middle.symm.trans p).cons y\n[GOAL]\ncase refine_3.cons.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2081 : List \u03b1\nu : \u03b1\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : a = y\ntail_eq\u271d : r\u2082 = t\u2082\nhead_eq\u271d : u = z\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ [u] ++ r\u2082\n[PROOFSTEP]\nsubsts r\u2082 y z t\u2081\n[GOAL]\ncase refine_3.cons.cons.cons.nil\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\nt\u2082 r\u2081 l\u2081 : List \u03b1\nu : \u03b1\np : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 a :: u :: l\u2081 ++ r\u2081 ~ [u] ++ t\u2082\n[PROOFSTEP]\nexact (swap _ _ _).trans ((perm_middle.symm.trans p).cons u)\n[GOAL]\ncase refine_3.cons.cons.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2081 : List \u03b1\nu v : \u03b1\nl\u2082 : List \u03b1\ntail_eq\u271d\u00b9 : l\u2081 ++ a :: r\u2081 = t\u2081\nhead_eq\u271d\u00b9 : v = y\ntail_eq\u271d : l\u2082 ++ a :: r\u2082 = t\u2082\nhead_eq\u271d : u = z\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ u :: v :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts u v t\u2081 t\u2082\n[GOAL]\ncase refine_3.cons.cons.cons.cons\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np\u271d : s\u2081 ~ s\u2082\ny : \u03b1\nr\u2081 r\u2082 : List \u03b1\nz : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ++ a :: r\u2081 ~ l\u2082 ++ a :: r\u2082\nIH : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 y :: z :: l\u2081 ++ r\u2081 ~ z :: y :: l\u2082 ++ r\u2082\n[PROOFSTEP]\nexact (IH _ _ _ _ rfl rfl).swap' _ _\n[GOAL]\ncase refine_4\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2081 t\u2082 t\u2083 : List \u03b1\np\u2081 : t\u2081 ~ t\u2082\np\u2082 : t\u2082 ~ t\u2083\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = t\u2083 \u2192 x ++ x_2 ~ x_1 ++ x_3\nl\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\ne\u2081 : l\u2081 ++ a :: r\u2081 = t\u2081\ne\u2082 : l\u2082 ++ a :: r\u2082 = t\u2083\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nsubsts t\u2081 t\u2083\n[GOAL]\ncase refine_4\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2082 l\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\np\u2081 : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\np\u2082 : t\u2082 ~ l\u2082 ++ a :: r\u2082\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nhave : a \u2208 t\u2082 := p\u2081.subset (by simp)\n[GOAL]\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2082 l\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\np\u2081 : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\np\u2082 : t\u2082 ~ l\u2082 ++ a :: r\u2082\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\n\u22a2 a \u2208 l\u2081 ++ a :: r\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_4\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2082 l\u2081 l\u2082 r\u2081 r\u2082 : List \u03b1\np\u2081 : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\np\u2082 : t\u2082 ~ l\u2082 ++ a :: r\u2082\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\nthis : a \u2208 t\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082 ++ r\u2082\n[PROOFSTEP]\nrcases mem_split this with \u27e8l\u2082, r\u2082, e\u2082\u27e9\n[GOAL]\ncase refine_4.intro.intro\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nt\u2082 l\u2081 l\u2082\u271d r\u2081 r\u2082\u271d : List \u03b1\np\u2081 : l\u2081 ++ a :: r\u2081 ~ t\u2082\nIH\u2081 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = t\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\np\u2082 : t\u2082 ~ l\u2082\u271d ++ a :: r\u2082\u271d\nIH\u2082 : \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = t\u2082 \u2192 x_1 ++ a :: x_3 = l\u2082\u271d ++ a :: r\u2082\u271d \u2192 x ++ x_2 ~ x_1 ++ x_3\nthis : a \u2208 t\u2082\nl\u2082 r\u2082 : List \u03b1\ne\u2082 : t\u2082 = l\u2082 ++ a :: r\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082\u271d ++ r\u2082\u271d\n[PROOFSTEP]\nsubst t\u2082\n[GOAL]\ncase refine_4.intro.intro\n\u03b1 : Type uu\na : \u03b1\ns\u2081 s\u2082 : List \u03b1\np : s\u2081 ~ s\u2082\nl\u2081 l\u2082\u271d r\u2081 r\u2082\u271d l\u2082 r\u2082 : List \u03b1\np\u2081 : l\u2081 ++ a :: r\u2081 ~ l\u2082 ++ a :: r\u2082\nIH\u2081 :\n \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2081 ++ a :: r\u2081 \u2192 x_1 ++ a :: x_3 = l\u2082 ++ a :: r\u2082 \u2192 x ++ x_2 ~ x_1 ++ x_3\np\u2082 : l\u2082 ++ a :: r\u2082 ~ l\u2082\u271d ++ a :: r\u2082\u271d\nIH\u2082 :\n \u2200 (x x_1 x_2 x_3 : List \u03b1), x ++ a :: x_2 = l\u2082 ++ a :: r\u2082 \u2192 x_1 ++ a :: x_3 = l\u2082\u271d ++ a :: r\u2082\u271d \u2192 x ++ x_2 ~ x_1 ++ x_3\nthis : a \u2208 l\u2082 ++ a :: r\u2082\n\u22a2 l\u2081 ++ r\u2081 ~ l\u2082\u271d ++ r\u2082\u271d\n[PROOFSTEP]\nexact (IH\u2081 _ _ _ _ rfl rfl).trans (IH\u2082 _ _ _ _ rfl rfl)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\no\u2081 o\u2082 : Option \u03b1\n\u22a2 Option.toList o\u2081 ~ Option.toList o\u2082 \u2194 o\u2081 = o\u2082\n[PROOFSTEP]\nrefine' \u27e8fun p => _, fun e => e \u25b8 Perm.refl _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\no\u2081 o\u2082 : Option \u03b1\np : Option.toList o\u2081 ~ Option.toList o\u2082\n\u22a2 o\u2081 = o\u2082\n[PROOFSTEP]\ncases' o\u2081 with a\n[GOAL]\ncase none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\no\u2082 : Option \u03b1\np : Option.toList none ~ Option.toList o\u2082\n\u22a2 none = o\u2082\n[PROOFSTEP]\ncases' o\u2082 with b\n[GOAL]\ncase some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\no\u2082 : Option \u03b1\na : \u03b1\np : Option.toList (some a) ~ Option.toList o\u2082\n\u22a2 some a = o\u2082\n[PROOFSTEP]\ncases' o\u2082 with b\n[GOAL]\ncase none.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\np : Option.toList none ~ Option.toList none\n\u22a2 none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nb : \u03b1\np : Option.toList none ~ Option.toList (some b)\n\u22a2 none = some b\n[PROOFSTEP]\ncases p.length_eq\n[GOAL]\ncase some.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\np : Option.toList (some a) ~ Option.toList none\n\u22a2 some a = none\n[PROOFSTEP]\ncases p.length_eq\n[GOAL]\ncase some.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\np : Option.toList (some a) ~ Option.toList (some b)\n\u22a2 some a = some b\n[PROOFSTEP]\nexact Option.mem_toList.1 (p.symm.subset <| by simp)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\np : Option.toList (some a) ~ Option.toList (some b)\n\u22a2 b \u2208 Option.toList (some b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nl : List \u03b1\np : l ~ a :: l\u2081\ns : l <+ a :: l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\ncases' s with _ _ _ s' u _ _ s'\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nl : List \u03b1\np : l ~ a :: l\u2081\ns' : l <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\nexact (p.subperm_left.2 <| (sublist_cons _ _).subperm).trans s'.subperm\n[GOAL]\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nu : List \u03b1\np : a :: u ~ a :: l\u2081\ns' : u <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\nexact \u27e8u, p.cons_inv, s'\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 l\u2082\ns : l\u2081 <+~ l\u2082\n\u22a2 a :: l\u2081 <+~ l\u2082\n[PROOFSTEP]\nrcases s with \u27e8l, p, s\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 l\u2082\nl : List \u03b1\np : l ~ l\u2081\ns : l <+ l\u2082\n\u22a2 a :: l\u2081 <+~ l\u2082\n[PROOFSTEP]\ninduction s generalizing l\u2081\n[GOAL]\ncase intro.intro.slnil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l l\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 []\np : [] ~ l\u2081\n\u22a2 a :: l\u2081 <+~ []\ncase intro.intro.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\na : \u03b1\nl\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 l\u2082\u271d \u2192 l\u2081\u271d ~ l\u2081 \u2192 a :: l\u2081 <+~ l\u2082\u271d\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 a\u271d\u00b9 :: l\u2082\u271d\np : l\u2081\u271d ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a\u271d\u00b9 :: l\u2082\u271d\ncase intro.intro.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\na : \u03b1\nl\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 l\u2082\u271d \u2192 l\u2081\u271d ~ l\u2081 \u2192 a :: l\u2081 <+~ l\u2082\u271d\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 a\u271d\u00b9 :: l\u2082\u271d\np : a\u271d\u00b9 :: l\u2081\u271d ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a\u271d\u00b9 :: l\u2082\u271d\n[PROOFSTEP]\ncase slnil => cases h\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l l\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 []\np : [] ~ l\u2081\n\u22a2 a :: l\u2081 <+~ []\n[PROOFSTEP]\ncase slnil => cases h\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l l\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 []\np : [] ~ l\u2081\n\u22a2 a :: l\u2081 <+~ []\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase intro.intro.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\na : \u03b1\nl\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 l\u2082\u271d \u2192 l\u2081\u271d ~ l\u2081 \u2192 a :: l\u2081 <+~ l\u2082\u271d\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 a\u271d\u00b9 :: l\u2082\u271d\np : l\u2081\u271d ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a\u271d\u00b9 :: l\u2082\u271d\ncase intro.intro.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\na : \u03b1\nl\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 l\u2082\u271d \u2192 l\u2081\u271d ~ l\u2081 \u2192 a :: l\u2081 <+~ l\u2082\u271d\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 a\u271d\u00b9 :: l\u2082\u271d\np : a\u271d\u00b9 :: l\u2081\u271d ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a\u271d\u00b9 :: l\u2082\u271d\n[PROOFSTEP]\ncase cons r\u2081 r\u2082 b s' ih =>\n simp at h\u2082 \n cases' h\u2082 with e m\n \u00b7 subst b\n exact \u27e8a :: r\u2081, p.cons a, s'.cons\u2082 _\u27e9\n \u00b7 rcases ih d\u2081 h\u2081 m p with \u27e8t, p', s'\u27e9\n exact \u27e8t, p', s'.cons _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : r\u2081 ~ l\u2081\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\ncase cons r\u2081 r\u2082 b s' ih =>\n simp at h\u2082 \n cases' h\u2082 with e m\n \u00b7 subst b\n exact \u27e8a :: r\u2081, p.cons a, s'.cons\u2082 _\u27e9\n \u00b7 rcases ih d\u2081 h\u2081 m p with \u27e8t, p', s'\u27e9\n exact \u27e8t, p', s'.cons _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : r\u2081 ~ l\u2081\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nsimp at h\u2082 \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : r\u2081 ~ l\u2081\nh\u2082 : a = b \u2228 a \u2208 r\u2082\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\ncases' h\u2082 with e m\n[GOAL]\ncase inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : r\u2081 ~ l\u2081\ne : a = b\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : r\u2081 ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a :: r\u2082\n[PROOFSTEP]\nexact \u27e8a :: r\u2081, p.cons a, s'.cons\u2082 _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns' : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : r\u2081 ~ l\u2081\nm : a \u2208 r\u2082\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nrcases ih d\u2081 h\u2081 m p with \u27e8t, p', s'\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\ns'\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : r\u2081 ~ l\u2081\nm : a \u2208 r\u2082\nt : List \u03b1\np' : t ~ a :: l\u2081\ns' : t <+ r\u2082\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nexact \u27e8t, p', s'.cons _\u27e9\n[GOAL]\ncase intro.intro.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\na : \u03b1\nl\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 l\u2082\u271d \u2192 l\u2081\u271d ~ l\u2081 \u2192 a :: l\u2081 <+~ l\u2082\u271d\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 a\u271d\u00b9 :: l\u2082\u271d\np : a\u271d\u00b9 :: l\u2081\u271d ~ l\u2081\n\u22a2 a :: l\u2081 <+~ a\u271d\u00b9 :: l\u2082\u271d\n[PROOFSTEP]\ncase cons\u2082 r\u2081 r\u2082 b _ ih =>\n have bm : b \u2208 l\u2081 := p.subset <| mem_cons_self _ _\n have am : a \u2208 r\u2082 := by\n simp only [find?, mem_cons] at h\u2082 \n exact h\u2082.resolve_left fun e => h\u2081 <| e.symm \u25b8 bm\n rcases mem_split bm with \u27e8t\u2081, t\u2082, rfl\u27e9\n have st : t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082 := by simp\n rcases ih (d\u2081.sublist st) (mt (fun x => st.subset x) h\u2081) am (Perm.cons_inv <| p.trans perm_middle) with \u27e8t, p', s'\u27e9\n exact \u27e8b :: t, (p'.cons b).trans <| (swap _ _ _).trans (perm_middle.symm.cons a), s'.cons\u2082 _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : b :: r\u2081 ~ l\u2081\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\ncase cons\u2082 r\u2081 r\u2082 b _ ih =>\n have bm : b \u2208 l\u2081 := p.subset <| mem_cons_self _ _\n have am : a \u2208 r\u2082 := by\n simp only [find?, mem_cons] at h\u2082 \n exact h\u2082.resolve_left fun e => h\u2081 <| e.symm \u25b8 bm\n rcases mem_split bm with \u27e8t\u2081, t\u2082, rfl\u27e9\n have st : t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082 := by simp\n rcases ih (d\u2081.sublist st) (mt (fun x => st.subset x) h\u2081) am (Perm.cons_inv <| p.trans perm_middle) with \u27e8t, p', s'\u27e9\n exact \u27e8b :: t, (p'.cons b).trans <| (swap _ _ _).trans (perm_middle.symm.cons a), s'.cons\u2082 _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : b :: r\u2081 ~ l\u2081\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nhave bm : b \u2208 l\u2081 := p.subset <| mem_cons_self _ _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : b :: r\u2081 ~ l\u2081\nbm : b \u2208 l\u2081\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nhave am : a \u2208 r\u2082 := by\n simp only [find?, mem_cons] at h\u2082 \n exact h\u2082.resolve_left fun e => h\u2081 <| e.symm \u25b8 bm\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : b :: r\u2081 ~ l\u2081\nbm : b \u2208 l\u2081\n\u22a2 a \u2208 r\u2082\n[PROOFSTEP]\nsimp only [find?, mem_cons] at h\u2082 \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\np : b :: r\u2081 ~ l\u2081\nbm : b \u2208 l\u2081\nh\u2082 : a = b \u2228 a \u2208 r\u2082\n\u22a2 a \u2208 r\u2082\n[PROOFSTEP]\nexact h\u2082.resolve_left fun e => h\u2081 <| e.symm \u25b8 bm\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nl\u2081 : List \u03b1\nd\u2081 : Nodup l\u2081\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : a \u2208 b :: r\u2082\np : b :: r\u2081 ~ l\u2081\nbm : b \u2208 l\u2081\nam : a \u2208 r\u2082\n\u22a2 a :: l\u2081 <+~ b :: r\u2082\n[PROOFSTEP]\nrcases mem_split bm with \u27e8t\u2081, t\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nh\u2082 : a \u2208 b :: r\u2082\nam : a \u2208 r\u2082\nt\u2081 t\u2082 : List \u03b1\nd\u2081 : Nodup (t\u2081 ++ b :: t\u2082)\nh\u2081 : \u00aca \u2208 t\u2081 ++ b :: t\u2082\np : b :: r\u2081 ~ t\u2081 ++ b :: t\u2082\nbm : b \u2208 t\u2081 ++ b :: t\u2082\n\u22a2 a :: (t\u2081 ++ b :: t\u2082) <+~ b :: r\u2082\n[PROOFSTEP]\nhave st : t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082 := by simp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nh\u2082 : a \u2208 b :: r\u2082\nam : a \u2208 r\u2082\nt\u2081 t\u2082 : List \u03b1\nd\u2081 : Nodup (t\u2081 ++ b :: t\u2082)\nh\u2081 : \u00aca \u2208 t\u2081 ++ b :: t\u2082\np : b :: r\u2081 ~ t\u2081 ++ b :: t\u2082\nbm : b \u2208 t\u2081 ++ b :: t\u2082\n\u22a2 t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nh\u2082 : a \u2208 b :: r\u2082\nam : a \u2208 r\u2082\nt\u2081 t\u2082 : List \u03b1\nd\u2081 : Nodup (t\u2081 ++ b :: t\u2082)\nh\u2081 : \u00aca \u2208 t\u2081 ++ b :: t\u2082\np : b :: r\u2081 ~ t\u2081 ++ b :: t\u2082\nbm : b \u2208 t\u2081 ++ b :: t\u2082\nst : t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082\n\u22a2 a :: (t\u2081 ++ b :: t\u2082) <+~ b :: r\u2082\n[PROOFSTEP]\nrcases ih (d\u2081.sublist st) (mt (fun x => st.subset x) h\u2081) am (Perm.cons_inv <| p.trans perm_middle) with \u27e8t, p', s'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2082 l r\u2081 r\u2082 : List \u03b1\nb : \u03b1\na\u271d : r\u2081 <+ r\u2082\nih : \u2200 {l\u2081 : List \u03b1}, Nodup l\u2081 \u2192 \u00aca \u2208 l\u2081 \u2192 a \u2208 r\u2082 \u2192 r\u2081 ~ l\u2081 \u2192 a :: l\u2081 <+~ r\u2082\nh\u2082 : a \u2208 b :: r\u2082\nam : a \u2208 r\u2082\nt\u2081 t\u2082 : List \u03b1\nd\u2081 : Nodup (t\u2081 ++ b :: t\u2082)\nh\u2081 : \u00aca \u2208 t\u2081 ++ b :: t\u2082\np : b :: r\u2081 ~ t\u2081 ++ b :: t\u2082\nbm : b \u2208 t\u2081 ++ b :: t\u2082\nst : t\u2081 ++ t\u2082 <+ t\u2081 ++ b :: t\u2082\nt : List \u03b1\np' : t ~ a :: (t\u2081 ++ t\u2082)\ns' : t <+ r\u2082\n\u22a2 a :: (t\u2081 ++ b :: t\u2082) <+~ b :: r\u2082\n[PROOFSTEP]\nexact \u27e8b :: t, (p'.cons b).trans <| (swap _ _ _).trans (perm_middle.symm.cons a), s'.cons\u2082 _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l : List \u03b1\np : l ~ l\u2081\ns : l <+ l\u2082\nh : length l\u2081 < length l\u2082\n\u22a2 \u2203 a, a :: l\u2081 <+~ l\u2082\n[PROOFSTEP]\nsuffices length l < length l\u2082 \u2192 \u2203 a : \u03b1, a :: l <+~ l\u2082 from\n (this <| p.symm.length_eq \u25b8 h).imp fun a => (p.cons a).subperm_right.1\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l : List \u03b1\np : l ~ l\u2081\ns : l <+ l\u2082\nh : length l\u2081 < length l\u2082\n\u22a2 length l < length l\u2082 \u2192 \u2203 a, a :: l <+~ l\u2082\n[PROOFSTEP]\nclear h p l\u2081\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d l\u2082 l : List \u03b1\ns : l <+ l\u2082\n\u22a2 length l < length l\u2082 \u2192 \u2203 a, a :: l <+~ l\u2082\n[PROOFSTEP]\ninduction' s with l\u2081 l\u2082 a s IH _ _ b _ IH\n[GOAL]\ncase slnil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d l\u2082 l : List \u03b1\n\u22a2 length [] < length [] \u2192 \u2203 a, [a] <+~ []\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2082\u271d l l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : length l\u2081 < length l\u2082 \u2192 \u2203 a, a :: l\u2081 <+~ l\u2082\n\u22a2 length l\u2081 < length (a :: l\u2082) \u2192 \u2203 a_2, a_2 :: l\u2081 <+~ a :: l\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d\u00b9 l\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\nb : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\nIH : length l\u2081\u271d < length l\u2082\u271d \u2192 \u2203 a, a :: l\u2081\u271d <+~ l\u2082\u271d\n\u22a2 length (b :: l\u2081\u271d) < length (b :: l\u2082\u271d) \u2192 \u2203 a, a :: b :: l\u2081\u271d <+~ b :: l\u2082\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase slnil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d l\u2082 l : List \u03b1\nh : length [] < length []\n\u22a2 \u2203 a, [a] <+~ []\n[PROOFSTEP]\ncases h\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2082\u271d l l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : length l\u2081 < length l\u2082 \u2192 \u2203 a, a :: l\u2081 <+~ l\u2082\nh : length l\u2081 < length (a :: l\u2082)\n\u22a2 \u2203 a_1, a_1 :: l\u2081 <+~ a :: l\u2082\n[PROOFSTEP]\ncases' lt_or_eq_of_le (Nat.le_of_lt_succ h : length l\u2081 \u2264 length l\u2082) with h h\n[GOAL]\ncase cons.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2082\u271d l l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : length l\u2081 < length l\u2082 \u2192 \u2203 a, a :: l\u2081 <+~ l\u2082\nh\u271d : length l\u2081 < length (a :: l\u2082)\nh : length l\u2081 < length l\u2082\n\u22a2 \u2203 a_1, a_1 :: l\u2081 <+~ a :: l\u2082\n[PROOFSTEP]\nexact (IH h).imp fun a s => s.trans (sublist_cons _ _).subperm\n[GOAL]\ncase cons.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2082\u271d l l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : length l\u2081 < length l\u2082 \u2192 \u2203 a, a :: l\u2081 <+~ l\u2082\nh\u271d : length l\u2081 < length (a :: l\u2082)\nh : length l\u2081 = length l\u2082\n\u22a2 \u2203 a_1, a_1 :: l\u2081 <+~ a :: l\u2082\n[PROOFSTEP]\nexact \u27e8a, s.eq_of_length h \u25b8 Subperm.refl _\u27e9\n[GOAL]\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d\u00b9 l\u2082 l l\u2081\u271d l\u2082\u271d : List \u03b1\nb : \u03b1\na\u271d : l\u2081\u271d <+ l\u2082\u271d\nIH : length l\u2081\u271d < length l\u2082\u271d \u2192 \u2203 a, a :: l\u2081\u271d <+~ l\u2082\u271d\nh : length (b :: l\u2081\u271d) < length (b :: l\u2082\u271d)\n\u22a2 \u2203 a, a :: b :: l\u2081\u271d <+~ b :: l\u2082\u271d\n[PROOFSTEP]\nexact (IH <| Nat.lt_of_succ_lt_succ h).imp fun a s => (swap _ _ _).subperm_right.1 <| (subperm_cons _).2 s\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nd : Nodup l\u2081\nH : l\u2081 \u2286 l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\ninduction' d with a l\u2081' h d IH\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nH\u271d : l\u2081 \u2286 l\u2082\nH : [] \u2286 l\u2082\n\u22a2 [] <+~ l\u2082\n[PROOFSTEP]\nexact \u27e8nil, Perm.nil, nil_sublist _\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nH\u271d : l\u2081 \u2286 l\u2082\na : \u03b1\nl\u2081' : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081' \u2192 a \u2260 a'\nd : Pairwise (fun x x_1 => x \u2260 x_1) l\u2081'\nIH : l\u2081' \u2286 l\u2082 \u2192 l\u2081' <+~ l\u2082\nH : a :: l\u2081' \u2286 l\u2082\n\u22a2 a :: l\u2081' <+~ l\u2082\n[PROOFSTEP]\ncases' forall_mem_cons.1 H with H\u2081 H\u2082\n[GOAL]\ncase cons.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nH\u271d : l\u2081 \u2286 l\u2082\na : \u03b1\nl\u2081' : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081' \u2192 a \u2260 a'\nd : Pairwise (fun x x_1 => x \u2260 x_1) l\u2081'\nIH : l\u2081' \u2286 l\u2082 \u2192 l\u2081' <+~ l\u2082\nH : a :: l\u2081' \u2286 l\u2082\nH\u2081 : a \u2208 l\u2082\nH\u2082 : \u2200 (x : \u03b1), x \u2208 l\u2081' \u2192 x \u2208 l\u2082\n\u22a2 a :: l\u2081' <+~ l\u2082\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nH\u271d : l\u2081 \u2286 l\u2082\na : \u03b1\nl\u2081' : List \u03b1\nd : Pairwise (fun x x_1 => x \u2260 x_1) l\u2081'\nIH : l\u2081' \u2286 l\u2082 \u2192 l\u2081' <+~ l\u2082\nH : a :: l\u2081' \u2286 l\u2082\nH\u2081 : a \u2208 l\u2082\nH\u2082 : \u2200 (x : \u03b1), x \u2208 l\u2081' \u2192 x \u2208 l\u2082\nh : \u00aca \u2208 l\u2081'\n\u22a2 a :: l\u2081' <+~ l\u2082\n[PROOFSTEP]\nexact cons_subperm_of_mem d h H\u2081 (IH H\u2082)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l : List \u03b1\nd : Nodup l\ns\u2081 : l\u2081 <+ l\ns\u2082 : l\u2082 <+ l\nh : l\u2081 ~ l\u2082\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\ninduction' s\u2082 with l\u2082 l a s\u2082 IH l\u2082 l a _ IH generalizing l\u2081\n[GOAL]\ncase slnil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d l\u2081\u271d l\u2082 l : List \u03b1\nd\u271d : Nodup l\ns\u2081\u271d : l\u2081\u271d <+ l\nh\u271d : l\u2081\u271d ~ l\u2082\nl\u2081 : List \u03b1\nd : Nodup []\ns\u2081 : l\u2081 <+ []\nh : l\u2081 ~ []\n\u22a2 l\u2081 = []\n[PROOFSTEP]\nexact h.eq_nil\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\ns\u2082 : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\nd : Nodup (a :: l)\ns\u2081 : l\u2081 <+ a :: l\nh : l\u2081 ~ l\u2082\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\nsimp at d \n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\ns\u2082 : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\ns\u2081 : l\u2081 <+ a :: l\nh : l\u2081 ~ l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\ncases' s\u2081 with _ _ _ s\u2081 l\u2081 _ _ s\u2081\n[GOAL]\ncase cons.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\ns\u2082 : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\nh : l\u2081 ~ l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\ns\u2081 : l\u2081 <+ l\n\u22a2 l\u2081 = l\u2082\n[PROOFSTEP]\nexact IH d.2 s\u2081 h\n[GOAL]\ncase cons.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\ns\u2082 : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\nl\u2081 : List \u03b1\nh : a :: l\u2081 ~ l\u2082\ns\u2081 : l\u2081 <+ l\n\u22a2 a :: l\u2081 = l\u2082\n[PROOFSTEP]\napply d.1.elim\n[GOAL]\ncase cons.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\ns\u2082 : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\nl\u2081 : List \u03b1\nh : a :: l\u2081 ~ l\u2082\ns\u2081 : l\u2081 <+ l\n\u22a2 a \u2208 l\n[PROOFSTEP]\nexact Subperm.subset \u27e8_, h.symm, s\u2082\u27e9 (mem_cons_self _ _)\n[GOAL]\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\na\u271d : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\nd : Nodup (a :: l)\ns\u2081 : l\u2081 <+ a :: l\nh : l\u2081 ~ a :: l\u2082\n\u22a2 l\u2081 = a :: l\u2082\n[PROOFSTEP]\nsimp at d \n[GOAL]\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\na\u271d : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\ns\u2081 : l\u2081 <+ a :: l\nh : l\u2081 ~ a :: l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\n\u22a2 l\u2081 = a :: l\u2082\n[PROOFSTEP]\ncases' s\u2081 with _ _ _ s\u2081 l\u2081 _ _ s\u2081\n[GOAL]\ncase cons\u2082.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\na\u271d : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\nh : l\u2081 ~ a :: l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\ns\u2081 : l\u2081 <+ l\n\u22a2 l\u2081 = a :: l\u2082\n[PROOFSTEP]\napply d.1.elim\n[GOAL]\ncase cons\u2082.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\na\u271d : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nl\u2081 : List \u03b1\nh : l\u2081 ~ a :: l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\ns\u2081 : l\u2081 <+ l\n\u22a2 a \u2208 l\n[PROOFSTEP]\nexact Subperm.subset \u27e8_, h, s\u2081\u27e9 (mem_cons_self _ _)\n[GOAL]\ncase cons\u2082.cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d l\u271d : List \u03b1\nd\u271d : Nodup l\u271d\ns\u2081\u271d : l\u2081\u271d <+ l\u271d\nh\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2082 l : List \u03b1\na : \u03b1\na\u271d : l\u2082 <+ l\nIH : \u2200 {l\u2081 : List \u03b1}, Nodup l \u2192 l\u2081 <+ l \u2192 l\u2081 ~ l\u2082 \u2192 l\u2081 = l\u2082\nd : \u00aca \u2208 l \u2227 Nodup l\nl\u2081 : List \u03b1\nh : a :: l\u2081 ~ a :: l\u2082\ns\u2081 : l\u2081 <+ l\n\u22a2 a :: l\u2081 = a :: l\u2082\n[PROOFSTEP]\nrw [IH d.2 s\u2081 h.cons_inv]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l : List \u03b1\nd : Nodup l\ns\u2081 : l\u2081 <+ l\ns\u2082 : l\u2082 <+ l\nh : l\u2081 = l\u2082\n\u22a2 l\u2081 ~ l\u2082\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nh\u2081 : \u00aca \u2208 l\u2081\n\u22a2 List.erase l\u2081 a ~ List.erase l\u2082 a\n[PROOFSTEP]\nhave h\u2082 : a \u2209 l\u2082 := mt p.mem_iff.2 h\u2081\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : \u00aca \u2208 l\u2082\n\u22a2 List.erase l\u2081 a ~ List.erase l\u2082 a\n[PROOFSTEP]\nrw [erase_of_not_mem h\u2081, erase_of_not_mem h\u2082]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nh\u2081 : \u00aca \u2208 l\u2081\nh\u2082 : \u00aca \u2208 l\u2082\n\u22a2 l\u2081 ~ l\u2082\n[PROOFSTEP]\nexact p\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 l <+~ a :: List.erase l a\n[PROOFSTEP]\nby_cases h : a \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : a \u2208 l\n\u22a2 l <+~ a :: List.erase l a\n[PROOFSTEP]\nexact (perm_cons_erase h).subperm\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : \u00aca \u2208 l\n\u22a2 l <+~ a :: List.erase l a\n[PROOFSTEP]\nrw [erase_of_not_mem h]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : \u00aca \u2208 l\n\u22a2 l <+~ a :: l\n[PROOFSTEP]\nexact (sublist_cons _ _).subperm\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 List.diff l\u2081 t ~ List.diff l\u2082 t\n[PROOFSTEP]\ninduction t generalizing l\u2081 l\u2082 h\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 List.diff l\u2081 [] ~ List.diff l\u2082 []\n[PROOFSTEP]\nsimp [*, Perm.erase]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 List.diff l\u2081 tail\u271d ~ List.diff l\u2082 tail\u271d\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 List.diff l\u2081 (head\u271d :: tail\u271d) ~ List.diff l\u2082 (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [*, Perm.erase]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl t\u2081 t\u2082 : List \u03b1\nh : t\u2081 ~ t\u2082\n\u22a2 List.diff l t\u2081 = List.diff l t\u2082\n[PROOFSTEP]\ninduction h generalizing l\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 l : List \u03b1\n\u22a2 List.diff l [] = List.diff l []\n[PROOFSTEP]\nfirst\n| simp [*, Perm.erase, erase_comm]\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 l : List \u03b1\n\u22a2 List.diff l [] = List.diff l []\n[PROOFSTEP]\nsimp [*, Perm.erase, erase_comm]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (l : List \u03b1), List.diff l l\u2081\u271d = List.diff l l\u2082\u271d\nl : List \u03b1\n\u22a2 List.diff l (x\u271d :: l\u2081\u271d) = List.diff l (x\u271d :: l\u2082\u271d)\n[PROOFSTEP]\nfirst\n| simp [*, Perm.erase, erase_comm]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nx\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (l : List \u03b1), List.diff l l\u2081\u271d = List.diff l l\u2082\u271d\nl : List \u03b1\n\u22a2 List.diff l (x\u271d :: l\u2081\u271d) = List.diff l (x\u271d :: l\u2082\u271d)\n[PROOFSTEP]\nsimp [*, Perm.erase, erase_comm]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d l : List \u03b1\n\u22a2 List.diff l (y\u271d :: x\u271d :: l\u271d) = List.diff l (x\u271d :: y\u271d :: l\u271d)\n[PROOFSTEP]\nfirst\n| simp [*, Perm.erase, erase_comm]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d l : List \u03b1\n\u22a2 List.diff l (y\u271d :: x\u271d :: l\u271d) = List.diff l (x\u271d :: y\u271d :: l\u271d)\n[PROOFSTEP]\nsimp [*, Perm.erase, erase_comm]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 (l : List \u03b1), List.diff l l\u2081\u271d = List.diff l l\u2082\u271d\na_ih\u271d : \u2200 (l : List \u03b1), List.diff l l\u2082\u271d = List.diff l l\u2083\u271d\nl : List \u03b1\n\u22a2 List.diff l l\u2081\u271d = List.diff l l\u2083\u271d\n[PROOFSTEP]\nfirst\n| simp [*, Perm.erase, erase_comm]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : \u2200 (l : List \u03b1), List.diff l l\u2081\u271d = List.diff l l\u2082\u271d\na_ih\u271d : \u2200 (l : List \u03b1), List.diff l l\u2082\u271d = List.diff l l\u2083\u271d\nl : List \u03b1\n\u22a2 List.diff l l\u2081\u271d = List.diff l l\u2083\u271d\n[PROOFSTEP]\nsimp [*, Perm.erase, erase_comm]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+~ l\u2082\nt : List \u03b1\n\u22a2 List.diff l\u2081 t <+~ List.diff l\u2082 t\n[PROOFSTEP]\ninduction t generalizing l\u2081 l\u2082 h\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+~ l\u2082\n\u22a2 List.diff l\u2081 [] <+~ List.diff l\u2082 []\n[PROOFSTEP]\nsimp [*, Subperm.erase]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+~ l\u2082 \u2192 List.diff l\u2081 tail\u271d <+~ List.diff l\u2082 tail\u271d\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+~ l\u2082\n\u22a2 List.diff l\u2081 (head\u271d :: tail\u271d) <+~ List.diff l\u2082 (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [*, Subperm.erase]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.erase (a :: l) b <+~ a :: List.erase l b\n[PROOFSTEP]\nby_cases h : a = b\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl : List \u03b1\nh : a = b\n\u22a2 List.erase (a :: l) b <+~ a :: List.erase l b\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 List.erase (a :: l) a <+~ a :: List.erase l a\n[PROOFSTEP]\nrw [erase_cons_head]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 l <+~ a :: List.erase l a\n[PROOFSTEP]\napply subperm_cons_erase\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl : List \u03b1\nh : \u00aca = b\n\u22a2 List.erase (a :: l) b <+~ a :: List.erase l b\n[PROOFSTEP]\nrw [erase_cons_tail _ h]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\n\u22a2 \u2203 x, a :: l\u2081 <+ a :: List.diff l\u2081 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\n\u22a2 List.diff (a :: l\u2081) (b :: l\u2082) <+~ a :: List.diff l\u2081 (b :: l\u2082)\n[PROOFSTEP]\nsimp only [diff_cons]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\n\u22a2 List.diff (List.erase (a :: l\u2081) b) l\u2082 <+~ a :: List.diff (List.erase l\u2081 b) l\u2082\n[PROOFSTEP]\nrefine' ((erase_cons_subperm_cons_erase a b l\u2081).diff_right l\u2082).trans _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\n\u22a2 List.diff (a :: List.erase l\u2081 b) l\u2082 <+~ a :: List.diff (List.erase l\u2081 b) l\u2082\n[PROOFSTEP]\napply subperm_cons_diff\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 List.bagInter l\u2081 t ~ List.bagInter l\u2082 t\n[PROOFSTEP]\ninduction' h with x _ _ _ _ x y _ _ _ _ _ _ ih_1 ih_2 generalizing t\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d t : List \u03b1\n\u22a2 List.bagInter [] t ~ List.bagInter [] t\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (t : List \u03b1), List.bagInter l\u2081\u271d t ~ List.bagInter l\u2082\u271d t\nt : List \u03b1\n\u22a2 List.bagInter (x :: l\u2081\u271d) t ~ List.bagInter (x :: l\u2082\u271d) t\n[PROOFSTEP]\nby_cases x \u2208 t\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (t : List \u03b1), List.bagInter l\u2081\u271d t ~ List.bagInter l\u2082\u271d t\nt : List \u03b1\n\u22a2 List.bagInter (x :: l\u2081\u271d) t ~ List.bagInter (x :: l\u2082\u271d) t\n[PROOFSTEP]\nby_cases x \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (t : List \u03b1), List.bagInter l\u2081\u271d t ~ List.bagInter l\u2082\u271d t\nt : List \u03b1\nh : x \u2208 t\n\u22a2 List.bagInter (x :: l\u2081\u271d) t ~ List.bagInter (x :: l\u2082\u271d) t\n[PROOFSTEP]\nsimp [*, Perm.cons]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : \u2200 (t : List \u03b1), List.bagInter l\u2081\u271d t ~ List.bagInter l\u2082\u271d t\nt : List \u03b1\nh : \u00acx \u2208 t\n\u22a2 List.bagInter (x :: l\u2081\u271d) t ~ List.bagInter (x :: l\u2082\u271d) t\n[PROOFSTEP]\nsimp [*, Perm.cons]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nby_cases h : x = y\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : x = y\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nby_cases xt : x \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : x \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nby_cases yt : y \u2208 t\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : \u00acx \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nby_cases yt : y \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : x \u2208 t\nyt : y \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nsimp [xt, yt, mem_erase_of_ne h, mem_erase_of_ne (Ne.symm h), erase_comm, swap]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : x \u2208 t\nyt : \u00acy \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nsimp [xt, yt, mt mem_of_mem_erase, Perm.cons]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : \u00acx \u2208 t\nyt : y \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nsimp [xt, yt, mt mem_of_mem_erase, Perm.cons]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d : List \u03b1\nx y : \u03b1\nl\u271d t : List \u03b1\nh : \u00acx = y\nxt : \u00acx \u2208 t\nyt : \u00acy \u2208 t\n\u22a2 List.bagInter (y :: x :: l\u271d) t ~ List.bagInter (x :: y :: l\u271d) t\n[PROOFSTEP]\nsimp [xt, yt]\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u271d l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih_1 : \u2200 (t : List \u03b1), List.bagInter l\u2081\u271d t ~ List.bagInter l\u2082\u271d t\nih_2 : \u2200 (t : List \u03b1), List.bagInter l\u2082\u271d t ~ List.bagInter l\u2083\u271d t\nt : List \u03b1\n\u22a2 List.bagInter l\u2081\u271d t ~ List.bagInter l\u2083\u271d t\n[PROOFSTEP]\nexact (ih_1 _).trans (ih_2 _)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl t\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\n\u22a2 List.bagInter l t\u2081 = List.bagInter l t\u2082\n[PROOFSTEP]\ninduction' l with a l IH generalizing t\u2081 t\u2082 p\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081\u271d t\u2082\u271d : List \u03b1\np\u271d : t\u2081\u271d ~ t\u2082\u271d\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\n\u22a2 List.bagInter [] t\u2081 = List.bagInter [] t\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081\u271d t\u2082\u271d : List \u03b1\np\u271d : t\u2081\u271d ~ t\u2082\u271d\na : \u03b1\nl : List \u03b1\nIH : \u2200 {t\u2081 t\u2082 : List \u03b1}, t\u2081 ~ t\u2082 \u2192 List.bagInter l t\u2081 = List.bagInter l t\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\n\u22a2 List.bagInter (a :: l) t\u2081 = List.bagInter (a :: l) t\u2082\n[PROOFSTEP]\nby_cases h : a \u2208 t\u2081\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081\u271d t\u2082\u271d : List \u03b1\np\u271d : t\u2081\u271d ~ t\u2082\u271d\na : \u03b1\nl : List \u03b1\nIH : \u2200 {t\u2081 t\u2082 : List \u03b1}, t\u2081 ~ t\u2082 \u2192 List.bagInter l t\u2081 = List.bagInter l t\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nh : a \u2208 t\u2081\n\u22a2 List.bagInter (a :: l) t\u2081 = List.bagInter (a :: l) t\u2082\n[PROOFSTEP]\nsimp [h, p.subset h, IH (p.erase _)]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081\u271d t\u2082\u271d : List \u03b1\np\u271d : t\u2081\u271d ~ t\u2082\u271d\na : \u03b1\nl : List \u03b1\nIH : \u2200 {t\u2081 t\u2082 : List \u03b1}, t\u2081 ~ t\u2082 \u2192 List.bagInter l t\u2081 = List.bagInter l t\u2082\nt\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\nh : \u00aca \u2208 t\u2081\n\u22a2 List.bagInter (a :: l) t\u2081 = List.bagInter (a :: l) t\u2082\n[PROOFSTEP]\nsimp [h, mt p.mem_iff.2 h, IH p]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nH : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\n\u22a2 l\u2081 ~ l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with a l\u2081 IH generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\u271d\nl\u2082 : List \u03b1\nH : \u2200 (a : \u03b1), count a [] = count a l\u2082\n\u22a2 [] ~ l\u2082\n[PROOFSTEP]\ncases' l\u2082 with b l\u2082\n[GOAL]\ncase nil.nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\nH : \u2200 (a : \u03b1), count a [] = count a []\n\u22a2 [] ~ []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase nil.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\u271d\nb : \u03b1\nl\u2082 : List \u03b1\nH : \u2200 (a : \u03b1), count a [] = count a (b :: l\u2082)\n\u22a2 [] ~ b :: l\u2082\n[PROOFSTEP]\nspecialize H b\n[GOAL]\ncase nil.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\u271d\nb : \u03b1\nl\u2082 : List \u03b1\nH : count b [] = count b (b :: l\u2082)\n\u22a2 [] ~ b :: l\u2082\n[PROOFSTEP]\nsimp at H \n[GOAL]\ncase nil.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081 = count a l\u2082\u271d\nb : \u03b1\nl\u2082 : List \u03b1\nH : 0 = count b l\u2082 + 1\n\u22a2 [] ~ b :: l\u2082\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nH : \u2200 (a_1 : \u03b1), count a_1 (a :: l\u2081) = count a_1 l\u2082\n\u22a2 a :: l\u2081 ~ l\u2082\n[PROOFSTEP]\nhave : a \u2208 l\u2082 := count_pos.1 (by rw [\u2190 H]; simp)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nH : \u2200 (a_1 : \u03b1), count a_1 (a :: l\u2081) = count a_1 l\u2082\n\u22a2 0 < count a l\u2082\n[PROOFSTEP]\nrw [\u2190 H]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nH : \u2200 (a_1 : \u03b1), count a_1 (a :: l\u2081) = count a_1 l\u2082\n\u22a2 0 < count a (a :: l\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nH : \u2200 (a_1 : \u03b1), count a_1 (a :: l\u2081) = count a_1 l\u2082\nthis : a \u2208 l\u2082\n\u22a2 a :: l\u2081 ~ l\u2082\n[PROOFSTEP]\nrefine' ((IH fun b => _).cons a).trans (perm_cons_erase this).symm\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nH : \u2200 (a_1 : \u03b1), count a_1 (a :: l\u2081) = count a_1 l\u2082\nthis : a \u2208 l\u2082\nb : \u03b1\n\u22a2 count b l\u2081 = count b (List.erase l\u2082 a)\n[PROOFSTEP]\nspecialize H b\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nthis : a \u2208 l\u2082\nb : \u03b1\nH : count b (a :: l\u2081) = count b l\u2082\n\u22a2 count b l\u2081 = count b (List.erase l\u2082 a)\n[PROOFSTEP]\nrw [(perm_cons_erase this).count_eq] at H \n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nthis : a \u2208 l\u2082\nb : \u03b1\nH : count b (a :: l\u2081) = count b (a :: List.erase l\u2082 a)\n\u22a2 count b l\u2081 = count b (List.erase l\u2082 a)\n[PROOFSTEP]\nby_cases h : b = a\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nthis : a \u2208 l\u2082\nb : \u03b1\nH : count b (a :: l\u2081) = count b (a :: List.erase l\u2082 a)\nh : b = a\n\u22a2 count b l\u2081 = count b (List.erase l\u2082 a)\n[PROOFSTEP]\nsimpa [h] using H\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : \u2200 (a : \u03b1), count a l\u2081\u271d = count a l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (a : \u03b1), count a l\u2081 = count a l\u2082) \u2192 l\u2081 ~ l\u2082\nl\u2082 : List \u03b1\nthis : a \u2208 l\u2082\nb : \u03b1\nH : count b (a :: l\u2081) = count b (a :: List.erase l\u2082 a)\nh : \u00acb = a\n\u22a2 count b l\u2081 = count b (List.erase l\u2082 a)\n[PROOFSTEP]\nsimpa [h] using H\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\n\u22a2 l ~ replicate m a ++ replicate n b \u2194 count a l = m \u2227 count b l = n \u2227 l \u2286 [a, b]\n[PROOFSTEP]\nrw [perm_iff_count, \u2190 Decidable.and_forall_ne a, \u2190 Decidable.and_forall_ne b]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\n\u22a2 (count a l = count a (replicate m a ++ replicate n b) \u2227\n (b \u2260 a \u2192 count b l = count b (replicate m a ++ replicate n b)) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2260 b \u2192 b_1 \u2260 a \u2192 count b_1 l = count b_1 (replicate m a ++ replicate n b)) \u2194\n count a l = m \u2227 count b l = n \u2227 l \u2286 [a, b]\n[PROOFSTEP]\nsuffices : l \u2286 [a, b] \u2194 \u2200 c, c \u2260 b \u2192 c \u2260 a \u2192 c \u2209 l\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\nthis : l \u2286 [a, b] \u2194 \u2200 (c : \u03b1), c \u2260 b \u2192 c \u2260 a \u2192 \u00acc \u2208 l\n\u22a2 (count a l = count a (replicate m a ++ replicate n b) \u2227\n (b \u2260 a \u2192 count b l = count b (replicate m a ++ replicate n b)) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2260 b \u2192 b_1 \u2260 a \u2192 count b_1 l = count b_1 (replicate m a ++ replicate n b)) \u2194\n count a l = m \u2227 count b l = n \u2227 l \u2286 [a, b]\ncase this\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\n\u22a2 l \u2286 [a, b] \u2194 \u2200 (c : \u03b1), c \u2260 b \u2192 c \u2260 a \u2192 \u00acc \u2208 l\n[PROOFSTEP]\n{simp (config := { contextual := true }) [count_replicate, h, h.symm, this]\n}\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\nthis : l \u2286 [a, b] \u2194 \u2200 (c : \u03b1), c \u2260 b \u2192 c \u2260 a \u2192 \u00acc \u2208 l\n\u22a2 (count a l = count a (replicate m a ++ replicate n b) \u2227\n (b \u2260 a \u2192 count b l = count b (replicate m a ++ replicate n b)) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2260 b \u2192 b_1 \u2260 a \u2192 count b_1 l = count b_1 (replicate m a ++ replicate n b)) \u2194\n count a l = m \u2227 count b l = n \u2227 l \u2286 [a, b]\n[PROOFSTEP]\nsimp (config := { contextual := true }) [count_replicate, h, h.symm, this]\n[GOAL]\ncase this\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na b : \u03b1\nm n : \u2115\nh : a \u2260 b\n\u22a2 l \u2286 [a, b] \u2194 \u2200 (c : \u03b1), c \u2260 b \u2192 c \u2260 a \u2192 \u00acc \u2208 l\n[PROOFSTEP]\nsimp_rw [Ne.def, \u2190 and_imp, \u2190 not_or, Decidable.not_imp_not, subset_def, mem_cons, not_mem_nil, or_false, or_comm]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\n\u22a2 l\u2081 ++ List.diff l\u2082 l\u2081 ~ l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with hd tl IH generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 [] \u2192 count x [] \u2264 count x l\u2082\n\u22a2 [] ++ List.diff l\u2082 [] ~ l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082\n[PROOFSTEP]\nhave : hd \u2208 l\u2082 := by\n rw [\u2190 count_pos]\n exact lt_of_lt_of_le (count_pos.mpr (mem_cons_self _ _)) (h hd (mem_cons_self _ _))\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 hd \u2208 l\u2082\n[PROOFSTEP]\nrw [\u2190 count_pos]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 0 < count hd l\u2082\n[PROOFSTEP]\nexact lt_of_lt_of_le (count_pos.mpr (mem_cons_self _ _)) (h hd (mem_cons_self _ _))\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : hd \u2208 l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082\n[PROOFSTEP]\nreplace := perm_cons_erase this\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082\n[PROOFSTEP]\nrefine' Perm.trans _ this.symm\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ hd :: List.erase l\u2082 hd\n[PROOFSTEP]\nrw [cons_append, diff_cons, perm_cons]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 tl ++ List.diff (List.erase l\u2082 hd) tl ~ List.erase l\u2082 hd\n[PROOFSTEP]\nrefine' IH fun x hx => _\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n[PROOFSTEP]\nspecialize h x (mem_cons_of_mem _ hx)\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n[PROOFSTEP]\nrw [perm_iff_count.mp this] at h \n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n[PROOFSTEP]\nby_cases hx : x = hd\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : x = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n[PROOFSTEP]\nsubst hd\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nx : \u03b1\nhx : x \u2208 tl\nthis : l\u2082 ~ x :: List.erase l\u2082 x\nh : count x (x :: tl) \u2264 count x (x :: List.erase l\u2082 x)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 x)\n[PROOFSTEP]\nsimpa [Nat.succ_le_succ_iff] using h\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : \u00acx = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n[PROOFSTEP]\nsimpa [hx] using h\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 l\u2081 <+~ l\u2082 \u2194 \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h x _ => Subperm.count_le h x, fun h => _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\nsuffices l\u2081 <+~ l\u2082.diff l\u2081 ++ l\u2081 by\n refine' this.trans (Perm.subperm _)\n exact perm_append_comm.trans (subperm_append_diff_self_of_count_le h)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nthis : l\u2081 <+~ List.diff l\u2082 l\u2081 ++ l\u2081\n\u22a2 l\u2081 <+~ l\u2082\n[PROOFSTEP]\nrefine' this.trans (Perm.subperm _)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nthis : l\u2081 <+~ List.diff l\u2082 l\u2081 ++ l\u2081\n\u22a2 List.diff l\u2082 l\u2081 ++ l\u2081 ~ l\u2082\n[PROOFSTEP]\nexact perm_append_comm.trans (subperm_append_diff_self_of_count_le h)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\n\u22a2 l\u2081 <+~ List.diff l\u2082 l\u2081 ++ l\u2081\n[PROOFSTEP]\nexact (subperm_append_right l\u2081).mpr nil_subperm\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nl : List \u03b1\na : \u03b1\nx\u271d : [a] <+~ l\ns : List \u03b1\nhla : s ~ [a]\nh : s <+ l\n\u22a2 a \u2208 l\n[PROOFSTEP]\nrwa [perm_singleton.mp hla, singleton_sublist] at h \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+~ l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\n\u22a2 x :: l\u2081 <+~ l\u2082\n[PROOFSTEP]\nrw [subperm_ext_iff] at h \u22a2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 x :: l\u2081 \u2192 count x_1 (x :: l\u2081) \u2264 count x_1 l\u2082\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\ny : \u03b1\nhy : y \u2208 x :: l\u2081\n\u22a2 count y (x :: l\u2081) \u2264 count y l\u2082\n[PROOFSTEP]\nby_cases hy' : y = x\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\ny : \u03b1\nhy : y \u2208 x :: l\u2081\nhy' : y = x\n\u22a2 count y (x :: l\u2081) \u2264 count y l\u2082\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\ny : \u03b1\nhx : count y l\u2081 < count y l\u2082\nhy : y \u2208 y :: l\u2081\n\u22a2 count y (y :: l\u2081) \u2264 count y l\u2082\n[PROOFSTEP]\nsimpa using Nat.succ_le_of_lt hx\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\ny : \u03b1\nhy : y \u2208 x :: l\u2081\nhy' : \u00acy = x\n\u22a2 count y (x :: l\u2081) \u2264 count y l\u2082\n[PROOFSTEP]\nrw [count_cons_of_ne hy']\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\ny : \u03b1\nhy : y \u2208 x :: l\u2081\nhy' : \u00acy = x\n\u22a2 count y l\u2081 \u2264 count y l\u2082\n[PROOFSTEP]\nrefine' h y _\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh\u271d : l\u2081 <+~ l\u2082\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\nx : \u03b1\nhx : count x l\u2081 < count x l\u2082\ny : \u03b1\nhy : y \u2208 x :: l\u2081\nhy' : \u00acy = x\n\u22a2 y \u2208 l\u2081\n[PROOFSTEP]\nsimpa [hy'] using hy\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nh : [] ~ b :: l\u2082\n\u22a2 False\n[PROOFSTEP]\nhave := h.nil_eq\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nh : [] ~ b :: l\u2082\nthis : [] = b :: l\u2082\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\na : \u03b1\nh : a \u2208 l\u2081\n\u22a2 count a (List.dedup l\u2081) = count a (List.dedup l\u2082)\n[PROOFSTEP]\nsimp [nodup_dedup, h, p.subset h]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\na : \u03b1\nh : \u00aca \u2208 l\u2081\n\u22a2 count a (List.dedup l\u2081) = count a (List.dedup l\u2082)\n[PROOFSTEP]\nsimp [h, mt p.mem_iff.2 h]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nh : a \u2208 l\u2081\n\u22a2 List.insert a l\u2081 ~ List.insert a l\u2082\n[PROOFSTEP]\nsimpa [h, p.subset h] using p\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nh : \u00aca \u2208 l\u2081\n\u22a2 List.insert a l\u2081 ~ List.insert a l\u2082\n[PROOFSTEP]\nsimpa [h, mt p.mem_iff.2 h] using p.cons a\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nby_cases xl : x \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : x \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nby_cases yl : y \u2208 l\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nby_cases yl : y \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : x \u2208 l\nyl : y \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nsimp [xl, yl]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : x \u2208 l\nyl : \u00acy \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nsimp [xl, yl]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : y \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nsimp [xl, yl]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : \u00acy \u2208 l\n\u22a2 List.insert x (List.insert y l) ~ List.insert y (List.insert x l)\n[PROOFSTEP]\nsimp [xl, yl]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : \u00acy \u2208 l\n\u22a2 List.insert x (y :: l) ~ List.insert y (x :: l)\n[PROOFSTEP]\nby_cases xy : x = y\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : \u00acy \u2208 l\nxy : x = y\n\u22a2 List.insert x (y :: l) ~ List.insert y (x :: l)\n[PROOFSTEP]\nsimp [xy]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : \u00acy \u2208 l\nxy : \u00acx = y\n\u22a2 List.insert x (y :: l) ~ List.insert y (x :: l)\n[PROOFSTEP]\nsimp [List.insert, xl, yl, xy, Ne.symm xy]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nl : List \u03b1\nxl : \u00acx \u2208 l\nyl : \u00acy \u2208 l\nxy : \u00acx = y\n\u22a2 x :: y :: l ~ y :: x :: l\n[PROOFSTEP]\nconstructor\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh : n \u2264 length l\n\u22a2 insertNth n x l ~ x :: l\n[PROOFSTEP]\ninduction' l with _ _ l_ih generalizing n\n[GOAL]\ncase nil\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length l\nn : \u2115\nh : n \u2264 length []\n\u22a2 insertNth n x [] ~ [x]\n[PROOFSTEP]\ncases n\n[GOAL]\ncase nil.zero\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nh : zero \u2264 length []\n\u22a2 insertNth zero x [] ~ [x]\ncase nil.succ\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nn\u271d : \u2115\nh : succ n\u271d \u2264 length []\n\u22a2 insertNth (succ n\u271d) x [] ~ [x]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase nil.succ\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nn\u271d : \u2115\nh : succ n\u271d \u2264 length []\n\u22a2 insertNth (succ n\u271d) x [] ~ [x]\n[PROOFSTEP]\ncases h\n[GOAL]\ncase cons\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nn : \u2115\nh : n \u2264 length (head\u271d :: tail\u271d)\n\u22a2 insertNth n x (head\u271d :: tail\u271d) ~ x :: head\u271d :: tail\u271d\n[PROOFSTEP]\ncases n\n[GOAL]\ncase cons.zero\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nh : zero \u2264 length (head\u271d :: tail\u271d)\n\u22a2 insertNth zero x (head\u271d :: tail\u271d) ~ x :: head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp [insertNth]\n[GOAL]\ncase cons.succ\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nn\u271d : \u2115\nh : succ n\u271d \u2264 length (head\u271d :: tail\u271d)\n\u22a2 insertNth (succ n\u271d) x (head\u271d :: tail\u271d) ~ x :: head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp only [insertNth, modifyNthTail]\n[GOAL]\ncase cons.succ\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nn\u271d : \u2115\nh : succ n\u271d \u2264 length (head\u271d :: tail\u271d)\n\u22a2 head\u271d :: modifyNthTail (cons x) n\u271d tail\u271d ~ x :: head\u271d :: tail\u271d\n[PROOFSTEP]\nrefine' Perm.trans (Perm.cons _ (l_ih _)) _\n[GOAL]\ncase cons.succ.refine'_1\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nn\u271d : \u2115\nh : succ n\u271d \u2264 length (head\u271d :: tail\u271d)\n\u22a2 n\u271d \u2264 length tail\u271d\n[PROOFSTEP]\napply Nat.le_of_succ_le_succ h\n[GOAL]\ncase cons.succ.refine'_2\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\ninst\u271d : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\nx : \u03b1\nl : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nl_ih : \u2200 {n : \u2115}, n \u2264 length tail\u271d \u2192 insertNth n x tail\u271d ~ x :: tail\u271d\nn\u271d : \u2115\nh : succ n\u271d \u2264 length (head\u271d :: tail\u271d)\n\u22a2 head\u271d :: x :: tail\u271d ~ x :: head\u271d :: tail\u271d\n[PROOFSTEP]\napply Perm.swap\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 l\u2081 \u222a t\u2081 ~ l\u2082 \u222a t\u2081\n[PROOFSTEP]\ninduction' h with a _ _ _ ih _ _ _ _ _ _ _ _ ih_1 ih_2\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\n\u22a2 [] \u222a t\u2081 ~ [] \u222a t\u2081\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\n\u22a2 [] \u222a t\u2081 ~ [] \u222a t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\na : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nih : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\n\u22a2 a :: l\u2081\u271d \u222a t\u2081 ~ a :: l\u2082\u271d \u222a t\u2081\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\na : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nih : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\n\u22a2 a :: l\u2081\u271d \u222a t\u2081 ~ a :: l\u2082\u271d \u222a t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 y\u271d :: x\u271d :: l\u271d \u222a t\u2081 ~ x\u271d :: y\u271d :: l\u271d \u222a t\u2081\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 y\u271d :: x\u271d :: l\u271d \u222a t\u2081 ~ x\u271d :: y\u271d :: l\u271d \u222a t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih_1 : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\nih_2 : l\u2082\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n\u22a2 l\u2081\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih_1 : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\nih_2 : l\u2082\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n\u22a2 l\u2081\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\na : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nih : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\n\u22a2 List.insert a (l\u2081\u271d \u222a t\u2081) ~ List.insert a (l\u2082\u271d \u222a t\u2081)\n[PROOFSTEP]\nexact ih.insert a\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 : List \u03b1\nx\u271d y\u271d : \u03b1\nl\u271d : List \u03b1\n\u22a2 List.insert y\u271d (List.insert x\u271d (l\u271d \u222a t\u2081)) ~ List.insert x\u271d (List.insert y\u271d (l\u271d \u222a t\u2081))\n[PROOFSTEP]\napply perm_insert_swap\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 t\u2081 l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih_1 : l\u2081\u271d \u222a t\u2081 ~ l\u2082\u271d \u222a t\u2081\nih_2 : l\u2082\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n\u22a2 l\u2081\u271d \u222a t\u2081 ~ l\u2083\u271d \u222a t\u2081\n[PROOFSTEP]\nexact ih_1.trans ih_2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl t\u2081 t\u2082 : List \u03b1\nh : t\u2081 ~ t\u2082\n\u22a2 l \u222a t\u2081 ~ l \u222a t\u2082\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : t\u2081 ~ t\u2082\n\u22a2 [] \u222a t\u2081 ~ [] \u222a t\u2082\n[PROOFSTEP]\nsimp [*, Perm.insert]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : t\u2081 ~ t\u2082\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : tail\u271d \u222a t\u2081 ~ tail\u271d \u222a t\u2082\n\u22a2 head\u271d :: tail\u271d \u222a t\u2081 ~ head\u271d :: tail\u271d \u222a t\u2082\n[PROOFSTEP]\nsimp [*, Perm.insert]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl t\u2081 t\u2082 : List \u03b1\np : t\u2081 ~ t\u2082\na : \u03b1\nx\u271d : a \u2208 l\n\u22a2 decide (a \u2208 t\u2081) = true \u2194 decide (a \u2208 t\u2082) = true\n[PROOFSTEP]\nsimpa using p.mem_iff\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl t\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\n\u22a2 l \u2229 (t\u2081 ++ t\u2082) ~ l \u2229 t\u2081 ++ l \u2229 t\u2082\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\n\u22a2 [] \u2229 (t\u2081 ++ t\u2082) ~ [] \u2229 t\u2081 ++ [] \u2229 t\u2082\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : tail\u271d \u2229 (t\u2081 ++ t\u2082) ~ tail\u271d \u2229 t\u2081 ++ tail\u271d \u2229 t\u2082\n\u22a2 (head\u271d :: tail\u271d) \u2229 (t\u2081 ++ t\u2082) ~ (head\u271d :: tail\u271d) \u2229 t\u2081 ++ (head\u271d :: tail\u271d) \u2229 t\u2082\n[PROOFSTEP]\ncase nil => simp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\n\u22a2 [] \u2229 (t\u2081 ++ t\u2082) ~ [] \u2229 t\u2081 ++ [] \u2229 t\u2082\n[PROOFSTEP]\ncase nil => simp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\n\u22a2 [] \u2229 (t\u2081 ++ t\u2082) ~ [] \u2229 t\u2081 ++ [] \u2229 t\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : tail\u271d \u2229 (t\u2081 ++ t\u2082) ~ tail\u271d \u2229 t\u2081 ++ tail\u271d \u2229 t\u2082\n\u22a2 (head\u271d :: tail\u271d) \u2229 (t\u2081 ++ t\u2082) ~ (head\u271d :: tail\u271d) \u2229 t\u2081 ++ (head\u271d :: tail\u271d) \u2229 t\u2082\n[PROOFSTEP]\ncase cons x xs l_ih =>\n by_cases h\u2081 : x \u2208 t\u2081\n \u00b7 have h\u2082 : x \u2209 t\u2082 := h h\u2081\n simp [*]\n by_cases h\u2082 : x \u2208 t\u2082\n \u00b7 simp only [*, inter_cons_of_not_mem, false_or_iff, mem_append, inter_cons_of_mem, not_false_iff]\n refine' Perm.trans (Perm.cons _ l_ih) _\n change [x] ++ xs \u2229 t\u2081 ++ xs \u2229 t\u2082 ~ xs \u2229 t\u2081 ++ ([x] ++ xs \u2229 t\u2082)\n rw [\u2190 List.append_assoc]\n solve_by_elim [Perm.append_right, perm_append_comm]\n \u00b7 simp [*]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\ncase cons x xs l_ih =>\n by_cases h\u2081 : x \u2208 t\u2081\n \u00b7 have h\u2082 : x \u2209 t\u2082 := h h\u2081\n simp [*]\n by_cases h\u2082 : x \u2208 t\u2082\n \u00b7 simp only [*, inter_cons_of_not_mem, false_or_iff, mem_append, inter_cons_of_mem, not_false_iff]\n refine' Perm.trans (Perm.cons _ l_ih) _\n change [x] ++ xs \u2229 t\u2081 ++ xs \u2229 t\u2082 ~ xs \u2229 t\u2081 ++ ([x] ++ xs \u2229 t\u2082)\n rw [\u2190 List.append_assoc]\n solve_by_elim [Perm.append_right, perm_append_comm]\n \u00b7 simp [*]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nby_cases h\u2081 : x \u2208 t\u2081\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : x \u2208 t\u2081\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nhave h\u2082 : x \u2209 t\u2082 := h h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : x \u2208 t\u2081\nh\u2082 : \u00acx \u2208 t\u2082\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nby_cases h\u2082 : x \u2208 t\u2082\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : x \u2208 t\u2082\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nsimp only [*, inter_cons_of_not_mem, false_or_iff, mem_append, inter_cons_of_mem, not_false_iff]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : x \u2208 t\u2082\n\u22a2 x :: xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ x :: xs \u2229 t\u2082\n[PROOFSTEP]\nrefine' Perm.trans (Perm.cons _ l_ih) _\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : x \u2208 t\u2082\n\u22a2 x :: (xs \u2229 t\u2081 ++ xs \u2229 t\u2082) ~ xs \u2229 t\u2081 ++ x :: xs \u2229 t\u2082\n[PROOFSTEP]\nchange [x] ++ xs \u2229 t\u2081 ++ xs \u2229 t\u2082 ~ xs \u2229 t\u2081 ++ ([x] ++ xs \u2229 t\u2082)\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : x \u2208 t\u2082\n\u22a2 [x] ++ xs \u2229 t\u2081 ++ xs \u2229 t\u2082 ~ xs \u2229 t\u2081 ++ ([x] ++ xs \u2229 t\u2082)\n[PROOFSTEP]\nrw [\u2190 List.append_assoc]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : x \u2208 t\u2082\n\u22a2 [x] ++ xs \u2229 t\u2081 ++ xs \u2229 t\u2082 ~ xs \u2229 t\u2081 ++ [x] ++ xs \u2229 t\u2082\n[PROOFSTEP]\nsolve_by_elim [Perm.append_right, perm_append_comm]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nt\u2081 t\u2082 : List \u03b1\nh : Disjoint t\u2081 t\u2082\nx : \u03b1\nxs : List \u03b1\nl_ih : xs \u2229 (t\u2081 ++ t\u2082) ~ xs \u2229 t\u2081 ++ xs \u2229 t\u2082\nh\u2081 : \u00acx \u2208 t\u2081\nh\u2082 : \u00acx \u2208 t\u2082\n\u22a2 (x :: xs) \u2229 (t\u2081 ++ t\u2082) ~ (x :: xs) \u2229 t\u2081 ++ (x :: xs) \u2229 t\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nd : Pairwise R l\u2081\n\u22a2 Pairwise R l\u2082\n[PROOFSTEP]\ninduction' d with a l\u2081 h _ IH generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b2 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2081 l\u2082\u271d : List \u03b1\np\u271d : l\u2081 ~ l\u2082\u271d\nl\u2082 : List \u03b1\np : [] ~ l\u2082\n\u22a2 Pairwise R l\u2082\n[PROOFSTEP]\nrw [\u2190 p.nil_eq]\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b2 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d l\u2082\u271d\u00b9 l\u2081 l\u2082\u271d : List \u03b1\np\u271d : l\u2081 ~ l\u2082\u271d\nl\u2082 : List \u03b1\np : [] ~ l\u2082\n\u22a2 Pairwise R []\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b2 l\u2082\u271d\u00b2 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 R a a'\na\u271d : Pairwise R l\u2081\nIH : \u2200 (l\u2082 : List \u03b1), l\u2081 ~ l\u2082 \u2192 Pairwise R l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\n\u22a2 Pairwise R l\u2082\n[PROOFSTEP]\nhave : a \u2208 l\u2082 := p.subset (mem_cons_self _ _)\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b2 l\u2082\u271d\u00b2 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 l\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 R a a'\na\u271d : Pairwise R l\u2081\nIH : \u2200 (l\u2082 : List \u03b1), l\u2081 ~ l\u2082 \u2192 Pairwise R l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\nthis : a \u2208 l\u2082\n\u22a2 Pairwise R l\u2082\n[PROOFSTEP]\nrcases mem_split this with \u27e8s\u2082, t\u2082, rfl\u27e9\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b2 l\u2082\u271d\u00b9 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d\u00b9 l\u2082\u271d l\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 R a a'\na\u271d : Pairwise R l\u2081\nIH : \u2200 (l\u2082 : List \u03b1), l\u2081 ~ l\u2082 \u2192 Pairwise R l\u2082\ns\u2082 t\u2082 : List \u03b1\np : a :: l\u2081 ~ s\u2082 ++ a :: t\u2082\nthis : a \u2208 s\u2082 ++ a :: t\u2082\n\u22a2 Pairwise R (s\u2082 ++ a :: t\u2082)\n[PROOFSTEP]\nhave p' := (p.trans perm_middle).cons_inv\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b2 l\u2082\u271d\u00b9 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d\u00b9 l\u2082\u271d l\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 R a a'\na\u271d : Pairwise R l\u2081\nIH : \u2200 (l\u2082 : List \u03b1), l\u2081 ~ l\u2082 \u2192 Pairwise R l\u2082\ns\u2082 t\u2082 : List \u03b1\np : a :: l\u2081 ~ s\u2082 ++ a :: t\u2082\nthis : a \u2208 s\u2082 ++ a :: t\u2082\np' : l\u2081 ~ s\u2082 ++ t\u2082\n\u22a2 Pairwise R (s\u2082 ++ a :: t\u2082)\n[PROOFSTEP]\nrefine' (pairwise_middle S).2 (pairwise_cons.2 \u27e8fun b m => _, IH _ p'\u27e9)\n[GOAL]\ncase cons.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b2 l\u2082\u271d\u00b9 : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nS : Symmetric R\nl\u2081\u271d\u00b9 l\u2082\u271d l\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 R a a'\na\u271d : Pairwise R l\u2081\nIH : \u2200 (l\u2082 : List \u03b1), l\u2081 ~ l\u2082 \u2192 Pairwise R l\u2082\ns\u2082 t\u2082 : List \u03b1\np : a :: l\u2081 ~ s\u2082 ++ a :: t\u2082\nthis : a \u2208 s\u2082 ++ a :: t\u2082\np' : l\u2081 ~ s\u2082 ++ t\u2082\nb : \u03b1\nm : b \u2208 s\u2082 ++ t\u2082\n\u22a2 R a b\n[PROOFSTEP]\nexact h _ (p'.symm.subset m)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nl\u2081 l\u2082 : List (List \u03b1)\nh : l\u2081 ~ l\u2082\nx\u2081 x\u2082 : List \u03b1\nxs : List (List \u03b1)\n\u22a2 List.join (x\u2082 :: x\u2081 :: xs) ~ List.join (x\u2081 :: x\u2082 :: xs)\n[PROOFSTEP]\nsimpa only [join, append_assoc] using perm_append_comm.append_right _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 f a ~ g a\n\u22a2 Forall\u2082 (fun x x_1 => x ~ x_1) (List.map f l) (List.map g l)\n[PROOFSTEP]\nrwa [List.forall\u2082_map_right_iff, List.forall\u2082_map_left_iff, List.forall\u2082_same]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\n\u22a2 List.bind l f ++ List.bind l g ~ List.bind l fun x => f x ++ g x\n[PROOFSTEP]\ninduction' l with a l IH\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\n\u22a2 List.bind [] f ++ List.bind [] g ~ List.bind [] fun x => f x ++ g x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : List.bind l f ++ List.bind l g ~ List.bind l fun x => f x ++ g x\n\u22a2 List.bind (a :: l) f ++ List.bind (a :: l) g ~ List.bind (a :: l) fun x => f x ++ g x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : List.bind l f ++ List.bind l g ~ List.bind l fun x => f x ++ g x\n\u22a2 f a ++ (List.bind l f ++ (g a ++ List.bind l g)) ~ f a ++ (g a ++ List.bind l fun x => f x ++ g x)\n[PROOFSTEP]\nrefine' (Perm.trans _ (IH.append_left _)).append_left _\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : List.bind l f ++ List.bind l g ~ List.bind l fun x => f x ++ g x\n\u22a2 List.bind l f ++ (g a ++ List.bind l g) ~ g a ++ (List.bind l f ++ List.bind l g)\n[PROOFSTEP]\nrw [\u2190 append_assoc, \u2190 append_assoc]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : List.bind l f ++ List.bind l g ~ List.bind l fun x => f x ++ g x\n\u22a2 List.bind l f ++ g a ++ List.bind l g ~ g a ++ List.bind l f ++ List.bind l g\n[PROOFSTEP]\nexact perm_append_comm.append_right _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 l : List \u03b1\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 List \u03b2\n\u22a2 map f l ++ List.bind l g ~ List.bind l fun x => f x :: g x\n[PROOFSTEP]\nsimpa [\u2190 map_eq_bind] using bind_append_perm l (fun x => [f x]) g\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\np : l\u2081 ~ l\u2082\n\u22a2 lookmap f l\u2081 ~ lookmap f l\u2082\n[PROOFSTEP]\ninduction' p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 _ IH\u2081 IH\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) []\n\u22a2 lookmap f [] ~ lookmap f []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (a :: l\u2081)\n\u22a2 lookmap f (a :: l\u2081) ~ lookmap f (a :: l\u2082)\n[PROOFSTEP]\ncases h : f a\n[GOAL]\ncase cons.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (a :: l\u2081)\nh : f a = none\n\u22a2 lookmap f (a :: l\u2081) ~ lookmap f (a :: l\u2082)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase cons.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (a :: l\u2081)\nh : f a = none\n\u22a2 lookmap f l\u2081 ~ lookmap f l\u2082\n[PROOFSTEP]\nexact IH (pairwise_cons.1 H).2\n[GOAL]\ncase cons.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (a :: l\u2081)\nval\u271d : \u03b1\nh : f a = some val\u271d\n\u22a2 lookmap f (a :: l\u2081) ~ lookmap f (a :: l\u2082)\n[PROOFSTEP]\nsimp [lookmap_cons_some _ _ h, p]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\ncases' h\u2081 : f a with c\n[GOAL]\ncase swap.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nh\u2081 : f a = none\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\ncases' h\u2082 : f b with d\n[GOAL]\ncase swap.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nc : \u03b1\nh\u2081 : f a = some c\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\ncases' h\u2082 : f b with d\n[GOAL]\ncase swap.none.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nh\u2081 : f a = none\nh\u2082 : f b = none\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase swap.none.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nh\u2081 : f a = none\nh\u2082 : f b = none\n\u22a2 b :: a :: lookmap f l ~ a :: b :: lookmap f l\n[PROOFSTEP]\napply swap\n[GOAL]\ncase swap.none.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nh\u2081 : f a = none\nd : \u03b1\nh\u2082 : f b = some d\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, lookmap_cons_some _ _ h\u2082]\n[GOAL]\ncase swap.none.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nh\u2081 : f a = none\nd : \u03b1\nh\u2082 : f b = some d\n\u22a2 d :: a :: l ~ a :: d :: l\n[PROOFSTEP]\napply swap\n[GOAL]\ncase swap.some.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nc : \u03b1\nh\u2081 : f a = some c\nh\u2082 : f b = none\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\nsimp [lookmap_cons_some _ _ h\u2081, h\u2082]\n[GOAL]\ncase swap.some.none\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nc : \u03b1\nh\u2081 : f a = some c\nh\u2082 : f b = none\n\u22a2 b :: c :: l ~ c :: b :: l\n[PROOFSTEP]\napply swap\n[GOAL]\ncase swap.some.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nc : \u03b1\nh\u2081 : f a = some c\nd : \u03b1\nh\u2082 : f b = some d\n\u22a2 lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)\n[PROOFSTEP]\nsimp [lookmap_cons_some _ _ h\u2081, lookmap_cons_some _ _ h\u2082]\n[GOAL]\ncase swap.some.some\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: a :: l)\nc : \u03b1\nh\u2081 : f a = some c\nd : \u03b1\nh\u2082 : f b = some d\n\u22a2 d :: a :: l ~ c :: b :: l\n[PROOFSTEP]\nrcases(pairwise_cons.1 H).1 _ (mem_cons.2 (Or.inl rfl)) _ h\u2082 _ h\u2081 with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase swap.some.some.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\nb : \u03b1\nl : List \u03b1\nd : \u03b1\nh\u2082 : f b = some d\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) (b :: b :: l)\nh\u2081 : f b = some d\n\u22a2 d :: b :: l ~ d :: b :: l\n[PROOFSTEP]\nexact Perm.refl _\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\nl\u2081 l\u2082 l\u2083 : List \u03b1\np\u2081 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nIH\u2082 : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2082 \u2192 lookmap f l\u2082 ~ lookmap f l\u2083\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\n\u22a2 lookmap f l\u2081 ~ lookmap f l\u2083\n[PROOFSTEP]\nrefine' (IH\u2081 H).trans (IH\u2082 ((p\u2081.pairwise_iff _).1 H))\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Option \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\u271d\nl\u2081 l\u2082 l\u2083 : List \u03b1\np\u2081 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081 \u2192 lookmap f l\u2081 ~ lookmap f l\u2082\nIH\u2082 : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2082 \u2192 lookmap f l\u2082 ~ lookmap f l\u2083\nH : Pairwise (fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d) l\u2081\n\u22a2 Symmetric fun a b => \u2200 (c : \u03b1), c \u2208 f a \u2192 \u2200 (d : \u03b1), d \u2208 f b \u2192 a = b \u2227 c = d\n[PROOFSTEP]\nexact fun a b h c h\u2081 d h\u2082 => (h d h\u2082 c h\u2081).imp Eq.symm Eq.symm\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\np : l\u2081 ~ l\u2082\n\u22a2 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\n[PROOFSTEP]\ninduction' p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 _ IH\u2081 IH\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) []\n\u22a2 eraseP (fun b => decide (f b)) [] ~ eraseP (fun b => decide (f b)) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (a :: l\u2081)\n\u22a2 eraseP (fun b => decide (f b)) (a :: l\u2081) ~ eraseP (fun b => decide (f b)) (a :: l\u2082)\n[PROOFSTEP]\nby_cases h : f a\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (a :: l\u2081)\nh : f a\n\u22a2 eraseP (fun b => decide (f b)) (a :: l\u2081) ~ eraseP (fun b => decide (f b)) (a :: l\u2082)\n[PROOFSTEP]\nsimp [h, p]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (a :: l\u2081)\nh : \u00acf a\n\u22a2 eraseP (fun b => decide (f b)) (a :: l\u2081) ~ eraseP (fun b => decide (f b)) (a :: l\u2082)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (a :: l\u2081)\nh : \u00acf a\n\u22a2 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\n[PROOFSTEP]\nexact IH (pairwise_cons.1 H).2\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nby_cases h\u2081 : f a\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : f a\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nby_cases h\u2082 : f b\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : \u00acf a\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nby_cases h\u2082 : f b\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : f a\nh\u2082 : f b\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : f a\nh\u2082 : \u00acf b\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : \u00acf a\nh\u2082 : f b\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : \u00acf a\nh\u2082 : \u00acf b\n\u22a2 eraseP (fun b => decide (f b)) (b :: a :: l) ~ eraseP (fun b => decide (f b)) (a :: b :: l)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : f a\nh\u2082 : f b\n\u22a2 a :: l ~ b :: l\n[PROOFSTEP]\ncases (pairwise_cons.1 H).1 _ (mem_cons.2 (Or.inl rfl)) h\u2082 h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081 l\u2082 : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\na b : \u03b1\nl : List \u03b1\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) (b :: a :: l)\nh\u2081 : \u00acf a\nh\u2082 : \u00acf b\n\u22a2 b :: a :: eraseP (fun b => decide (f b)) l ~ a :: b :: eraseP (fun b => decide (f b)) l\n[PROOFSTEP]\napply swap\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\nl\u2081 l\u2082 l\u2083 : List \u03b1\np\u2081 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nIH\u2082 : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2082 \u2192 eraseP (fun b => decide (f b)) l\u2082 ~ eraseP (fun b => decide (f b)) l\u2083\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\n\u22a2 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2083\n[PROOFSTEP]\nrefine' (IH\u2081 H).trans (IH\u2082 ((p\u2081.pairwise_iff _).1 H))\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nf : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred f\nl\u2081\u271d l\u2082\u271d : List \u03b1\nH\u271d : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\u271d\nl\u2081 l\u2082 l\u2083 : List \u03b1\np\u2081 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081 \u2192 eraseP (fun b => decide (f b)) l\u2081 ~ eraseP (fun b => decide (f b)) l\u2082\nIH\u2082 : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2082 \u2192 eraseP (fun b => decide (f b)) l\u2082 ~ eraseP (fun b => decide (f b)) l\u2083\nH : Pairwise (fun a b => f a \u2192 f b \u2192 False) l\u2081\n\u22a2 Symmetric fun a b => f a \u2192 f b \u2192 False\n[PROOFSTEP]\nexact fun a b h h\u2081 h\u2082 => h h\u2082 h\u2081\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.inter ys (take n xs)\n[PROOFSTEP]\nsimp only [List.inter]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.filter (fun x => decide (x \u2208 take n xs)) ys\n[PROOFSTEP]\nexact\n Perm.trans\n (show xs.take n ~ xs.filter (\u00b7 \u2208 xs.take n) by conv_lhs => rw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')])\n (Perm.filter _ h)\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.filter (fun x => decide (x \u2208 take n xs)) xs\n[PROOFSTEP]\nconv_lhs => rw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n| take n xs\n[PROOFSTEP]\nrw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n| take n xs\n[PROOFSTEP]\nrw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n| take n xs\n[PROOFSTEP]\nrw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nby_cases h'' : n \u2264 xs.length\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nlet n' := xs.length - n\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nhave h\u2080 : n = xs.length - n' := by rwa [tsub_tsub_cancel_of_le]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\n\u22a2 n = length xs - n'\n[PROOFSTEP]\nrwa [tsub_tsub_cancel_of_le]\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nhave h\u2081 : n' \u2264 xs.length := by apply tsub_le_self\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\n\u22a2 n' \u2264 length xs\n[PROOFSTEP]\napply tsub_le_self\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nhave h\u2082 : xs.drop n = (xs.reverse.take n').reverse := by rw [reverse_take _ h\u2081, h\u2080, reverse_reverse]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\n\u22a2 drop n xs = reverse (take n' (reverse xs))\n[PROOFSTEP]\nrw [reverse_take _ h\u2081, h\u2080, reverse_reverse]\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 reverse (take n' (reverse xs)) ~ List.inter ys (reverse (take n' (reverse xs)))\n[PROOFSTEP]\napply (reverse_perm _).trans\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 take n' (reverse xs) ~ List.inter ys (reverse (take n' (reverse xs)))\n[PROOFSTEP]\nrw [inter_reverse]\n[GOAL]\ncase pos\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 take n' (reverse xs) ~ List.inter ys (take n' (reverse xs))\n[PROOFSTEP]\napply Perm.take_inter _ _ h'\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 reverse xs ~ ys\n[PROOFSTEP]\napply (reverse_perm _).trans\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : n \u2264 length xs\nn' : \u2115 := length xs - n\nh\u2080 : n = length xs - n'\nh\u2081 : n' \u2264 length xs\nh\u2082 : drop n xs = reverse (take n' (reverse xs))\n\u22a2 xs ~ ys\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : \u00acn \u2264 length xs\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nhave : drop n xs = [] := by\n apply eq_nil_of_length_eq_zero\n rw [length_drop, tsub_eq_zero_iff_le]\n apply le_of_not_ge h''\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : \u00acn \u2264 length xs\n\u22a2 drop n xs = []\n[PROOFSTEP]\napply eq_nil_of_length_eq_zero\n[GOAL]\ncase x\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : \u00acn \u2264 length xs\n\u22a2 length (drop n xs) = 0\n[PROOFSTEP]\nrw [length_drop, tsub_eq_zero_iff_le]\n[GOAL]\ncase x\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : \u00acn \u2264 length xs\n\u22a2 length xs \u2264 n\n[PROOFSTEP]\napply le_of_not_ge h''\n[GOAL]\ncase neg\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\nh'' : \u00acn \u2264 length xs\nthis : drop n xs = []\n\u22a2 drop n xs ~ List.inter ys (drop n xs)\n[PROOFSTEP]\nsimp [this, List.inter]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 dropSlice n m xs ~ ys \u2229 dropSlice n m xs\n[PROOFSTEP]\nsimp only [dropSlice_eq]\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ++ drop (n + m) xs ~ ys \u2229 (take n xs ++ drop (n + m) xs)\n[PROOFSTEP]\nhave : n \u2264 n + m := Nat.le_add_right _ _\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\nthis : n \u2264 n + m\n\u22a2 take n xs ++ drop (n + m) xs ~ ys \u2229 (take n xs ++ drop (n + m) xs)\n[PROOFSTEP]\nhave h\u2082 := h.nodup_iff.2 h'\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\nthis : n \u2264 n + m\nh\u2082 : Nodup xs\n\u22a2 take n xs ++ drop (n + m) xs ~ ys \u2229 (take n xs ++ drop (n + m) xs)\n[PROOFSTEP]\napply Perm.trans _ (Perm.inter_append _).symm\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\nthis : n \u2264 n + m\nh\u2082 : Nodup xs\n\u22a2 take n xs ++ drop (n + m) xs ~ ys \u2229 take n xs ++ ys \u2229 drop (n + m) xs\n[PROOFSTEP]\nexact Perm.append (Perm.take_inter _ h h') (Perm.drop_inter _ h h')\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn m : \u2115\nh : xs ~ ys\nh' : Nodup ys\nthis : n \u2264 n + m\nh\u2082 : Nodup xs\n\u22a2 Disjoint (take n xs) (drop (n + m) xs)\n[PROOFSTEP]\nexact disjoint_take_drop h\u2082 this\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 {ts is l : List \u03b1}, l \u2208 permutationsAux ts is \u2192 l ~ ts ++ is\n[PROOFSTEP]\nshow \u2200 (ts is l : List \u03b1), l \u2208 permutationsAux ts is \u2192 l ~ ts ++ is\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (ts is l : List \u03b1), l \u2208 permutationsAux ts is \u2192 l ~ ts ++ is\n[PROOFSTEP]\nrefine' permutationsAux.rec (by simp) _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (is l : List \u03b1), l \u2208 permutationsAux [] is \u2192 l ~ [] ++ is\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (t : \u03b1) (ts is : List \u03b1),\n (\u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is) \u2192\n (\u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []) \u2192\n \u2200 (l : List \u03b1), l \u2208 permutationsAux (t :: ts) is \u2192 l ~ t :: ts ++ is\n[PROOFSTEP]\nintrov IH1 IH2 m\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl : List \u03b1\nm : l \u2208 permutationsAux (t :: ts) is\n\u22a2 l ~ t :: ts ++ is\n[PROOFSTEP]\nrw [permutationsAux_cons, permutations, mem_foldr_permutationsAux2] at m \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl : List \u03b1\nm :\n l \u2208 permutationsAux ts (t :: is) \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 is :: permutationsAux is [] \u2227 l\u2082 \u2260 [] \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n\u22a2 l ~ t :: ts ++ is\n[PROOFSTEP]\nrcases m with (m | \u27e8l\u2081, l\u2082, m, _, e\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl : List \u03b1\nm : l \u2208 permutationsAux ts (t :: is)\n\u22a2 l ~ t :: ts ++ is\n[PROOFSTEP]\nexact (IH1 _ m).trans perm_middle\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl l\u2081 l\u2082 : List \u03b1\nm : l\u2081 ++ l\u2082 \u2208 is :: permutationsAux is []\nleft\u271d : l\u2082 \u2260 []\ne : l = l\u2081 ++ t :: l\u2082 ++ ts\n\u22a2 l ~ t :: ts ++ is\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nm : l\u2081 ++ l\u2082 \u2208 is :: permutationsAux is []\nleft\u271d : l\u2082 \u2260 []\n\u22a2 l\u2081 ++ t :: l\u2082 ++ ts ~ t :: ts ++ is\n[PROOFSTEP]\nhave p : l\u2081 ++ l\u2082 ~ is := by\n simp [permutations] at m \n cases' m with e m\n \u00b7 simp [e]\n exact is.append_nil \u25b8 IH2 _ m\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nm : l\u2081 ++ l\u2082 \u2208 is :: permutationsAux is []\nleft\u271d : l\u2082 \u2260 []\n\u22a2 l\u2081 ++ l\u2082 ~ is\n[PROOFSTEP]\nsimp [permutations] at m \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nleft\u271d : l\u2082 \u2260 []\nm : l\u2081 ++ l\u2082 = is \u2228 l\u2081 ++ l\u2082 \u2208 permutationsAux is []\n\u22a2 l\u2081 ++ l\u2082 ~ is\n[PROOFSTEP]\ncases' m with e m\n[GOAL]\ncase inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nleft\u271d : l\u2082 \u2260 []\ne : l\u2081 ++ l\u2082 = is\n\u22a2 l\u2081 ++ l\u2082 ~ is\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nleft\u271d : l\u2082 \u2260 []\nm : l\u2081 ++ l\u2082 \u2208 permutationsAux is []\n\u22a2 l\u2081 ++ l\u2082 ~ is\n[PROOFSTEP]\nexact is.append_nil \u25b8 IH2 _ m\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l \u2208 permutationsAux ts (t :: is) \u2192 l ~ ts ++ t :: is\nIH2 : \u2200 (l : List \u03b1), l \u2208 permutationsAux is [] \u2192 l ~ is ++ []\nl\u2081 l\u2082 : List \u03b1\nm : l\u2081 ++ l\u2082 \u2208 is :: permutationsAux is []\nleft\u271d : l\u2082 \u2260 []\np : l\u2081 ++ l\u2082 ~ is\n\u22a2 l\u2081 ++ t :: l\u2082 ++ ts ~ t :: ts ++ is\n[PROOFSTEP]\nexact ((perm_middle.trans (p.cons _)).append_right _).trans (perm_append_comm.cons _)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (ts is : List \u03b1), length (permutationsAux ts is) + (length is)! = (length ts + length is)!\n[PROOFSTEP]\nrefine' permutationsAux.rec (by simp) _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (is : List \u03b1), length (permutationsAux [] is) + (length is)! = (length [] + length is)!\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (t : \u03b1) (ts is : List \u03b1),\n length (permutationsAux ts (t :: is)) + (length (t :: is))! = (length ts + length (t :: is))! \u2192\n length (permutationsAux is []) + (length [])! = (length is + length [])! \u2192\n length (permutationsAux (t :: ts) is) + (length is)! = (length (t :: ts) + length is)!\n[PROOFSTEP]\nintro t ts is IH1 IH2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : length (permutationsAux ts (t :: is)) + (length (t :: is))! = (length ts + length (t :: is))!\nIH2 : length (permutationsAux is []) + (length [])! = (length is + length [])!\n\u22a2 length (permutationsAux (t :: ts) is) + (length is)! = (length (t :: ts) + length is)!\n[PROOFSTEP]\nhave IH2 : length (permutationsAux is nil) + 1 = is.length ! := by simpa using IH2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : length (permutationsAux ts (t :: is)) + (length (t :: is))! = (length ts + length (t :: is))!\nIH2 : length (permutationsAux is []) + (length [])! = (length is + length [])!\n\u22a2 length (permutationsAux is []) + 1 = (length is)!\n[PROOFSTEP]\nsimpa using IH2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : length (permutationsAux ts (t :: is)) + (length (t :: is))! = (length ts + length (t :: is))!\nIH2\u271d : length (permutationsAux is []) + (length [])! = (length is + length [])!\nIH2 : length (permutationsAux is []) + 1 = (length is)!\n\u22a2 length (permutationsAux (t :: ts) is) + (length is)! = (length (t :: ts) + length is)!\n[PROOFSTEP]\nsimp [Nat.factorial, Nat.add_succ, mul_comm] at IH1 \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH2\u271d : length (permutationsAux is []) + (length [])! = (length is + length [])!\nIH2 : length (permutationsAux is []) + 1 = (length is)!\nIH1 :\n length (permutationsAux ts (t :: is)) + (length is)! * succ (length is) =\n (length ts + length is)! * succ (length ts + length is)\n\u22a2 length (permutationsAux (t :: ts) is) + (length is)! = (length (t :: ts) + length is)!\n[PROOFSTEP]\nrw [permutationsAux_cons, length_foldr_permutationsAux2' _ _ _ _ _ fun l m => (perm_of_mem_permutations m).length_eq,\n permutations, length, length, IH2, Nat.succ_add, Nat.factorial_succ, mul_comm (_ + 1), \u2190 Nat.succ_eq_add_one, \u2190 IH1,\n add_comm (_ * _), add_assoc, Nat.mul_succ, mul_comm]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 is l : List \u03b1\nH : l ~ [] ++ is \u2192 (\u2203 ts' x, l = ts' ++ is) \u2228 l \u2208 permutationsAux is []\n\u22a2 l ~ is \u2192 l \u2208 permutations is\n[PROOFSTEP]\nsimpa [permutations, perm_nil] using H\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 {ts is l : List \u03b1}, l ~ is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts is\n[PROOFSTEP]\nshow \u2200 (ts is l : List \u03b1), l ~ is ++ ts \u2192 (\u2203 (is' : _) (_ : is' ~ is), l = is' ++ ts) \u2228 l \u2208 permutationsAux ts is\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (ts is l : List \u03b1), l ~ is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts is\n[PROOFSTEP]\nrefine' permutationsAux.rec (by simp) _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (is l : List \u03b1), l ~ is ++ [] \u2192 (\u2203 is' x, l = is' ++ []) \u2228 l \u2208 permutationsAux [] is\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\n\u22a2 \u2200 (t : \u03b1) (ts is : List \u03b1),\n (\u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)) \u2192\n (\u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []) \u2192\n \u2200 (l : List \u03b1), l ~ is ++ t :: ts \u2192 (\u2203 is' x, l = is' ++ t :: ts) \u2228 l \u2208 permutationsAux (t :: ts) is\n[PROOFSTEP]\nintro t ts is IH1 IH2 l p\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl : List \u03b1\np : l ~ is ++ t :: ts\n\u22a2 (\u2203 is' x, l = is' ++ t :: ts) \u2228 l \u2208 permutationsAux (t :: ts) is\n[PROOFSTEP]\nrw [permutationsAux_cons, mem_foldr_permutationsAux2]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl : List \u03b1\np : l ~ is ++ t :: ts\n\u22a2 (\u2203 is' x, l = is' ++ t :: ts) \u2228\n l \u2208 permutationsAux ts (t :: is) \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrcases IH1 _ (p.trans perm_middle) with (\u27e8is', p', e\u27e9 | m)\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl : List \u03b1\np : l ~ is ++ t :: ts\nis' : List \u03b1\np' : is' ~ t :: is\ne : l = is' ++ ts\n\u22a2 (\u2203 is' x, l = is' ++ t :: ts) \u2228\n l \u2208 permutationsAux ts (t :: is) \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nclear p\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl is' : List \u03b1\np' : is' ~ t :: is\ne : l = is' ++ ts\n\u22a2 (\u2203 is' x, l = is' ++ t :: ts) \u2228\n l \u2208 permutationsAux ts (t :: is) \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nis' : List \u03b1\np' : is' ~ t :: is\n\u22a2 (\u2203 is'_1 x, is' ++ ts = is'_1 ++ t :: ts) \u2228\n is' ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 is' ++ ts = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrcases mem_split (p'.symm.subset (mem_cons_self _ _)) with \u27e8l\u2081, l\u2082, e\u27e9\n[GOAL]\ncase inl.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nis' : List \u03b1\np' : is' ~ t :: is\nl\u2081 l\u2082 : List \u03b1\ne : is' = l\u2081 ++ t :: l\u2082\n\u22a2 (\u2203 is'_1 x, is' ++ ts = is'_1 ++ t :: ts) \u2228\n is' ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 is' ++ ts = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsubst is'\n[GOAL]\ncase inl.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 l\u2082 : List \u03b1\np' : l\u2081 ++ t :: l\u2082 ~ t :: is\n\u22a2 (\u2203 is' x, l\u2081 ++ t :: l\u2082 ++ ts = is' ++ t :: ts) \u2228\n l\u2081 ++ t :: l\u2082 ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081_1 l\u2082_1, l\u2081_1 ++ l\u2082_1 \u2208 permutations is \u2227 l\u2082_1 \u2260 [] \u2227 l\u2081 ++ t :: l\u2082 ++ ts = l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nhave p := (perm_middle.symm.trans p').cons_inv\n[GOAL]\ncase inl.intro.intro.intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 l\u2082 : List \u03b1\np' : l\u2081 ++ t :: l\u2082 ~ t :: is\np : l\u2081 ++ l\u2082 ~ is\n\u22a2 (\u2203 is' x, l\u2081 ++ t :: l\u2082 ++ ts = is' ++ t :: ts) \u2228\n l\u2081 ++ t :: l\u2082 ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081_1 l\u2082_1, l\u2081_1 ++ l\u2082_1 \u2208 permutations is \u2227 l\u2082_1 \u2260 [] \u2227 l\u2081 ++ t :: l\u2082 ++ ts = l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\ncases' l\u2082 with a l\u2082'\n[GOAL]\ncase inl.intro.intro.intro.intro.nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 : List \u03b1\np' : l\u2081 ++ [t] ~ t :: is\np : l\u2081 ++ [] ~ is\n\u22a2 (\u2203 is' x, l\u2081 ++ [t] ++ ts = is' ++ t :: ts) \u2228\n l\u2081 ++ [t] ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081_1 l\u2082, l\u2081_1 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l\u2081 ++ [t] ++ ts = l\u2081_1 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nexact Or.inl \u27e8l\u2081, by simpa using p\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 : List \u03b1\np' : l\u2081 ++ [t] ~ t :: is\np : l\u2081 ++ [] ~ is\n\u22a2 \u2203 x, l\u2081 ++ [t] ++ ts = l\u2081 ++ t :: ts\n[PROOFSTEP]\nsimpa using p\n[GOAL]\ncase inl.intro.intro.intro.intro.cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082' : List \u03b1\np' : l\u2081 ++ t :: a :: l\u2082' ~ t :: is\np : l\u2081 ++ a :: l\u2082' ~ is\n\u22a2 (\u2203 is' x, l\u2081 ++ t :: a :: l\u2082' ++ ts = is' ++ t :: ts) \u2228\n l\u2081 ++ t :: a :: l\u2082' ++ ts \u2208 permutationsAux ts (t :: is) \u2228\n \u2203 l\u2081_1 l\u2082, l\u2081_1 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l\u2081 ++ t :: a :: l\u2082' ++ ts = l\u2081_1 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8l\u2081, a :: l\u2082', mem_permutations_of_perm_lemma (IH2 _) p, by simp\u27e9)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082' : List \u03b1\np' : l\u2081 ++ t :: a :: l\u2082' ~ t :: is\np : l\u2081 ++ a :: l\u2082' ~ is\n\u22a2 a :: l\u2082' \u2260 [] \u2227 l\u2081 ++ t :: a :: l\u2082' ++ ts = l\u2081 ++ t :: a :: l\u2082' ++ ts\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nt : \u03b1\nts is : List \u03b1\nIH1 : \u2200 (l : List \u03b1), l ~ t :: is ++ ts \u2192 (\u2203 is' x, l = is' ++ ts) \u2228 l \u2208 permutationsAux ts (t :: is)\nIH2 : \u2200 (l : List \u03b1), l ~ [] ++ is \u2192 (\u2203 is' x, l = is' ++ is) \u2228 l \u2208 permutationsAux is []\nl : List \u03b1\np : l ~ is ++ t :: ts\nm : l \u2208 permutationsAux ts (t :: is)\n\u22a2 (\u2203 is' x, l = is' ++ t :: ts) \u2228\n l \u2208 permutationsAux ts (t :: is) \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 permutations is \u2227 l\u2082 \u2260 [] \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nexact Or.inr (Or.inl m)\n[GOAL]\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 List.beq = fun a b => decide (a = b)\n[PROOFSTEP]\nfunext l\u2081 l\u2082\n[GOAL]\ncase h.h\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 List.beq l\u2081 l\u2082 = decide (l\u2081 = l\u2082)\n[PROOFSTEP]\nshow (l\u2081 == l\u2082) = _\n[GOAL]\ncase h.h\n\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (l\u2081 == l\u2082) = decide (l\u2081 = l\u2082)\n[PROOFSTEP]\nrw [Bool.eq_iff_eq_true_iff, @beq_iff_eq _ (_), decide_eq_true_iff]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\n[PROOFSTEP]\ninduction' l with c l ih\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\n\u22a2 List.bind (permutations'Aux a []) (permutations'Aux b) ~ List.bind (permutations'Aux b []) (permutations'Aux a)\n[PROOFSTEP]\nsimp [swap]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\n\u22a2 List.bind (permutations'Aux a (c :: l)) (permutations'Aux b) ~\n List.bind (permutations'Aux b (c :: l)) (permutations'Aux a)\n[PROOFSTEP]\nsimp [permutations'Aux]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\n\u22a2 (b :: a :: c :: l) ::\n (a :: b :: c :: l) ::\n (map (cons a \u2218 cons c) (permutations'Aux b l) ++\n List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b)) ~\n (a :: b :: c :: l) ::\n (b :: a :: c :: l) ::\n (map (cons b \u2218 cons c) (permutations'Aux a l) ++\n List.bind (map (cons c) (permutations'Aux b l)) (permutations'Aux a))\n[PROOFSTEP]\napply Perm.swap'\n[GOAL]\ncase cons.p\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\n\u22a2 map (cons a \u2218 cons c) (permutations'Aux b l) ++ List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++ List.bind (map (cons c) (permutations'Aux b l)) (permutations'Aux a)\n[PROOFSTEP]\nhave :\n \u2200 a b,\n (map (cons c) (permutations'Aux a l)).bind (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++ map (cons c) ((permutations'Aux a l).bind (permutations'Aux b)) :=\n by\n intros a' b'\n simp only [map_bind, permutations'Aux]\n show List.bind (permutations'Aux _ l) (fun a => ([b' :: c :: a] ++ map (cons c) (permutations'Aux _ a))) ~ _\n refine' (bind_append_perm _ (fun x => [b' :: c :: x]) _).symm.trans _\n rw [\u2190 map_eq_bind, \u2190 bind_map]\n exact Perm.refl _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\n\u22a2 \u2200 (a b : \u03b1),\n List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b))\n[PROOFSTEP]\nintros a' b'\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\na' b' : \u03b1\n\u22a2 List.bind (map (cons c) (permutations'Aux a' l)) (permutations'Aux b') ~\n map (cons b' \u2218 cons c) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) (permutations'Aux b'))\n[PROOFSTEP]\nsimp only [map_bind, permutations'Aux]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\na' b' : \u03b1\n\u22a2 (List.bind (permutations'Aux a' l) fun a => (b' :: c :: a) :: map (cons c) (permutations'Aux b' a)) ~\n map (cons b' \u2218 cons c) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) (permutations'Aux b'))\n[PROOFSTEP]\nshow List.bind (permutations'Aux _ l) (fun a => ([b' :: c :: a] ++ map (cons c) (permutations'Aux _ a))) ~ _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\na' b' : \u03b1\n\u22a2 (List.bind (permutations'Aux a' l) fun a => [b' :: c :: a] ++ map (cons c) (permutations'Aux b' a)) ~\n map (cons b' \u2218 cons c) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) (permutations'Aux b'))\n[PROOFSTEP]\nrefine' (bind_append_perm _ (fun x => [b' :: c :: x]) _).symm.trans _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\na' b' : \u03b1\n\u22a2 ((List.bind (permutations'Aux a' l) fun x => [b' :: c :: x]) ++\n List.bind (permutations'Aux a' l) fun a => map (cons c) (permutations'Aux b' a)) ~\n map (cons b' \u2218 cons c) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) (permutations'Aux b'))\n[PROOFSTEP]\nrw [\u2190 map_eq_bind, \u2190 bind_map]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\na' b' : \u03b1\n\u22a2 map (fun x => b' :: c :: x) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) fun a => permutations'Aux b' a) ~\n map (cons b' \u2218 cons c) (permutations'Aux a' l) ++\n map (cons c) (List.bind (permutations'Aux a' l) (permutations'Aux b'))\n[PROOFSTEP]\nexact Perm.refl _\n[GOAL]\ncase cons.p\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\nthis :\n \u2200 (a b : \u03b1),\n List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b))\n\u22a2 map (cons a \u2218 cons c) (permutations'Aux b l) ++ List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++ List.bind (map (cons c) (permutations'Aux b l)) (permutations'Aux a)\n[PROOFSTEP]\nrefine' (((this _ _).append_left _).trans _).trans ((this _ _).append_left _).symm\n[GOAL]\ncase cons.p\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\nthis :\n \u2200 (a b : \u03b1),\n List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b))\n\u22a2 map (cons a \u2218 cons c) (permutations'Aux b l) ++\n (map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b))) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++\n (map (cons a \u2218 cons c) (permutations'Aux b l) ++\n map (cons c) (List.bind (permutations'Aux b l) (permutations'Aux a)))\n[PROOFSTEP]\nrw [\u2190 append_assoc, \u2190 append_assoc]\n[GOAL]\ncase cons.p\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\na b c : \u03b1\nl : List \u03b1\nih : List.bind (permutations'Aux a l) (permutations'Aux b) ~ List.bind (permutations'Aux b l) (permutations'Aux a)\nthis :\n \u2200 (a b : \u03b1),\n List.bind (map (cons c) (permutations'Aux a l)) (permutations'Aux b) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b))\n\u22a2 map (cons a \u2218 cons c) (permutations'Aux b l) ++ map (cons b \u2218 cons c) (permutations'Aux a l) ++\n map (cons c) (List.bind (permutations'Aux a l) (permutations'Aux b)) ~\n map (cons b \u2218 cons c) (permutations'Aux a l) ++ map (cons a \u2218 cons c) (permutations'Aux b l) ++\n map (cons c) (List.bind (permutations'Aux b l) (permutations'Aux a))\n[PROOFSTEP]\nexact perm_append_comm.append (ih.map _)\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\np : s ~ t\n\u22a2 List.permutations' s ~ List.permutations' t\n[PROOFSTEP]\ninduction' p with a s t _ IH a b l s t u _ _ IH\u2081 IH\u2082\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\n\u22a2 List.permutations' [] ~ List.permutations' []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d t\u271d : List \u03b1\na : \u03b1\ns t : List \u03b1\na\u271d : s ~ t\nIH : List.permutations' s ~ List.permutations' t\n\u22a2 List.permutations' (a :: s) ~ List.permutations' (a :: t)\n[PROOFSTEP]\nexact IH.bind_right _\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.permutations' (b :: a :: l) ~ List.permutations' (a :: b :: l)\n[PROOFSTEP]\ndsimp [permutations']\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.bind (List.bind (List.permutations' l) (permutations'Aux a)) (permutations'Aux b) ~\n List.bind (List.bind (List.permutations' l) (permutations'Aux b)) (permutations'Aux a)\n[PROOFSTEP]\nrw [bind_assoc, bind_assoc]\n[GOAL]\ncase swap\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 (List.bind (List.permutations' l) fun x => List.bind (permutations'Aux a x) (permutations'Aux b)) ~\n List.bind (List.permutations' l) fun x => List.bind (permutations'Aux b x) (permutations'Aux a)\n[PROOFSTEP]\napply Perm.bind_left\n[GOAL]\ncase swap.h\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 \u2200 (a_1 : List \u03b1),\n a_1 \u2208 List.permutations' l \u2192\n List.bind (permutations'Aux a a_1) (permutations'Aux b) ~ List.bind (permutations'Aux b a_1) (permutations'Aux a)\n[PROOFSTEP]\nintro l' _\n[GOAL]\ncase swap.h\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s t : List \u03b1\na b : \u03b1\nl l' : List \u03b1\na\u271d : l' \u2208 List.permutations' l\n\u22a2 List.bind (permutations'Aux a l') (permutations'Aux b) ~ List.bind (permutations'Aux b l') (permutations'Aux a)\n[PROOFSTEP]\napply perm_permutations'Aux_comm\n[GOAL]\ncase trans\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d t\u271d s t u : List \u03b1\na\u271d\u00b9 : s ~ t\na\u271d : t ~ u\nIH\u2081 : List.permutations' s ~ List.permutations' t\nIH\u2082 : List.permutations' t ~ List.permutations' u\n\u22a2 List.permutations' s ~ List.permutations' u\n[PROOFSTEP]\nexact IH\u2081.trans IH\u2082\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts : List \u03b1\n\u22a2 permutations ts ~ permutations' ts\n[PROOFSTEP]\nobtain \u27e8n, h\u27e9 : \u2203 n, length ts < n := \u27e8_, Nat.lt_succ_self _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts : List \u03b1\nn : \u2115\nh : length ts < n\n\u22a2 permutations ts ~ permutations' ts\n[PROOFSTEP]\ninduction' n with n IH generalizing ts\n[GOAL]\ncase intro.zero\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d : List \u03b1\nn : \u2115\nh\u271d : length ts\u271d < n\nts : List \u03b1\nh : length ts < zero\n\u22a2 permutations ts ~ permutations' ts\n[PROOFSTEP]\ncases h\n[GOAL]\ncase intro.succ\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d : List \u03b1\nn\u271d : \u2115\nh\u271d : length ts\u271d < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts : List \u03b1\nh : length ts < succ n\n\u22a2 permutations ts ~ permutations' ts\n[PROOFSTEP]\nrefine' List.reverseRecOn ts (fun _ => _) (fun ts t _ h => _) h\n[GOAL]\ncase intro.succ.refine'_1\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d : List \u03b1\nn\u271d : \u2115\nh\u271d : length ts\u271d < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts : List \u03b1\nh : length ts < succ n\nx\u271d : length [] < succ n\n\u22a2 permutations [] ~ permutations' []\n[PROOFSTEP]\nsimp [permutations]\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length (ts ++ [t]) < succ n\n\u22a2 permutations (ts ++ [t]) ~ permutations' (ts ++ [t])\n[PROOFSTEP]\nrw [\u2190 concat_eq_append, length_concat, Nat.succ_lt_succ_iff] at h \n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\n\u22a2 permutations (ts ++ [t]) ~ permutations' (ts ++ [t])\n[PROOFSTEP]\nhave IH\u2082 := (IH ts.reverse (by rwa [length_reverse])).trans (reverse_perm _).permutations'\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\n\u22a2 length (reverse ts) < n\n[PROOFSTEP]\nrwa [length_reverse]\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 permutations (ts ++ [t]) ~ permutations' (ts ++ [t])\n[PROOFSTEP]\nsimp only [permutations_append, foldr_permutationsAux2, permutationsAux_nil, permutationsAux_cons, append_nil]\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 (map (fun x => x ++ [t]) (permutations ts) ++\n List.bind (permutations (reverse ts)) fun y => (permutationsAux2 t [] [] y id).snd) ~\n permutations' (ts ++ [t])\n[PROOFSTEP]\nrefine'\n (perm_append_comm.trans ((IH\u2082.bind_right _).append ((IH _ h).map _))).trans\n (Perm.trans _ perm_append_comm.permutations')\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 (List.bind (permutations' ts) fun y => (permutationsAux2 t [] [] y id).snd) ++\n map (fun x => x ++ [t]) (permutations' ts) ~\n permutations' ([t] ++ ts)\n[PROOFSTEP]\nrw [map_eq_bind, singleton_append, permutations']\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 ((List.bind (permutations' ts) fun y => (permutationsAux2 t [] [] y id).snd) ++\n List.bind (permutations' ts) fun x => [x ++ [t]]) ~\n List.bind (permutations' ts) (permutations'Aux t)\n[PROOFSTEP]\nrefine' (bind_append_perm _ _ _).trans _\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 (List.bind (permutations' ts) fun x => (permutationsAux2 t [] [] x id).snd ++ [x ++ [t]]) ~\n List.bind (permutations' ts) (permutations'Aux t)\n[PROOFSTEP]\nrefine' Perm.of_eq _\n[GOAL]\ncase intro.succ.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 (List.bind (permutations' ts) fun x => (permutationsAux2 t [] [] x id).snd ++ [x ++ [t]]) =\n List.bind (permutations' ts) (permutations'Aux t)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.succ.refine'_2.e_b\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\n\u22a2 (fun x => (permutationsAux2 t [] [] x id).snd ++ [x ++ [t]]) = permutations'Aux t\n[PROOFSTEP]\nfunext _\n[GOAL]\ncase intro.succ.refine'_2.e_b.h\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 ts\u271d\u00b9 : List \u03b1\nn\u271d : \u2115\nh\u271d\u00b9 : length ts\u271d\u00b9 < n\u271d\nn : \u2115\nIH : \u2200 (ts : List \u03b1), length ts < n \u2192 permutations ts ~ permutations' ts\nts\u271d : List \u03b1\nh\u271d : length ts\u271d < succ n\nts : List \u03b1\nt : \u03b1\nx\u271d\u00b9 : length ts < succ n \u2192 permutations ts ~ permutations' ts\nh : length ts < n\nIH\u2082 : permutations (reverse ts) ~ permutations' ts\nx\u271d : List \u03b1\n\u22a2 (permutationsAux2 t [] [] x\u271d id).snd ++ [x\u271d ++ [t]] = permutations'Aux t x\u271d\n[PROOFSTEP]\nrw [permutations'Aux_eq_permutationsAux2, permutationsAux2_append]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nn : \u2115\nhn : n < length (permutations'Aux x s)\n\u22a2 nthLe (permutations'Aux x s) n hn = insertNth n x s\n[PROOFSTEP]\ninduction' s with y s IH generalizing n\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nn\u271d : \u2115\nhn\u271d : n\u271d < length (permutations'Aux x s)\nn : \u2115\nhn : n < length (permutations'Aux x [])\n\u22a2 nthLe (permutations'Aux x []) n hn = insertNth n x []\n[PROOFSTEP]\nsimp only [length, zero_add, lt_one_iff] at hn \n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nn\u271d : \u2115\nhn\u271d\u00b9 : n\u271d < length (permutations'Aux x s)\nn : \u2115\nhn\u271d : n < length (permutations'Aux x [])\nhn : n = 0\n\u22a2 nthLe (permutations'Aux x []) n hn\u271d = insertNth n x []\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nn\u271d : \u2115\nhn\u271d : n\u271d < length (permutations'Aux x s\u271d)\ny : \u03b1\ns : List \u03b1\nIH : \u2200 (n : \u2115) (hn : n < length (permutations'Aux x s)), nthLe (permutations'Aux x s) n hn = insertNth n x s\nn : \u2115\nhn : n < length (permutations'Aux x (y :: s))\n\u22a2 nthLe (permutations'Aux x (y :: s)) n hn = insertNth n x (y :: s)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase cons.zero\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nn : \u2115\nhn\u271d : n < length (permutations'Aux x s\u271d)\ny : \u03b1\ns : List \u03b1\nIH : \u2200 (n : \u2115) (hn : n < length (permutations'Aux x s)), nthLe (permutations'Aux x s) n hn = insertNth n x s\nhn : zero < length (permutations'Aux x (y :: s))\n\u22a2 nthLe (permutations'Aux x (y :: s)) zero hn = insertNth zero x (y :: s)\n[PROOFSTEP]\nsimp [nthLe]\n[GOAL]\ncase cons.succ\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nn : \u2115\nhn\u271d : n < length (permutations'Aux x s\u271d)\ny : \u03b1\ns : List \u03b1\nIH : \u2200 (n : \u2115) (hn : n < length (permutations'Aux x s)), nthLe (permutations'Aux x s) n hn = insertNth n x s\nn\u271d : \u2115\nhn : succ n\u271d < length (permutations'Aux x (y :: s))\n\u22a2 nthLe (permutations'Aux x (y :: s)) (succ n\u271d) hn = insertNth (succ n\u271d) x (y :: s)\n[PROOFSTEP]\nsimpa [nthLe] using IH _ _\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\n\u22a2 count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\n[PROOFSTEP]\ninduction' l with y l IH generalizing x\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d x : \u03b1\n\u22a2 count [x] (permutations'Aux x []) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) []) + 1\n[PROOFSTEP]\nsimp [takeWhile, count]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d y : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\n\u22a2 count (x :: y :: l) (permutations'Aux x (y :: l)) =\n length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) (y :: l)) + 1\n[PROOFSTEP]\nrw [permutations'Aux, DecEq_eq, count_cons_self]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d y : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\n\u22a2 count (x :: y :: l) (map (cons y) (permutations'Aux x l)) + 1 =\n length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) (y :: l)) + 1\n[PROOFSTEP]\nby_cases hx : x = y\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d y : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\nhx : x = y\n\u22a2 count (x :: y :: l) (map (cons y) (permutations'Aux x l)) + 1 =\n length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) (y :: l)) + 1\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\n\u22a2 count (x :: x :: l) (map (cons x) (permutations'Aux x l)) + 1 =\n length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) (x :: l)) + 1\n[PROOFSTEP]\nsimpa [takeWhile, Nat.succ_inj', DecEq_eq] using IH _\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d y : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\nhx : \u00acx = y\n\u22a2 count (x :: y :: l) (map (cons y) (permutations'Aux x l)) + 1 =\n length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) (y :: l)) + 1\n[PROOFSTEP]\nrw [takeWhile]\n[GOAL]\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d y : \u03b1\nl : List \u03b1\nIH :\n \u2200 (x : \u03b1),\n count (x :: l) (permutations'Aux x l) = length (takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l) + 1\nx : \u03b1\nhx : \u00acx = y\n\u22a2 count (x :: y :: l) (map (cons y) (permutations'Aux x l)) + 1 =\n length\n (match decide ((fun x x_1 => x = x_1) x y) with\n | true => y :: takeWhile (fun b => decide ((fun x x_1 => x = x_1) x b)) l\n | false => []) +\n 1\n[PROOFSTEP]\nsimp only [mem_map, cons.injEq, Ne.symm hx, false_and, and_false, exists_false, not_false_iff, count_eq_zero_of_not_mem,\n zero_add, hx, decide_False, length_nil]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\n\u22a2 length (permutations'Aux x s) = length s + 1\n[PROOFSTEP]\ninduction' s with y s IH\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\n\u22a2 length (permutations'Aux x []) = length [] + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx y : \u03b1\ns : List \u03b1\nIH : length (permutations'Aux x s) = length s + 1\n\u22a2 length (permutations'Aux x (y :: s)) = length (y :: s) + 1\n[PROOFSTEP]\nsimpa using IH\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\n\u22a2 0 < length (permutations'Aux x s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\n\u22a2 Function.Injective (permutations'Aux x)\n[PROOFSTEP]\nintro s t h\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\n\u22a2 s = t\n[PROOFSTEP]\napply insertNth_injective s.length x\n[GOAL]\ncase a\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\n\u22a2 insertNth (length s) x s = insertNth (length s) x t\n[PROOFSTEP]\nhave hl : s.length = t.length := by simpa using congr_arg length h\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\n\u22a2 length s = length t\n[PROOFSTEP]\nsimpa using congr_arg length h\n[GOAL]\ncase a\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\nhl : length s = length t\n\u22a2 insertNth (length s) x s = insertNth (length s) x t\n[PROOFSTEP]\nrw [\u2190 nthLe_permutations'Aux s x s.length (by simp), \u2190 nthLe_permutations'Aux t x s.length (by simp [hl])]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\nhl : length s = length t\n\u22a2 length s < length (permutations'Aux x s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\nhl : length s = length t\n\u22a2 length s < length (permutations'Aux x t)\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\ncase a\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\nx : \u03b1\ns t : List \u03b1\nh : permutations'Aux x s = permutations'Aux x t\nhl : length s = length t\n\u22a2 nthLe (permutations'Aux x s) (length s) (_ : length s < length (permutations'Aux x s)) =\n nthLe (permutations'Aux x t) (length s) (_ : length s < length (permutations'Aux x t))\n[PROOFSTEP]\nsimp [h, hl]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 Nodup (permutations'Aux x s)\n[PROOFSTEP]\ninduction' s with y s IH\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\nhx : \u00acx \u2208 []\n\u22a2 Nodup (permutations'Aux x [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx \u2208 y :: s\n\u22a2 Nodup (permutations'Aux x (y :: s))\n[PROOFSTEP]\nsimp only [not_or, mem_cons] at hx \n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx = y \u2227 \u00acx \u2208 s\n\u22a2 Nodup (permutations'Aux x (y :: s))\n[PROOFSTEP]\nsimp only [permutations'Aux, nodup_cons, mem_map, cons.injEq, exists_eq_right_right, not_and]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx = y \u2227 \u00acx \u2208 s\n\u22a2 (y :: s \u2208 permutations'Aux x s \u2192 \u00acy = x) \u2227 Nodup (map (cons y) (permutations'Aux x s))\n[PROOFSTEP]\nrefine' \u27e8fun _ => Ne.symm hx.left, _\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx = y \u2227 \u00acx \u2208 s\n\u22a2 Nodup (map (cons y) (permutations'Aux x s))\n[PROOFSTEP]\nrw [nodup_map_iff]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx = y \u2227 \u00acx \u2208 s\n\u22a2 Nodup (permutations'Aux x s)\n[PROOFSTEP]\nexact IH hx.right\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s\u271d : List \u03b1\nx : \u03b1\nhx\u271d : \u00acx \u2208 s\u271d\ny : \u03b1\ns : List \u03b1\nIH : \u00acx \u2208 s \u2192 Nodup (permutations'Aux x s)\nhx : \u00acx = y \u2227 \u00acx \u2208 s\n\u22a2 Function.Injective (cons y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\n\u22a2 Nodup (permutations'Aux x s) \u2194 \u00acx \u2208 s\n[PROOFSTEP]\nrefine' \u27e8fun h => _, nodup_permutations'Aux_of_not_mem _ _\u27e9\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh : Nodup (permutations'Aux x s)\n\u22a2 \u00acx \u2208 s\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh : Nodup (permutations'Aux x s)\nH : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8k, hk, hk'\u27e9 := nthLe_of_mem H\n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh : Nodup (permutations'Aux x s)\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 False\n[PROOFSTEP]\nrw [nodup_iff_nthLe_inj] at h \n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 False\n[PROOFSTEP]\nsuffices k = k + 1 by simp at this \n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nthis : k = k + 1\n\u22a2 False\n[PROOFSTEP]\nsimp at this \n[GOAL]\ncase intro.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 k = k + 1\n[PROOFSTEP]\nrefine' h k (k + 1) _ _ _\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 k < length (permutations'Aux x s)\n[PROOFSTEP]\nsimpa [Nat.lt_succ_iff] using hk.le\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 k + 1 < length (permutations'Aux x s)\n[PROOFSTEP]\nsimpa using hk\n[GOAL]\ncase intro.intro.refine'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 nthLe (permutations'Aux x s) k (_ : k < length (permutations'Aux x s)) =\n nthLe (permutations'Aux x s) (k + 1) (_ : k + 1 < length (permutations'Aux x s))\n[PROOFSTEP]\nrw [nthLe_permutations'Aux, nthLe_permutations'Aux]\n[GOAL]\ncase intro.intro.refine'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 insertNth k x s = insertNth (k + 1) x s\n[PROOFSTEP]\nhave hl : length (insertNth k x s) = length (insertNth (k + 1) x s) := by\n rw [length_insertNth _ _ hk.le, length_insertNth _ _ (Nat.succ_le_of_lt hk)]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\n\u22a2 length (insertNth k x s) = length (insertNth (k + 1) x s)\n[PROOFSTEP]\nrw [length_insertNth _ _ hk.le, length_insertNth _ _ (Nat.succ_le_of_lt hk)]\n[GOAL]\ncase intro.intro.refine'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\n\u22a2 insertNth k x s = insertNth (k + 1) x s\n[PROOFSTEP]\nrefine' ext_nthLe hl fun n hn hn' => _\n[GOAL]\ncase intro.intro.refine'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nn : \u2115\nhn : n < length (insertNth k x s)\nhn' : n < length (insertNth (k + 1) x s)\n\u22a2 nthLe (insertNth k x s) n hn = nthLe (insertNth (k + 1) x s) n hn'\n[PROOFSTEP]\nrcases lt_trichotomy n k with (H | rfl | H)\n[GOAL]\ncase intro.intro.refine'_3.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nn : \u2115\nhn : n < length (insertNth k x s)\nhn' : n < length (insertNth (k + 1) x s)\nH : n < k\n\u22a2 nthLe (insertNth k x s) n hn = nthLe (insertNth (k + 1) x s) n hn'\n[PROOFSTEP]\nrw [nthLe_insertNth_of_lt _ _ _ _ H (H.trans hk), nthLe_insertNth_of_lt _ _ _ _ (H.trans (Nat.lt_succ_self _))]\n[GOAL]\ncase intro.intro.refine'_3.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH : x \u2208 s\nn : \u2115\nhk : n < length s\nhk' : nthLe s n hk = x\nhl : length (insertNth n x s) = length (insertNth (n + 1) x s)\nhn : n < length (insertNth n x s)\nhn' : n < length (insertNth (n + 1) x s)\n\u22a2 nthLe (insertNth n x s) n hn = nthLe (insertNth (n + 1) x s) n hn'\n[PROOFSTEP]\nrw [nthLe_insertNth_self _ _ _ hk.le, nthLe_insertNth_of_lt _ _ _ _ (Nat.lt_succ_self _) hk, hk']\n[GOAL]\ncase intro.intro.refine'_3.inr.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nn : \u2115\nhn : n < length (insertNth k x s)\nhn' : n < length (insertNth (k + 1) x s)\nH : k < n\n\u22a2 nthLe (insertNth k x s) n hn = nthLe (insertNth (k + 1) x s) n hn'\n[PROOFSTEP]\nrcases(Nat.succ_le_of_lt H).eq_or_lt with (rfl | H')\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nhn : succ k < length (insertNth k x s)\nhn' : succ k < length (insertNth (k + 1) x s)\nH : k < succ k\n\u22a2 nthLe (insertNth k x s) (succ k) hn = nthLe (insertNth (k + 1) x s) (succ k) hn'\n[PROOFSTEP]\nrw [nthLe_insertNth_self _ _ _ (Nat.succ_le_of_lt hk)]\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nhn : succ k < length (insertNth k x s)\nhn' : succ k < length (insertNth (k + 1) x s)\nH : k < succ k\n\u22a2 nthLe (insertNth k x s) (succ k) hn = x\n[PROOFSTEP]\nconvert hk' using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nhn : succ k < length (insertNth k x s)\nhn' : succ k < length (insertNth (k + 1) x s)\nH : k < succ k\n\u22a2 nthLe (insertNth k x s) (succ k) hn = nthLe s k hk\n[PROOFSTEP]\nexact nthLe_insertNth_add_succ _ _ _ 0 _\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nn : \u2115\nhn : n < length (insertNth k x s)\nhn' : n < length (insertNth (k + 1) x s)\nH : k < n\nH' : succ k < n\n\u22a2 nthLe (insertNth k x s) n hn = nthLe (insertNth (k + 1) x s) n hn'\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := Nat.exists_eq_add_of_lt H'\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inr.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn : succ k + m + 1 < length (insertNth k x s)\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 nthLe (insertNth k x s) (succ k + m + 1) hn = nthLe (insertNth (k + 1) x s) (succ k + m + 1) hn'\n[PROOFSTEP]\nerw [length_insertNth _ _ hk.le, Nat.succ_lt_succ_iff, Nat.succ_add] at hn \n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inr.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 nthLe (insertNth k x s) (succ k + m + 1) hn\u271d = nthLe (insertNth (k + 1) x s) (succ k + m + 1) hn'\n[PROOFSTEP]\nrw [nthLe_insertNth_add_succ]\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inr.intro\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 nthLe (insertNth k x s) (succ k + m + 1) hn\u271d = nthLe s (k + 1 + m) ?intro.intro.refine'_3.inr.inr.inr.intro.hk'\ncase intro.intro.refine'_3.inr.inr.inr.intro.hk'\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 k + 1 + m < length s\n[PROOFSTEP]\nconvert nthLe_insertNth_add_succ s x k m.succ (by simpa using hn) using 2\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 k + succ m < length s\n[PROOFSTEP]\nsimpa using hn\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 succ k + m + 1 = k + succ m + 1\n[PROOFSTEP]\nsimp [Nat.add_succ, Nat.succ_add]\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 k + 1 + m = k + succ m\n[PROOFSTEP]\nsimp [add_left_comm, add_comm]\n[GOAL]\ncase intro.intro.refine'_3.inr.inr.inr.intro.hk'\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nh :\n \u2200 (i j : \u2115) (h\u2081 : i < length (permutations'Aux x s)) (h\u2082 : j < length (permutations'Aux x s)),\n nthLe (permutations'Aux x s) i h\u2081 = nthLe (permutations'Aux x s) j h\u2082 \u2192 i = j\nH\u271d : x \u2208 s\nk : \u2115\nhk : k < length s\nhk' : nthLe s k hk = x\nhl : length (insertNth k x s) = length (insertNth (k + 1) x s)\nm : \u2115\nhn\u271d : succ k + m + 1 < length (insertNth k x s)\nhn : succ (k + m) < length s\nhn' : succ k + m + 1 < length (insertNth (k + 1) x s)\nH : k < succ k + m + 1\nH' : succ k < succ k + m + 1\n\u22a2 k + 1 + m < length s\n[PROOFSTEP]\nsimpa [Nat.succ_add] using hn\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nhs : Nodup s\n\u22a2 Nodup (permutations s)\n[PROOFSTEP]\nrw [(permutations_perm_permutations' s).nodup_iff]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nhs : Nodup s\n\u22a2 Nodup (permutations' s)\n[PROOFSTEP]\ninduction' hs with x l h h' IH\n[GOAL]\ncase nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\n\u22a2 Nodup (permutations' [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\n\u22a2 Nodup (permutations' (x :: l))\n[PROOFSTEP]\nrw [permutations']\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\n\u22a2 Nodup (List.bind (permutations' l) (permutations'Aux x))\n[PROOFSTEP]\nrw [nodup_bind]\n[GOAL]\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\n\u22a2 (\u2200 (x_1 : List \u03b1), x_1 \u2208 permutations' l \u2192 Nodup (permutations'Aux x x_1)) \u2227\n Pairwise (fun a b => Disjoint (permutations'Aux x a) (permutations'Aux x b)) (permutations' l)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.left\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\n\u22a2 \u2200 (x_1 : List \u03b1), x_1 \u2208 permutations' l \u2192 Nodup (permutations'Aux x x_1)\n[PROOFSTEP]\nintro ys hy\n[GOAL]\ncase cons.left\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nys : List \u03b1\nhy : ys \u2208 permutations' l\n\u22a2 Nodup (permutations'Aux x ys)\n[PROOFSTEP]\nrw [mem_permutations'] at hy \n[GOAL]\ncase cons.left\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nys : List \u03b1\nhy : ys ~ l\n\u22a2 Nodup (permutations'Aux x ys)\n[PROOFSTEP]\nrw [nodup_permutations'Aux_iff, hy.mem_iff]\n[GOAL]\ncase cons.left\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nys : List \u03b1\nhy : ys ~ l\n\u22a2 \u00acx \u2208 l\n[PROOFSTEP]\nexact fun H => h x H rfl\n[GOAL]\ncase cons.right\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\n\u22a2 Pairwise (fun a b => Disjoint (permutations'Aux x a) (permutations'Aux x b)) (permutations' l)\n[PROOFSTEP]\nrefine' IH.pairwise_of_forall_ne fun as ha bs hb H => _\n[GOAL]\ncase cons.right\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as \u2208 permutations' l\nbs : List \u03b1\nhb : bs \u2208 permutations' l\nH : as \u2260 bs\n\u22a2 Disjoint (permutations'Aux x as) (permutations'Aux x bs)\n[PROOFSTEP]\nrw [disjoint_iff_ne]\n[GOAL]\ncase cons.right\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as \u2208 permutations' l\nbs : List \u03b1\nhb : bs \u2208 permutations' l\nH : as \u2260 bs\n\u22a2 \u2200 (a : List \u03b1), a \u2208 permutations'Aux x as \u2192 \u2200 (b : List \u03b1), b \u2208 permutations'Aux x bs \u2192 a \u2260 b\n[PROOFSTEP]\nrintro a ha' b hb' rfl\n[GOAL]\ncase cons.right\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as \u2208 permutations' l\nbs : List \u03b1\nhb : bs \u2208 permutations' l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8n, hn\u27e9, hn'\u27e9 := get_of_mem ha'\n[GOAL]\ncase cons.right.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as \u2208 permutations' l\nbs : List \u03b1\nhb : bs \u2208 permutations' l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn : n < length (permutations'Aux x as)\nhn' : get (permutations'Aux x as) { val := n, isLt := hn } = a\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8m, hm\u27e9, hm'\u27e9 := get_of_mem hb'\n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as \u2208 permutations' l\nbs : List \u03b1\nhb : bs \u2208 permutations' l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn : n < length (permutations'Aux x as)\nhn' : get (permutations'Aux x as) { val := n, isLt := hn } = a\nm : \u2115\nhm : m < length (permutations'Aux x bs)\nhm' : get (permutations'Aux x bs) { val := m, isLt := hm } = a\n\u22a2 False\n[PROOFSTEP]\nrw [mem_permutations'] at ha hb \n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn : n < length (permutations'Aux x as)\nhn' : get (permutations'Aux x as) { val := n, isLt := hn } = a\nm : \u2115\nhm : m < length (permutations'Aux x bs)\nhm' : get (permutations'Aux x bs) { val := m, isLt := hm } = a\n\u22a2 False\n[PROOFSTEP]\nhave hl : as.length = bs.length := (ha.trans hb.symm).length_eq\n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn : n < length (permutations'Aux x as)\nhn' : get (permutations'Aux x as) { val := n, isLt := hn } = a\nm : \u2115\nhm : m < length (permutations'Aux x bs)\nhm' : get (permutations'Aux x bs) { val := m, isLt := hm } = a\nhl : length as = length bs\n\u22a2 False\n[PROOFSTEP]\nsimp only [Nat.lt_succ_iff, length_permutations'Aux] at hn hm \n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : get (permutations'Aux x as) { val := n, isLt := hn\u271d } = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : get (permutations'Aux x bs) { val := m, isLt := hm\u271d } = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 nthLe, nthLe_permutations'Aux] at hn' hm' \n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\n\u22a2 False\n[PROOFSTEP]\nhave hx : nthLe (insertNth n x as) m (by rwa [length_insertNth _ _ hn, Nat.lt_succ_iff, hl]) = x := by\n simp [hn', \u2190 hm', hm]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\n\u22a2 m < length (insertNth n x as)\n[PROOFSTEP]\nrwa [length_insertNth _ _ hn, Nat.lt_succ_iff, hl]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\n\u22a2 nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\n[PROOFSTEP]\nsimp [hn', \u2190 hm', hm]\n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\n\u22a2 False\n[PROOFSTEP]\nhave hx' : nthLe (insertNth m x bs) n (by rwa [length_insertNth _ _ hm, Nat.lt_succ_iff, \u2190 hl]) = x := by\n simp [hm', \u2190 hn', hn]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\n\u22a2 n < length (insertNth m x bs)\n[PROOFSTEP]\nrwa [length_insertNth _ _ hm, Nat.lt_succ_iff, \u2190 hl]\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\n\u22a2 nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\n[PROOFSTEP]\nsimp [hm', \u2190 hn', hn]\n[GOAL]\ncase cons.right.intro.mk.intro.mk\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\n\u22a2 False\n[PROOFSTEP]\nrcases lt_trichotomy n m with (ht | ht | ht)\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n < m\n\u22a2 False\n[PROOFSTEP]\nsuffices x \u2208 bs by exact h x (hb.subset this) rfl\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n < m\nthis : x \u2208 bs\n\u22a2 False\n[PROOFSTEP]\nexact h x (hb.subset this) rfl\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n < m\n\u22a2 x \u2208 bs\n[PROOFSTEP]\nrw [\u2190 hx', nthLe_insertNth_of_lt _ _ _ _ ht (ht.trans_le hm)]\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n < m\n\u22a2 nthLe bs n (_ : n < length bs) \u2208 bs\n[PROOFSTEP]\nexact nthLe_mem _ _ _\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n = m\n\u22a2 False\n[PROOFSTEP]\nsimp only [ht] at hm' hn' \n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n = m\nhn' : insertNth m x as = a\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hm'] at hn' \n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inl\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : n = m\nhn' : insertNth m x as = insertNth m x bs\n\u22a2 False\n[PROOFSTEP]\nexact H (insertNth_injective _ _ hn')\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : m < n\n\u22a2 False\n[PROOFSTEP]\nsuffices x \u2208 as by exact h x (ha.subset this) rfl\n[GOAL]\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : m < n\nthis : x \u2208 as\n\u22a2 False\n[PROOFSTEP]\nexact h x (ha.subset this) rfl\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : m < n\n\u22a2 x \u2208 as\n[PROOFSTEP]\nrw [\u2190 hx, nthLe_insertNth_of_lt _ _ _ _ ht (ht.trans_le hn)]\n[GOAL]\ncase cons.right.intro.mk.intro.mk.inr.inr\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 s : List \u03b1\nx : \u03b1\nl : List \u03b1\nh : \u2200 (a' : \u03b1), a' \u2208 l \u2192 x \u2260 a'\nh' : Pairwise (fun x x_1 => x \u2260 x_1) l\nIH : Nodup (permutations' l)\nas : List \u03b1\nha : as ~ l\nbs : List \u03b1\nhb : bs ~ l\nH : as \u2260 bs\na : List \u03b1\nha' : a \u2208 permutations'Aux x as\nhb' : a \u2208 permutations'Aux x bs\nn : \u2115\nhn\u271d : n < length (permutations'Aux x as)\nhn' : insertNth n x as = a\nm : \u2115\nhm\u271d : m < length (permutations'Aux x bs)\nhm' : insertNth m x bs = a\nhl : length as = length bs\nhn : n \u2264 length as\nhm : m \u2264 length bs\nhx : nthLe (insertNth n x as) m (_ : m < length (insertNth n x as)) = x\nhx' : nthLe (insertNth m x bs) n (_ : n < length (insertNth m x bs)) = x\nht : m < n\n\u22a2 nthLe as m (_ : m < length as) \u2208 as\n[PROOFSTEP]\nexact nthLe_mem _ _ _\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Perm", "llama_tokens": 136597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.2920626789105918}} {"text": "[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q f : R[X]\n\u22a2 { toFinsupp := f.toFinsupp } = f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := { toFinsupp := toFinsupp\u271d }.toFinsupp } = { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na b : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a + b } = Polynomial.add { toFinsupp := a } { toFinsupp := b }\n[PROOFSTEP]\nrw [add_def]\n[GOAL]\nR\u271d : Type u\na\u271d b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := -a } = Polynomial.neg { toFinsupp := a }\n[PROOFSTEP]\nrw [neg_def]\n[GOAL]\nR\u271d : Type u\na\u271d b\u271d : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na b : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }\n[PROOFSTEP]\nrw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg]\n[GOAL]\nR\u271d : Type u\na\u271d b\u271d : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na b : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na b : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a * b } = Polynomial.mul { toFinsupp := a } { toFinsupp := b }\n[PROOFSTEP]\nrw [mul_def]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\nn : \u2115\n\u22a2 { toFinsupp := a ^ n } = { toFinsupp := a } ^ n\n[PROOFSTEP]\nchange _ = npowRec n _\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\nn : \u2115\n\u22a2 { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }\n[PROOFSTEP]\ninduction n with\n| zero => simp [npowRec]\n| succ n n_ih => simp [npowRec, n_ih, pow_succ]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\nn : \u2115\n\u22a2 { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }\n[PROOFSTEP]\ninduction n with\n| zero => simp [npowRec]\n| succ n n_ih => simp [npowRec, n_ih, pow_succ]\n[GOAL]\ncase zero\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a ^ Nat.zero } = npowRec Nat.zero { toFinsupp := a }\n[PROOFSTEP]\n\n| zero => simp [npowRec]\n[GOAL]\ncase zero\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a ^ Nat.zero } = npowRec Nat.zero { toFinsupp := a }\n[PROOFSTEP]\nsimp [npowRec]\n[GOAL]\ncase succ\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\nn : \u2115\nn_ih : { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }\n\u22a2 { toFinsupp := a ^ Nat.succ n } = npowRec (Nat.succ n) { toFinsupp := a }\n[PROOFSTEP]\n\n| succ n n_ih => simp [npowRec, n_ih, pow_succ]\n[GOAL]\ncase succ\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\nn : \u2115\nn_ih : { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }\n\u22a2 { toFinsupp := a ^ Nat.succ n } = npowRec (Nat.succ n) { toFinsupp := a }\n[PROOFSTEP]\nsimp [npowRec, n_ih, pow_succ]\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q a b : R[X]\n\u22a2 (a + b).toFinsupp = a.toFinsupp + b.toFinsupp\n[PROOFSTEP]\ncases a\n[GOAL]\ncase ofFinsupp\nR : Type u\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q b : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d } + b).toFinsupp = { toFinsupp := toFinsupp\u271d }.toFinsupp + b.toFinsupp\n[PROOFSTEP]\ncases b\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d\u00b9 } + { toFinsupp := toFinsupp\u271d }).toFinsupp =\n { toFinsupp := toFinsupp\u271d\u00b9 }.toFinsupp + { toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_add]\n[GOAL]\nR\u271d : Type u\na\u271d b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na : R[X]\n\u22a2 (-a).toFinsupp = -a.toFinsupp\n[PROOFSTEP]\ncases a\n[GOAL]\ncase ofFinsupp\nR\u271d : Type u\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (-{ toFinsupp := toFinsupp\u271d }).toFinsupp = -{ toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_neg]\n[GOAL]\nR\u271d : Type u\na\u271d b\u271d : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na b : R[X]\n\u22a2 (a - b).toFinsupp = a.toFinsupp - b.toFinsupp\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 toFinsupp_neg, \u2190 toFinsupp_add]\n[GOAL]\nR\u271d : Type u\na\u271d b\u271d : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q : R\u271d[X]\nR : Type u\ninst\u271d : Ring R\na b : R[X]\n\u22a2 (a - b).toFinsupp = (a + -b).toFinsupp\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q a b : R[X]\n\u22a2 (a * b).toFinsupp = a.toFinsupp * b.toFinsupp\n[PROOFSTEP]\ncases a\n[GOAL]\ncase ofFinsupp\nR : Type u\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q b : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d } * b).toFinsupp = { toFinsupp := toFinsupp\u271d }.toFinsupp * b.toFinsupp\n[PROOFSTEP]\ncases b\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d\u00b9 } * { toFinsupp := toFinsupp\u271d }).toFinsupp =\n { toFinsupp := toFinsupp\u271d\u00b9 }.toFinsupp * { toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_mul]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q a : R[X]\nn : \u2115\n\u22a2 (a ^ n).toFinsupp = a.toFinsupp ^ n\n[PROOFSTEP]\ncases a\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d } ^ n).toFinsupp = { toFinsupp := toFinsupp\u271d }.toFinsupp ^ n\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_pow]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q a : R[X]\n\u22a2 a.toFinsupp = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [\u2190 toFinsupp_zero, toFinsupp_inj]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q a : R[X]\n\u22a2 a.toFinsupp = 1 \u2194 a = 1\n[PROOFSTEP]\nrw [\u2190 toFinsupp_one, toFinsupp_inj]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a } = 0 \u2194 a = 0\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_zero, ofFinsupp_inj]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := a } = 1 \u2194 a = 1\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_one, ofFinsupp_inj]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\nS\u2081 : Type ?u.114071\nS\u2082 : Type ?u.114238\ninst\u271d\u00b2 : SMulZeroClass S\u2081 R\ninst\u271d\u00b9 : SMulZeroClass S\u2082 R\ninst\u271d : SMulCommClass S\u2081 S\u2082 R\n\u22a2 \u2200 (m : S\u2081) (n : S\u2082) (a : R[X]), m \u2022 n \u2022 a = n \u2022 m \u2022 a\n[PROOFSTEP]\nrintro m n \u27e8f\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\nS\u2081 : Type ?u.114071\nS\u2082 : Type ?u.114238\ninst\u271d\u00b2 : SMulZeroClass S\u2081 R\ninst\u271d\u00b9 : SMulZeroClass S\u2082 R\ninst\u271d : SMulCommClass S\u2081 S\u2082 R\nm : S\u2081\nn : S\u2082\nf : AddMonoidAlgebra R \u2115\n\u22a2 m \u2022 n \u2022 { toFinsupp := f } = n \u2022 m \u2022 { toFinsupp := f }\n[PROOFSTEP]\nsimp_rw [\u2190 ofFinsupp_smul, smul_comm m n f]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u2074 : Semiring R\np q : R[X]\nS\u2081 : Type ?u.117123\nS\u2082 : Type ?u.117122\ninst\u271d\u00b3 : SMul S\u2081 S\u2082\ninst\u271d\u00b2 : SMulZeroClass S\u2081 R\ninst\u271d\u00b9 : SMulZeroClass S\u2082 R\ninst\u271d : IsScalarTower S\u2081 S\u2082 R\n\u22a2 \u2200 (x : S\u2081) (y : S\u2082) (z : R[X]), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n[PROOFSTEP]\nrintro _ _ \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u2074 : Semiring R\np q : R[X]\nS\u2081 : Type ?u.117123\nS\u2082 : Type ?u.117122\ninst\u271d\u00b3 : SMul S\u2081 S\u2082\ninst\u271d\u00b2 : SMulZeroClass S\u2081 R\ninst\u271d\u00b9 : SMulZeroClass S\u2082 R\ninst\u271d : IsScalarTower S\u2081 S\u2082 R\nx\u271d : S\u2081\ny\u271d : S\u2082\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (x\u271d \u2022 y\u271d) \u2022 { toFinsupp := toFinsupp\u271d } = x\u271d \u2022 y\u271d \u2022 { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nsimp_rw [\u2190 ofFinsupp_smul, smul_assoc]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\n\u03b1 : Type u_1\nK : Type u_2\ninst\u271d\u00b2 : Semiring K\ninst\u271d\u00b9 : DistribSMul \u03b1 K\ninst\u271d : IsScalarTower \u03b1 K K\n\u22a2 \u2200 (x : \u03b1) (y z : K[X]), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n[PROOFSTEP]\nrintro _ \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\n\u03b1 : Type u_1\nK : Type u_2\ninst\u271d\u00b2 : Semiring K\ninst\u271d\u00b9 : DistribSMul \u03b1 K\ninst\u271d : IsScalarTower \u03b1 K K\nx\u271d : \u03b1\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra K \u2115\n\u22a2 (x\u271d \u2022 { toFinsupp := toFinsupp\u271d\u00b9 }) \u2022 { toFinsupp := toFinsupp\u271d } =\n x\u271d \u2022 { toFinsupp := toFinsupp\u271d\u00b9 } \u2022 { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nsimp_rw [smul_eq_mul, \u2190 ofFinsupp_smul, \u2190 ofFinsupp_mul, \u2190 ofFinsupp_smul, smul_mul_assoc]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\nS : Type ?u.129006\ninst\u271d\u00b2 : SMulZeroClass S R\ninst\u271d\u00b9 : SMulZeroClass S\u1d50\u1d52\u1d56 R\ninst\u271d : IsCentralScalar S R\n\u22a2 \u2200 (m : S) (a : R[X]), MulOpposite.op m \u2022 a = m \u2022 a\n[PROOFSTEP]\nrintro _ \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np q : R[X]\nS : Type ?u.129006\ninst\u271d\u00b2 : SMulZeroClass S R\ninst\u271d\u00b9 : SMulZeroClass S\u1d50\u1d52\u1d56 R\ninst\u271d : IsCentralScalar S R\nm\u271d : S\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 MulOpposite.op m\u271d \u2022 { toFinsupp := toFinsupp\u271d } = m\u271d \u2022 { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nsimp_rw [\u2190 ofFinsupp_smul, op_smul_eq_smul]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\ninst\u271d : Subsingleton R\nsrc\u271d : Inhabited R[X] := inhabited\n\u22a2 \u2200 (a : R[X]), a = default\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\ninst\u271d : Subsingleton R\nsrc\u271d : Inhabited R[X] := inhabited\nx : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := x } = default\n[PROOFSTEP]\nrefine' congr_arg ofFinsupp _\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\ninst\u271d : Subsingleton R\nsrc\u271d : Inhabited R[X] := inhabited\nx : AddMonoidAlgebra R \u2115\n\u22a2 x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\np : AddMonoidAlgebra R \u2115\n\u22a2 support { toFinsupp := p } = p.support\n[PROOFSTEP]\nrw [support]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\n\u22a2 p.toFinsupp.support = support p\n[PROOFSTEP]\nrw [support]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 support p = \u2205 \u2194 p = 0\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support { toFinsupp := toFinsupp\u271d } = \u2205 \u2194 { toFinsupp := toFinsupp\u271d } = 0\n[PROOFSTEP]\nsimp [support]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 Finset.card (support p) = 0 \u2194 p = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nx y : R\n\u22a2 (fun t => { toFinsupp := Finsupp.single n t }) (x + y) =\n (fun t => { toFinsupp := Finsupp.single n t }) x + (fun t => { toFinsupp := Finsupp.single n t }) y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nx y : R\n\u22a2 { toFinsupp := Finsupp.single n x + Finsupp.single n y } =\n { toFinsupp := Finsupp.single n x } + { toFinsupp := Finsupp.single n y }\n[PROOFSTEP]\nrw [ofFinsupp_add]\n -- Porting note: Was `simp [\u2190 ofFinsupp_smul]`.\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr x : R\n\u22a2 AddHom.toFun\n { toFun := fun t => { toFinsupp := Finsupp.single n t },\n map_add' :=\n (_ :\n \u2200 (x y : R),\n (fun t => { toFinsupp := Finsupp.single n t }) (x + y) =\n (fun t => { toFinsupp := Finsupp.single n t }) x + (fun t => { toFinsupp := Finsupp.single n t }) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun t => { toFinsupp := Finsupp.single n t },\n map_add' :=\n (_ :\n \u2200 (x y : R),\n (fun t => { toFinsupp := Finsupp.single n t }) (x + y) =\n (fun t => { toFinsupp := Finsupp.single n t }) x + (fun t => { toFinsupp := Finsupp.single n t }) y) }\n x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr x : R\n\u22a2 { toFinsupp := Finsupp.single n (r * x) } = r \u2022 { toFinsupp := Finsupp.single n x }\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_smul, smul_single']\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 (\u2191(monomial n) r).toFinsupp = Finsupp.single n r\n[PROOFSTEP]\nsimp [monomial]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 { toFinsupp := Finsupp.single n r } = \u2191(monomial n) r\n[PROOFSTEP]\nsimp [monomial]\n[GOAL]\nR : Type u\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn m : \u2115\nr s : R\n\u22a2 (\u2191(monomial n) r * \u2191(monomial m) s).toFinsupp = (\u2191(monomial (n + m)) (r * s)).toFinsupp\n[PROOFSTEP]\nsimp only [toFinsupp_monomial, toFinsupp_mul, AddMonoidAlgebra.single_mul_single]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\nk : \u2115\n\u22a2 \u2191(monomial n) r ^ k = \u2191(monomial (n * k)) (r ^ k)\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(monomial n) r ^ Nat.zero = \u2191(monomial (n * Nat.zero)) (r ^ Nat.zero)\n[PROOFSTEP]\nsimp [pow_zero, monomial_zero_one]\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\nk : \u2115\nih : \u2191(monomial n) r ^ k = \u2191(monomial (n * k)) (r ^ k)\n\u22a2 \u2191(monomial n) r ^ Nat.succ k = \u2191(monomial (n * Nat.succ k)) (r ^ Nat.succ k)\n[PROOFSTEP]\nsimp [pow_succ, ih, monomial_mul_monomial, Nat.succ_eq_add_one, mul_add, add_comm]\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nS : Type u_1\ninst\u271d : SMulZeroClass S R\na : S\nn : \u2115\nb : R\n\u22a2 (a \u2022 \u2191(monomial n) b).toFinsupp = (\u2191(monomial n) (a \u2022 b)).toFinsupp\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na\u271d b\u271d : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nS : Type u_1\ninst\u271d : SMulZeroClass S R\na : S\nn : \u2115\nb : R\n\u22a2 a \u2022 Finsupp.single n b = Finsupp.single n (a \u2022 b)\n[PROOFSTEP]\nrw [smul_single]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 support (p + q) \u2286 support p \u222a support q\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support ({ toFinsupp := toFinsupp\u271d } + q) \u2286 support { toFinsupp := toFinsupp\u271d } \u222a support q\n[PROOFSTEP]\nrcases q with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support ({ toFinsupp := toFinsupp\u271d\u00b9 } + { toFinsupp := toFinsupp\u271d }) \u2286\n support { toFinsupp := toFinsupp\u271d\u00b9 } \u222a support { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nsimp only [\u2190 ofFinsupp_add, support]\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (toFinsupp\u271d\u00b9 + toFinsupp\u271d).support \u2286 toFinsupp\u271d\u00b9.support \u222a toFinsupp\u271d.support\n[PROOFSTEP]\nexact support_add\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nsrc\u271d : R \u2192\u2097[R] R[X] := monomial 0\n\u22a2 AddHom.toFun src\u271d.toAddHom 1 = 1\n[PROOFSTEP]\nsimp [monomial_zero_one]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nsrc\u271d : R \u2192\u2097[R] R[X] := monomial 0\n\u22a2 \u2200 (x y : R),\n OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : 1 = 1) } (x * y) =\n OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : 1 = 1) } x *\n OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : 1 = 1) } y\n[PROOFSTEP]\nsimp [monomial_mul_monomial]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nsrc\u271d : R \u2192\u2097[R] R[X] := monomial 0\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : 1 = 1) },\n map_mul' := (_ : \u2200 (a a_1 : R), \u2191(monomial 0) (a * a_1) = \u2191(monomial 0) a * \u2191(monomial 0) a_1) })\n 0 =\n 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191C 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191C (bit1 a) = bit1 (\u2191C a)\n[PROOFSTEP]\nsimp [bit1, C_bit0]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191C a * \u2191(monomial n) b = \u2191(monomial n) (a * b)\n[PROOFSTEP]\nsimp only [\u2190 monomial_zero_left, monomial_mul_monomial, zero_add]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191(monomial n) a * \u2191C b = \u2191(monomial n) (a * b)\n[PROOFSTEP]\nsimp only [\u2190 monomial_zero_left, monomial_mul_monomial, add_zero]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 \u2191(monomial n) 1 = X ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191(monomial Nat.zero) 1 = X ^ Nat.zero\n[PROOFSTEP]\nsimp [monomial_zero_one]\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : \u2191(monomial n) 1 = X ^ n\n\u22a2 \u2191(monomial (Nat.succ n)) 1 = X ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ, \u2190 ih, \u2190 monomial_one_one_eq_X, monomial_mul_monomial, add_comm, one_mul]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 X * p = p * X\n[PROOFSTEP]\nrcases p with\n \u27e8\u27e9\n -- Porting note: `ofFinsupp.injEq` is required.\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 X * { toFinsupp := toFinsupp\u271d } = { toFinsupp := toFinsupp\u271d } * X\n[PROOFSTEP]\nsimp only [X, \u2190 ofFinsupp_single, \u2190 ofFinsupp_mul, LinearMap.coe_mk, ofFinsupp.injEq]\n -- Porting note: Was `ext`.\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 Finsupp.single 1 1 * toFinsupp\u271d = toFinsupp\u271d * Finsupp.single 1 1\n[PROOFSTEP]\nrefine Finsupp.ext fun _ => ?_\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\nx\u271d : \u2115\n\u22a2 \u2191(Finsupp.single 1 1 * toFinsupp\u271d) x\u271d = \u2191(toFinsupp\u271d * Finsupp.single 1 1) x\u271d\n[PROOFSTEP]\nsimp [AddMonoidAlgebra.mul_apply, AddMonoidAlgebra.sum_single_index, add_comm]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 X ^ n * p = p * X ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 X ^ Nat.zero * p = p * X ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : X ^ n * p = p * X ^ n\n\u22a2 X ^ Nat.succ n * p = p * X ^ Nat.succ n\n[PROOFSTEP]\nconv_lhs => rw [pow_succ']\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : X ^ n * p = p * X ^ n\n| X ^ Nat.succ n * p\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : X ^ n * p = p * X ^ n\n| X ^ Nat.succ n * p\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : X ^ n * p = p * X ^ n\n| X ^ Nat.succ n * p\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nih : X ^ n * p = p * X ^ n\n\u22a2 X ^ n * X * p = p * X ^ Nat.succ n\n[PROOFSTEP]\nrw [mul_assoc, X_mul, \u2190 mul_assoc, ih, mul_assoc, \u2190 pow_succ']\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 p * X ^ n * q = p * q * X ^ n\n[PROOFSTEP]\nrw [mul_assoc, X_pow_mul, \u2190 mul_assoc]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(monomial n) r * X = \u2191(monomial (n + 1)) r\n[PROOFSTEP]\nerw [monomial_mul_monomial, mul_one]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\nk : \u2115\n\u22a2 \u2191(monomial n) r * X ^ k = \u2191(monomial (n + k)) r\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(monomial n) r * X ^ Nat.zero = \u2191(monomial (n + Nat.zero)) r\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\nk : \u2115\nih : \u2191(monomial n) r * X ^ k = \u2191(monomial (n + k)) r\n\u22a2 \u2191(monomial n) r * X ^ Nat.succ k = \u2191(monomial (n + Nat.succ k)) r\n[PROOFSTEP]\nsimp [ih, pow_succ', \u2190 mul_assoc, add_assoc]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nr : R\n\u22a2 X * \u2191(monomial n) r = \u2191(monomial (n + 1)) r\n[PROOFSTEP]\nrw [X_mul, monomial_mul_X]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nk n : \u2115\nr : R\n\u22a2 X ^ k * \u2191(monomial n) r = \u2191(monomial (n + k)) r\n[PROOFSTEP]\nrw [X_pow_mul, monomial_mul_X_pow]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\np : AddMonoidAlgebra R \u2115\n\u22a2 coeff { toFinsupp := p } = \u2191p\n[PROOFSTEP]\nrw [coeff]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 Injective coeff\n[PROOFSTEP]\nrintro \u27e8p\u27e9\n \u27e8q\u27e9\n -- Porting note: `ofFinsupp.injEq` is required.\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d : R[X]\np q : AddMonoidAlgebra R \u2115\n\u22a2 coeff { toFinsupp := p } = coeff { toFinsupp := q } \u2192 { toFinsupp := p } = { toFinsupp := q }\n[PROOFSTEP]\nsimp only [coeff, FunLike.coe_fn_eq, imp_self, ofFinsupp.injEq]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q f : R[X]\ni : \u2115\n\u22a2 \u2191f.toFinsupp i = coeff f i\n[PROOFSTEP]\ncases f\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ni : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 \u2191{ toFinsupp := toFinsupp\u271d }.toFinsupp i = coeff { toFinsupp := toFinsupp\u271d } i\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff (\u2191(monomial n) a) m = if n = m then a else 0\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_single]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff { toFinsupp := Finsupp.single n a } m = if n = m then a else 0\n[PROOFSTEP]\nsimp only [coeff, LinearMap.coe_mk]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191(Finsupp.single n a) m = if n = m then a else 0\n[PROOFSTEP]\nrw [Finsupp.single_apply]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff 1 0 = 1\n[PROOFSTEP]\nrw [\u2190 monomial_zero_one, coeff_monomial]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (if 0 = 0 then 1 else 0) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff (\u2191(monomial (n + 1)) a) 0 = 0\n[PROOFSTEP]\nsimp [coeff_monomial]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nhn : n \u2260 1\n\u22a2 coeff X n = 0\n[PROOFSTEP]\nrw [coeff_X, if_neg hn.symm]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 n \u2208 support p \u2194 coeff p n \u2260 0\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 n \u2208 support { toFinsupp := toFinsupp\u271d } \u2194 coeff { toFinsupp := toFinsupp\u271d } n \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u00acn \u2208 support p \u2194 coeff p n = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff (\u2191C a) n = if n = 0 then a else 0\n[PROOFSTEP]\nconvert coeff_monomial (a := a) (m := n) (n := 0) using 2\n[GOAL]\ncase h.e'_3.h\u2081.a\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 n = 0 \u2194 0 = n\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nh : n \u2260 0\n\u22a2 coeff (\u2191C a) n = 0\n[PROOFSTEP]\nrw [coeff_C, if_neg h]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nr : R\nn : \u2115\n\u22a2 coeff (\u2191C r) (n + 1) = 0\n[PROOFSTEP]\nsimp [coeff_C]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 coeff (\u2191m) n = \u2191(if n = 0 then m else 0)\n[PROOFSTEP]\nsimp only [\u2190 C_eq_nat_cast, coeff_C, Nat.cast_ite, Nat.cast_zero]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 \u2191C a * X ^ (n + 1) = \u2191(monomial (n + 1)) a\n[PROOFSTEP]\nrw [pow_succ', \u2190 mul_assoc, C_mul_X_pow_eq_monomial, X, monomial_mul_monomial, mul_one]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : R\nn : \u2115\n\u22a2 (\u2191C a * X ^ n).toFinsupp = Finsupp.single n a\n[PROOFSTEP]\nrw [C_mul_X_pow_eq_monomial, toFinsupp_monomial]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2191C a * X = \u2191(monomial 1) a\n[PROOFSTEP]\nrw [\u2190 C_mul_X_pow_eq_monomial, pow_one]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : R\n\u22a2 (\u2191C a * X).toFinsupp = Finsupp.single 1 a\n[PROOFSTEP]\nrw [C_mul_X_eq_monomial, toFinsupp_monomial]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 Subsingleton R \u2192 Subsingleton R[X]\n[PROOFSTEP]\nintro\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na\u271d : Subsingleton R\n\u22a2 Subsingleton R[X]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (\u2200 (f g : R[X]), f = g) \u2194 \u2200 (a b : R), a = b\n[PROOFSTEP]\nsimpa only [\u2190 subsingleton_iff] using subsingleton_iff_subsingleton\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d p q : R[X]\n\u22a2 p = q \u2194 \u2200 (n : \u2115), coeff p n = coeff q n\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q\u271d q : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := toFinsupp\u271d } = q \u2194 \u2200 (n : \u2115), coeff { toFinsupp := toFinsupp\u271d } n = coeff q n\n[PROOFSTEP]\nrcases q with\n \u27e8\u27e9\n -- Porting note: Was `simp [coeff, FunLike.ext_iff]`\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 { toFinsupp := toFinsupp\u271d\u00b9 } = { toFinsupp := toFinsupp\u271d } \u2194\n \u2200 (n : \u2115), coeff { toFinsupp := toFinsupp\u271d\u00b9 } n = coeff { toFinsupp := toFinsupp\u271d } n\n[PROOFSTEP]\nsimp [coeff]\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 toFinsupp\u271d\u00b9 = toFinsupp\u271d \u2194 \u2200 (n : \u2115), \u2191toFinsupp\u271d\u00b9 n = \u2191toFinsupp\u271d n\n[PROOFSTEP]\nexact FunLike.ext_iff (F := \u2115 \u2192\u2080 R)\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 AddSubmonoid.closure {p | \u2203 n a, p = \u2191(monomial n) a} = \u22a4\n[PROOFSTEP]\napply top_unique\n[GOAL]\ncase h\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u22a4 \u2264 AddSubmonoid.closure {p | \u2203 n a, p = \u2191(monomial n) a}\n[PROOFSTEP]\nrw [\u2190 AddSubmonoid.map_equiv_top (toFinsuppIso R).symm.toAddEquiv, \u2190 Finsupp.add_closure_setOf_eq_single,\n AddMonoidHom.map_mclosure]\n[GOAL]\ncase h\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 AddSubmonoid.closure\n (\u2191(AddEquiv.toAddMonoidHom (RingEquiv.toAddEquiv (RingEquiv.symm (toFinsuppIso R)))) ''\n {f | \u2203 a b, f = Finsupp.single a b}) \u2264\n AddSubmonoid.closure {p | \u2203 n a, p = \u2191(monomial n) a}\n[PROOFSTEP]\nrefine' AddSubmonoid.closure_mono (Set.image_subset_iff.2 _)\n[GOAL]\ncase h\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 {f | \u2203 a b, f = Finsupp.single a b} \u2286\n \u2191(AddEquiv.toAddMonoidHom (RingEquiv.toAddEquiv (RingEquiv.symm (toFinsuppIso R)))) \u207b\u00b9'\n {p | \u2203 n a, p = \u2191(monomial n) a}\n[PROOFSTEP]\nrintro _ \u27e8n, a, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\n\u22a2 Finsupp.single n a \u2208\n \u2191(AddEquiv.toAddMonoidHom (RingEquiv.toAddEquiv (RingEquiv.symm (toFinsuppIso R)))) \u207b\u00b9'\n {p | \u2203 n a, p = \u2191(monomial n) a}\n[PROOFSTEP]\nexact \u27e8n, a, Polynomial.ofFinsupp_single _ _\u27e9\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nM : Type u_1\ninst\u271d : AddMonoid M\nf g : R[X] \u2192+ M\nh : \u2200 (n : \u2115) (a : R), \u2191f (\u2191(monomial n) a) = \u2191g (\u2191(monomial n) a)\n\u22a2 Set.EqOn \u2191f \u2191g {p | \u2203 n a, p = \u2191(monomial n) a}\n[PROOFSTEP]\nrintro p \u27e8n, a, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nM : Type u_1\ninst\u271d : AddMonoid M\nf g : R[X] \u2192+ M\nh : \u2200 (n : \u2115) (a : R), \u2191f (\u2191(monomial n) a) = \u2191g (\u2191(monomial n) a)\nn : \u2115\na : R\n\u22a2 \u2191f (\u2191(monomial n) a) = \u2191g (\u2191(monomial n) a)\n[PROOFSTEP]\nexact h n a\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\nh : 0 = 1\np : R[X]\n\u22a2 p = 0\n[PROOFSTEP]\nrw [\u2190 one_smul R p, \u2190 h, zero_smul]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\nH : a \u2260 0\n\u22a2 support (\u2191(monomial n) a) = {n}\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_single, support]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\nH : a \u2260 0\n\u22a2 (match { toFinsupp := Finsupp.single n a } with\n | { toFinsupp := p } => p.support) =\n {n}\n[PROOFSTEP]\nexact Finsupp.support_single_ne_zero _ H\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\n\u22a2 support (\u2191(monomial n) a) \u2286 {n}\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_single, support]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\n\u22a2 (match { toFinsupp := Finsupp.single n a } with\n | { toFinsupp := p } => p.support) \u2286\n {n}\n[PROOFSTEP]\nexact Finsupp.support_single_subset\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nc : R\nh : c \u2260 0\n\u22a2 support (\u2191C c * X) = {1}\n[PROOFSTEP]\nrw [C_mul_X_eq_monomial, support_monomial 1 h]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nc : R\n\u22a2 support (\u2191C c * X) \u2286 {1}\n[PROOFSTEP]\nsimpa only [C_mul_X_eq_monomial] using support_monomial' 1 c\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nc : R\nh : c \u2260 0\n\u22a2 support (\u2191C c * X ^ n) = {n}\n[PROOFSTEP]\nrw [C_mul_X_pow_eq_monomial, support_monomial n h]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nc : R\n\u22a2 support (\u2191C c * X ^ n) \u2286 {n}\n[PROOFSTEP]\nsimpa only [C_mul_X_pow_eq_monomial] using support_monomial' n c\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 X ^ n = \u2191(monomial n) 1\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 X ^ Nat.zero = \u2191(monomial Nat.zero) 1\n[PROOFSTEP]\nrw [pow_zero, monomial_zero_one]\n[GOAL]\ncase succ\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nhn : X ^ n = \u2191(monomial n) 1\n\u22a2 X ^ Nat.succ n = \u2191(monomial (Nat.succ n)) 1\n[PROOFSTEP]\nrw [pow_succ', hn, X, monomial_mul_monomial, one_mul]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 (X ^ n).toFinsupp = Finsupp.single n 1\n[PROOFSTEP]\nrw [X_pow_eq_monomial, toFinsupp_monomial]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 a \u2022 X ^ n = \u2191(monomial n) a\n[PROOFSTEP]\nrw [X_pow_eq_monomial, smul_monomial, smul_eq_mul, mul_one]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nH : \u00ac1 = 0\nn : \u2115\n\u22a2 support (X ^ n) = {n}\n[PROOFSTEP]\nconvert support_monomial n H\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nH : \u00ac1 = 0\nn : \u2115\n\u22a2 X ^ n = \u2191(monomial n) 1\n[PROOFSTEP]\nexact X_pow_eq_monomial n\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nH : 1 = 0\n\u22a2 support X = \u2205\n[PROOFSTEP]\nrw [X, H, monomial_zero_right, support_zero]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\nH : \u00ac1 = 0\n\u22a2 support X = {1}\n[PROOFSTEP]\nrw [\u2190 pow_one X, support_X_pow H 1]\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\na : R\nha : a \u2260 0\ni j : \u2115\n\u22a2 \u2191(monomial i) a = \u2191(monomial j) a \u2194 i = j\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_single, \u2190 ofFinsupp_single, ofFinsupp.injEq, Finsupp.single_left_inj ha]\n[GOAL]\nR : Type u\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nk l m n : \u2115\nu v : R\nhu : u \u2260 0\nhv : v \u2260 0\n\u22a2 \u2191C u * X ^ k + \u2191C v * X ^ l = \u2191C u * X ^ m + \u2191C v * X ^ n \u2194\n k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nsimp_rw [C_mul_X_pow_eq_monomial, \u2190 toFinsupp_inj, toFinsupp_add, toFinsupp_monomial]\n[GOAL]\nR : Type u\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nk l m n : \u2115\nu v : R\nhu : u \u2260 0\nhv : v \u2260 0\n\u22a2 Finsupp.single k u + Finsupp.single l v = Finsupp.single m u + Finsupp.single n v \u2194\n k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nexact Finsupp.single_add_single_eq_single_add_single hu hv\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 p * q = \u2211 i in support p, sum q fun j a => \u2191(monomial (i + j)) (coeff p i * a)\n[PROOFSTEP]\napply toFinsupp_injective\n[GOAL]\ncase a\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (p * q).toFinsupp = (\u2211 i in support p, sum q fun j a => \u2191(monomial (i + j)) (coeff p i * a)).toFinsupp\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase a.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\nq : R[X]\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d } * q).toFinsupp =\n (\u2211 i in support { toFinsupp := toFinsupp\u271d },\n sum q fun j a => \u2191(monomial (i + j)) (coeff { toFinsupp := toFinsupp\u271d } i * a)).toFinsupp\n[PROOFSTEP]\nrcases q with \u27e8\u27e9\n[GOAL]\ncase a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 ({ toFinsupp := toFinsupp\u271d\u00b9 } * { toFinsupp := toFinsupp\u271d }).toFinsupp =\n (\u2211 i in support { toFinsupp := toFinsupp\u271d\u00b9 },\n sum { toFinsupp := toFinsupp\u271d } fun j a =>\n \u2191(monomial (i + j)) (coeff { toFinsupp := toFinsupp\u271d\u00b9 } i * a)).toFinsupp\n[PROOFSTEP]\nsimp [support, sum, coeff, toFinsupp_sum]\n[GOAL]\ncase a.ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 toFinsupp\u271d\u00b9 * toFinsupp\u271d =\n \u2211 x in toFinsupp\u271d\u00b9.support, \u2211 x_1 in toFinsupp\u271d.support, Finsupp.single (x + x_1) (\u2191toFinsupp\u271d\u00b9 x * \u2191toFinsupp\u271d x_1)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nS : Type u_1\ninst\u271d : AddCommMonoid S\nf : \u2115 \u2192 R \u2192 S\n\u22a2 sum 0 f = 0\n[PROOFSTEP]\nsimp [sum]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d : R[X]\nS : Type u_1\ninst\u271d : AddCommMonoid S\np q : R[X]\nf : \u2115 \u2192 R \u2192 S\nhf : \u2200 (i : \u2115), f i 0 = 0\nh_add : \u2200 (a : \u2115) (b\u2081 b\u2082 : R), f a (b\u2081 + b\u2082) = f a b\u2081 + f a b\u2082\n\u22a2 sum (p + q) f = sum p f + sum q f\n[PROOFSTEP]\nrw [show p + q = \u27e8p.toFinsupp + q.toFinsupp\u27e9 from add_def p q]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d : R[X]\nS : Type u_1\ninst\u271d : AddCommMonoid S\np q : R[X]\nf : \u2115 \u2192 R \u2192 S\nhf : \u2200 (i : \u2115), f i 0 = 0\nh_add : \u2200 (a : \u2115) (b\u2081 b\u2082 : R), f a (b\u2081 + b\u2082) = f a b\u2081 + f a b\u2082\n\u22a2 sum { toFinsupp := p.toFinsupp + q.toFinsupp } f = sum p f + sum q f\n[PROOFSTEP]\nexact Finsupp.sum_add_index (fun i _ \u21a6 hf i) (fun a _ b\u2081 b\u2082 \u21a6 h_add a b\u2081 b\u2082)\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q : R[X]\nS : Type u_1\ninst\u271d : AddCommMonoid S\np : R[X]\nf g : \u2115 \u2192 R \u2192 S\n\u22a2 sum p (f + g) = sum p f + sum p g\n[PROOFSTEP]\nsimp [sum_def, Finset.sum_add_distrib]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\n\u22a2 (sum p fun n a => \u2191C a * X ^ n) = p\n[PROOFSTEP]\nsimp_rw [C_mul_X_pow_eq_monomial, sum_monomial_eq]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 (erase n p).toFinsupp = Finsupp.erase n p.toFinsupp\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (erase n { toFinsupp := toFinsupp\u271d }).toFinsupp = Finsupp.erase n { toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nsimp only [erase_def]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\np : AddMonoidAlgebra R \u2115\nn : \u2115\n\u22a2 { toFinsupp := Finsupp.erase n p } = erase n { toFinsupp := p }\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase mk\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nsupport\u271d : Finset \u2115\ntoFun\u271d : \u2115 \u2192 R\nmem_support_toFun\u271d : \u2200 (a : \u2115), a \u2208 support\u271d \u2194 toFun\u271d a \u2260 0\n\u22a2 { toFinsupp := Finsupp.erase n { support := support\u271d, toFun := toFun\u271d, mem_support_toFun := mem_support_toFun\u271d } } =\n erase n { toFinsupp := { support := support\u271d, toFun := toFun\u271d, mem_support_toFun := mem_support_toFun\u271d } }\n[PROOFSTEP]\nsimp only [erase_def]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 support (erase n p) = Finset.erase (support p) n\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (erase n { toFinsupp := toFinsupp\u271d }) = Finset.erase (support { toFinsupp := toFinsupp\u271d }) n\n[PROOFSTEP]\nsimp only [support, erase_def, Finsupp.support_erase]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 (\u2191(monomial n) (coeff p n) + erase n p).toFinsupp = p.toFinsupp\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (\u2191(monomial n) (coeff { toFinsupp := toFinsupp\u271d } n) + erase n { toFinsupp := toFinsupp\u271d }).toFinsupp =\n { toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nrw [toFinsupp_add, toFinsupp_monomial, toFinsupp_erase, coeff]\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 Finsupp.single n\n ((match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d } with\n | { toFinsupp := p } => \u2191p)\n n) +\n Finsupp.erase n { toFinsupp := toFinsupp\u271d }.toFinsupp =\n { toFinsupp := toFinsupp\u271d }.toFinsupp\n[PROOFSTEP]\nexact Finsupp.single_add_erase _ _\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn i : \u2115\n\u22a2 coeff (erase n p) i = if i = n then 0 else coeff p i\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn i : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (erase n { toFinsupp := toFinsupp\u271d }) i = if i = n then 0 else coeff { toFinsupp := toFinsupp\u271d } i\n[PROOFSTEP]\nsimp only [erase_def, coeff]\n -- Porting note: Was `convert rfl`.\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn i : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 \u2191(Finsupp.erase n toFinsupp\u271d) i = if i = n then 0 else \u2191toFinsupp\u271d i\n[PROOFSTEP]\nexact ite_congr rfl (fun _ => rfl) (fun _ => rfl)\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 (erase n 0).toFinsupp = 0.toFinsupp\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\n\u22a2 (erase n (\u2191(monomial n) a)).toFinsupp = 0.toFinsupp\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 coeff (erase n p) n = 0\n[PROOFSTEP]\nsimp [coeff_erase]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn i : \u2115\nh : i \u2260 n\n\u22a2 coeff (erase n p) i = coeff p i\n[PROOFSTEP]\nsimp [coeff_erase, h]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\n\u22a2 coeff (update p n a) = Function.update (coeff p) n a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\nx\u271d : \u2115\n\u22a2 coeff (update p n a) x\u271d = Function.update (coeff p) n a x\u271d\n[PROOFSTEP]\ncases p\n[GOAL]\ncase h.ofFinsupp\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\na : R\nx\u271d : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (update { toFinsupp := toFinsupp\u271d } n a) x\u271d = Function.update (coeff { toFinsupp := toFinsupp\u271d }) n a x\u271d\n[PROOFSTEP]\nsimp only [coeff, update, Function.update_apply, coe_update]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\ni : \u2115\n\u22a2 coeff (update p n a) i = if i = n then a else coeff p i\n[PROOFSTEP]\nrw [coeff_update, Function.update_apply]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\n\u22a2 coeff (update p n a) n = a\n[PROOFSTEP]\nrw [p.coeff_update_apply, if_pos rfl]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\ni : \u2115\nh : i \u2260 n\n\u22a2 coeff (update p n a) i = coeff p i\n[PROOFSTEP]\nrw [p.coeff_update_apply, if_neg h]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 update p n 0 = erase n p\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\na b : R\nm n\u271d\u00b9 : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn n\u271d : \u2115\n\u22a2 coeff (update p n 0) n\u271d = coeff (erase n p) n\u271d\n[PROOFSTEP]\nrw [coeff_update_apply, coeff_erase]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\n\u22a2 support (update p n a) = if a = 0 then Finset.erase (support p) n else insert n (support p)\n[PROOFSTEP]\nclassical\ncases p\nsimp only [support, update, Finsupp.support_update]\ncongr\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\n\u22a2 support (update p n a) = if a = 0 then Finset.erase (support p) n else insert n (support p)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase ofFinsupp\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (update { toFinsupp := toFinsupp\u271d } n a) =\n if a = 0 then Finset.erase (support { toFinsupp := toFinsupp\u271d }) n\n else insert n (support { toFinsupp := toFinsupp\u271d })\n[PROOFSTEP]\nsimp only [support, update, Finsupp.support_update]\n[GOAL]\ncase ofFinsupp\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nn : \u2115\na : R\ninst\u271d : Decidable (a = 0)\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (if a = 0 then Finset.erase toFinsupp\u271d.support n else insert n toFinsupp\u271d.support) =\n if a = 0 then Finset.erase toFinsupp\u271d.support n else insert n toFinsupp\u271d.support\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\n\u22a2 support (update p n 0) = Finset.erase (support p) n\n[PROOFSTEP]\nrw [update_zero_eq_erase, support_erase]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\nha : a \u2260 0\n\u22a2 support (update p n a) = insert n (support p)\n[PROOFSTEP]\nclassical rw [support_update, if_neg ha]\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn : \u2115\na : R\nha : a \u2260 0\n\u22a2 support (update p n a) = insert n (support p)\n[PROOFSTEP]\nrw [support_update, if_neg ha]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\np : R[X]\nn : \u2115\n\u22a2 coeff (-p) n = -coeff p n\n[PROOFSTEP]\nrcases p with\n \u27e8\u27e9\n -- Porting note: The last rule should be `apply`ed.\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (-{ toFinsupp := toFinsupp\u271d }) n = -coeff { toFinsupp := toFinsupp\u271d } n\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_neg, coeff, coeff]\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := -toFinsupp\u271d } with\n | { toFinsupp := p } => \u2191p)\n n =\n -(match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d } with\n | { toFinsupp := p } => \u2191p)\n n\n[PROOFSTEP]\napply Finsupp.neg_apply\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\np q : R[X]\nn : \u2115\n\u22a2 coeff (p - q) n = coeff p n - coeff q n\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nq : R[X]\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff ({ toFinsupp := toFinsupp\u271d } - q) n = coeff { toFinsupp := toFinsupp\u271d } n - coeff q n\n[PROOFSTEP]\nrcases q with\n \u27e8\u27e9\n -- Porting note: The last rule should be `apply`ed.\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff ({ toFinsupp := toFinsupp\u271d\u00b9 } - { toFinsupp := toFinsupp\u271d }) n =\n coeff { toFinsupp := toFinsupp\u271d\u00b9 } n - coeff { toFinsupp := toFinsupp\u271d } n\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_sub, coeff, coeff, coeff]\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d\u00b9 toFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d\u00b9 - toFinsupp\u271d } with\n | { toFinsupp := p } => \u2191p)\n n =\n (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d\u00b9 } with\n | { toFinsupp := p } => \u2191p)\n n -\n (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d } with\n | { toFinsupp := p } => \u2191p)\n n\n[PROOFSTEP]\napply Finsupp.sub_apply\n[GOAL]\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\na : R\n\u22a2 \u2191(monomial n) (-a) = -\u2191(monomial n) a\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero, \u2190 monomial_add, neg_add_self, monomial_zero_right]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Ring R\np : R[X]\n\u22a2 support (-p) = support p\n[PROOFSTEP]\nrcases p with\n \u27e8\u27e9\n -- Porting note: The last rule should be `apply`ed.\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Ring R\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (-{ toFinsupp := toFinsupp\u271d }) = support { toFinsupp := toFinsupp\u271d }\n[PROOFSTEP]\nrw [\u2190 ofFinsupp_neg, support, support]\n[GOAL]\ncase ofFinsupp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Ring R\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (match { toFinsupp := -toFinsupp\u271d } with\n | { toFinsupp := p } => p.support) =\n match { toFinsupp := toFinsupp\u271d } with\n | { toFinsupp := p } => p.support\n[PROOFSTEP]\napply Finsupp.support_neg\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\n\u22a2 \u2191C \u2191n = \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\n\u22a2 Nontrivial R[X]\n[PROOFSTEP]\nhave h : Nontrivial (AddMonoidAlgebra R \u2115) := by infer_instance\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\n\u22a2 Nontrivial (AddMonoidAlgebra R \u2115)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nh : Nontrivial (AddMonoidAlgebra R \u2115)\n\u22a2 Nontrivial R[X]\n[PROOFSTEP]\nrcases h.exists_pair_ne with \u27e8x, y, hxy\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nh : Nontrivial (AddMonoidAlgebra R \u2115)\nx y : AddMonoidAlgebra R \u2115\nhxy : x \u2260 y\n\u22a2 Nontrivial R[X]\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8x\u27e9, \u27e8y\u27e9, _\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\nh : Nontrivial (AddMonoidAlgebra R \u2115)\nx y : AddMonoidAlgebra R \u2115\nhxy : x \u2260 y\n\u22a2 { toFinsupp := x } \u2260 { toFinsupp := y }\n[PROOFSTEP]\nsimp [hxy]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\n\u22a2 \u00accoeff X 1 = coeff 0 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\na\u271d b : R\nm n : \u2115\ninst\u271d : DivisionRing R\na : \u211a\nf : R[X]\n\u22a2 a \u2022 f = \u2191C \u2191a * f\n[PROOFSTEP]\nrw [\u2190 Rat.smul_one_eq_coe, \u2190 Polynomial.smul_C, C_1, smul_one_mul]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Basic", "llama_tokens": 24363, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.2918504294154786}} {"text": "[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\n\u22a2 SMulCommClass { x // x \u2208 center M } { x // x \u2208 center M } M\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Submonoid.Center", "llama_tokens": 58, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.29170346893537613}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh : IsAdmissible abv\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : \u03b9), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 % b - A i\u2080 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nlet e := Fintype.equivFin \u03b9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh : IsAdmissible abv\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : \u03b9), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 % b - A i\u2080 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := h.exists_partition' (Fintype.card \u03b9) h\u03b5 hb (A \u2218 e.symm)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh : IsAdmissible abv\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\nt : Fin (Fintype.card \u03b9) \u2192 Fin (IsAdmissible.card h \u03b5)\nht : \u2200 (i\u2080 i\u2081 : Fin (Fintype.card \u03b9)), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv ((A \u2218 \u2191e.symm) i\u2081 % b - (A \u2218 \u2191e.symm) i\u2080 % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : \u03b9), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 % b - A i\u2080 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrefine' \u27e8t \u2218 e, fun i\u2080 i\u2081 h \u21a6 _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh\u271d : IsAdmissible abv\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\nt : Fin (Fintype.card \u03b9) \u2192 Fin (IsAdmissible.card h\u271d \u03b5)\nht : \u2200 (i\u2080 i\u2081 : Fin (Fintype.card \u03b9)), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv ((A \u2218 \u2191e.symm) i\u2081 % b - (A \u2218 \u2191e.symm) i\u2080 % b)) < \u2191abv b \u2022 \u03b5\ni\u2080 i\u2081 : \u03b9\nh : (t \u2218 \u2191e) i\u2080 = (t \u2218 \u2191e) i\u2081\n\u22a2 \u2191(\u2191abv (A i\u2081 % b - A i\u2080 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nconvert (config := { transparency := .default }) ht (e i\u2080) (e i\u2081) h\n[GOAL]\ncase h.e'_3.h.e'_3.h.e'_6.h.e'_5.h.e'_5.h.e'_1\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh\u271d : IsAdmissible abv\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\nt : Fin (Fintype.card \u03b9) \u2192 Fin (IsAdmissible.card h\u271d \u03b5)\nht : \u2200 (i\u2080 i\u2081 : Fin (Fintype.card \u03b9)), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv ((A \u2218 \u2191e.symm) i\u2081 % b - (A \u2218 \u2191e.symm) i\u2080 % b)) < \u2191abv b \u2022 \u03b5\ni\u2080 i\u2081 : \u03b9\nh : (t \u2218 \u2191e) i\u2080 = (t \u2218 \u2191e) i\u2081\n\u22a2 i\u2081 = \u2191e.symm (\u2191e i\u2081)\n[PROOFSTEP]\nsimp only [e.symm_apply_apply]\n[GOAL]\ncase h.e'_3.h.e'_3.h.e'_6.h.e'_6.h.e'_5.h.e'_1\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : \u03b9 \u2192 R\nh\u271d : IsAdmissible abv\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\nt : Fin (Fintype.card \u03b9) \u2192 Fin (IsAdmissible.card h\u271d \u03b5)\nht : \u2200 (i\u2080 i\u2081 : Fin (Fintype.card \u03b9)), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv ((A \u2218 \u2191e.symm) i\u2081 % b - (A \u2218 \u2191e.symm) i\u2080 % b)) < \u2191abv b \u2022 \u03b5\ni\u2080 i\u2081 : \u03b9\nh : (t \u2218 \u2191e) i\u2080 = (t \u2218 \u2191e) i\u2081\n\u22a2 i\u2080 = \u2191e.symm (\u2191e i\u2080)\n[PROOFSTEP]\nsimp only [e.symm_apply_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nn : \u2115\nh : IsAdmissible abv\n\u22a2 \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nhaveI := Classical.decEq R\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nn : \u2115\nh : IsAdmissible abv\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.zero)) \u2192 Fin Nat.zero \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin Nat.zero), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nintro \u03b5 _h\u03b5 b _hb A\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.zero)) \u2192 Fin Nat.zero \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin Nat.zero), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrefine' \u27e80, 1, _, _\u27e9\n[GOAL]\ncase zero.refine'_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.zero)) \u2192 Fin Nat.zero \u2192 R\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.refine'_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.zero)) \u2192 Fin Nat.zero \u2192 R\n\u22a2 \u2200 (k : Fin Nat.zero), \u2191(\u2191abv (A 1 k % b - A 0 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrintro \u27e8i, \u27e8\u27e9\u27e9\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (Nat.succ n)), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5 b hb A\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (Nat.succ n)), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nlet M := h.card \u03b5\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (Nat.succ n)), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8s, s_inj, hs\u27e9 :\n \u2203 s : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ,\n Function.Injective s \u2227 \u2200 i\u2080 i\u2081, (abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b) : \u211d) < abv b \u2022 \u03b5 :=\n by\n -- We can partition the `A`s into `M` subsets where\n -- the first components lie close together:\n obtain \u27e8t, ht\u27e9 :\n \u2203 t : Fin (M ^ n.succ).succ \u2192 Fin M, \u2200 i\u2080 i\u2081, t i\u2080 = t i\u2081 \u2192 (abv (A i\u2081 0 % b - A i\u2080 0 % b) : \u211d) < abv b \u2022 \u03b5 :=\n h.exists_partition h\u03b5 hb fun x \u21a6\n A x\n 0\n -- Since the `M` subsets contain more than `M * M^n` elements total,\n -- there must be a subset that contains more than `M^n` elements.\n obtain \u27e8s, hs\u27e9 :=\n @Fintype.exists_lt_card_fiber_of_mul_lt_card _ _ _ _ _ t (M ^ n)\n (by simpa only [Fintype.card_fin, pow_succ] using Nat.lt_succ_self (M ^ n.succ))\n refine' \u27e8fun i \u21a6 (Finset.univ.filter fun x \u21a6 t x = s).toList.nthLe i _, _, fun i\u2080 i\u2081 \u21a6 ht _ _ _\u27e9\n \u00b7 refine' i.2.trans_le _\n rwa [Finset.length_toList]\n \u00b7 intro i j h\n ext\n exact Fin.mk.inj_iff.mp (List.nodup_iff_injective_get.mp (Finset.nodup_toList _) h)\n have : \u2200 i h, (Finset.univ.filter fun x \u21a6 t x = s).toList.nthLe i h \u2208 Finset.univ.filter fun x \u21a6 t x = s :=\n by\n intro i h\n exact Finset.mem_toList.mp (List.get_mem _ i h)\n obtain \u27e8_, h\u2080\u27e9 := Finset.mem_filter.mp (this i\u2080 _)\n obtain \u27e8_, h\u2081\u27e9 := Finset.mem_filter.mp (this i\u2081 _)\n exact h\u2080.trans h\u2081.symm\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 :\n \u2203 t : Fin (M ^ n.succ).succ \u2192 Fin M, \u2200 i\u2080 i\u2081, t i\u2080 = t i\u2081 \u2192 (abv (A i\u2081 0 % b - A i\u2080 0 % b) : \u211d) < abv b \u2022 \u03b5 :=\n h.exists_partition h\u03b5 hb fun x \u21a6\n A x\n 0\n -- Since the `M` subsets contain more than `M * M^n` elements total,\n -- there must be a subset that contains more than `M^n` elements.\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 :=\n @Fintype.exists_lt_card_fiber_of_mul_lt_card _ _ _ _ _ t (M ^ n)\n (by simpa only [Fintype.card_fin, pow_succ] using Nat.lt_succ_self (M ^ n.succ))\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 Fintype.card (Fin M) * M ^ n < Fintype.card (Fin (Nat.succ (M ^ Nat.succ n)))\n[PROOFSTEP]\nsimpa only [Fintype.card_fin, pow_succ] using Nat.lt_succ_self (M ^ n.succ)\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrefine' \u27e8fun i \u21a6 (Finset.univ.filter fun x \u21a6 t x = s).toList.nthLe i _, _, fun i\u2080 i\u2081 \u21a6 ht _ _ _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni : Fin (Nat.succ (M ^ n))\n\u22a2 \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))\n[PROOFSTEP]\nrefine' i.2.trans_le _\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni : Fin (Nat.succ (M ^ n))\n\u22a2 Nat.succ (M ^ n) \u2264 List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))\n[PROOFSTEP]\nrwa [Finset.length_toList]\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\n\u22a2 Function.Injective fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)))\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni j : Fin (Nat.succ (M ^ n))\nh :\n (fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i =\n (fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n j\n\u22a2 i = j\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.refine'_2.h\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni j : Fin (Nat.succ (M ^ n))\nh :\n (fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i =\n (fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n j\n\u22a2 \u2191i = \u2191j\n[PROOFSTEP]\nexact Fin.mk.inj_iff.mp (List.nodup_iff_injective_get.mp (Finset.nodup_toList _) h)\n[GOAL]\ncase intro.intro.refine'_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\n\u22a2 t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2080) =\n t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2081)\n[PROOFSTEP]\nhave : \u2200 i h, (Finset.univ.filter fun x \u21a6 t x = s).toList.nthLe i h \u2208 Finset.univ.filter fun x \u21a6 t x = s :=\n by\n intro i h\n exact Finset.mem_toList.mp (List.get_mem _ i h)\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\n\u22a2 \u2200 (i : \u2115) (h : i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))),\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) i h \u2208\n Finset.filter (fun x => t x = s) Finset.univ\n[PROOFSTEP]\nintro i h\n[GOAL]\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\ni : \u2115\nh : i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))\n\u22a2 List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) i h \u2208\n Finset.filter (fun x => t x = s) Finset.univ\n[PROOFSTEP]\nexact Finset.mem_toList.mp (List.get_mem _ i h)\n[GOAL]\ncase intro.intro.refine'_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\nthis :\n \u2200 (i : \u2115) (h : i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))),\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) i h \u2208\n Finset.filter (fun x => t x = s) Finset.univ\n\u22a2 t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2080) =\n t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2081)\n[PROOFSTEP]\nobtain \u27e8_, h\u2080\u27e9 := Finset.mem_filter.mp (this i\u2080 _)\n[GOAL]\ncase intro.intro.refine'_3.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\nthis :\n \u2200 (i : \u2115) (h : i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))),\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) i h \u2208\n Finset.filter (fun x => t x = s) Finset.univ\nleft\u271d : List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2080 ?m.22999 \u2208 Finset.univ\nh\u2080 : t (List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2080 ?m.22999) = s\n\u22a2 t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2080) =\n t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2081)\n[PROOFSTEP]\nobtain \u27e8_, h\u2081\u27e9 := Finset.mem_filter.mp (this i\u2081 _)\n[GOAL]\ncase intro.intro.refine'_3.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\nt : Fin (Nat.succ (M ^ Nat.succ n)) \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ Nat.succ n))), t i\u2080 = t i\u2081 \u2192 \u2191(\u2191abv (A i\u2081 0 % b - A i\u2080 0 % b)) < \u2191abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < Finset.card (Finset.filter (fun x => t x = s) Finset.univ)\ni\u2080 i\u2081 : Fin (Nat.succ (M ^ n))\nthis :\n \u2200 (i : \u2115) (h : i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))),\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) i h \u2208\n Finset.filter (fun x => t x = s) Finset.univ\nleft\u271d\u00b9 : List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2080 ?m.22999 \u2208 Finset.univ\nh\u2080 : t (List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2080 ?m.22999) = s\nleft\u271d : List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2081 ?m.23393 \u2208 Finset.univ\nh\u2081 : t (List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\u2081 ?m.23393) = s\n\u22a2 t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2080) =\n t\n ((fun i =>\n List.nthLe (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ)) \u2191i\n (_ : \u2191i < List.length (Finset.toList (Finset.filter (fun x => t x = s) Finset.univ))))\n i\u2081)\n[PROOFSTEP]\nexact h\u2080.trans h\u2081.symm\n[GOAL]\ncase succ.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h \u03b5\ns : Fin (Nat.succ (M ^ n)) \u2192 Fin (Nat.succ (M ^ Nat.succ n))\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (Nat.succ n)), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8k\u2080, k\u2081, hk, h\u27e9 := ih h\u03b5 hb fun x \u21a6 Fin.tail (A (s x))\n[GOAL]\ncase succ.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\ns : Fin (Nat.succ (M ^ n)) \u2192 Fin (Nat.succ (M ^ Nat.succ n))\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n))\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(\u2191abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (Nat.succ n)), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrefine' \u27e8s k\u2080, s k\u2081, fun h \u21a6 hk (s_inj h), fun i \u21a6 Fin.cases _ (fun i \u21a6 _) i\u27e9\n[GOAL]\ncase succ.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\ns : Fin (Nat.succ (M ^ n)) \u2192 Fin (Nat.succ (M ^ Nat.succ n))\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n))\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(\u2191abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < \u2191abv b \u2022 \u03b5\ni : Fin (Nat.succ n)\n\u22a2 \u2191(\u2191abv (A (s k\u2081) 0 % b - A (s k\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nexact hs k\u2080 k\u2081\n[GOAL]\ncase succ.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : IsAdmissible abv\nthis : DecidableEq R\nn : \u2115\nih :\n \u2200 {\u03b5 : \u211d},\n 0 < \u03b5 \u2192\n \u2200 {b : R},\n b \u2260 0 \u2192\n \u2200 (A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n)) \u2192 Fin n \u2192 R),\n \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Nat.succ n)) \u2192 Fin (Nat.succ n) \u2192 R\nM : \u2115 := IsAdmissible.card h\u271d \u03b5\ns : Fin (Nat.succ (M ^ n)) \u2192 Fin (Nat.succ (M ^ Nat.succ n))\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (Nat.succ (M ^ n))), \u2191(\u2191abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < \u2191abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ n))\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(\u2191abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < \u2191abv b \u2022 \u03b5\ni\u271d : Fin (Nat.succ n)\ni : Fin n\n\u22a2 \u2191(\u2191abv (A (s k\u2081) (Fin.succ i) % b - A (s k\u2080) (Fin.succ i) % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nexact h i\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nlet e := Fintype.equivFin \u03b9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nobtain \u27e8i\u2080, i\u2081, ne, h\u27e9 := h.exists_approx_aux (Fintype.card \u03b9) h\u03b5 hb fun x y \u21a6 A x (e.symm y)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh\u271d : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\ni\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9))\nne : i\u2080 \u2260 i\u2081\nh : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nrefine' \u27e8i\u2080, i\u2081, ne, fun k \u21a6 _\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh\u271d : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\ni\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9))\nne : i\u2080 \u2260 i\u2081\nh : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5\nk : \u03b9\n\u22a2 \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5\n[PROOFSTEP]\nconvert h (e k)\n[GOAL]\ncase h.e'_3.h.e'_3.h.e'_6.h.e'_5.h.e'_5.h.e'_2\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh\u271d : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\ni\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9))\nne : i\u2080 \u2260 i\u2081\nh : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5\nk : \u03b9\n\u22a2 k = \u2191e.symm (\u2191e k)\n[PROOFSTEP]\nsimp only [e.symm_apply_apply]\n[GOAL]\ncase h.e'_3.h.e'_3.h.e'_6.h.e'_6.h.e'_5.h.e'_2\nR : Type u_1\ninst\u271d\u00b9 : EuclideanDomain R\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nh\u271d : IsAdmissible abv\nA : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R\ne : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9\ni\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9))\nne : i\u2080 \u2260 i\u2081\nh : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5\nk : \u03b9\n\u22a2 k = \u2191e.symm (\u2191e k)\n[PROOFSTEP]\nsimp only [e.symm_apply_apply]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue", "llama_tokens": 17136, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.2916030714396655}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LT \u03b1\na : \u03b1\n\u22a2 \u00acIsSuccLimit a \u2194 \u2203 b, b \u22d6 a\n[PROOFSTEP]\nsimp [IsSuccLimit]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nh : IsSuccLimit (succ a)\n\u22a2 IsMax a\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nh : IsSuccLimit (succ a)\nH : \u00acIsMax a\n\u22a2 False\n[PROOFSTEP]\nexact h a (covby_succ_of_not_isMax H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nha : \u00acIsMax a\n\u22a2 \u00acIsSuccLimit (succ a)\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nha : IsSuccLimit (succ a)\n\u22a2 IsMax a\n[PROOFSTEP]\nexact ha.isMax\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh : IsSuccLimit a\nb : \u03b1\n\u22a2 succ b \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nh : IsSuccLimit (succ b)\n\u22a2 False\n[PROOFSTEP]\nexact not_isMax _ h.isMax\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh : IsSuccLimit a\nb : \u03b1\nhb : b \u2264 a\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrcases hb.exists_succ_iterate with \u27e8_ | n, rfl\u27e9\n[GOAL]\ncase intro.zero\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nh : IsSuccLimit (succ^[Nat.zero] b)\nhb : b \u2264 succ^[Nat.zero] b\n\u22a2 succ^[Nat.zero] b \u2264 b\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit (succ^[Nat.succ n] b)\nhb : b \u2264 succ^[Nat.succ n] b\n\u22a2 succ^[Nat.succ n] b \u2264 b\n[PROOFSTEP]\nrw [iterate_succ_apply'] at h \n[GOAL]\ncase intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit (succ (succ^[n] b))\nhb : b \u2264 succ^[Nat.succ n] b\n\u22a2 succ^[Nat.succ n] b \u2264 b\n[PROOFSTEP]\nexact (not_isSuccLimit_succ _ h).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b3 : SuccOrder \u03b1\ninst\u271d\u00b2 : IsSuccArchimedean \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u00acIsSuccLimit a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\n\u22a2 \u00acIsSuccLimit a \u2194 \u2203 b, \u00acIsMax b \u2227 succ b = a\n[PROOFSTEP]\nrw [not_isSuccLimit_iff_exists_covby]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\n\u22a2 (\u2203 b, b \u22d6 a) \u2194 \u2203 b, \u00acIsMax b \u2227 succ b = a\n[PROOFSTEP]\nrefine' exists_congr fun b => \u27e8fun hba => \u27e8hba.lt.not_isMax, (Covby.succ_eq hba)\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\n\u22a2 \u00acIsMax b \u2227 succ b = a \u2192 b \u22d6 a\n[PROOFSTEP]\nrintro \u27e8h, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nh : \u00acIsMax b\n\u22a2 b \u22d6 succ b\n[PROOFSTEP]\nexact covby_succ_of_not_isMax h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nh : \u00acIsSuccLimit a\n\u22a2 a \u2208 range succ\n[PROOFSTEP]\ncases' not_isSuccLimit_iff.1 h with b hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nh : \u00acIsSuccLimit a\nb : \u03b1\nhb : \u00acIsMax b \u2227 succ b = a\n\u22a2 a \u2208 range succ\n[PROOFSTEP]\nexact \u27e8b, hb.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhb : IsSuccLimit b\nha : a < b\n\u22a2 succ a < b\n[PROOFSTEP]\nby_cases h : IsMax a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhb : IsSuccLimit b\nha : a < b\nh : IsMax a\n\u22a2 succ a < b\n[PROOFSTEP]\nrwa [h.succ_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhb : IsSuccLimit b\nha : a < b\nh : \u00acIsMax a\n\u22a2 succ a < b\n[PROOFSTEP]\nrw [lt_iff_le_and_ne, succ_le_iff_of_not_isMax h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhb : IsSuccLimit b\nha : a < b\nh : \u00acIsMax a\n\u22a2 a < b \u2227 succ a \u2260 b\n[PROOFSTEP]\nrefine' \u27e8ha, fun hab => _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhb : IsSuccLimit b\nha : a < b\nh : \u00acIsMax a\nhab : succ a = b\n\u22a2 False\n[PROOFSTEP]\nsubst hab\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\nC : \u03b1 \u2192 Sort u_2\nh : \u00acIsMax a\nhb : IsSuccLimit (succ a)\nha : a < succ a\n\u22a2 False\n[PROOFSTEP]\nexact (h hb.isMax).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\n\u22a2 C b\n[PROOFSTEP]\nby_cases hb : IsSuccLimit b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : IsSuccLimit b\n\u22a2 C b\n[PROOFSTEP]\nexact hl b hb\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : \u00acIsSuccLimit b\n\u22a2 C b\n[PROOFSTEP]\nhave H := Classical.choose_spec (not_isSuccLimit_iff.1 hb)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : \u00acIsSuccLimit b\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = b)) = b\n\u22a2 C b\n[PROOFSTEP]\nrw [\u2190 H.2]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nb : \u03b1\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : \u00acIsSuccLimit b\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = b)) = b\n\u22a2 C (succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = b)))\n[PROOFSTEP]\nexact hs _ H.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : IsSuccLimit b\n\u22a2 isSuccLimitRecOn b hs hl = hl b hb\n[PROOFSTEP]\nclassical exact dif_pos hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nhb : IsSuccLimit b\n\u22a2 isSuccLimitRecOn b hs hl = hl b hb\n[PROOFSTEP]\nexact dif_pos hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb\n[PROOFSTEP]\nhave hb' := not_isSuccLimit_succ_of_not_isMax hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb\n[PROOFSTEP]\nhave H := Classical.choose_spec (not_isSuccLimit_iff.1 hb')\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb\n[PROOFSTEP]\nrw [isSuccLimitRecOn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 (if hb : IsSuccLimit (succ b) then hl (succ b) hb\n else\n let_fun H :=\n (_ :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b);\n Eq.mpr (_ : C (succ b) = C (succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))) =\n hs b hb\n[PROOFSTEP]\nsimp only [cast_eq_iff_heq, hb', not_false_iff, eq_mpr_eq_cast, dif_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq\n (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n (hs b hb)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b) = b\n[PROOFSTEP]\nfirst\n| exact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right\n| exact proof_irrel_heq H.left hb\n[GOAL]\ncase e_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b) = b\n[PROOFSTEP]\nexact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right\n[GOAL]\ncase e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb\n[PROOFSTEP]\nfirst\n| exact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right\n| exact proof_irrel_heq H.left hb\n[GOAL]\ncase e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb\n[PROOFSTEP]\nexact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right\n[GOAL]\ncase e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb\n[PROOFSTEP]\nexact proof_irrel_heq H.left hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u00acIsSuccLimit a \u2194 a \u2208 range succ\n[PROOFSTEP]\nsimp_rw [isSuccLimit_iff_succ_ne, not_forall, not_ne_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 (\u2203 x, succ x = a) \u2194 a \u2208 range succ\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : IsSuccArchimedean \u03b1\nh : IsSuccLimit a\nb : \u03b1\nhb : b \u2264 a\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : IsSuccArchimedean \u03b1\nb : \u03b1\nhb : b \u2264 a\n\u22a2 IsSuccLimit a \u2192 a \u2264 b\n[PROOFSTEP]\nrefine' Succ.rec (fun _ => le_rfl) (fun c _ H hc => _) hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : IsSuccArchimedean \u03b1\nb : \u03b1\nhb : b \u2264 a\nc : \u03b1\nx\u271d : b \u2264 c\nH : IsSuccLimit c \u2192 c \u2264 b\nhc : IsSuccLimit (succ c)\n\u22a2 succ c \u2264 b\n[PROOFSTEP]\nhave := hc.isMax.succ_eq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : IsSuccArchimedean \u03b1\nb : \u03b1\nhb : b \u2264 a\nc : \u03b1\nx\u271d : b \u2264 c\nH : IsSuccLimit c \u2192 c \u2264 b\nhc : IsSuccLimit (succ c)\nthis : succ c = c\n\u22a2 succ c \u2264 b\n[PROOFSTEP]\nrw [this] at hc \u22a2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d : IsSuccArchimedean \u03b1\nb : \u03b1\nhb : b \u2264 a\nc : \u03b1\nx\u271d : b \u2264 c\nH : IsSuccLimit c \u2192 c \u2264 b\nhc : IsSuccLimit c\nthis : succ c = c\n\u22a2 c \u2264 b\n[PROOFSTEP]\nexact H hc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 \u00acIsSuccLimit a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LT \u03b1\na\u271d a : \u03b1\n\u22a2 \u00acIsPredLimit a \u2194 \u2203 b, a \u22d6 b\n[PROOFSTEP]\nsimp [IsPredLimit]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LT \u03b1\na : \u03b1\n\u22a2 IsSuccLimit (\u2191toDual a) \u2194 IsPredLimit a\n[PROOFSTEP]\nsimp [IsSuccLimit, IsPredLimit]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LT \u03b1\na : \u03b1\n\u22a2 IsPredLimit (\u2191toDual a) \u2194 IsSuccLimit a\n[PROOFSTEP]\nsimp [IsSuccLimit, IsPredLimit]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nh : IsPredLimit (pred a)\n\u22a2 IsMin a\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nh : IsPredLimit (pred a)\nH : \u00acIsMin a\n\u22a2 False\n[PROOFSTEP]\nexact h a (pred_covby_of_not_isMin H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nha : \u00acIsMin a\n\u22a2 \u00acIsPredLimit (pred a)\n[PROOFSTEP]\ncontrapose! ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nha : IsPredLimit (pred a)\n\u22a2 IsMin a\n[PROOFSTEP]\nexact ha.isMin\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh : IsPredLimit a\nb : \u03b1\n\u22a2 pred b \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\ninst\u271d : NoMinOrder \u03b1\nb : \u03b1\nh : IsPredLimit (pred b)\n\u22a2 False\n[PROOFSTEP]\nexact not_isMin _ h.isMin\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b3 : PredOrder \u03b1\ninst\u271d\u00b2 : IsPredArchimedean \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u00acIsPredLimit a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\n\u22a2 \u00acIsPredLimit a \u2194 \u2203 b, \u00acIsMin b \u2227 pred b = a\n[PROOFSTEP]\nrw [\u2190 isSuccLimit_toDual_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\n\u22a2 \u00acIsSuccLimit (\u2191toDual a) \u2194 \u2203 b, \u00acIsMin b \u2227 pred b = a\n[PROOFSTEP]\nexact not_isSuccLimit_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\nh : \u00acIsPredLimit a\n\u22a2 a \u2208 range pred\n[PROOFSTEP]\ncases' not_isPredLimit_iff.1 h with b hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nh : \u00acIsPredLimit a\nb : \u03b1\nhb : \u00acIsMin b \u2227 pred b = a\n\u22a2 a \u2208 range pred\n[PROOFSTEP]\nexact \u27e8b, hb.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PredOrder \u03b1\na b : \u03b1\nC : \u03b1 \u2192 Sort u_2\ninst\u271d\u00b9 : IsPredArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u00acIsPredLimit a\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Order.SuccPred.Limit", "llama_tokens": 8241, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.291603056486972}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\nJ : Type v\ninst\u271d : Category.{v, v} J\nF : J \u2964 SheafedSpace C\nX Y : SheafedSpace C\nf g : X \u27f6 Y\n\u22a2 Epi (coequalizer.\u03c0 f g).base\n[PROOFSTEP]\nerw [\u2190 show _ = (coequalizer.\u03c0 f g).base from \u03b9_comp_coequalizerComparison f g (SheafedSpace.forget C)]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\nJ : Type v\ninst\u271d : Category.{v, v} J\nF : J \u2964 SheafedSpace C\nX Y : SheafedSpace C\nf g : X \u27f6 Y\n\u22a2 Epi (coequalizer.\u03c0 ((forget C).map f) ((forget C).map g) \u226b coequalizerComparison f g (forget C))\n[PROOFSTEP]\nrw [\u2190 PreservesCoequalizer.iso_hom]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasLimits C\nJ : Type v\ninst\u271d : Category.{v, v} J\nF : J \u2964 SheafedSpace C\nX Y : SheafedSpace C\nf g : X \u27f6 Y\n\u22a2 Epi (coequalizer.\u03c0 ((forget C).map f) ((forget C).map g) \u226b (PreservesCoequalizer.iso (forget C) f g).hom)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\nx : \u2191\u2191(colimit (F \u22d9 forgetToSheafedSpace)).toPresheafedSpace\n\u22a2 LocalRing \u2191(TopCat.Presheaf.stalk (colimit (F \u22d9 forgetToSheafedSpace)).toPresheafedSpace.presheaf x)\n[PROOFSTEP]\nobtain \u27e8i, y, \u27e8\u27e9\u27e9 := SheafedSpace.colimit_exists_rep (F \u22d9 forgetToSheafedSpace) x\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\n\u22a2 LocalRing\n \u2191(TopCat.Presheaf.stalk (colimit (F \u22d9 forgetToSheafedSpace)).toPresheafedSpace.presheaf\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y))\n[PROOFSTEP]\nhaveI : LocalRing (((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) := (F.obj i).localRing _\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\nthis : LocalRing \u2191(PresheafedSpace.stalk ((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace y)\n\u22a2 LocalRing\n \u2191(TopCat.Presheaf.stalk (colimit (F \u22d9 forgetToSheafedSpace)).toPresheafedSpace.presheaf\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y))\n[PROOFSTEP]\nexact\n (asIso\n (PresheafedSpace.stalkMap\n (colimit.\u03b9 (C := SheafedSpace.{u + 1, u, u} CommRingCatMax.{u, u}) (F \u22d9 forgetToSheafedSpace) i : _)\n y)).symm.commRingCatIsoToRingEquiv.localRing\n[GOAL]\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\nx\u271d\u00b2 x\u271d\u00b9 : Discrete \u03b9\nj j' : \u03b9\nx\u271d : { as := j } \u27f6 { as := j' }\nf : j = j'\n\u22a2 F.map { down := { down := f } } \u226b\n (fun j =>\n { val := colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(F.obj j).toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j) x)) })\n { as := j' } =\n (fun j =>\n { val := colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(F.obj j).toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j) x)) })\n { as := j } \u226b\n ((Functor.const (Discrete \u03b9)).obj (coproduct F)).map { down := { down := f } }\n[PROOFSTEP]\nsubst f\n[GOAL]\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\nx\u271d\u00b2 x\u271d\u00b9 : Discrete \u03b9\nj : \u03b9\nx\u271d : { as := j } \u27f6 { as := j }\n\u22a2 F.map { down := { down := (_ : j = j) } } \u226b\n (fun j =>\n { val := colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(F.obj j).toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j) x)) })\n { as := j } =\n (fun j =>\n { val := colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(F.obj j).toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) j) x)) })\n { as := j } \u226b\n ((Functor.const (Discrete \u03b9)).obj (coproduct F)).map { down := { down := (_ : j = j) } }\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\n\u22a2 \u2200 (x : \u2191\u2191(coproductCofan F).pt.toPresheafedSpace),\n IsLocalRingHom\n (PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) x)\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\nx : \u2191\u2191(coproductCofan F).pt.toPresheafedSpace\n\u22a2 IsLocalRingHom\n (PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) x)\n[PROOFSTEP]\nobtain \u27e8i, y, \u27e8\u27e9\u27e9 := SheafedSpace.colimit_exists_rep (F \u22d9 forgetToSheafedSpace) x\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\n\u22a2 IsLocalRingHom\n (PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y))\n[PROOFSTEP]\nhave :=\n PresheafedSpace.stalkMap.comp\n (colimit.\u03b9 (C := SheafedSpace.{u + 1, u, u} CommRingCatMax.{u, u}) (F \u22d9 forgetToSheafedSpace) i)\n (colimit.desc (C := SheafedSpace.{u + 1, u, u} CommRingCatMax.{u, u}) (F \u22d9 forgetToSheafedSpace)\n (forgetToSheafedSpace.mapCocone s))\n y\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\nthis :\n PresheafedSpace.stalkMap\n (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n y =\n PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y) \u226b\n PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y\n\u22a2 IsLocalRingHom\n (PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y))\n[PROOFSTEP]\nrw [\u2190 IsIso.comp_inv_eq] at this \n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\nthis :\n PresheafedSpace.stalkMap\n (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n y \u226b\n inv (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y) =\n PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y)\n\u22a2 IsLocalRingHom\n (PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y))\n[PROOFSTEP]\nerw [\u2190 this,\n PresheafedSpace.stalkMap.congr_hom _ _\n (colimit.\u03b9_desc (C := SheafedSpace.{u + 1, u, u} CommRingCatMax.{u, u}) (forgetToSheafedSpace.mapCocone s) i : _)]\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\nthis :\n PresheafedSpace.stalkMap\n (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n y \u226b\n inv (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y) =\n PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y)\n\u22a2 IsLocalRingHom\n ((eqToHom\n (_ :\n PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).base\n y) =\n PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace\n (\u2191(NatTrans.app (forgetToSheafedSpace.mapCocone s).\u03b9 i).base y)) \u226b\n PresheafedSpace.stalkMap (NatTrans.app (forgetToSheafedSpace.mapCocone s).\u03b9 i) y) \u226b\n inv (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y))\n[PROOFSTEP]\nhaveI : IsLocalRingHom (PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).\u03b9.app i) y) := (s.\u03b9.app i).2 y\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u\nF : Discrete \u03b9 \u2964 LocallyRingedSpace\ns : Cocone F\ni : Discrete \u03b9\ny : \u2191\u2191((F \u22d9 forgetToSheafedSpace).obj i).toPresheafedSpace\nthis\u271d :\n PresheafedSpace.stalkMap\n (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n y \u226b\n inv (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y) =\n PresheafedSpace.stalkMap (colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i).base y)\nthis : IsLocalRingHom (PresheafedSpace.stalkMap (NatTrans.app (forgetToSheafedSpace.mapCocone s).\u03b9 i) y)\n\u22a2 IsLocalRingHom\n ((eqToHom\n (_ :\n PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace\n (\u2191(colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i \u226b\n colimit.desc (F \u22d9 forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).base\n y) =\n PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace\n (\u2191(NatTrans.app (forgetToSheafedSpace.mapCocone s).\u03b9 i).base y)) \u226b\n PresheafedSpace.stalkMap (NatTrans.app (forgetToSheafedSpace.mapCocone s).\u03b9 i) y) \u226b\n inv (PresheafedSpace.stalkMap (colimit.\u03b9 (F \u22d9 forgetToSheafedSpace) i) y))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\n\u22a2 IsLocalRingHom (NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U))\n[PROOFSTEP]\nhave := \u03b9_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nthis :\n coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n coequalizerComparison f.val g.val SheafedSpace.forgetToPresheafedSpace =\n SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)\n\u22a2 IsLocalRingHom (NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U))\n[PROOFSTEP]\nrw [\u2190 PreservesCoequalizer.iso_hom] at this \n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nthis :\n coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =\n SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)\n\u22a2 IsLocalRingHom (NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U))\n[PROOFSTEP]\nerw [SheafedSpace.congr_app this.symm (op U)]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nthis :\n coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =\n SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)\n\u22a2 IsLocalRingHom\n (NatTrans.app\n (coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).c\n (op U) \u226b\n Y.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj\n (op U) =\n (Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)).base).op.obj (op U))))\n[PROOFSTEP]\nrw [PresheafedSpace.comp_c_app, \u2190 PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_\u03c0]\n -- Porting note : this instance has to be manually added\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nthis :\n coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =\n SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)\n\u22a2 IsLocalRingHom\n ((NatTrans.app (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c (op U) \u226b\n (PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit\n (parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val))\n ((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj\n (op U).unop)).hom \u226b\n limit.\u03c0\n (PresheafedSpace.componentwiseDiagram\n (parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val))\n ((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj\n (op U).unop))\n (op WalkingParallelPair.one)) \u226b\n Y.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj\n (op U) =\n (Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)).base).op.obj (op U))))\n[PROOFSTEP]\nhaveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=\n PresheafedSpace.c_isIso_of_iso _\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nthis\u271d :\n coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =\n SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)\nthis : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c\n\u22a2 IsLocalRingHom\n ((NatTrans.app (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c (op U) \u226b\n (PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit\n (parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val))\n ((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj\n (op U).unop)).hom \u226b\n limit.\u03c0\n (PresheafedSpace.componentwiseDiagram\n (parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val))\n ((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj\n (op U).unop))\n (op WalkingParallelPair.one)) \u226b\n Y.presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (coequalizer.\u03c0 (SheafedSpace.forgetToPresheafedSpace.map f.val)\n (SheafedSpace.forgetToPresheafedSpace.map g.val) \u226b\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj\n (op U) =\n (Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.\u03c0 f.val g.val)).base).op.obj (op U))))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier) =\n (imageBasicOpen f g U s).carrier\n[PROOFSTEP]\nfapply\n Types.coequalizer_preimage_image_eq_of_preimage_eq (f.1.base : X.carrier.1 \u27f6 Y.carrier.1)\n (g.1.base : X.carrier.1 \u27f6 Y.carrier.1)\n[GOAL]\ncase e\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191f.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base = \u2191g.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base\n[PROOFSTEP]\next\n[GOAL]\ncase e.h\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\na\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 (\u2191f.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base) a\u271d = (\u2191g.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base) a\u271d\n[PROOFSTEP]\nsimp_rw [types_comp_apply, \u2190 TopCat.comp_app, \u2190 PresheafedSpace.comp_base]\n[GOAL]\ncase e.h\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\na\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 \u2191(f.val \u226b coequalizer.\u03c0 f.val g.val).base a\u271d = \u2191(g.val \u226b coequalizer.\u03c0 f.val g.val).base a\u271d\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e.h.e_a.e_self\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\na\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace\n\u22a2 f.val \u226b coequalizer.\u03c0 f.val g.val = g.val \u226b coequalizer.\u03c0 f.val g.val\n[PROOFSTEP]\nexact coequalizer.condition f.1 g.1\n[GOAL]\ncase h\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsColimit\n (Cofork.of\u03c0 \u2191(coequalizer.\u03c0 f.val g.val).base\n (_ : \u2191f.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base = \u2191g.val.base \u226b \u2191(coequalizer.\u03c0 f.val g.val).base))\n[PROOFSTEP]\napply isColimitCoforkMapOfIsColimit (forget TopCat)\n[GOAL]\ncase h.l\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsColimit (Cofork.of\u03c0 (coequalizer.\u03c0 f.val g.val).base ?h.w)\ncase h.w\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 f.val.base \u226b (coequalizer.\u03c0 f.val g.val).base = g.val.base \u226b (coequalizer.\u03c0 f.val g.val).base\n[PROOFSTEP]\napply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)\n[GOAL]\ncase h.l.l\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsColimit (Cofork.of\u03c0 (coequalizer.\u03c0 f.val g.val) ?h.l.w)\ncase h.l.w\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 f.val \u226b coequalizer.\u03c0 f.val g.val = g.val \u226b coequalizer.\u03c0 f.val g.val\n[PROOFSTEP]\nexact coequalizerIsCoequalizer f.1 g.1\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191f.val.base \u207b\u00b9' (imageBasicOpen f g U s).carrier = \u2191g.val.base \u207b\u00b9' (imageBasicOpen f g U s).carrier\n[PROOFSTEP]\nsuffices\n (TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =\n (TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)\n by injection this\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nthis : (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s)\n\u22a2 \u2191f.val.base \u207b\u00b9' (imageBasicOpen f g U s).carrier = \u2191g.val.base \u207b\u00b9' (imageBasicOpen f g U s).carrier\n[PROOFSTEP]\ninjection this\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s)\n[PROOFSTEP]\ndelta imageBasicOpen\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 (Opens.map f.val.base).obj\n (RingedSpace.basicOpen (toRingedSpace Y)\n (let_fun this := \u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s;\n this)) =\n (Opens.map g.val.base).obj\n (RingedSpace.basicOpen (toRingedSpace Y)\n (let_fun this := \u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s;\n this))\n[PROOFSTEP]\nrw [preimage_basicOpen f, preimage_basicOpen g]\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 RingedSpace.basicOpen (toRingedSpace X)\n (\u2191(NatTrans.app f.val.c\n (op\n {\n unop :=\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))\n (let_fun this := \u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s;\n this)) =\n RingedSpace.basicOpen (toRingedSpace X)\n (\u2191(NatTrans.app g.val.c\n (op\n {\n unop :=\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop,\n is_open' := (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop)) } }.unop))\n (let_fun this := \u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s;\n this))\n[PROOFSTEP]\ndsimp only [Functor.op, unop_op]\n -- Porting note : change `rw` to `erw`\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 RingedSpace.basicOpen (toRingedSpace X)\n (\u2191(NatTrans.app f.val.c\n (op\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191U,\n is_open' := (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop)) }))\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s)) =\n RingedSpace.basicOpen (toRingedSpace X)\n (\u2191(NatTrans.app g.val.c\n (op\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191U,\n is_open' := (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' \u2191(op U).unop)) }))\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\n[PROOFSTEP]\nerw [\u2190 comp_apply, \u2190 SheafedSpace.comp_c_app', \u2190 comp_apply, \u2190 SheafedSpace.comp_c_app',\n SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply, X.toRingedSpace.basicOpen_res]\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 ((Opens.map (f.val \u226b coequalizer.\u03c0 f.val g.val).base).op.obj (op U)).unop \u2293\n RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app (g.val \u226b coequalizer.\u03c0 f.val g.val).c (op U)) s) =\n RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app (g.val \u226b coequalizer.\u03c0 f.val g.val).c (op U)) s)\n[PROOFSTEP]\napply inf_eq_right.mpr\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app (g.val \u226b coequalizer.\u03c0 f.val g.val).c (op U)) s) \u2264\n ((Opens.map (f.val \u226b coequalizer.\u03c0 f.val g.val).base).op.obj (op U)).unop\n[PROOFSTEP]\nrefine' (RingedSpace.basicOpen_le _ _).trans _\n[GOAL]\ncase H\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 ((Opens.map (g.val \u226b coequalizer.\u03c0 f.val g.val).base).op.obj (op U)).unop \u2264\n ((Opens.map (f.val \u226b coequalizer.\u03c0 f.val g.val).base).op.obj (op U)).unop\n[PROOFSTEP]\nrw [coequalizer.condition f.1 g.1]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier)\n[PROOFSTEP]\nrw [\u2190 (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1 g.1)).isOpen_preimage,\n TopCat.coequalizer_isOpen_iff, \u2190 Set.preimage_comp]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen\n (\u2191(TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget CommRingCat) f.val g.val)) \u2218\n \u2191(colimit.\u03b9\n (parallelPair ((SheafedSpace.forget CommRingCat).map f.val) ((SheafedSpace.forget CommRingCat).map g.val))\n WalkingParallelPair.one) \u207b\u00b9'\n (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier))\n[PROOFSTEP]\nerw [\u2190 coe_comp]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen\n (\u2191(colimit.\u03b9\n (parallelPair ((SheafedSpace.forget CommRingCat).map f.val) ((SheafedSpace.forget CommRingCat).map g.val))\n WalkingParallelPair.one \u226b\n (PreservesCoequalizer.iso (SheafedSpace.forget CommRingCat) f.val g.val).hom) \u207b\u00b9'\n (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier))\n[PROOFSTEP]\nrw [PreservesCoequalizer.iso_hom, \u03b9_comp_coequalizerComparison]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen\n (\u2191((SheafedSpace.forget CommRingCat).map (coequalizer.\u03c0 f.val g.val)) \u207b\u00b9'\n (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier))\n[PROOFSTEP]\ndsimp only [SheafedSpace.forget]\n -- Porting note : change `rw` to `erw`\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' (imageBasicOpen f g U s).carrier))\n[PROOFSTEP]\nerw [imageBasicOpen_image_preimage]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 IsOpen (imageBasicOpen f g U s).carrier\n[PROOFSTEP]\nexact (imageBasicOpen f g U s).2\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nx : \u2191(toTopCat Y)\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val) x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase map_nonunit\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nx : \u2191(toTopCat Y)\n\u22a2 \u2200 (a : \u2191(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base x))),\n IsUnit (\u2191(PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val) x) a) \u2192 IsUnit a\n[PROOFSTEP]\nrintro a ha\n[GOAL]\ncase map_nonunit\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nx : \u2191(toTopCat Y)\na : \u2191(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base x))\nha : IsUnit (\u2191(PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val) x) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nrcases TopCat.Presheaf.germ_exist _ _ a with \u27e8U, hU, s, rfl\u27e9\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val) x)\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s))\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap_germ_apply (coequalizer.\u03c0 f.1 g.1 : _) U \u27e8_, hU\u27e9] at ha \n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nlet V := imageBasicOpen f g U s\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nhave hV : (coequalizer.\u03c0 f.1 g.1).base \u207b\u00b9' ((coequalizer.\u03c0 f.1 g.1).base '' V.1) = V.1 :=\n imageBasicOpen_image_preimage f g U s\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nhave hV' : V = \u27e8(coequalizer.\u03c0 f.1 g.1).base \u207b\u00b9' ((coequalizer.\u03c0 f.1 g.1).base '' V.1), hV.symm \u25b8 V.2\u27e9 :=\n SetLike.ext' hV.symm\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nhave V_open : IsOpen ((coequalizer.\u03c0 f.val g.val).base '' V.1) := imageBasicOpen_image_open f g U s\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nhave VleU : (\u27e8(coequalizer.\u03c0 f.val g.val).base '' V.1, V_open\u27e9 : TopologicalSpace.Opens _) \u2264 U :=\n Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nhave hxV : x \u2208 V := \u27e8\u27e8_, hU\u27e9, ha, rfl\u27e9\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\nhxV : x \u2208 V\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x, property := hU })\n s)\n[PROOFSTEP]\nerw [\u2190\n (coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)\n \u27e8_, @Set.mem_image_of_mem _ _ (coequalizer.\u03c0 f.val g.val).base x V.1 hxV\u27e9 s]\n[GOAL]\ncase map_nonunit.intro.intro.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\nhxV : x \u2208 V\n\u22a2 IsUnit\n (\u2191(TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf\n { val := \u2191(coequalizer.\u03c0 f.val g.val).base x,\n property := (_ : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) })\n (\u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.map (homOfLE VleU).op) s))\n[PROOFSTEP]\napply RingHom.isUnit_map\n[GOAL]\ncase map_nonunit.intro.intro.intro.a\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\nhxV : x \u2208 V\n\u22a2 IsUnit (\u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.map (homOfLE VleU).op) s)\n[PROOFSTEP]\nrw [\u2190 isUnit_map_iff ((coequalizer.\u03c0 f.val g.val : _).c.app _), \u2190 comp_apply, NatTrans.naturality, comp_apply,\n TopCat.Presheaf.pushforwardObj_map, \u2190 isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]\n -- Porting note : change `rw` to `erw`\n[GOAL]\ncase map_nonunit.intro.intro.intro.a\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\nhxV : x \u2208 V\n\u22a2 IsUnit\n (\u2191(Y.presheaf.map (eqToHom hV').op)\n (\u2191(Y.presheaf.map ((Opens.map (coequalizer.\u03c0 f.val g.val).base).op.map (homOfLE VleU).op))\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s)))\n[PROOFSTEP]\nerw [\u2190 comp_apply, \u2190 comp_apply, Category.assoc, \u2190 Y.presheaf.map_comp]\n[GOAL]\ncase map_nonunit.intro.intro.intro.a\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nU\u271d : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\ns\u271d : \u2191((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U\u271d))\nx : \u2191(toTopCat Y)\nU : Opens \u2191\u2191(coequalizer f.val g.val).toPresheafedSpace\nhU : \u2191(coequalizer.\u03c0 f.val g.val).base x \u2208 U\ns : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))\nha :\n IsUnit\n (\u2191(TopCat.Presheaf.germ Y.presheaf { val := x, property := hU })\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U)) s))\nV : Opens \u2191(toTopCat Y) := imageBasicOpen f g U s\nhV : \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier) = V.carrier\nhV' :\n V =\n { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier),\n is_open' :=\n (_ : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base \u207b\u00b9' (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier))) }\nV_open : IsOpen (\u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier)\nVleU : { carrier := \u2191(coequalizer.\u03c0 f.val g.val).base '' V.carrier, is_open' := V_open } \u2264 U\nhxV : x \u2208 V\n\u22a2 IsUnit\n (\u2191(NatTrans.app (coequalizer.\u03c0 f.val g.val).c (op U) \u226b\n Y.presheaf.map ((Opens.map (coequalizer.\u03c0 f.val g.val).base).op.map (homOfLE VleU).op \u226b (eqToHom hV').op))\n s)\n[PROOFSTEP]\nconvert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _) (((coequalizer.\u03c0 f.val g.val).c.app (op U)) s)\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nx : \u2191\u2191(Limits.coequalizer f.val g.val).toPresheafedSpace\n\u22a2 LocalRing \u2191(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf x)\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := (TopCat.epi_iff_surjective (coequalizer.\u03c0 f.val g.val).base).mp inferInstance x\n[GOAL]\ncase intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 LocalRing\n \u2191(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf\n (\u2191(coequalizer.\u03c0 f.val g.val).base y))\n[PROOFSTEP]\nexact (PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val : _) y).domain_localRing\n[GOAL]\nX\u271d Y\u271d : LocallyRingedSpace\nf\u271d g\u271d : X\u271d \u27f6 Y\u271d\nX Y : RingedSpace\nf g : X \u27f6 Y\nH : f = g\nx : \u2191\u2191X.toPresheafedSpace\nh : IsLocalRingHom (PresheafedSpace.stalkMap f x)\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap g x)\n[PROOFSTEP]\nrw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]\n[GOAL]\nX\u271d Y\u271d : LocallyRingedSpace\nf\u271d g\u271d : X\u271d \u27f6 Y\u271d\nX Y : RingedSpace\nf g : X \u27f6 Y\nH : f = g\nx : \u2191\u2191X.toPresheafedSpace\nh : IsLocalRingHom (PresheafedSpace.stalkMap f x)\n\u22a2 IsLocalRingHom\n (eqToHom\n (_ :\n PresheafedSpace.stalk Y.toPresheafedSpace (\u2191g.base x) =\n PresheafedSpace.stalk Y.toPresheafedSpace (\u2191f.base x)) \u226b\n PresheafedSpace.stalkMap f x)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\n\u22a2 IsColimit (coequalizerCofork f g)\n[PROOFSTEP]\napply Cofork.IsColimit.mk'\n[GOAL]\ncase create\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\n\u22a2 (s : Cofork f g) \u2192\n { l //\n Cofork.\u03c0 (coequalizerCofork f g) \u226b l = Cofork.\u03c0 s \u2227\n \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s \u2192 m = l }\n[PROOFSTEP]\nintro s\n[GOAL]\ncase create\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\n\u22a2 { l //\n Cofork.\u03c0 (coequalizerCofork f g) \u226b l = Cofork.\u03c0 s \u2227\n \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s \u2192 m = l }\n[PROOFSTEP]\nhave e : f.val \u226b s.\u03c0.val = g.val \u226b s.\u03c0.val := by injection s.condition\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\n\u22a2 f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n[PROOFSTEP]\ninjection s.condition\n[GOAL]\ncase create\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n\u22a2 { l //\n Cofork.\u03c0 (coequalizerCofork f g) \u226b l = Cofork.\u03c0 s \u2227\n \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s \u2192 m = l }\n[PROOFSTEP]\nrefine \u27e8\u27e8coequalizer.desc s.\u03c0.1 e, ?_\u27e9, ?_\u27e9\n[GOAL]\ncase create.refine_1\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n\u22a2 \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase create.refine_1\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nx : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)\n[PROOFSTEP]\nrcases(TopCat.epi_iff_surjective (coequalizer.\u03c0 f.val g.val).base).mp inferInstance x with\n \u27e8y, rfl\u27e9\n -- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this\n -- used to allow you to provide the proof that `... \u226b ...` is a local ring homomorphism later,\n -- but this is no longer possible\n[GOAL]\ncase create.refine_1.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y))\n[PROOFSTEP]\nset h := _\n[GOAL]\ncase create.refine_1.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : ?m.154167 := ?m.154168\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y))\n[PROOFSTEP]\nchange IsLocalRingHom h\n[GOAL]\ncase create.refine_1.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\n\u22a2 IsLocalRingHom h\n[PROOFSTEP]\nsuffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).\u03c0.1 _).comp h)\n[GOAL]\ncase create.refine_1.intro\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\nthis : IsLocalRingHom (RingHom.comp (PresheafedSpace.stalkMap (Cofork.\u03c0 (coequalizerCofork f g)).val y) h)\n\u22a2 IsLocalRingHom h\n[PROOFSTEP]\napply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).\u03c0.1 _)\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\n\u22a2 IsLocalRingHom (RingHom.comp (PresheafedSpace.stalkMap (Cofork.\u03c0 (coequalizerCofork f g)).val y) h)\n[PROOFSTEP]\nchange IsLocalRingHom (_ \u226b PresheafedSpace.stalkMap (coequalizerCofork f g).\u03c0.val y)\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\n\u22a2 IsLocalRingHom (h \u226b PresheafedSpace.stalkMap (Cofork.\u03c0 (coequalizerCofork f g)).val y)\n[PROOFSTEP]\nerw [\u2190 PresheafedSpace.stalkMap.comp]\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.\u03c0 f.val g.val \u226b coequalizer.desc (Cofork.\u03c0 s).val e) y)\n[PROOFSTEP]\napply isLocalRingHom_stalkMap_congr _ _ (coequalizer.\u03c0_desc s.\u03c0.1 e).symm y\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\ny : (forget TopCat).obj \u2191Y.toPresheafedSpace\nh : \u2191(PresheafedSpace.stalk s.pt.toPresheafedSpace\n (\u2191(coequalizer.desc (Cofork.\u03c0 s).val e).base (\u2191(coequalizer.\u03c0 f.val g.val).base y))) \u2192+*\n \u2191(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace (\u2191(coequalizer.\u03c0 f.val g.val).base y)) :=\n PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) (\u2191(coequalizer.\u03c0 f.val g.val).base y)\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (Cofork.\u03c0 s).val y)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase create.refine_2\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n\u22a2 Cofork.\u03c0 (coequalizerCofork f g) \u226b\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) } =\n Cofork.\u03c0 s \u2227\n \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s \u2192\n m =\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase create.refine_2.left\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n\u22a2 Cofork.\u03c0 (coequalizerCofork f g) \u226b\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) } =\n Cofork.\u03c0 s\n[PROOFSTEP]\nexact LocallyRingedSpace.Hom.ext _ _ (coequalizer.\u03c0_desc _ _)\n[GOAL]\ncase create.refine_2.right\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\n\u22a2 \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s \u2192\n m =\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) }\n[PROOFSTEP]\nintro m h\n[GOAL]\ncase create.refine_2.right\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 m =\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) }\n[PROOFSTEP]\nreplace h : (coequalizerCofork f g).\u03c0.1 \u226b m.1 = s.\u03c0.1 := by rw [\u2190 h]; rfl\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : Cofork.\u03c0 (coequalizerCofork f g) \u226b m = Cofork.\u03c0 s\n\u22a2 (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 (coequalizerCofork f g) \u226b m).val\n[PROOFSTEP]\nrfl\n[GOAL]\ncase create.refine_2.right\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 m =\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) }\n[PROOFSTEP]\napply LocallyRingedSpace.Hom.ext\n[GOAL]\ncase create.refine_2.right.val\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 m.val =\n { val := coequalizer.desc (Cofork.\u03c0 s).val e,\n prop :=\n (_ :\n \u2200 (x : \u2191\u2191(coequalizerCofork f g).pt.toPresheafedSpace),\n IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.\u03c0 s).val e) x)) }.val\n[PROOFSTEP]\napply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.of\u03c0 s.\u03c0.1 e) m.1\n[GOAL]\ncase create.refine_2.right.val\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 \u2200 (j : WalkingParallelPair),\n NatTrans.app (colimit.cocone (parallelPair f.val g.val)).\u03b9 j \u226b m.val =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s).val e).\u03b9 j\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase create.refine_2.right.val.zero\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 NatTrans.app (colimit.cocone (parallelPair f.val g.val)).\u03b9 WalkingParallelPair.zero \u226b m.val =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s).val e).\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nrw [\u2190 (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left, Category.assoc]\n[GOAL]\ncase create.refine_2.right.val.zero\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 (parallelPair f.val g.val).map WalkingParallelPairHom.left \u226b\n NatTrans.app (colimit.cocone (parallelPair f.val g.val)).\u03b9 WalkingParallelPair.one \u226b m.val =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s).val e).\u03b9 WalkingParallelPair.zero\n[PROOFSTEP]\nchange _ \u226b _ \u226b _ = _ \u226b _\n[GOAL]\ncase create.refine_2.right.val.zero\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 (parallelPair f.val g.val).map WalkingParallelPairHom.left \u226b\n NatTrans.app (colimit.cocone (parallelPair f.val g.val)).\u03b9 WalkingParallelPair.one \u226b m.val =\n f.val \u226b (Cofork.\u03c0 s).val\n[PROOFSTEP]\ncongr\n[GOAL]\ncase create.refine_2.right.val.one\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\ns : Cofork f g\ne : f.val \u226b (Cofork.\u03c0 s).val = g.val \u226b (Cofork.\u03c0 s).val\nm :\n ((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one \u27f6\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nh : (Cofork.\u03c0 (coequalizerCofork f g)).val \u226b m.val = (Cofork.\u03c0 s).val\n\u22a2 NatTrans.app (colimit.cocone (parallelPair f.val g.val)).\u03b9 WalkingParallelPair.one \u226b m.val =\n NatTrans.app (Cofork.of\u03c0 (Cofork.\u03c0 s).val e).\u03b9 WalkingParallelPair.one\n[PROOFSTEP]\nexact h\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\n\u22a2 PreservesColimit F forgetToSheafedSpace\n[PROOFSTEP]\nsuffices :\n PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))\n forgetToSheafedSpace\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\nthis :\n PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))\n forgetToSheafedSpace\n\u22a2 PreservesColimit F forgetToSheafedSpace\n[PROOFSTEP]\napply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\n\u22a2 PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))\n forgetToSheafedSpace\n[PROOFSTEP]\napply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)\n[GOAL]\ncase this\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\n\u22a2 IsColimit\n (forgetToSheafedSpace.mapCocone\n (coequalizerCofork (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right)))\n[PROOFSTEP]\napply (isColimitMapCoconeCoforkEquiv _ _).symm _\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\n\u22a2 IsColimit\n (Cofork.of\u03c0\n (forgetToSheafedSpace.map\n { val := coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,\n prop :=\n (_ :\n \u2200 (x : \u2191(toTopCat (F.obj WalkingParallelPair.one))),\n IsLocalRingHom\n (PresheafedSpace.stalkMap\n (coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val)\n x)) })\n (_ :\n forgetToSheafedSpace.map (F.map WalkingParallelPairHom.left) \u226b\n forgetToSheafedSpace.map\n { val := coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,\n prop :=\n (_ :\n \u2200 (x : \u2191(toTopCat (F.obj WalkingParallelPair.one))),\n IsLocalRingHom\n (PresheafedSpace.stalkMap\n (coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val\n (F.map WalkingParallelPairHom.right).val)\n x)) } =\n forgetToSheafedSpace.map (F.map WalkingParallelPairHom.right) \u226b\n forgetToSheafedSpace.map\n { val := coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,\n prop :=\n (_ :\n \u2200 (x : \u2191(toTopCat (F.obj WalkingParallelPair.one))),\n IsLocalRingHom\n (PresheafedSpace.stalkMap\n (coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val\n (F.map WalkingParallelPairHom.right).val)\n x)) }))\n[PROOFSTEP]\ndsimp only [forgetToSheafedSpace]\n[GOAL]\nX Y : LocallyRingedSpace\nf g : X \u27f6 Y\nF : WalkingParallelPair \u2964 LocallyRingedSpace\n\u22a2 IsColimit\n (Cofork.of\u03c0 (coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val)\n (_ :\n (F.map WalkingParallelPairHom.left).val \u226b\n coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val =\n (F.map WalkingParallelPairHom.right).val \u226b\n coequalizer.\u03c0 (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val))\n[PROOFSTEP]\nexact coequalizerIsCoequalizer _ _\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.LocallyRingedSpace.HasColimits", "llama_tokens": 28174, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982179521105, "lm_q2_score": 0.4493926344647597, "lm_q1q2_score": 0.2911157477670755}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA : Mon_ C\nM : Mod_ A\n\u22a2 (\ud835\udfd9 A.X \u2297 M.act) \u226b M.act = (\u03b1_ A.X A.X M.X).inv \u226b (A.mul \u2297 \ud835\udfd9 M.X) \u226b M.act\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA : Mon_ C\nM\u271d M : Mod_ A\n\u22a2 (\ud835\udfd9 M).hom = \ud835\udfd9 M.X\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.one \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act = (\u03bb_ M.X).hom\n[PROOFSTEP]\nslice_lhs 1 2 => rw [\u2190 comp_tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (A.one \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (A.one \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (A.one \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.one \u226b f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act = (\u03bb_ M.X).hom\n[PROOFSTEP]\nrw [f.one_hom, one_act]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b (\ud835\udfd9 A.X \u2297 (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act\n[PROOFSTEP]\nslice_rhs 2 3 => rw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 A.X \u2297 (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 A.X \u2297 (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 A.X \u2297 (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act) \u226b (f.hom \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b ((f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\ud835\udfd9 B.X \u2297 (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act)) \u226b M.act\n[PROOFSTEP]\nrw [id_tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b ((f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X) \u226b (\ud835\udfd9 B.X \u2297 M.act)) \u226b M.act\n[PROOFSTEP]\nslice_rhs 4 5 => rw [Mod_.assoc_flip]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 M.act) \u226b M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| \ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X\n[PROOFSTEP]\nrw [Mod_.assoc_flip]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 M.act) \u226b M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| \ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X\n[PROOFSTEP]\nrw [Mod_.assoc_flip]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 M.act) \u226b M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| \ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X\n[PROOFSTEP]\nrw [Mod_.assoc_flip]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b\n (f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X) \u226b (\u03b1_ B.X B.X M.X).inv \u226b (B.mul \u2297 \ud835\udfd9 M.X) \u226b M.act\n[PROOFSTEP]\nslice_rhs 3 4 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X) \u226b (\u03b1_ B.X B.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X) \u226b (\u03b1_ B.X B.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom \u2297 \ud835\udfd9 M.X) \u226b (\u03b1_ B.X B.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b\n (f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (((\u03b1_ B.X A.X M.X).inv \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)) \u226b (B.mul \u2297 \ud835\udfd9 M.X)) \u226b M.act\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\u03b1_ B.X A.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\u03b1_ B.X A.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (f.hom \u2297 \ud835\udfd9 (A.X \u2297 M.X)) \u226b (\u03b1_ B.X A.X M.X).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n (\u03b1_ A.X A.X M.X).hom \u226b\n ((((\u03b1_ A.X A.X M.X).inv \u226b ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X)) \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)) \u226b (B.mul \u2297 \ud835\udfd9 M.X)) \u226b M.act\n[PROOFSTEP]\nslice_rhs 1 3 => rw [Iso.hom_inv_id_assoc]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom \u226b (\u03b1_ A.X A.X M.X).inv \u226b ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [Iso.hom_inv_id_assoc]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom \u226b (\u03b1_ A.X A.X M.X).inv \u226b ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [Iso.hom_inv_id_assoc]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\u03b1_ A.X A.X M.X).hom \u226b (\u03b1_ A.X A.X M.X).inv \u226b ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| (\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act =\n ((((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X) \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)) \u226b (B.mul \u2297 \ud835\udfd9 M.X)) \u226b M.act\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 comp_tensor_id, tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X) \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X) \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 \ud835\udfd9 A.X) \u2297 \ud835\udfd9 M.X) \u226b ((\ud835\udfd9 B.X \u2297 f.hom) \u2297 \ud835\udfd9 M.X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| B.mul \u2297 \ud835\udfd9 M.X\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, tensor_id_comp_id_tensor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act = (((f.hom \u2297 f.hom) \u2297 \ud835\udfd9 M.X) \u226b (B.mul \u2297 \ud835\udfd9 M.X)) \u226b M.act\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 comp_tensor_id, \u2190 f.mul_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 f.hom) \u2297 \ud835\udfd9 M.X) \u226b (B.mul \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, \u2190 f.mul_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 f.hom) \u2297 \ud835\udfd9 M.X) \u226b (B.mul \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, \u2190 f.mul_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| ((f.hom \u2297 f.hom) \u2297 \ud835\udfd9 M.X) \u226b (B.mul \u2297 \ud835\udfd9 M.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n| M.act\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, \u2190 f.mul_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM\u271d : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nM : Mod_ B\n\u22a2 (A.mul \u2297 \ud835\udfd9 M.X) \u226b (f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act = (A.mul \u226b f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act\n[PROOFSTEP]\nrw [comp_tensor_id, Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n\u22a2 ((fun M => mk M.X ((f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act)) X\u271d).act \u226b g.hom =\n (\ud835\udfd9 A.X \u2297 g.hom) \u226b ((fun M => mk M.X ((f.hom \u2297 \ud835\udfd9 M.X) \u226b M.act)) Y\u271d).act\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n\u22a2 ((f.hom \u2297 \ud835\udfd9 X\u271d.X) \u226b X\u271d.act) \u226b g.hom = (\ud835\udfd9 A.X \u2297 g.hom) \u226b (f.hom \u2297 \ud835\udfd9 Y\u271d.X) \u226b Y\u271d.act\n[PROOFSTEP]\nslice_rhs 1 2 => rw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 A.X \u2297 g.hom) \u226b (f.hom \u2297 \ud835\udfd9 Y\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| Y\u271d.act\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 A.X \u2297 g.hom) \u226b (f.hom \u2297 \ud835\udfd9 Y\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| Y\u271d.act\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 A.X \u2297 g.hom) \u226b (f.hom \u2297 \ud835\udfd9 Y\u271d.X)\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| Y\u271d.act\n[PROOFSTEP]\nrw [id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n\u22a2 ((f.hom \u2297 \ud835\udfd9 X\u271d.X) \u226b X\u271d.act) \u226b g.hom = ((f.hom \u2297 \ud835\udfd9 X\u271d.X) \u226b (\ud835\udfd9 B.X \u2297 g.hom)) \u226b Y\u271d.act\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 g.act_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 B.X \u2297 g.hom) \u226b Y\u271d.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| f.hom \u2297 \ud835\udfd9 X\u271d.X\n[PROOFSTEP]\nrw [\u2190 g.act_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 B.X \u2297 g.hom) \u226b Y\u271d.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| f.hom \u2297 \ud835\udfd9 X\u271d.X\n[PROOFSTEP]\nrw [\u2190 g.act_hom]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| (\ud835\udfd9 B.X \u2297 g.hom) \u226b Y\u271d.act\ncase a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n| f.hom \u2297 \ud835\udfd9 X\u271d.X\n[PROOFSTEP]\nrw [\u2190 g.act_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : MonoidalCategory C\nA\u271d : Mon_ C\nM : Mod_ A\u271d\nA B : Mon_ C\nf : A \u27f6 B\nX\u271d Y\u271d : Mod_ B\ng : X\u271d \u27f6 Y\u271d\n\u22a2 ((f.hom \u2297 \ud835\udfd9 X\u271d.X) \u226b X\u271d.act) \u226b g.hom = (f.hom \u2297 \ud835\udfd9 X\u271d.X) \u226b X\u271d.act \u226b g.hom\n[PROOFSTEP]\nrw [Category.assoc]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Mod_", "llama_tokens": 14196, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.29108612035508524}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ns : \u2115 \u2192 Set (NullMeasurableSpace \u03b1)\nhs : \u2200 (i : \u2115), (fun s => \u2203 t, MeasurableSet t \u2227 s =\u1d50[\u03bc] t) (s i)\n\u22a2 (fun s => \u2203 t, MeasurableSet t \u2227 s =\u1d50[\u03bc] t) (\u22c3 (i : \u2115), s i)\n[PROOFSTEP]\nchoose t htm hts using hs\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ns : \u2115 \u2192 Set (NullMeasurableSpace \u03b1)\nt : \u2115 \u2192 Set \u03b1\nhtm : \u2200 (i : \u2115), MeasurableSet (t i)\nhts : \u2200 (i : \u2115), s i =\u1d50[\u03bc] t i\n\u22a2 \u2203 t, MeasurableSet t \u2227 \u22c3 (i : \u2115), s i =\u1d50[\u03bc] t\n[PROOFSTEP]\nexact \u27e8\u22c3 i, t i, MeasurableSet.iUnion htm, EventuallyEq.countable_iUnion hts\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 NullMeasurableSet t\n\u22a2 NullMeasurableSet (\u22c3\u2080 s)\n[PROOFSTEP]\nrw [sUnion_eq_biUnion]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ns : Set (Set \u03b1)\nhs : Set.Countable s\nh : \u2200 (t : Set \u03b1), t \u2208 s \u2192 NullMeasurableSet t\n\u22a2 NullMeasurableSet (\u22c3 (i : Set \u03b1) (_ : i \u2208 s), i)\n[PROOFSTEP]\nexact MeasurableSet.biUnion hs h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nh : NullMeasurableSet s\n\u22a2 \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\n[PROOFSTEP]\nrcases h with \u27e8t, htm, hst\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d t : Set \u03b1\nhtm : MeasurableSet t\nhst : s =\u1d50[\u03bc] t\n\u22a2 \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\n[PROOFSTEP]\nrefine' \u27e8t \u222a toMeasurable \u03bc (s \\ t), _, htm.union (measurableSet_toMeasurable _ _), _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d t : Set \u03b1\nhtm : MeasurableSet t\nhst : s =\u1d50[\u03bc] t\n\u22a2 t \u222a toMeasurable \u03bc (s \\ t) \u2287 s\n[PROOFSTEP]\nexact diff_subset_iff.1 (subset_toMeasurable _ _)\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d t : Set \u03b1\nhtm : MeasurableSet t\nhst : s =\u1d50[\u03bc] t\n\u22a2 t \u222a toMeasurable \u03bc (s \\ t) =\u1d50[\u03bc] s\n[PROOFSTEP]\nhave : toMeasurable \u03bc (s \\ t) =\u1d50[\u03bc] (\u2205 : Set \u03b1) := by simp [ae_le_set.1 hst.le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d t : Set \u03b1\nhtm : MeasurableSet t\nhst : s =\u1d50[\u03bc] t\n\u22a2 toMeasurable \u03bc (s \\ t) =\u1d50[\u03bc] \u2205\n[PROOFSTEP]\nsimp [ae_le_set.1 hst.le]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d t : Set \u03b1\nhtm : MeasurableSet t\nhst : s =\u1d50[\u03bc] t\nthis : toMeasurable \u03bc (s \\ t) =\u1d50[\u03bc] \u2205\n\u22a2 t \u222a toMeasurable \u03bc (s \\ t) =\u1d50[\u03bc] s\n[PROOFSTEP]\nsimpa only [union_empty] using hst.symm.union this\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nh : NullMeasurableSet s\n\u22a2 toMeasurable \u03bc s =\u1d50[\u03bc] s\n[PROOFSTEP]\nrw [toMeasurable_def, dif_pos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nh : NullMeasurableSet s\n\u22a2 Exists.choose ?hc =\u1d50[\u03bc] s\ncase hc\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nh : NullMeasurableSet s\n\u22a2 \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\n[PROOFSTEP]\nexact (exists_measurable_superset_ae_eq h).choose_spec.snd.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)\n[PROOFSTEP]\nchoose t ht_sub htm ht_eq using fun i => exists_measurable_subset_ae_eq (h i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)\n[PROOFSTEP]\nrcases exists_null_pairwise_disjoint_diff hd with \u27e8u, hum, hu\u2080, hud\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nh : \u2200 (i : \u03b9), NullMeasurableSet (s i)\nhd : Pairwise (AEDisjoint \u03bc on s)\nt : \u03b9 \u2192 Set \u03b1\nht_sub : \u2200 (i : \u03b9), t i \u2286 s i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nht_eq : \u2200 (i : \u03b9), t i =\u1d50[\u03bc] s i\nu : \u03b9 \u2192 Set \u03b1\nhum : \u2200 (i : \u03b9), MeasurableSet (u i)\nhu\u2080 : \u2200 (i : \u03b9), \u2191\u2191\u03bc (u i) = 0\nhud : Pairwise (Disjoint on fun i => s i \\ u i)\n\u22a2 \u2203 t, (\u2200 (i : \u03b9), t i \u2286 s i) \u2227 (\u2200 (i : \u03b9), s i =\u1d50[\u03bc] t i) \u2227 (\u2200 (i : \u03b9), MeasurableSet (t i)) \u2227 Pairwise (Disjoint on t)\n[PROOFSTEP]\nexact\n \u27e8fun i => t i \\ u i, fun i => (diff_subset _ _).trans (ht_sub _), fun i =>\n (ht_eq _).symm.trans (diff_null_ae_eq_self (hu\u2080 i)).symm, fun i => (htm i).diff (hum i),\n hud.mono fun i j h => h.mono (diff_subset_diff_left (ht_sub i)) (diff_subset_diff_left (ht_sub j))\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhn : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b9), MeasurableSet (f i)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)\n[PROOFSTEP]\nrw [measure_eq_extend (MeasurableSet.iUnion h), extend_iUnion MeasurableSet.empty _ MeasurableSet.iUnion _ hn h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhn : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b9), MeasurableSet (f i)\n\u22a2 \u2211' (i : \u03b9), extend (fun t _ht => \u2191\u2191\u03bc t) (f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)\n[PROOFSTEP]\nsimp [measure_eq_extend, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhn : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b9), MeasurableSet (f i)\n\u22a2 \u2191\u2191\u03bc \u2205 = 0\n[PROOFSTEP]\nexact \u03bc.empty\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0\u271d : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhn : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b9), MeasurableSet (f i)\n\u22a2 \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984,\n (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 Pairwise (Disjoint on f) \u2192 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f i) = \u2211' (i : \u2115), \u2191\u2191\u03bc (f i)\n[PROOFSTEP]\nexact \u03bc.m_iUnion\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on f)\nh : \u2200 (i : \u03b9), NullMeasurableSet (f i)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)\n[PROOFSTEP]\nrcases exists_subordinate_pairwise_disjoint h hd with \u27e8t, _ht_sub, ht_eq, htm, htd\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nhd : Pairwise (AEDisjoint \u03bc on f)\nh : \u2200 (i : \u03b9), NullMeasurableSet (f i)\nt : \u03b9 \u2192 Set \u03b1\n_ht_sub : \u2200 (i : \u03b9), t i \u2286 f i\nht_eq : \u2200 (i : \u03b9), f i =\u1d50[\u03bc] t i\nhtm : \u2200 (i : \u03b9), MeasurableSet (t i)\nhtd : Pairwise (Disjoint on t)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), f i) = \u2211' (i : \u03b9), \u2191\u2191\u03bc (f i)\n[PROOFSTEP]\ncalc\n \u03bc (\u22c3 i, f i) = \u03bc (\u22c3 i, t i) := measure_congr (EventuallyEq.countable_iUnion ht_eq)\n _ = \u2211' i, \u03bc (t i) := (measure_iUnion htd htm)\n _ = \u2211' i, \u03bc (f i) := tsum_congr fun i => measure_congr (ht_eq _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [union_eq_iUnion, measure_iUnion\u2080, tsum_fintype, Fintype.sum_bool, cond, cond]\n[GOAL]\ncase hd\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 Pairwise (AEDisjoint \u03bc on fun b => bif b then s else t)\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2200 (i : Bool), NullMeasurableSet (bif i then s else t)\n[PROOFSTEP]\nexacts [(pairwise_on_bool AEDisjoint.symmetric).2 hd, fun b => Bool.casesOn b ht hs]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrefine' le_antisymm _ _\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nrcases exists_measurable_superset \u03bc s with \u27e8s', hsub, hs'm, hs'\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs'm : MeasurableSet s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nreplace hs'm : NullMeasurableSet s' \u03bc := hs'm.nullMeasurableSet\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\ns' : Set \u03b1\nhsub : s \u2286 s'\nhs' : \u2191\u2191\u03bc s' = \u2191\u2191\u03bc s\nhs'm : NullMeasurableSet s'\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t) \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\ncalc\n \u03bc (s \u2229 t) + \u03bc (s \\ t) \u2264 \u03bc (s' \u2229 t) + \u03bc (s' \\ t) :=\n add_le_add (measure_mono <| inter_subset_inter_left _ hsub) (measure_mono <| diff_subset_diff_left hsub)\n _ = \u03bc (s' \u2229 t \u222a s' \\ t) :=\n (measure_union\u2080_aux (hs'm.inter ht) (hs'm.diff ht) <| (@disjoint_inf_sdiff _ s' t _).aedisjoint).symm\n _ = \u03bc s' := (congr_arg \u03bc (inter_union_diff _ _))\n _ = \u03bc s := hs'\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\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (s \u2229 t) + \u2191\u2191\u03bc (s \\ t)\n[PROOFSTEP]\ncalc\n \u03bc s = \u03bc (s \u2229 t \u222a s \\ t) := by rw [inter_union_diff]\n _ \u2264 \u03bc (s \u2229 t) + \u03bc (s \\ t) := measure_union_le _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet t\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc (s \u2229 t \u222a s \\ t)\n[PROOFSTEP]\nrw [inter_union_diff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nht : NullMeasurableSet 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\u2080 (s \u222a t) ht, union_inter_cancel_right, union_diff_right, \u2190 measure_inter_add_diff\u2080 s ht,\n add_comm, \u2190 add_assoc, add_right_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nhs : NullMeasurableSet 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\u2080 t hs, add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nht : NullMeasurableSet t\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [\u2190 measure_union_add_inter\u2080 s ht, hd, add_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhs : NullMeasurableSet s\nhd : AEDisjoint \u03bc s t\n\u22a2 \u2191\u2191\u03bc (s \u222a t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [union_comm, measure_union\u2080 hs (AEDisjoint.symm hd), add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t s : Set \u03b1\nhs : NullMeasurableSet s\n\u22a2 \u2191\u2191\u03bc s + \u2191\u2191\u03bc s\u1d9c = \u2191\u2191\u03bc univ\n[PROOFSTEP]\nrw [\u2190 measure_union\u2080' hs aedisjoint_compl_right, union_compl_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\ninst\u271d : MeasurableSingletonClass (NullMeasurableSpace \u03b1)\ns : Finset \u03b1\n\u22a2 NullMeasurableSet \u2191s\n[PROOFSTEP]\napply Finset.measurableSet\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 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\n\u03bc : Measure \u03b1\ng : \u03b1 \u2192 \u03b2\nhf : NullMeasurable f\nhg : f =\u1d50[\u03bc] g\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b1\nhx : f x = g x\n\u22a2 x \u2208 f \u207b\u00b9' s \u2194 x \u2208 g \u207b\u00b9' s\n[PROOFSTEP]\nrw [mem_preimage, mem_preimage, hx]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u22a2 OuterMeasure.trim \u2191\u03bc = \u2191\u03bc\n[PROOFSTEP]\nrefine' le_antisymm (fun s => _) (@OuterMeasure.le_trim (NullMeasurableSpace \u03b1 \u03bc) _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\n\u22a2 \u2191(OuterMeasure.trim \u2191\u03bc) s \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [@OuterMeasure.trim_eq_iInf (NullMeasurableSpace \u03b1 \u03bc) _]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\n\u22a2 \u2a05 (t : Set (NullMeasurableSpace \u03b1)) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave : \u2200 s, \u03bc.toOuterMeasure s = \u03bc s := by simp only [forall_const]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\n\u22a2 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [forall_const]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\nthis : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s = \u2191\u2191\u03bc s\n\u22a2 \u2a05 (t : Set (NullMeasurableSpace \u03b1)) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [this, measure_eq_iInf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\nthis : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s = \u2191\u2191\u03bc s\n\u22a2 \u2a05 (t : Set (NullMeasurableSpace \u03b1)) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191\u2191\u03bc t \u2264\n \u2a05 (t : Set \u03b1) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191\u2191\u03bc t\n[PROOFSTEP]\napply iInf\u2082_mono\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d : Measure \u03b1\ns\u271d t : Set \u03b1\nx\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set (NullMeasurableSpace \u03b1)\nthis : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s = \u2191\u2191\u03bc s\n\u22a2 \u2200 (i : Set (NullMeasurableSpace \u03b1)), s \u2286 i \u2192 \u2a05 (_ : MeasurableSet i), \u2191\u2191\u03bc i \u2264 \u2a05 (_ : MeasurableSet i), \u2191\u2191\u03bc i\n[PROOFSTEP]\nexact fun t _ht => iInf_mono' fun h => \u27e8MeasurableSet.nullMeasurableSet h, le_rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.NullMeasurable", "llama_tokens": 7692, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.2910861073815311}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b1\ninst\u271d\u00b9 : IsSimpleOrder \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Finset.univ = {\u22a4, \u22a5}\n[PROOFSTEP]\nchange Finset.map _ (Finset.univ : Finset Bool) = _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b1\ninst\u271d\u00b9 : IsSimpleOrder \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Finset.map { toFun := \u2191IsSimpleOrder.equivBool.symm, inj' := (_ : Function.Injective \u2191IsSimpleOrder.equivBool.symm) }\n Finset.univ =\n {\u22a4, \u22a5}\n[PROOFSTEP]\nrw [Fintype.univ_bool]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b1\ninst\u271d\u00b9 : IsSimpleOrder \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Finset.map { toFun := \u2191IsSimpleOrder.equivBool.symm, inj' := (_ : Function.Injective \u2191IsSimpleOrder.equivBool.symm) }\n {true, false} =\n {\u22a4, \u22a5}\n[PROOFSTEP]\nsimp only [Finset.map_insert, Function.Embedding.coeFn_mk, Finset.map_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : BoundedOrder \u03b1\ninst\u271d\u00b9 : IsSimpleOrder \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 {\u2191IsSimpleOrder.equivBool.symm true, \u2191IsSimpleOrder.equivBool.symm false} = {\u22a4, \u22a5}\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : Bool\n\u22a2 a = \u22a5 \u2228 a = \u22a4\n[PROOFSTEP]\nrw [\u2190 Finset.mem_singleton, Or.comm, \u2190 Finset.mem_insert, top_eq_true, bot_eq_false, \u2190 Fintype.univ_bool]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : Bool\n\u22a2 a \u2208 Finset.univ\n[PROOFSTEP]\napply Finset.mem_univ\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\n\u22a2 IsCoatomic \u03b1\n[PROOFSTEP]\nrefine' IsCoatomic.mk fun b => or_iff_not_imp_left.2 fun ht => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\nb : \u03b1\nht : \u00acb = \u22a4\n\u22a2 \u2203 a, IsCoatom a \u2227 b \u2264 a\n[PROOFSTEP]\nobtain \u27e8c, hc, hmax\u27e9 := Set.Finite.exists_maximal_wrt id {x : \u03b1 | b \u2264 x \u2227 x \u2260 \u22a4} (Set.toFinite _) \u27e8b, le_rfl, ht\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\nb : \u03b1\nht : \u00acb = \u22a4\nc : \u03b1\nhc : c \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4}\nhmax : \u2200 (a' : \u03b1), a' \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4} \u2192 id c \u2264 id a' \u2192 id c = id a'\n\u22a2 \u2203 a, IsCoatom a \u2227 b \u2264 a\n[PROOFSTEP]\nrefine' \u27e8c, \u27e8hc.2, fun y hcy => _\u27e9, hc.1\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\nb : \u03b1\nht : \u00acb = \u22a4\nc : \u03b1\nhc : c \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4}\nhmax : \u2200 (a' : \u03b1), a' \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4} \u2192 id c \u2264 id a' \u2192 id c = id a'\ny : \u03b1\nhcy : c < y\n\u22a2 y = \u22a4\n[PROOFSTEP]\nby_contra hyt\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\nb : \u03b1\nht : \u00acb = \u22a4\nc : \u03b1\nhc : c \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4}\nhmax : \u2200 (a' : \u03b1), a' \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4} \u2192 id c \u2264 id a' \u2192 id c = id a'\ny : \u03b1\nhcy : c < y\nhyt : \u00acy = \u22a4\n\u22a2 False\n[PROOFSTEP]\nobtain rfl : c = y := hmax y \u27e8hc.1.trans hcy.le, hyt\u27e9 hcy.le\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : Finite \u03b1\nb : \u03b1\nht : \u00acb = \u22a4\nc : \u03b1\nhc : c \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4}\nhmax : \u2200 (a' : \u03b1), a' \u2208 {x | b \u2264 x \u2227 x \u2260 \u22a4} \u2192 id c \u2264 id a' \u2192 id c = id a'\nhcy : c < c\nhyt : \u00acc = \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact (lt_self_iff_false _).mp hcy\n", "meta": {"mathlib_filename": "Mathlib.Order.Atoms.Finite", "llama_tokens": 1636, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.2910034779690752}} {"text": "[GOAL]\nJ : Type w\nX\u271d Y\u271d Z\u271d : WidePullbackShape J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 X\u271d \u27f6 Z\u271d\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nJ : Type w\nX\u271d Z\u271d : WidePullbackShape J\ng : X\u271d \u27f6 Z\u271d\n\u22a2 X\u271d \u27f6 Z\u271d\ncase term J : Type w Z\u271d : WidePullbackShape J j\u271d : J g : none \u27f6 Z\u271d \u22a2 some j\u271d \u27f6 Z\u271d\n[PROOFSTEP]\nexact g\n[GOAL]\ncase term\nJ : Type w\nZ\u271d : WidePullbackShape J\nj\u271d : J\ng : none \u27f6 Z\u271d\n\u22a2 some j\u271d \u27f6 Z\u271d\n[PROOFSTEP]\ncases g\n[GOAL]\ncase term.id\nJ : Type w\nj\u271d : J\n\u22a2 some j\u271d \u27f6 none\n[PROOFSTEP]\napply Hom.term _\n[GOAL]\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePullbackShape J\n\u22a2 Subsingleton (x\u271d\u00b9 \u27f6 x\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase allEq\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePullbackShape J\n\u22a2 \u2200 (a b : x\u271d\u00b9 \u27f6 x\u271d), a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase allEq\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePullbackShape J\na b : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 a = b\n[PROOFSTEP]\ncasesm*WidePullbackShape _, (_ : WidePullbackShape _) \u27f6 (_ : WidePullbackShape _)\n[GOAL]\ncase allEq.none.none.id.id\nJ : Type w\n\u22a2 Hom.id none = Hom.id none\ncase allEq.some.none.term.term\nJ : Type w\nval\u271d : J\n\u22a2 Hom.term val\u271d = Hom.term val\u271d\ncase allEq.some.some.id.id J : Type w val\u271d : J \u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase allEq.some.none.term.term\nJ : Type w\nval\u271d : J\n\u22a2 Hom.term val\u271d = Hom.term val\u271d\ncase allEq.some.some.id.id J : Type w val\u271d : J \u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase allEq.some.some.id.id\nJ : Type w\nval\u271d : J\n\u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 objs j \u27f6 B\nX\u271d Y\u271d : WidePullbackShape J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (fun j => Option.casesOn j B objs) X\u271d \u27f6 (fun j => Option.casesOn j B objs) Y\u271d\n[PROOFSTEP]\ncases' f with _ j\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 objs j \u27f6 B\nX\u271d : WidePullbackShape J\n\u22a2 (fun j => Option.casesOn j B objs) X\u271d \u27f6 (fun j => Option.casesOn j B objs) X\u271d\n[PROOFSTEP]\napply \ud835\udfd9 _\n[GOAL]\ncase term\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 objs j \u27f6 B\nj : J\n\u22a2 (fun j => Option.casesOn j B objs) (some j) \u27f6 (fun j => Option.casesOn j B objs) none\n[PROOFSTEP]\nexact arrows j\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nj : WidePullbackShape J\n\u22a2 F.obj j = (wideCospan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.term j)).obj j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nj j' : WidePullbackShape J\nf : j \u27f6 j'\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n j \u226b\n F.map f\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nj' : WidePullbackShape J\nf : none \u27f6 j'\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n none \u226b\n F.map f\n[PROOFSTEP]\ncases j'\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nj' : WidePullbackShape J\nval\u271d : J\nf : some val\u271d \u27f6 j'\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map f\n[PROOFSTEP]\ncases j'\n[GOAL]\ncase none.none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nf : none \u27f6 none\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n none \u226b\n F.map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nval\u271d : J\nf : none \u27f6 some val\u271d\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n none \u226b\n F.map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nval\u271d : J\nf : some val\u271d \u27f6 none\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf\u271d : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\u271d\nval\u271d\u00b9 val\u271d : J\nf : some val\u271d\u00b9 \u27f6 some val\u271d\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03c0 j)\n (some val\u271d\u00b9) \u226b\n F.map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.id none) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none \u226b\n F.map (Hom.id none)\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase some.none.term\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.term val\u271d) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map (Hom.term val\u271d)\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.id (some val\u271d)) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map (Hom.id (some val\u271d))\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.id none) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none \u226b\n F.map (Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.none.term\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.term val\u271d) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map (Hom.term val\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 ((Functor.const (WidePullbackShape J)).obj X).map (Hom.id (some val\u271d)) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03c0 j)\n (some val\u271d) \u226b\n F.map (Hom.id (some val\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\n\u22a2 \ud835\udfd9 X \u226b f = f \u226b F.map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp [w]\n[GOAL]\ncase some.none.term\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 \ud835\udfd9 X \u226b f = \u03c0 val\u271d \u226b F.map (Hom.term val\u271d)\n[PROOFSTEP]\nsimp [w]\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePullbackShape J \u2964 C\nX : C\nf : X \u27f6 F.obj none\n\u03c0 : (j : J) \u2192 X \u27f6 F.obj (some j)\nw : \u2200 (j : J), \u03c0 j \u226b F.map (Hom.term j) = f\nval\u271d : J\n\u22a2 \ud835\udfd9 X \u226b \u03c0 val\u271d = \u03c0 val\u271d \u226b F.map (\ud835\udfd9 (some val\u271d))\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nj : WidePullbackShape J\n\u22a2 (\ud835\udfed (WidePullbackShape J)).obj j \u2245\n ((wideCospan none (fun j => some (\u2191h j)) fun j => Hom.term (\u2191h j)) \u22d9\n wideCospan none (fun j => some (Equiv.invFun h j)) fun j => Hom.term (Equiv.invFun h j)).obj\n j\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nval\u271d : J\n\u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nX\u271d Y\u271d : WidePullbackShape J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (WidePullbackShape J)).map f \u226b\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (some val \u2245\n Option.rec none (fun val => some (\u2191h.symm val))\n (Option.rec none (fun val => some (\u2191h val)) (some val))) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n Y\u271d).hom =\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (some val \u2245\n Option.rec none (fun val => some (\u2191h.symm val))\n (Option.rec none (fun val => some (\u2191h val)) (some val))) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n X\u271d).hom \u226b\n ((wideCospan none (fun j => some (\u2191h j)) fun j => Hom.term (\u2191h j)) \u22d9\n wideCospan none (fun j => some (Equiv.invFun h j)) fun j => Hom.term (Equiv.invFun h j)).map\n f\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nj : WidePullbackShape J'\n\u22a2 ((wideCospan none (fun j => some (Equiv.invFun h j)) fun j => Hom.term (Equiv.invFun h j)) \u22d9\n wideCospan none (fun j => some (\u2191h j)) fun j => Hom.term (\u2191h j)).obj\n j \u2245\n (\ud835\udfed (WidePullbackShape J')).obj j\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J' \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J' \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nval\u271d : J'\n\u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nX\u271d Y\u271d : WidePullbackShape J'\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((wideCospan none (fun j => some (Equiv.invFun h j)) fun j => Hom.term (Equiv.invFun h j)) \u22d9\n wideCospan none (fun j => some (\u2191h j)) fun j => Hom.term (\u2191h j)).map\n f \u226b\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (Option.rec none (fun val => some (\u2191h val))\n (Option.rec none (fun val => some (\u2191h.symm val)) (some val)) \u2245\n some val) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n Y\u271d).hom =\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (Option.rec none (fun val => some (\u2191h val))\n (Option.rec none (fun val => some (\u2191h.symm val)) (some val)) \u2245\n some val) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n X\u271d).hom \u226b\n (\ud835\udfed (WidePullbackShape J')).map f\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nJ : Type w\nX\u271d Y\u271d Z\u271d : WidePushoutShape J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 X\u271d \u27f6 Z\u271d\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nJ : Type w\nX\u271d Z\u271d : WidePushoutShape J\ng : X\u271d \u27f6 Z\u271d\n\u22a2 X\u271d \u27f6 Z\u271d\ncase init J : Type w Z\u271d : WidePushoutShape J j\u271d : J g : some j\u271d \u27f6 Z\u271d \u22a2 none \u27f6 Z\u271d\n[PROOFSTEP]\nexact g\n[GOAL]\ncase init\nJ : Type w\nZ\u271d : WidePushoutShape J\nj\u271d : J\ng : some j\u271d \u27f6 Z\u271d\n\u22a2 none \u27f6 Z\u271d\n[PROOFSTEP]\ncases g\n[GOAL]\ncase init.id\nJ : Type w\nj\u271d : J\n\u22a2 none \u27f6 some j\u271d\n[PROOFSTEP]\napply Hom.init _\n[GOAL]\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePushoutShape J\n\u22a2 Subsingleton (x\u271d\u00b9 \u27f6 x\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase allEq\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePushoutShape J\n\u22a2 \u2200 (a b : x\u271d\u00b9 \u27f6 x\u271d), a = b\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase allEq\nJ : Type w\nx\u271d\u00b9 x\u271d : WidePushoutShape J\na b : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 a = b\n[PROOFSTEP]\ncasesm*WidePushoutShape _, (_ : WidePushoutShape _) \u27f6 (_ : WidePushoutShape _)\n[GOAL]\ncase allEq.none.none.id.id\nJ : Type w\n\u22a2 Hom.id none = Hom.id none\ncase allEq.none.some.init.init\nJ : Type w\nval\u271d : J\n\u22a2 Hom.init val\u271d = Hom.init val\u271d\ncase allEq.some.some.id.id J : Type w val\u271d : J \u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase allEq.none.none.id.id\nJ : Type w\n\u22a2 Hom.id none = Hom.id none\ncase allEq.none.some.init.init\nJ : Type w\nval\u271d : J\n\u22a2 Hom.init val\u271d = Hom.init val\u271d\ncase allEq.some.some.id.id J : Type w val\u271d : J \u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase allEq.none.some.init.init\nJ : Type w\nval\u271d : J\n\u22a2 Hom.init val\u271d = Hom.init val\u271d\ncase allEq.some.some.id.id J : Type w val\u271d : J \u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase allEq.some.some.id.id\nJ : Type w\nval\u271d : J\n\u22a2 Hom.id (some val\u271d) = Hom.id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nX\u271d Y\u271d : WidePushoutShape J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (fun j => Option.casesOn j B objs) X\u271d \u27f6 (fun j => Option.casesOn j B objs) Y\u271d\n[PROOFSTEP]\ncases' f with _ j\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nX\u271d : WidePushoutShape J\n\u22a2 (fun j => Option.casesOn j B objs) X\u271d \u27f6 (fun j => Option.casesOn j B objs) X\u271d\n[PROOFSTEP]\napply \ud835\udfd9 _\n[GOAL]\ncase init\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nj : J\n\u22a2 (fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) (some j)\n[PROOFSTEP]\nexact arrows j\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nX\u271d Y\u271d Z\u271d : WidePushoutShape J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192 Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (f \u226b g) =\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n f \u226b\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nX\u271d Z\u271d : WidePushoutShape J\ng : X\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192 Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.id X\u271d \u226b g) =\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.id X\u271d) \u226b\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\nsimp only [Eq.ndrec, hom_id, eq_rec_constant, Category.id_comp]\n[GOAL]\ncase id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nX\u271d Z\u271d : WidePushoutShape J\ng : X\u271d \u27f6 Z\u271d\n\u22a2 Hom.rec (motive := fun a a_1 t =>\n X\u271d = a \u2192\n Z\u271d = a_1 \u2192 HEq (\ud835\udfd9 X\u271d \u226b g) t \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun X h =>\n Eq.rec (motive := fun x x_1 =>\n Z\u271d = x \u2192\n HEq (\ud835\udfd9 X\u271d \u226b g) (\ud835\udfd9 x) \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun h =>\n Eq.rec (motive := fun x x_1 =>\n (f : X\u271d \u27f6 x) \u2192\n HEq f (\ud835\udfd9 X\u271d) \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) x))\n (fun f h => \ud835\udfd9 (Option.rec B (fun val => objs val) X\u271d)) (_ : X\u271d = Z\u271d) (\ud835\udfd9 X\u271d \u226b g))\n h)\n (fun j h =>\n Eq.rec (motive := fun x x_1 =>\n (f : x \u27f6 Z\u271d) \u2192\n Z\u271d = some j \u2192\n HEq f (Hom.init j) \u2192 (Option.rec B (fun val => objs val) x \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun f h =>\n Eq.rec (motive := fun x x_1 =>\n (f : none \u27f6 x) \u2192 HEq f (Hom.init j) \u2192 (B \u27f6 Option.rec B (fun val => objs val) x)) (fun f h => arrows j)\n (_ : some j = Z\u271d) f)\n (_ : none = X\u271d) (\ud835\udfd9 X\u271d \u226b g))\n (\ud835\udfd9 X\u271d \u226b g) (_ : X\u271d = X\u271d) (_ : Z\u271d = Z\u271d) (_ : HEq (Hom.id X\u271d \u226b g) (Hom.id X\u271d \u226b g)) =\n Hom.rec (motive := fun a a_1 t =>\n X\u271d = a \u2192 Z\u271d = a_1 \u2192 HEq g t \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun X h =>\n Eq.rec (motive := fun x x_1 =>\n Z\u271d = x \u2192 HEq g (\ud835\udfd9 x) \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun h =>\n Eq.rec (motive := fun x x_1 =>\n (f : X\u271d \u27f6 x) \u2192\n HEq f (\ud835\udfd9 X\u271d) \u2192 (Option.rec B (fun val => objs val) X\u271d \u27f6 Option.rec B (fun val => objs val) x))\n (fun f h => \ud835\udfd9 (Option.rec B (fun val => objs val) X\u271d)) (_ : X\u271d = Z\u271d) g)\n h)\n (fun j h =>\n Eq.rec (motive := fun x x_1 =>\n (f : x \u27f6 Z\u271d) \u2192\n Z\u271d = some j \u2192\n HEq f (Hom.init j) \u2192 (Option.rec B (fun val => objs val) x \u27f6 Option.rec B (fun val => objs val) Z\u271d))\n (fun f h =>\n Eq.rec (motive := fun x x_1 =>\n (f : none \u27f6 x) \u2192 HEq f (Hom.init j) \u2192 (B \u27f6 Option.rec B (fun val => objs val) x)) (fun f h => arrows j)\n (_ : some j = Z\u271d) f)\n (_ : none = X\u271d) g)\n g (_ : X\u271d = X\u271d) (_ : Z\u271d = Z\u271d) (_ : HEq g g)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase init\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nZ\u271d : WidePushoutShape J\nj\u271d : J\ng : some j\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192 Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.init j\u271d \u226b g) =\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.init j\u271d) \u226b\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase init.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nj\u271d : J\n\u22a2 { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192 Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.init j\u271d \u226b Hom.id (some j\u271d)) =\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.init j\u271d) \u226b\n { obj := fun j => Option.casesOn j B objs,\n map := fun {X Y} f =>\n Hom.casesOn (motive := fun a a_1 t =>\n X = a \u2192\n Y = a_1 \u2192 HEq f t \u2192 ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n f\n (fun X_1 h =>\n Eq.ndrec (motive := fun X_2 =>\n Y = X_2 \u2192\n HEq f (Hom.id X_2) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : X \u27f6 Y) \u2192\n HEq f (Hom.id X) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.id X = f) \u25b8 \ud835\udfd9 ((fun j => Option.casesOn j B objs) X)) (_ : X = Y) f)\n h)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = some j \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) X \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : none \u27f6 Y) \u2192\n HEq f (Hom.init j) \u2192\n ((fun j => Option.casesOn j B objs) none \u27f6 (fun j => Option.casesOn j B objs) Y))\n (fun f h => (_ : Hom.init j = f) \u25b8 arrows j) (_ : some j = Y) f)\n (_ : none = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (Hom.id (some j\u271d))\n[PROOFSTEP]\nsimp only [Eq.ndrec, hom_id, eq_rec_constant, Category.comp_id]\n[GOAL]\ncase init.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nB : C\nobjs : J \u2192 C\narrows : (j : J) \u2192 B \u27f6 objs j\nj\u271d : J\n\u22a2 Hom.rec (motive := fun a a_1 t => none = a \u2192 some j\u271d = a_1 \u2192 HEq (Hom.init j\u271d \u226b \ud835\udfd9 (some j\u271d)) t \u2192 (B \u27f6 objs j\u271d))\n (fun X h =>\n Eq.rec (motive := fun x x_1 => some j\u271d = x \u2192 HEq (Hom.init j\u271d \u226b \ud835\udfd9 (some j\u271d)) (\ud835\udfd9 x) \u2192 (B \u27f6 objs j\u271d))\n (fun h =>\n Eq.rec (motive := fun x x_1 => (f : none \u27f6 x) \u2192 HEq f (\ud835\udfd9 none) \u2192 (B \u27f6 Option.rec B (fun val => objs val) x))\n (fun f h => \ud835\udfd9 B) (_ : none = some j\u271d) (Hom.init j\u271d \u226b \ud835\udfd9 (some j\u271d)))\n h)\n (fun j h h =>\n Eq.rec (motive := fun x x_1 => (f : none \u27f6 x) \u2192 HEq f (Hom.init j) \u2192 (B \u27f6 Option.rec B (fun val => objs val) x))\n (fun f h => arrows j) (_ : some j = some j\u271d) (Hom.init j\u271d \u226b \ud835\udfd9 (some j\u271d)))\n (Hom.init j\u271d \u226b \ud835\udfd9 (some j\u271d)) (_ : none = none) (_ : some j\u271d = some j\u271d)\n (_ : HEq (Hom.init j\u271d \u226b Hom.id (some j\u271d)) (Hom.init j\u271d \u226b Hom.id (some j\u271d))) =\n arrows j\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nj : WidePushoutShape J\n\u22a2 F.obj j = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\n\u22a2 F.obj none = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj none\ncase some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nval\u271d : J\n\u22a2 F.obj (some val\u271d) = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj (some val\u271d)\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\n\u22a2 F.obj none = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj none\ncase some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nval\u271d : J\n\u22a2 F.obj (some val\u271d) = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nval\u271d : J\n\u22a2 F.obj (some val\u271d) = (wideSpan (F.obj none) (fun j => F.obj (some j)) fun j => F.map (Hom.init j)).obj (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nj j' : WidePushoutShape J\nf : j \u27f6 j'\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n j \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases j\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nj' : WidePushoutShape J\nf : none \u27f6 j'\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases j'\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nj' : WidePushoutShape J\nval\u271d : J\nf : some val\u271d \u27f6 j'\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n j' =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n (some val\u271d) \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases j'\n[GOAL]\ncase none.none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nf : none \u27f6 none\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n none =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nval\u271d : J\nf : none \u27f6 some val\u271d\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.none\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nval\u271d : J\nf : some val\u271d \u27f6 none\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n none =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n (some val\u271d) \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase some.some\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf\u271d : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\u271d\nval\u271d\u00b9 val\u271d : J\nf : some val\u271d\u00b9 \u27f6 some val\u271d\n\u22a2 F.map f \u226b\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\u271d\n | some j => \u03b9 j)\n (some val\u271d\u00b9) \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\n\u22a2 F.map (Hom.id none) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.id none)\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase none.some.init\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (Hom.init val\u271d) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.init val\u271d)\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (Hom.id (some val\u271d)) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.id (some val\u271d))\n[PROOFSTEP]\nrefine id _\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\n\u22a2 F.map (Hom.id none) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.some.init\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (Hom.init val\u271d) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n none \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.init val\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (Hom.id (some val\u271d)) \u226b\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) =\n (fun j =>\n match j with\n | none => f\n | some j => \u03b9 j)\n (some val\u271d) \u226b\n ((Functor.const (WidePushoutShape J)).obj X).map (Hom.id (some val\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase none.none.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\n\u22a2 F.map (\ud835\udfd9 none) \u226b f = f \u226b \ud835\udfd9 X\n[PROOFSTEP]\nsimp [w]\n[GOAL]\ncase none.some.init\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (Hom.init val\u271d) \u226b \u03b9 val\u271d = f \u226b \ud835\udfd9 X\n[PROOFSTEP]\nsimp [w]\n[GOAL]\ncase some.some.id\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nF : WidePushoutShape J \u2964 C\nX : C\nf : F.obj none \u27f6 X\n\u03b9 : (j : J) \u2192 F.obj (some j) \u27f6 X\nw : \u2200 (j : J), F.map (Hom.init j) \u226b \u03b9 j = f\nval\u271d : J\n\u22a2 F.map (\ud835\udfd9 (some val\u271d)) \u226b \u03b9 val\u271d = \u03b9 val\u271d \u226b \ud835\udfd9 X\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nj : WidePushoutShape J\n\u22a2 (\ud835\udfed (WidePushoutShape J)).obj j \u2245\n ((wideSpan none (fun j => some (\u2191h j)) fun j => Hom.init (\u2191h j)) \u22d9\n wideSpan none (fun j => some (Equiv.invFun h j)) fun j => Hom.init (Equiv.invFun h j)).obj\n j\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nval\u271d : J\n\u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nX\u271d Y\u271d : WidePushoutShape J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (WidePushoutShape J)).map f \u226b\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (some val \u2245\n Option.rec none (fun val => some (\u2191h.symm val))\n (Option.rec none (fun val => some (\u2191h val)) (some val))) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n Y\u271d).hom =\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (some val \u2245\n Option.rec none (fun val => some (\u2191h.symm val))\n (Option.rec none (fun val => some (\u2191h val)) (some val))) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n X\u271d).hom \u226b\n ((wideSpan none (fun j => some (\u2191h j)) fun j => Hom.init (\u2191h j)) \u22d9\n wideSpan none (fun j => some (Equiv.invFun h j)) fun j => Hom.init (Equiv.invFun h j)).map\n f\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nj : WidePushoutShape J'\n\u22a2 ((wideSpan none (fun j => some (Equiv.invFun h j)) fun j => Hom.init (Equiv.invFun h j)) \u22d9\n wideSpan none (fun j => some (\u2191h j)) fun j => Hom.init (\u2191h j)).obj\n j \u2245\n (\ud835\udfed (WidePushoutShape J')).obj j\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J' \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrepeat rfl\n[GOAL]\ncase none\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\n\u22a2 none \u2245 none\ncase some J : Type w C : Type u inst : Category.{v, u} C J' : Type w' h : J \u2243 J' val\u271d : J' \u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nJ : Type w\nC : Type u\ninst : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nval\u271d : J'\n\u22a2 some val\u271d \u2245 some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d : Category.{v, u} C\nJ' : Type w'\nh : J \u2243 J'\nX\u271d Y\u271d : WidePushoutShape J'\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((wideSpan none (fun j => some (Equiv.invFun h j)) fun j => Hom.init (Equiv.invFun h j)) \u22d9\n wideSpan none (fun j => some (\u2191h j)) fun j => Hom.init (\u2191h j)).map\n f \u226b\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (Option.rec none (fun val => some (\u2191h val))\n (Option.rec none (fun val => some (\u2191h.symm val)) (some val)) \u2245\n some val) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n Y\u271d).hom =\n ((fun j =>\n id\n (Option.casesOn j (id (Iso.refl none)) fun val =>\n Eq.mpr\n (_ :\n (Option.rec none (fun val => some (\u2191h val))\n (Option.rec none (fun val => some (\u2191h.symm val)) (some val)) \u2245\n some val) =\n (some val \u2245 some val))\n (Iso.refl (some val))))\n X\u271d).hom \u226b\n (\ud835\udfed (WidePushoutShape J')).map f\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nj : J\n\u22a2 \u03c0 arrows j \u226b arrows j = base arrows\n[PROOFSTEP]\napply limit.w (WidePullbackShape.wideCospan _ _ _) (WidePullbackShape.Hom.term j)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\nj : J\n\u22a2 lift f fs w \u226b \u03c0 arrows j = fs j\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, WidePullbackShape.mkCone_pt, WidePullbackShape.mkCone_\u03c0_app]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\n\u22a2 lift f fs w \u226b base arrows = f\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, WidePullbackShape.mkCone_pt, WidePullbackShape.mkCone_\u03c0_app]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 (\u2200 (j : J), g \u226b \u03c0 arrows j = fs j) \u2192 g \u226b base arrows = f \u2192 g = lift f fs w\n[PROOFSTEP]\nintro h1 h2\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g \u226b \u03c0 arrows j = fs j\nh2 : g \u226b base arrows = f\n\u22a2 g = lift f fs w\n[PROOFSTEP]\napply (limit.isLimit (WidePullbackShape.wideCospan B objs arrows)).uniq (WidePullbackShape.mkCone f fs <| w)\n[GOAL]\ncase x\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g \u226b \u03c0 arrows j = fs j\nh2 : g \u226b base arrows = f\n\u22a2 \u2200 (j : WidePullbackShape J),\n g \u226b NatTrans.app (limit.cone (WidePullbackShape.wideCospan B objs arrows)).\u03c0 j =\n NatTrans.app (WidePullbackShape.mkCone f fs w).\u03c0 j\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase x.none\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g \u226b \u03c0 arrows j = fs j\nh2 : g \u226b base arrows = f\n\u22a2 g \u226b NatTrans.app (limit.cone (WidePullbackShape.wideCospan B objs arrows)).\u03c0 none =\n NatTrans.app (WidePullbackShape.mkCone f fs w).\u03c0 none\n[PROOFSTEP]\napply h2\n[GOAL]\ncase x.some\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g \u226b \u03c0 arrows j = fs j\nh2 : g \u226b base arrows = f\nval\u271d : J\n\u22a2 g \u226b NatTrans.app (limit.cone (WidePullbackShape.wideCospan B objs arrows)).\u03c0 (some val\u271d) =\n NatTrans.app (WidePullbackShape.mkCone f fs w).\u03c0 (some val\u271d)\n[PROOFSTEP]\napply h1\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type ?u.192408\ninst\u271d\u00b9 : Category.{v\u2082, ?u.192408} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 \u2200 (j : J), (fun j => g \u226b \u03c0 arrows j) j \u226b arrows j = g \u226b base arrows\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g = lift (g \u226b base arrows) (fun j => g \u226b \u03c0 arrows j) (_ : \u2200 (j : J), (g \u226b \u03c0 arrows j) \u226b arrows j = g \u226b base arrows)\n[PROOFSTEP]\napply eq_lift_of_comp_eq\n[GOAL]\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 \u2200 (j : J), g \u226b \u03c0 arrows j = g \u226b \u03c0 arrows j\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows\n[PROOFSTEP]\nrfl\n -- Porting note: quite a few missing refl's in aesop_cat now\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng1 g2 : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 (\u2200 (j : J), g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j) \u2192 g1 \u226b base arrows = g2 \u226b base arrows \u2192 g1 = g2\n[PROOFSTEP]\nintro h1 h2\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng1 g2 : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j\nh2 : g1 \u226b base arrows = g2 \u226b base arrows\n\u22a2 g1 = g2\n[PROOFSTEP]\napply limit.hom_ext\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng1 g2 : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j\nh2 : g1 \u226b base arrows = g2 \u226b base arrows\n\u22a2 \u2200 (j : WidePullbackShape J),\n g1 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j =\n g2 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase w.none\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng1 g2 : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j\nh2 : g1 \u226b base arrows = g2 \u226b base arrows\n\u22a2 g1 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none =\n g2 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none\n[PROOFSTEP]\napply h2\n[GOAL]\ncase w.some\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng1 g2 : X \u27f6 widePullback B (fun j => objs j) arrows\nh1 : \u2200 (j : J), g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j\nh2 : g1 \u226b base arrows = g2 \u226b base arrows\nval\u271d : J\n\u22a2 g1 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val\u271d) =\n g2 \u226b limit.\u03c0 (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val\u271d)\n[PROOFSTEP]\napply h1\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nj : J\n\u22a2 arrows j \u226b \u03b9 arrows j = head arrows\n[PROOFSTEP]\napply colimit.w (WidePushoutShape.wideSpan _ _ _) (WidePushoutShape.Hom.init j)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\nj : J\n\u22a2 \u03b9 arrows j \u226b desc f fs w = fs j\n[PROOFSTEP]\nsimp only [colimit.\u03b9_desc, WidePushoutShape.mkCocone_pt, WidePushoutShape.mkCocone_\u03b9_app]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\n\u22a2 head arrows \u226b desc f fs w = f\n[PROOFSTEP]\nsimp only [colimit.\u03b9_desc, WidePushoutShape.mkCocone_pt, WidePushoutShape.mkCocone_\u03b9_app]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 (\u2200 (j : J), \u03b9 arrows j \u226b g = fs j) \u2192 head arrows \u226b g = f \u2192 g = desc f fs w\n[PROOFSTEP]\nintro h1 h2\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g = fs j\nh2 : head arrows \u226b g = f\n\u22a2 g = desc f fs w\n[PROOFSTEP]\napply (colimit.isColimit (WidePushoutShape.wideSpan B objs arrows)).uniq (WidePushoutShape.mkCocone f fs <| w)\n[GOAL]\ncase x\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g = fs j\nh2 : head arrows \u226b g = f\n\u22a2 \u2200 (j : WidePushoutShape J),\n NatTrans.app (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).\u03b9 j \u226b g =\n NatTrans.app (WidePushoutShape.mkCocone f fs w).\u03b9 j\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase x.none\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g = fs j\nh2 : head arrows \u226b g = f\n\u22a2 NatTrans.app (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).\u03b9 none \u226b g =\n NatTrans.app (WidePushoutShape.mkCocone f fs w).\u03b9 none\n[PROOFSTEP]\napply h2\n[GOAL]\ncase x.some\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g = fs j\nh2 : head arrows \u226b g = f\nval\u271d : J\n\u22a2 NatTrans.app (colimit.cocone (WidePushoutShape.wideSpan B objs arrows)).\u03b9 (some val\u271d) \u226b g =\n NatTrans.app (WidePushoutShape.mkCocone f fs w).\u03b9 (some val\u271d)\n[PROOFSTEP]\napply h1\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type ?u.203710\ninst\u271d\u00b9 : Category.{v\u2082, ?u.203710} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nj : J\n\u22a2 arrows j \u226b (fun j => \u03b9 arrows j \u226b g) j = head arrows \u226b g\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type ?u.203710\ninst\u271d\u00b9 : Category.{v\u2082, ?u.203710} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\nj : J\n\u22a2 (arrows j \u226b \u03b9 arrows j) \u226b g = head arrows \u226b g\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 g =\n desc (head arrows \u226b g) (fun j => \u03b9 arrows j \u226b g)\n (_ : \u2200 (j : J), arrows j \u226b (fun j => \u03b9 arrows j \u226b g) j = head arrows \u226b g)\n[PROOFSTEP]\napply eq_desc_of_comp_eq\n[GOAL]\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 \u2200 (j : J), \u03b9 arrows j \u226b g = \u03b9 arrows j \u226b g\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 head arrows \u226b g = head arrows \u226b g\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 head arrows \u226b g = head arrows \u226b g\n[PROOFSTEP]\nrfl\n -- Porting note: another missing rfl\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng1 g2 : widePushout B (fun j => objs j) arrows \u27f6 X\n\u22a2 (\u2200 (j : J), \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2) \u2192 head arrows \u226b g1 = head arrows \u226b g2 \u2192 g1 = g2\n[PROOFSTEP]\nintro h1 h2\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng1 g2 : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2\nh2 : head arrows \u226b g1 = head arrows \u226b g2\n\u22a2 g1 = g2\n[PROOFSTEP]\napply colimit.hom_ext\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng1 g2 : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2\nh2 : head arrows \u226b g1 = head arrows \u226b g2\n\u22a2 \u2200 (j : WidePushoutShape J),\n colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) j \u226b g1 =\n colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) j \u226b g2\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase w.none\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng1 g2 : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2\nh2 : head arrows \u226b g1 = head arrows \u226b g2\n\u22a2 colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) none \u226b g1 =\n colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) none \u226b g2\n[PROOFSTEP]\napply h2\n[GOAL]\ncase w.some\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 B \u27f6 objs j\ninst\u271d : HasWidePushout B objs arrows\nX : D\nf : B \u27f6 X\nfs : (j : J) \u2192 objs j \u27f6 X\nw : \u2200 (j : J), arrows j \u226b fs j = f\ng1 g2 : widePushout B (fun j => objs j) arrows \u27f6 X\nh1 : \u2200 (j : J), \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2\nh2 : head arrows \u226b g1 = head arrows \u226b g2\nval\u271d : J\n\u22a2 colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) (some val\u271d) \u226b g1 =\n colimit.\u03b9 (WidePushoutShape.wideSpan B (fun j => objs j) arrows) (some val\u271d) \u226b g2\n[PROOFSTEP]\napply h1\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "llama_tokens": 31988, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.29071928111249895}} {"text": "[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 : S\nhb : IsInteger R b\n\u22a2 IsInteger R (a \u2022 b)\n[PROOFSTEP]\nrcases hb with \u27e8b', hb\u27e9\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\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 IsInteger R (a \u2022 b)\n[PROOFSTEP]\nuse a * b'\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\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 \u2191(algebraMap R S) (a * b') = a \u2022 b\n[PROOFSTEP]\nrw [\u2190 hb, (algebraMap R S).map_mul, Algebra.smul_def]\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\ninst\u271d : IsLocalization M S\na : S\n\u22a2 \u2203 b, IsInteger R (\u2191b \u2022 a)\n[PROOFSTEP]\nsimp_rw [Algebra.smul_def, mul_comm _ a]\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\ninst\u271d : IsLocalization M S\na : S\n\u22a2 \u2203 b, IsInteger R (a * \u2191(algebraMap R S) \u2191b)\n[PROOFSTEP]\napply exists_integer_multiple'\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 IsInteger R (\u2191b \u2022 f i)\n[PROOFSTEP]\nhaveI := Classical.propDecidable\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 IsInteger R (\u2191b \u2022 f i)\n[PROOFSTEP]\nrefine' \u27e8\u220f i in s, (sec M (f i)).2, fun i hi => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 R\n[PROOFSTEP]\nexact (\u220f j in s.erase i, (sec M (f j)).2) * (sec M (f i)).1\n[GOAL]\ncase refine'_2\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(algebraMap R S) (\u2191(\u220f j in Finset.erase s i, (sec M (f j)).snd) * (sec M (f i)).fst) =\n \u2191(\u220f i in s, (sec M (f i)).snd) \u2022 f i\n[PROOFSTEP]\nrw [RingHom.map_mul, sec_spec', \u2190 mul_assoc, \u2190 (algebraMap R S).map_mul, \u2190 Algebra.smul_def]\n[GOAL]\ncase refine'_2\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (\u2191(\u220f j in Finset.erase s i, (sec M (f j)).snd) * \u2191(sec M (f i)).snd) \u2022 f i = \u2191(\u220f i in s, (sec M (f i)).snd) \u2022 f i\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase refine'_2.e_a\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u220f j in Finset.erase s i, (sec M (f j)).snd) * \u2191(sec M (f i)).snd = \u2191(\u220f i in s, (sec M (f i)).snd)\n[PROOFSTEP]\nrefine' _root_.trans _ ((Submonoid.subtype M).map_prod _ _).symm\n[GOAL]\ncase refine'_2.e_a\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(\u220f j in Finset.erase s i, (sec M (f j)).snd) * \u2191(sec M (f i)).snd =\n \u220f x in s, \u2191(Submonoid.subtype M) (sec M (f x)).snd\n[PROOFSTEP]\nrw [mul_comm, Submonoid.coe_finset_prod,\n -- Porting note: explicitly supplied `f`\u2190\n Finset.prod_insert (f := fun i => ((sec M (f i)).snd : R)) (s.not_mem_erase i), Finset.insert_erase hi]\n[GOAL]\ncase refine'_2.e_a\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\ninst\u271d : IsLocalization M S\n\u03b9 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 S\nthis : (a : Prop) \u2192 Decidable a\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u220f x in s, \u2191(sec M (f x)).snd = \u220f x in s, \u2191(Submonoid.subtype M) (sec M (f x)).snd\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 S\n\u22a2 \u2203 b, \u2200 (i : \u03b9), IsInteger R (\u2191b \u2022 f i)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 S\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 b, \u2200 (i : \u03b9), IsInteger R (\u2191b \u2022 f i)\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := exist_integer_multiples M Finset.univ f\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\n\u03b9 : Type u_4\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 S\nval\u271d : Fintype \u03b9\nb : { x // x \u2208 M }\nhb : \u2200 (i : \u03b9), i \u2208 Finset.univ \u2192 IsInteger R (\u2191b \u2022 f i)\n\u22a2 \u2203 b, \u2200 (i : \u03b9), IsInteger R (\u2191b \u2022 f i)\n[PROOFSTEP]\nexact \u27e8b, fun i => hb i (Finset.mem_univ _)\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\n\u22a2 \u2191(algebraMap R S) '' \u2191(finsetIntegerMultiple M s) = commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\ndelta finsetIntegerMultiple commonDenom\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\n\u22a2 \u2191(algebraMap R S) '' \u2191(Finset.image (fun t => integerMultiple M s id t) (Finset.attach s)) =\n commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\nrw [Finset.coe_image]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\n\u22a2 \u2191(algebraMap R S) '' ((fun t => integerMultiple M s id t) '' \u2191(Finset.attach s)) = commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx\u271d : S\n\u22a2 x\u271d \u2208 \u2191(algebraMap R S) '' ((fun t => integerMultiple M s id t) '' \u2191(Finset.attach s)) \u2194\n x\u271d \u2208 commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx\u271d : S\n\u22a2 x\u271d \u2208 \u2191(algebraMap R S) '' ((fun t => integerMultiple M s id t) '' \u2191(Finset.attach s)) \u2192\n x\u271d \u2208 commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\nrintro \u27e8_, \u27e8x, -, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx : { x // x \u2208 s }\n\u22a2 \u2191(algebraMap R S) ((fun t => integerMultiple M s id t) x) \u2208 commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\nrw [map_integerMultiple]\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx : { x // x \u2208 s }\n\u22a2 commonDenom M s id \u2022 id \u2191x \u2208 commonDenomOfFinset M s \u2022 \u2191s\n[PROOFSTEP]\nexact Set.mem_image_of_mem _ x.prop\n[GOAL]\ncase h.mpr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx\u271d : S\n\u22a2 x\u271d \u2208 commonDenomOfFinset M s \u2022 \u2191s \u2192\n x\u271d \u2208 \u2191(algebraMap R S) '' ((fun t => integerMultiple M s id t) '' \u2191(Finset.attach s))\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R S\nP : Type u_3\ninst\u271d\u00b2 : CommRing P\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : DecidableEq R\ns : Finset S\nx : S\nhx : x \u2208 \u2191s\n\u22a2 (fun x => \u2191(Submonoid.subtype M) (commonDenomOfFinset M s) \u2022 x) x \u2208\n \u2191(algebraMap R S) '' ((fun t => integerMultiple M s id t) '' \u2191(Finset.attach s))\n[PROOFSTEP]\nexact \u27e8_, \u27e8\u27e8x, hx\u27e9, s.mem_attach _, rfl\u27e9, map_integerMultiple M s id _\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.Integer", "llama_tokens": 4704, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.29071928111249895}} {"text": "[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nB : Bicone F\nj : J\n\u22a2 Bicone.\u03b9 B j \u226b Bicone.\u03c0 B j = \ud835\udfd9 (F j)\n[PROOFSTEP]\nsimpa using B.\u03b9_\u03c0 j j\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nB : Bicone F\nj j' : J\nh : j \u2260 j'\n\u22a2 Bicone.\u03b9 B j \u226b Bicone.\u03c0 B j' = 0\n[PROOFSTEP]\nsimpa [h] using B.\u03b9_\u03c0 j j'\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nB : Bicone F\n\u22a2 \u2200 \u2983X Y : Discrete J\u2984 (f : X \u27f6 Y),\n (Discrete.functor F).map f \u226b (fun j => \u03b9 B j.as) Y = (fun j => \u03b9 B j.as) X \u226b ((const (Discrete J)).obj B.pt).map f\n[PROOFSTEP]\nintro \u27e8j\u27e9 \u27e8j'\u27e9 \u27e8\u27e8f\u27e9\u27e9\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nB : Bicone F\nj j' : J\nf : { as := j }.as = { as := j' }.as\n\u22a2 (Discrete.functor F).map { down := { down := f } } \u226b (fun j => \u03b9 B j.as) { as := j' } =\n (fun j => \u03b9 B j.as) { as := j } \u226b ((const (Discrete J)).obj B.pt).map { down := { down := f } }\n[PROOFSTEP]\ncases f\n[GOAL]\ncase refl\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nB : Bicone F\nj : J\n\u22a2 (Discrete.functor F).map { down := { down := (_ : { as := j }.as = { as := j }.as) } } \u226b\n (fun j => \u03b9 B j.as) { as := j } =\n (fun j => \u03b9 B j.as) { as := j } \u226b\n ((const (Discrete J)).obj B.pt).map { down := { down := (_ : { as := j }.as = { as := j }.as) } }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Cone (Discrete.functor f)\nht : IsLimit t\nj j' : J\n\u22a2 (fun j => IsLimit.lift ht (Fan.mk (f j) fun j' => if h : j = j' then eqToHom (_ : f j = f j') else 0)) j \u226b\n (fun j => NatTrans.app t.\u03c0 { as := j }) j' =\n if h : j = j' then eqToHom (_ : f j = f j') else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Bicone f\nht : IsLimit (toCone t)\nj : J\nj' : Discrete J\n\u22a2 \u03b9 t j \u226b NatTrans.app (toCone t).\u03c0 j' =\n IsLimit.lift ht (Fan.mk (f j) fun j' => if h : j = j' then eqToHom (_ : f j = f j') else 0) \u226b\n NatTrans.app (toCone t).\u03c0 j'\n[PROOFSTEP]\nrw [ht.fac]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Bicone f\nht : IsLimit (toCone t)\nj : J\nj' : Discrete J\n\u22a2 \u03b9 t j \u226b NatTrans.app (toCone t).\u03c0 j' =\n NatTrans.app (Fan.mk (f j) fun j' => if h : j = j' then eqToHom (_ : f j = f j') else 0).\u03c0 j'\n[PROOFSTEP]\nsimp [t.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Cocone (Discrete.functor f)\nht : IsColimit t\nj j' : J\n\u22a2 (fun j => NatTrans.app t.\u03b9 { as := j }) j \u226b\n (fun j => IsColimit.desc ht (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom (_ : f j' = f j) else 0)) j' =\n if h : j = j' then eqToHom (_ : f j = f j') else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Bicone f\nht : IsColimit (toCocone t)\nj : J\nj' : Discrete J\n\u22a2 NatTrans.app (toCocone t).\u03b9 j' \u226b \u03c0 t j =\n NatTrans.app (toCocone t).\u03b9 j' \u226b\n IsColimit.desc ht (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom (_ : f j' = f j) else 0)\n[PROOFSTEP]\nrw [ht.fac]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF f : J \u2192 C\nt : Bicone f\nht : IsColimit (toCocone t)\nj : J\nj' : Discrete J\n\u22a2 NatTrans.app (toCocone t).\u03b9 j' \u226b \u03c0 t j =\n NatTrans.app (Cofan.mk (f j) fun j' => if h : j' = j then eqToHom (_ : f j' = f j) else 0).\u03b9 j'\n[PROOFSTEP]\nsimp [t.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\n\u22a2 (fun k => \u03b9 c (\u2191g k)) k \u226b (fun k => \u03c0 c (\u2191g k)) k' = if h : k = k' then eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k') else 0\n[PROOFSTEP]\nsimp only [c.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\n\u22a2 (if h : \u2191g k = \u2191g k' then eqToHom (_ : f (\u2191g k) = f (\u2191g k')) else 0) =\n if h : k = k' then eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k') else 0\n[PROOFSTEP]\nsplit_ifs with h h' h'\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh : \u2191g k = \u2191g k'\nh' : k = k'\n\u22a2 eqToHom (_ : f (\u2191g k) = f (\u2191g k')) = eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k')\n[PROOFSTEP]\nsimp [Equiv.apply_eq_iff_eq g] at h h' \n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh : \u2191g k = \u2191g k'\nh' : \u00ack = k'\n\u22a2 eqToHom (_ : f (\u2191g k) = f (\u2191g k')) = 0\n[PROOFSTEP]\nsimp [Equiv.apply_eq_iff_eq g] at h h' \n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh : \u00ac\u2191g k = \u2191g k'\nh' : k = k'\n\u22a2 0 = eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k')\n[PROOFSTEP]\nsimp [Equiv.apply_eq_iff_eq g] at h h' \n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh : \u00ac\u2191g k = \u2191g k'\nh' : \u00ack = k'\n\u22a2 0 = 0\n[PROOFSTEP]\nsimp [Equiv.apply_eq_iff_eq g] at h h' \n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh\u271d : \u2191g k = \u2191g k'\nh' h : k = k'\n\u22a2 eqToHom (_ : f (\u2191g k) = f (\u2191g k')) = eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k')\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh\u271d : \u2191g k = \u2191g k'\nh' : \u00ack = k'\nh : k = k'\n\u22a2 eqToHom (_ : f (\u2191g k) = f (\u2191g k')) = 0\n[PROOFSTEP]\ntauto\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh' : k = k'\nh : \u00ack = k'\n\u22a2 0 = eqToHom (_ : (f \u2218 \u2191g) k = (f \u2218 \u2191g) k')\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nk k' : K\nh' h : \u00ack = k'\n\u22a2 0 = 0\n[PROOFSTEP]\ntauto\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\n\u22a2 \u2200 (j : Discrete K),\n NatTrans.app (toCone (whisker c g)).\u03c0 j =\n (Iso.refl (toCone (whisker c g)).pt).hom \u226b\n NatTrans.app\n ((Cones.postcompose (Discrete.functorComp f \u2191g).inv).obj\n (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c))).\u03c0\n j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\n\u22a2 \u2200 (j : Discrete K),\n NatTrans.app (toCocone (whisker c g)).\u03b9 j \u226b (Iso.refl (toCocone (whisker c g)).pt).hom =\n NatTrans.app\n ((Cocones.precompose (Discrete.functorComp f \u2191g).hom).obj\n (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c))).\u03b9\n j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\n\u22a2 IsBilimit (whisker c g) \u2243 IsBilimit c\n[PROOFSTEP]\nrefine' equivOfSubsingletonOfSubsingleton (fun hc => \u27e8_, _\u27e9) fun hc => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\n\u22a2 IsLimit (toCone c)\n[PROOFSTEP]\nlet this := IsLimit.ofIsoLimit hc.isLimit (Bicone.whiskerToCone c g)\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\nthis : IsLimit\n ((Cones.postcompose (Discrete.functorComp f \u2191g).inv).obj\n (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c))) :=\n IsLimit.ofIsoLimit hc.isLimit (whiskerToCone c g)\n\u22a2 IsLimit (toCone c)\n[PROOFSTEP]\nlet this := (IsLimit.postcomposeHomEquiv (Discrete.functorComp f g).symm _) this\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\nthis\u271d : IsLimit\n ((Cones.postcompose (Discrete.functorComp f \u2191g).inv).obj\n (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c))) :=\n IsLimit.ofIsoLimit hc.isLimit (whiskerToCone c g)\nthis : (fun x => IsLimit (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c))) this\u271d :=\n \u2191(IsLimit.postcomposeHomEquiv (Discrete.functorComp f \u2191g).symm\n (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c)))\n this\u271d\n\u22a2 IsLimit (toCone c)\n[PROOFSTEP]\nexact IsLimit.ofWhiskerEquivalence (Discrete.equivalence g) this\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\n\u22a2 IsColimit (toCocone c)\n[PROOFSTEP]\nlet this := IsColimit.ofIsoColimit hc.isColimit (Bicone.whiskerToCocone c g)\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\nthis : IsColimit\n ((Cocones.precompose (Discrete.functorComp f \u2191g).hom).obj\n (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c))) :=\n IsColimit.ofIsoColimit hc.isColimit (whiskerToCocone c g)\n\u22a2 IsColimit (toCocone c)\n[PROOFSTEP]\nlet this := (IsColimit.precomposeHomEquiv (Discrete.functorComp f g) _) this\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit (whisker c g)\nthis\u271d : IsColimit\n ((Cocones.precompose (Discrete.functorComp f \u2191g).hom).obj\n (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c))) :=\n IsColimit.ofIsoColimit hc.isColimit (whiskerToCocone c g)\nthis : (fun x => IsColimit (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c))) this\u271d :=\n \u2191(IsColimit.precomposeHomEquiv (Discrete.functorComp f \u2191g)\n (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c)))\n this\u271d\n\u22a2 IsColimit (toCocone c)\n[PROOFSTEP]\nexact IsColimit.ofWhiskerEquivalence (Discrete.equivalence g) this\n[GOAL]\ncase refine'_3\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsLimit (toCone (whisker c g))\n[PROOFSTEP]\napply IsLimit.ofIsoLimit _ (Bicone.whiskerToCone c g).symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsLimit\n ((Cones.postcompose (Discrete.functorComp f \u2191g).inv).obj\n (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c)))\n[PROOFSTEP]\napply (IsLimit.postcomposeHomEquiv (Discrete.functorComp f g).symm _).symm _\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsLimit (Cone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCone c))\n[PROOFSTEP]\nexact IsLimit.whiskerEquivalence hc.isLimit (Discrete.equivalence g)\n[GOAL]\ncase refine'_4\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsColimit (toCocone (whisker c g))\n[PROOFSTEP]\napply IsColimit.ofIsoColimit _ (Bicone.whiskerToCocone c g).symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsColimit\n ((Cocones.precompose (Discrete.functorComp f \u2191g).hom).obj\n (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c)))\n[PROOFSTEP]\napply (IsColimit.precomposeHomEquiv (Discrete.functorComp f g) _).symm _\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2192 C\nK : Type w'\nf : J \u2192 C\nc : Bicone f\ng : K \u2243 J\nhc : IsBilimit c\n\u22a2 IsColimit (Cocone.whisker (Discrete.functor (Discrete.mk \u2218 \u2191g)) (toCocone c))\n[PROOFSTEP]\nexact IsColimit.whiskerEquivalence hc.isColimit (Discrete.equivalence g)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nF\u271d : J \u2192 C\nK : Type w'\ninst\u271d : HasBiproductsOfShape K C\ne : J \u2243 K\nF : J \u2192 C\nh : LimitBicone (F \u2218 \u2191e.symm)\nc : Bicone (F \u2218 \u2191e.symm)\nhc : Bicone.IsBilimit c\n\u22a2 LimitBicone F\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), e.symm_apply_apply] using LimitBicone.mk (c.whisker e) ((c.whiskerIsBilimitIff _).2 hc)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nF : J \u2192 C\ninst\u271d\u00b9 : HasFiniteBiproducts C\ninst\u271d : Finite J\n\u22a2 HasBiproductsOfShape J C\n[PROOFSTEP]\nrcases Finite.exists_equiv_fin J with \u27e8n, \u27e8e\u27e9\u27e9\n[GOAL]\ncase intro.intro\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nF : J \u2192 C\ninst\u271d\u00b9 : HasFiniteBiproducts C\ninst\u271d : Finite J\nn : \u2115\ne : J \u2243 Fin n\n\u22a2 HasBiproductsOfShape J C\n[PROOFSTEP]\nhaveI : HasBiproductsOfShape (Fin n) C := HasFiniteBiproducts.out n\n[GOAL]\ncase intro.intro\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nF : J \u2192 C\ninst\u271d\u00b9 : HasFiniteBiproducts C\ninst\u271d : Finite J\nn : \u2115\ne : J \u2243 Fin n\nthis : HasBiproductsOfShape (Fin n) C\n\u22a2 HasBiproductsOfShape J C\n[PROOFSTEP]\nexact hasBiproductsOfShape_of_equiv C e\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : DecidableEq J\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\n\u22a2 \u03b9 f j \u226b \u03c0 f j' = if h : j = j' then eqToHom (_ : f j = f j') else 0\n[PROOFSTEP]\nconvert (biproduct.bicone f).\u03b9_\u03c0 j j'\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : J\n\u22a2 \u03b9 f j \u226b \u03c0 f j = \ud835\udfd9 (f j)\n[PROOFSTEP]\nsimp [biproduct.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\nh : j \u2260 j'\n\u22a2 \u03b9 f j \u226b \u03c0 f j' = 0\n[PROOFSTEP]\nsimp [biproduct.\u03b9_\u03c0, h]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\nw : j = j'\n\u22a2 eqToHom (_ : f j = f j') \u226b \u03b9 f j' = \u03b9 f j\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : J\n\u22a2 eqToHom (_ : f j = f j) \u226b \u03b9 f j = \u03b9 f j\n[PROOFSTEP]\nsimp\n -- The `simpNF` linter incorrectly identifies these as simp lemmas that could never apply.\n -- https://github.com/leanprover-community/mathlib4/issues/5049\n -- They are used by `simp` in `biproduct.whisker_equiv` below.\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj j' : J\nw : j = j'\n\u22a2 \u03c0 f j \u226b eqToHom (_ : f j = f j') = \u03c0 f j'\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : J\n\u22a2 \u03c0 f j \u226b eqToHom (_ : f j = f j) = \u03c0 f j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : Discrete J\n\u22a2 (isoProduct f).hom \u226b limit.\u03c0 (Discrete.functor f) j = Pi.lift (\u03c0 f) \u226b limit.\u03c0 (Discrete.functor f) j\n[PROOFSTEP]\nsimp [biproduct.isoProduct]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : J\n\u22a2 (isoProduct f).inv \u226b \u03c0 f j = lift (Pi.\u03c0 f) \u226b \u03c0 f j\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : Discrete J\n\u22a2 colimit.\u03b9 (Discrete.functor f) j \u226b (isoCoproduct f).inv = colimit.\u03b9 (Discrete.functor f) j \u226b Sigma.desc (\u03b9 f)\n[PROOFSTEP]\nsimp [biproduct.isoCoproduct]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nj : J\n\u22a2 \u03b9 f j \u226b (isoCoproduct f).hom = \u03b9 f j \u226b desc (Sigma.\u03b9 f)\n[PROOFSTEP]\nsimp [\u2190 Iso.eq_comp_inv]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\n\u22a2 map p = map' p\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\n\u22a2 \u03b9 (fun b => f b) j\u271d \u226b map p \u226b \u03c0 (fun b => g b) j\u271d\u00b9 = \u03b9 (fun b => f b) j\u271d \u226b map' p \u226b \u03c0 (fun b => g b) j\u271d\u00b9\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\n\u22a2 \u03b9 (fun b => f b) j\u271d \u226b map p \u226b \u03c0 (fun b => g b) j\u271d\u00b9 = \u03b9 (fun b => f b) j\u271d \u226b map' p \u226b \u03c0 (fun b => g b) j\u271d\u00b9\n[PROOFSTEP]\nsimp only [Discrete.natTrans_app, Limits.IsColimit.\u03b9_map_assoc, Limits.IsLimit.map_\u03c0, Category.assoc, \u2190\n Bicone.toCone_\u03c0_app_mk, \u2190 biproduct.bicone_\u03c0, \u2190 Bicone.toCocone_\u03b9_app_mk, \u2190 biproduct.bicone_\u03b9]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\n\u22a2 NatTrans.app (Bicone.toCocone (bicone fun b => f b)).\u03b9 { as := j\u271d } \u226b\n NatTrans.app (Bicone.toCone (bicone fun b => f b)).\u03c0 { as := j\u271d\u00b9 } \u226b p j\u271d\u00b9 =\n p j\u271d \u226b\n NatTrans.app (Bicone.toCocone (bicone fun b => g b)).\u03b9 { as := j\u271d } \u226b\n NatTrans.app (Bicone.toCone (bicone fun b => g b)).\u03c0 { as := j\u271d\u00b9 }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\n\u22a2 \u03b9 (fun b => f b) j\u271d \u226b \u03c0 (fun b => f b) j\u271d\u00b9 \u226b p j\u271d\u00b9 = p j\u271d \u226b \u03b9 (fun b => g b) j\u271d \u226b \u03c0 (fun b => g b) j\u271d\u00b9\n[PROOFSTEP]\nrw [biproduct.\u03b9_\u03c0_assoc, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\n\u22a2 (if h : j\u271d = j\u271d\u00b9 then eqToHom (_ : f j\u271d = f j\u271d\u00b9) else 0) \u226b p j\u271d\u00b9 =\n p j\u271d \u226b if h : j\u271d = j\u271d\u00b9 then eqToHom (_ : g j\u271d = g j\u271d\u00b9) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\nh : j\u271d = j\u271d\u00b9\n\u22a2 eqToHom (_ : f j\u271d = f j\u271d\u00b9) \u226b p j\u271d\u00b9 = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d\u00b9)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d : J\n\u22a2 eqToHom (_ : f j\u271d = f j\u271d) \u226b p j\u271d = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d)\n[PROOFSTEP]\nrw [eqToHom_refl, Category.id_comp]\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d : J\n\u22a2 p j\u271d = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d)\n[PROOFSTEP]\nerw [Category.comp_id]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (b : J) \u2192 f b \u27f6 g b\nj\u271d\u00b9 j\u271d : J\nh : \u00acj\u271d = j\u271d\u00b9\n\u22a2 0 \u226b p j\u271d\u00b9 = p j\u271d \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (j : J) \u2192 f j \u27f6 g j\nj : J\n\u22a2 \u03b9 f j \u226b map p = p j \u226b \u03b9 g j\n[PROOFSTEP]\nrw [biproduct.map_eq_map']\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (j : J) \u2192 f j \u27f6 g j\nj : J\n\u22a2 \u03b9 f j \u226b map' p = p j \u226b \u03b9 g j\n[PROOFSTEP]\napply\n Limits.IsColimit.\u03b9_map (biproduct.isColimit f) (biproduct.bicone g).toCocone (Discrete.natTrans fun j => p j.as)\n (Discrete.mk j)\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (j : J) \u2192 f j \u27f6 g j\nP : C\nk : (j : J) \u2192 g j \u27f6 P\n\u22a2 map p \u226b desc k = desc fun j => p j \u226b k j\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\np : (j : J) \u2192 f j \u27f6 g j\nP : C\nk : (j : J) \u2192 g j \u27f6 P\nj\u271d : J\n\u22a2 \u03b9 (fun b => f b) j\u271d \u226b map p \u226b desc k = \u03b9 (fun b => f b) j\u271d \u226b desc fun j => p j \u226b k j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nP : C\nk : (j : J) \u2192 P \u27f6 f j\np : (j : J) \u2192 f j \u27f6 g j\n\u22a2 lift k \u226b map p = lift fun j => k j \u226b p j\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nP : C\nk : (j : J) \u2192 P \u27f6 f j\np : (j : J) \u2192 f j \u27f6 g j\nj\u271d : J\n\u22a2 (lift k \u226b map p) \u226b \u03c0 (fun b => g b) j\u271d = (lift fun j => k j \u226b p j) \u226b \u03c0 (fun b => g b) j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type ?u.108203\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\n\u22a2 g k = g (\u2191e (\u2191e.symm k))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type ?u.124125\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\n\u22a2 g (\u2191e (\u2191e.symm k)) = g k\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (whisker_equiv e w).hom = lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)\n[PROOFSTEP]\nsimp only [whisker_equiv_hom]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (desc fun j => (w j).inv \u226b \u03b9 g (\u2191e j)) =\n lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)\n[PROOFSTEP]\next k j\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\n\u22a2 \u03b9 (fun j => f j) j \u226b (desc fun j => (w j).inv \u226b \u03b9 g (\u2191e j)) \u226b \u03c0 g k =\n \u03b9 (fun j => f j) j \u226b\n (lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)) \u226b \u03c0 g k\n[PROOFSTEP]\nby_cases h : k = e j\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : k = \u2191e j\n\u22a2 \u03b9 (fun j => f j) j \u226b (desc fun j => (w j).inv \u226b \u03b9 g (\u2191e j)) \u226b \u03c0 g k =\n \u03b9 (fun j => f j) j \u226b\n (lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)) \u226b \u03c0 g k\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\n\u22a2 \u03b9 (fun j => f j) j \u226b (desc fun j => (w j).inv \u226b \u03b9 g (\u2191e j)) \u226b \u03c0 g (\u2191e j) =\n \u03b9 (fun j => f j) j \u226b\n (lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)) \u226b \u03c0 g (\u2191e j)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : \u00ack = \u2191e j\n\u22a2 \u03b9 (fun j => f j) j \u226b (desc fun j => (w j).inv \u226b \u03b9 g (\u2191e j)) \u226b \u03c0 g k =\n \u03b9 (fun j => f j) j \u226b\n (lift fun k => \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)) \u226b \u03c0 g k\n[PROOFSTEP]\nsimp only [\u03b9_desc_assoc, Category.assoc, ne_eq, lift_\u03c0]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : \u00ack = \u2191e j\n\u22a2 (w j).inv \u226b \u03b9 g (\u2191e j) \u226b \u03c0 g k =\n \u03b9 (fun j => f j) j \u226b \u03c0 f (\u2191e.symm k) \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)\n[PROOFSTEP]\nrw [biproduct.\u03b9_\u03c0_ne, biproduct.\u03b9_\u03c0_ne_assoc]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : \u00ack = \u2191e j\n\u22a2 (w j).inv \u226b 0 = 0 \u226b (w (\u2191e.symm k)).inv \u226b eqToHom (_ : g (\u2191e (\u2191e.symm k)) = g k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : \u00ack = \u2191e j\n\u22a2 j \u2260 \u2191e.symm k\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nh : \u00ack = \u2191e (\u2191e.symm k)\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nj : J\nh : \u00ack = \u2191e j\n\u22a2 \u2191e j \u2260 k\n[PROOFSTEP]\nexact Ne.symm h\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (whisker_equiv e w).inv = lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom\n[PROOFSTEP]\nsimp only [whisker_equiv_inv]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (desc fun k => eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k)) =\n lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom\n[PROOFSTEP]\next j k\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\n\u22a2 \u03b9 g k \u226b\n (desc fun k => eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k)) \u226b\n \u03c0 (fun j => f j) j =\n \u03b9 g k \u226b (lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom) \u226b \u03c0 (fun j => f j) j\n[PROOFSTEP]\nby_cases h : k = e j\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : k = \u2191e j\n\u22a2 \u03b9 g k \u226b\n (desc fun k => eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k)) \u226b\n \u03c0 (fun j => f j) j =\n \u03b9 g k \u226b (lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom) \u226b \u03c0 (fun j => f j) j\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\n\u22a2 \u03b9 g (\u2191e j) \u226b\n (desc fun k => eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k)) \u226b\n \u03c0 (fun j => f j) j =\n \u03b9 g (\u2191e j) \u226b (lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom) \u226b \u03c0 (fun j => f j) j\n[PROOFSTEP]\nsimp only [\u03b9_desc_assoc, \u2190 eqToHom_iso_hom_naturality_assoc w (e.symm_apply_apply j).symm, Equiv.symm_apply_apply,\n eqToHom_comp_\u03b9, Category.assoc, bicone_\u03b9_\u03c0_self, Category.comp_id, lift_\u03c0, bicone_\u03b9_\u03c0_self_assoc]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = \u2191e j\n\u22a2 \u03b9 g k \u226b\n (desc fun k => eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k)) \u226b\n \u03c0 (fun j => f j) j =\n \u03b9 g k \u226b (lift fun j => \u03c0 g (\u2191e j) \u226b (w j).hom) \u226b \u03c0 (fun j => f j) j\n[PROOFSTEP]\nsimp only [\u03b9_desc_assoc, Category.assoc, ne_eq, lift_\u03c0]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = \u2191e j\n\u22a2 eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b \u03b9 (fun j => f j) (\u2191e.symm k) \u226b \u03c0 (fun j => f j) j =\n \u03b9 g k \u226b \u03c0 g (\u2191e j) \u226b (w j).hom\n[PROOFSTEP]\nrw [biproduct.\u03b9_\u03c0_ne, biproduct.\u03b9_\u03c0_ne_assoc]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = \u2191e j\n\u22a2 eqToHom (_ : g k = g (\u2191e (\u2191e.symm k))) \u226b (w (\u2191e.symm k)).hom \u226b 0 = 0 \u226b (w j).hom\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = \u2191e j\n\u22a2 k \u2260 \u2191e j\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = \u2191e j\n\u22a2 \u2191e.symm k \u2260 j\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.h\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type u_1\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nh : \u00ack = \u2191e (\u2191e.symm k)\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n if h : { fst := j, snd := x } = { fst := j', snd := y } then\n eqToHom\n (_ :\n g { fst := j, snd := x }.fst { fst := j, snd := x }.snd =\n g { fst := j', snd := y }.fst { fst := j', snd := y }.snd)\n else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\nh : { fst := j, snd := x } = { fst := j', snd := y }\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n eqToHom\n (_ :\n g { fst := j, snd := x }.fst { fst := j, snd := x }.snd =\n g { fst := j', snd := y }.fst { fst := j', snd := y }.snd)\n[PROOFSTEP]\nobtain \u27e8rfl, rfl\u27e9 := h\n[GOAL]\ncase pos.refl\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j, snd := x } =\n eqToHom\n (_ :\n g { fst := j, snd := x }.fst { fst := j, snd := x }.snd =\n g { fst := j, snd := x }.fst { fst := j, snd := x }.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\nh : \u00ac{ fst := j, snd := x } = { fst := j', snd := y }\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n 0\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\nh : j = j' \u2192 \u00acHEq x y\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n 0\n[PROOFSTEP]\nby_cases w : j = j'\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\nh : j = j' \u2192 \u00acHEq x y\nw : j = j'\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n 0\n[PROOFSTEP]\ncases w\n[GOAL]\ncase pos.refl\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx y : f j\nh : j = j \u2192 \u00acHEq x y\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j, snd := y } =\n 0\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase pos.refl\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx y : f j\nh : \u00acx = y\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j, snd := y } =\n 0\n[PROOFSTEP]\nsimp [biproduct.\u03b9_\u03c0_ne _ h]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type ?u.135486\nf : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 f i \u2192 C\ninst\u271d\u00b9 : \u2200 (i : \u03b9), HasBiproduct (g i)\ninst\u271d : HasBiproduct fun i => \u2a01 g i\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u00d7 f i\nj : \u03b9\nx : f j\nj' : \u03b9\ny : f j'\nh : j = j' \u2192 \u00acHEq x y\nw : \u00acj = j'\n\u22a2 (fun X => biproduct.\u03b9 (g X.fst) X.snd \u226b biproduct.\u03b9 (fun i => \u2a01 g i) X.fst) { fst := j, snd := x } \u226b\n (fun X => biproduct.\u03c0 (fun i => \u2a01 g i) X.fst \u226b biproduct.\u03c0 (g X.fst) X.snd) { fst := j', snd := y } =\n 0\n[PROOFSTEP]\nsimp [biproduct.\u03b9_\u03c0_ne_assoc _ w]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\n\u22a2 fromSubtype f p \u226b \u03c0 f j = if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\next i\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\n\u22a2 \u03b9 (Subtype.restrict p f) i \u226b fromSubtype f p \u226b \u03c0 f j =\n \u03b9 (Subtype.restrict p f) i \u226b if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\n\u22a2 \u03b9 (Subtype.restrict p f) i \u226b fromSubtype f p \u226b \u03c0 f j =\n \u03b9 (Subtype.restrict p f) i \u226b if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\nrw [biproduct.fromSubtype, biproduct.\u03b9_desc_assoc, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\n\u22a2 (if h : \u2191i = j then eqToHom (_ : f \u2191i = f j) else 0) =\n \u03b9 (Subtype.restrict p f) i \u226b if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\nby_cases h : p j\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n\u22a2 (if h : \u2191i = j then eqToHom (_ : f \u2191i = f j) else 0) =\n \u03b9 (Subtype.restrict p f) i \u226b if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\nrw [dif_pos h, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n\u22a2 (if h : \u2191i = j then eqToHom (_ : f \u2191i = f j) else 0) =\n if h_1 : i = { val := j, property := h } then\n eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n else 0\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u2191i = j\nh\u2082 : i = { val := j, property := h }\n\u22a2 eqToHom (_ : f \u2191i = f j) = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u2191i = j\nh\u2082 : \u00aci = { val := j, property := h }\n\u22a2 eqToHom (_ : f \u2191i = f j) = 0\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u00ac\u2191i = j\nh\u2082 : i = { val := j, property := h }\n\u22a2 0 = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u00ac\u2191i = j\nh\u2082 : \u00aci = { val := j, property := h }\n\u22a2 0 = 0\n[PROOFSTEP]\nexacts [rfl, False.elim (h\u2082 (Subtype.ext h\u2081)), False.elim (h\u2081 (congr_arg Subtype.val h\u2082)), rfl]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : \u00acp j\n\u22a2 (if h : \u2191i = j then eqToHom (_ : f \u2191i = f j) else 0) =\n \u03b9 (Subtype.restrict p f) i \u226b if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\nrw [dif_neg h, dif_neg (show (i : J) \u2260 j from fun h\u2082 => h (h\u2082 \u25b8 i.2)), comp_zero]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\n\u22a2 \u2200 (j : J),\n fromSubtype f p \u226b \u03c0 f j =\n (lift fun j => if h : p j then \u03c0 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 f j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj : Subtype p\n\u22a2 fromSubtype f p \u226b \u03c0 f \u2191j = \u03c0 (Subtype.restrict p f) j\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\n\u22a2 \u03b9 (Subtype.restrict p f) j\u271d \u226b fromSubtype f p \u226b \u03c0 f \u2191j = \u03b9 (Subtype.restrict p f) j\u271d \u226b \u03c0 (Subtype.restrict p f) j\n[PROOFSTEP]\nrw [biproduct.fromSubtype, biproduct.\u03b9_desc_assoc, biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\n\u22a2 (if h : \u2191j\u271d = \u2191j then eqToHom (_ : f \u2191j\u271d = f \u2191j) else 0) =\n if h : j\u271d = j then eqToHom (_ : Subtype.restrict p f j\u271d = Subtype.restrict p f j) else 0\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u2191j\u271d = \u2191j\nh\u2082 : j\u271d = j\n\u22a2 eqToHom (_ : f \u2191j\u271d = f \u2191j) = eqToHom (_ : Subtype.restrict p f j\u271d = Subtype.restrict p f j)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u2191j\u271d = \u2191j\nh\u2082 : \u00acj\u271d = j\n\u22a2 eqToHom (_ : f \u2191j\u271d = f \u2191j) = 0\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u00ac\u2191j\u271d = \u2191j\nh\u2082 : j\u271d = j\n\u22a2 0 = eqToHom (_ : Subtype.restrict p f j\u271d = Subtype.restrict p f j)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u00ac\u2191j\u271d = \u2191j\nh\u2082 : \u00acj\u271d = j\n\u22a2 0 = 0\n[PROOFSTEP]\nexacts [rfl, False.elim (h\u2082 (Subtype.ext h\u2081)), False.elim (h\u2081 (congr_arg Subtype.val h\u2082)), rfl]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\n\u22a2 \u03b9 f j \u226b toSubtype f p = if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\next i\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\n\u22a2 (\u03b9 f j \u226b toSubtype f p) \u226b \u03c0 (Subtype.restrict p f) i =\n (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i\n[PROOFSTEP]\nrw [biproduct.toSubtype, Category.assoc, biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\n\u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) =\n (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i\n[PROOFSTEP]\nby_cases h : p j\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n\u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) =\n (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i\n[PROOFSTEP]\nrw [dif_pos h, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n\u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) =\n if h_1 : { val := j, property := h } = i then\n eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i)\n else 0\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : j = \u2191i\nh\u2082 : { val := j, property := h } = i\n\u22a2 eqToHom (_ : f j = f \u2191i) = eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : j = \u2191i\nh\u2082 : \u00ac{ val := j, property := h } = i\n\u22a2 eqToHom (_ : f j = f \u2191i) = 0\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u00acj = \u2191i\nh\u2082 : { val := j, property := h } = i\n\u22a2 0 = eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh\u2081 : \u00acj = \u2191i\nh\u2082 : \u00ac{ val := j, property := h } = i\n\u22a2 0 = 0\n[PROOFSTEP]\nexacts [rfl, False.elim (h\u2082 (Subtype.ext h\u2081)), False.elim (h\u2081 (congr_arg Subtype.val h\u2082)), rfl]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\nj : J\ni : Subtype p\nh : \u00acp j\n\u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) =\n (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i\n[PROOFSTEP]\nrw [dif_neg h, dif_neg (show j \u2260 i from fun h\u2082 => h (h\u2082.symm \u25b8 i.2)), zero_comp]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\n\u22a2 \u2200 (j : J),\n \u03b9 f j \u226b toSubtype f p =\n \u03b9 f j \u226b desc fun j => if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj : Subtype p\n\u22a2 \u03b9 f \u2191j \u226b toSubtype f p = \u03b9 (Subtype.restrict p f) j\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\n\u22a2 (\u03b9 f \u2191j \u226b toSubtype f p) \u226b \u03c0 (Subtype.restrict p f) j\u271d = \u03b9 (Subtype.restrict p f) j \u226b \u03c0 (Subtype.restrict p f) j\u271d\n[PROOFSTEP]\nrw [biproduct.toSubtype, Category.assoc, biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\n\u22a2 (if h : \u2191j = \u2191j\u271d then eqToHom (_ : f \u2191j = f \u2191j\u271d) else 0) =\n if h : j = j\u271d then eqToHom (_ : Subtype.restrict p f j = Subtype.restrict p f j\u271d) else 0\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u2191j = \u2191j\u271d\nh\u2082 : j = j\u271d\n\u22a2 eqToHom (_ : f \u2191j = f \u2191j\u271d) = eqToHom (_ : Subtype.restrict p f j = Subtype.restrict p f j\u271d)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u2191j = \u2191j\u271d\nh\u2082 : \u00acj = j\u271d\n\u22a2 eqToHom (_ : f \u2191j = f \u2191j\u271d) = 0\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u00ac\u2191j = \u2191j\u271d\nh\u2082 : j = j\u271d\n\u22a2 0 = eqToHom (_ : Subtype.restrict p f j = Subtype.restrict p f j\u271d)\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj j\u271d : Subtype p\nh\u2081 : \u00ac\u2191j = \u2191j\u271d\nh\u2082 : \u00acj = j\u271d\n\u22a2 0 = 0\n[PROOFSTEP]\nexacts [rfl, False.elim (h\u2082 (Subtype.ext h\u2081)), False.elim (h\u2081 (congr_arg Subtype.val h\u2082)), rfl]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\n\u22a2 fromSubtype f p \u226b toSubtype f p = \ud835\udfd9 (\u2a01 Subtype.restrict p f)\n[PROOFSTEP]\nrefine' biproduct.hom_ext _ _ fun j => _\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d : HasBiproduct (Subtype.restrict p f)\nj : Subtype p\n\u22a2 (fromSubtype f p \u226b toSubtype f p) \u226b \u03c0 (Subtype.restrict p f) j =\n \ud835\udfd9 (\u2a01 Subtype.restrict p f) \u226b \u03c0 (Subtype.restrict p f) j\n[PROOFSTEP]\nrw [Category.assoc, biproduct.toSubtype_\u03c0, biproduct.fromSubtype_\u03c0_subtype, Category.id_comp]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\n\u22a2 toSubtype f p \u226b fromSubtype f p = map fun j => if p j then \ud835\udfd9 (f j) else 0\n[PROOFSTEP]\next1 i\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\ni : J\n\u22a2 (toSubtype f p \u226b fromSubtype f p) \u226b \u03c0 f i = (map fun j => if p j then \ud835\udfd9 (f j) else 0) \u226b \u03c0 f i\n[PROOFSTEP]\nby_cases h : p i\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\ni : J\nh : p i\n\u22a2 (toSubtype f p \u226b fromSubtype f p) \u226b \u03c0 f i = (map fun j => if p j then \ud835\udfd9 (f j) else 0) \u226b \u03c0 f i\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\np : J \u2192 Prop\ninst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f)\ninst\u271d : DecidablePred p\ni : J\nh : \u00acp i\n\u22a2 (toSubtype f p \u226b fromSubtype f p) \u226b \u03c0 f i = (map fun j => if p j then \ud835\udfd9 (f j) else 0) \u226b \u03c0 f i\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\n\u22a2 (fromSubtype f fun j => j \u2260 i) \u226b \u03c0 f i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\n\u22a2 (Fork.\u03b9 s \u226b toSubtype f fun j => j \u2260 i) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)) =\n Fork.\u03b9 s\n[PROOFSTEP]\napply biproduct.hom_ext\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\n\u22a2 \u2200 (j : J),\n ((Fork.\u03b9 s \u226b toSubtype f fun j => j \u2260 i) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0))) \u226b\n \u03c0 f j =\n Fork.\u03b9 s \u226b \u03c0 f j\n[PROOFSTEP]\nintro j\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\nj : J\n\u22a2 ((Fork.\u03b9 s \u226b toSubtype f fun j => j \u2260 i) \u226b\n Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0))) \u226b\n \u03c0 f j =\n Fork.\u03b9 s \u226b \u03c0 f j\n[PROOFSTEP]\nrw [KernelFork.\u03b9_of\u03b9, Category.assoc, Category.assoc, biproduct.toSubtype_fromSubtype_assoc, biproduct.map_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\nj : J\n\u22a2 (Fork.\u03b9 s \u226b \u03c0 (fun j => f j) j \u226b if j \u2260 i then \ud835\udfd9 (f j) else 0) = Fork.\u03b9 s \u226b \u03c0 f j\n[PROOFSTEP]\nrcases Classical.em (i = j) with (rfl | h)\n[GOAL]\ncase w.inl\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\n\u22a2 (Fork.\u03b9 s \u226b \u03c0 (fun j => f j) i \u226b if i \u2260 i then \ud835\udfd9 (f i) else 0) = Fork.\u03b9 s \u226b \u03c0 f i\n[PROOFSTEP]\nrw [if_neg (Classical.not_not.2 rfl), comp_zero, comp_zero, KernelFork.condition]\n[GOAL]\ncase w.inr\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\nj : J\nh : \u00aci = j\n\u22a2 (Fork.\u03b9 s \u226b \u03c0 (fun j => f j) j \u226b if j \u2260 i then \ud835\udfd9 (f j) else 0) = Fork.\u03b9 s \u226b \u03c0 f j\n[PROOFSTEP]\nrw [if_pos (Ne.symm h), Category.comp_id]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\n\u22a2 \u2200\n {m :\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((const WalkingParallelPair).obj\n (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)).pt).obj\n WalkingParallelPair.zero},\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)) =\n Fork.\u03b9 s \u2192\n m = Fork.\u03b9 s \u226b toSubtype f fun j => j \u2260 i\n[PROOFSTEP]\nintro m hm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\nm :\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((const WalkingParallelPair).obj\n (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)).pt).obj\n WalkingParallelPair.zero\nhm :\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)) =\n Fork.\u03b9 s\n\u22a2 m = Fork.\u03b9 s \u226b toSubtype f fun j => j \u2260 i\n[PROOFSTEP]\nrw [\u2190 hm, KernelFork.\u03b9_of\u03b9, Category.assoc, biproduct.fromSubtype_toSubtype]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Fork (\u03c0 f i) 0\nm :\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((const WalkingParallelPair).obj\n (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)).pt).obj\n WalkingParallelPair.zero\nhm :\n m \u226b Fork.\u03b9 (KernelFork.of\u03b9 (fromSubtype f fun j => j \u2260 i) (_ : (fromSubtype f fun j => \u00acj = i) \u226b \u03c0 f i = 0)) =\n Fork.\u03b9 s\n\u22a2 m = m \u226b \ud835\udfd9 (\u2a01 Subtype.restrict (fun j => j \u2260 i) f)\n[PROOFSTEP]\nexact (Category.comp_id _).symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\n\u22a2 (\u03b9 f i \u226b toSubtype f fun j => j \u2260 i) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\n\u22a2 Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b\n (fromSubtype f fun j => j \u2260 i) \u226b Cofork.\u03c0 s =\n Cofork.\u03c0 s\n[PROOFSTEP]\napply biproduct.hom_ext'\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\n\u22a2 \u2200 (j : J),\n \u03b9 f j \u226b\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b\n (fromSubtype f fun j => j \u2260 i) \u226b Cofork.\u03c0 s =\n \u03b9 f j \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nintro j\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\nj : J\n\u22a2 \u03b9 f j \u226b\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b\n (fromSubtype f fun j => j \u2260 i) \u226b Cofork.\u03c0 s =\n \u03b9 f j \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [CokernelCofork.\u03c0_of\u03c0, biproduct.toSubtype_fromSubtype_assoc, biproduct.\u03b9_map_assoc]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\nj : J\n\u22a2 (if j \u2260 i then \ud835\udfd9 (f j) else 0) \u226b \u03b9 (fun j => f j) j \u226b Cofork.\u03c0 s = \u03b9 f j \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrcases Classical.em (i = j) with (rfl | h)\n[GOAL]\ncase w.inl\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\n\u22a2 (if i \u2260 i then \ud835\udfd9 (f i) else 0) \u226b \u03b9 (fun j => f j) i \u226b Cofork.\u03c0 s = \u03b9 f i \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [if_neg (Classical.not_not.2 rfl), zero_comp, CokernelCofork.condition]\n[GOAL]\ncase w.inr\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\nj : J\nh : \u00aci = j\n\u22a2 (if j \u2260 i then \ud835\udfd9 (f j) else 0) \u226b \u03b9 (fun j => f j) j \u226b Cofork.\u03c0 s = \u03b9 f j \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [if_pos (Ne.symm h), Category.id_comp]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\n\u22a2 \u2200\n {m :\n ((const WalkingParallelPair).obj\n (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)).pt).obj\n WalkingParallelPair.one \u27f6\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b m =\n Cofork.\u03c0 s \u2192\n m = (fromSubtype f fun j => j \u2260 i) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nintro m hm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\nm :\n ((const WalkingParallelPair).obj\n (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)).pt).obj\n WalkingParallelPair.one \u27f6\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nhm :\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b m =\n Cofork.\u03c0 s\n\u22a2 m = (fromSubtype f fun j => j \u2260 i) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [\u2190 hm, CokernelCofork.\u03c0_of\u03c0, \u2190 Category.assoc, biproduct.fromSubtype_toSubtype]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ni : J\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct (Subtype.restrict (fun j => j \u2260 i) f)\ns : Cofork (\u03b9 f i) 0\nm :\n ((const WalkingParallelPair).obj\n (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)).pt).obj\n WalkingParallelPair.one \u27f6\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nhm :\n Cofork.\u03c0 (CokernelCofork.of\u03c0 (toSubtype f fun j => j \u2260 i) (_ : (\u03b9 f i \u226b toSubtype f fun j => \u00acj = i) = 0)) \u226b m =\n Cofork.\u03c0 s\n\u22a2 m = \ud835\udfd9 (\u2a01 Subtype.restrict (fun j => j \u2260 i) f) \u226b m\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 biproduct.fromSubtype f p\u1d9c \u226b biproduct.toSubtype f p = 0\n[PROOFSTEP]\next j k\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\nk : Subtype p\u1d9c\n\u22a2 biproduct.\u03b9 (Subtype.restrict p\u1d9c f) k \u226b\n (biproduct.fromSubtype f p\u1d9c \u226b biproduct.toSubtype f p) \u226b biproduct.\u03c0 (Subtype.restrict p f) j =\n biproduct.\u03b9 (Subtype.restrict p\u1d9c f) k \u226b 0 \u226b biproduct.\u03c0 (Subtype.restrict p f) j\n[PROOFSTEP]\nsimp only [Category.assoc, biproduct.\u03b9_fromSubtype_assoc, biproduct.\u03b9_toSubtype_assoc, comp_zero, zero_comp]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\nk : Subtype p\u1d9c\n\u22a2 (if h : p \u2191k then biproduct.\u03b9 (Subtype.restrict p f) { val := \u2191k, property := h } else 0) \u226b\n biproduct.\u03c0 (Subtype.restrict p f) j =\n 0\n[PROOFSTEP]\nerw [dif_neg k.2]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\nk : Subtype p\u1d9c\n\u22a2 0 \u226b biproduct.\u03c0 (Subtype.restrict p f) j = 0\n[PROOFSTEP]\nsimp only [zero_comp]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 \u2200 {W' : C} (g' : W' \u27f6 \u2a01 f) (eq' : g' \u226b biproduct.toSubtype f p = 0),\n (fun {W} g x => g \u226b biproduct.toSubtype f p\u1d9c) g' eq' \u226b biproduct.fromSubtype f p\u1d9c = g'\n[PROOFSTEP]\nintro W' g' w\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nw : g' \u226b biproduct.toSubtype f p = 0\n\u22a2 (fun {W} g x => g \u226b biproduct.toSubtype f p\u1d9c) g' w \u226b biproduct.fromSubtype f p\u1d9c = g'\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nw : g' \u226b biproduct.toSubtype f p = 0\nj : K\n\u22a2 ((fun {W} g x => g \u226b biproduct.toSubtype f p\u1d9c) g' w \u226b biproduct.fromSubtype f p\u1d9c) \u226b biproduct.\u03c0 f j =\n g' \u226b biproduct.\u03c0 f j\n[PROOFSTEP]\nsimp only [Category.assoc, biproduct.toSubtype_fromSubtype, Pi.compl_apply, biproduct.map_\u03c0]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nw : g' \u226b biproduct.toSubtype f p = 0\nj : K\n\u22a2 (g' \u226b biproduct.\u03c0 (fun b => f b) j \u226b if (p j)\u1d9c then \ud835\udfd9 (f j) else 0) = g' \u226b biproduct.\u03c0 f j\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nw : g' \u226b biproduct.toSubtype f p = 0\nj : K\nh : (p j)\u1d9c\n\u22a2 g' \u226b biproduct.\u03c0 (fun b => f b) j \u226b \ud835\udfd9 (f j) = g' \u226b biproduct.\u03c0 f j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nw : g' \u226b biproduct.toSubtype f p = 0\nj : K\nh : \u00ac(p j)\u1d9c\n\u22a2 g' \u226b biproduct.\u03c0 (fun b => f b) j \u226b 0 = g' \u226b biproduct.\u03c0 f j\n[PROOFSTEP]\nreplace w := w =\u226b biproduct.\u03c0 _ \u27e8j, not_not.mp h\u27e9\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW' : C\ng' : W' \u27f6 \u2a01 f\nj : K\nh : \u00ac(p j)\u1d9c\nw :\n (g' \u226b biproduct.toSubtype f p) \u226b biproduct.\u03c0 (Subtype.restrict p f) { val := j, property := (_ : p j) } =\n 0 \u226b biproduct.\u03c0 (Subtype.restrict p f) { val := j, property := (_ : p j) }\n\u22a2 g' \u226b biproduct.\u03c0 (fun b => f b) j \u226b 0 = g' \u226b biproduct.\u03c0 f j\n[PROOFSTEP]\nsimpa using w.symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 \u2200 {W' : C} (g' : W' \u27f6 \u2a01 f) (eq' : g' \u226b biproduct.toSubtype f p = 0) (m : W' \u27f6 \u2a01 Subtype.restrict p\u1d9c f),\n m \u226b biproduct.fromSubtype f p\u1d9c = g' \u2192 m = (fun {W} g x => g \u226b biproduct.toSubtype f p\u1d9c) g' eq'\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 biproduct.fromSubtype f p \u226b biproduct.toSubtype f p\u1d9c = 0\n[PROOFSTEP]\next j k\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\u1d9c\nk : Subtype p\n\u22a2 biproduct.\u03b9 (Subtype.restrict p f) k \u226b\n (biproduct.fromSubtype f p \u226b biproduct.toSubtype f p\u1d9c) \u226b biproduct.\u03c0 (Subtype.restrict p\u1d9c f) j =\n biproduct.\u03b9 (Subtype.restrict p f) k \u226b 0 \u226b biproduct.\u03c0 (Subtype.restrict p\u1d9c f) j\n[PROOFSTEP]\nsimp only [Category.assoc, Pi.compl_apply, biproduct.\u03b9_fromSubtype_assoc, biproduct.\u03b9_toSubtype_assoc, comp_zero,\n zero_comp]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\u1d9c\nk : Subtype p\n\u22a2 (if h : (p \u2191k)\u1d9c then biproduct.\u03b9 (Subtype.restrict p\u1d9c f) { val := \u2191k, property := (_ : (p \u2191k)\u1d9c) } else 0) \u226b\n biproduct.\u03c0 (Subtype.restrict p\u1d9c f) j =\n 0\n[PROOFSTEP]\nrw [dif_neg]\n[GOAL]\ncase w.w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\u1d9c\nk : Subtype p\n\u22a2 0 \u226b biproduct.\u03c0 (Subtype.restrict p\u1d9c f) j = 0\ncase w.w.hnc\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\u1d9c\nk : Subtype p\n\u22a2 \u00ac(p \u2191k)\u1d9c\n[PROOFSTEP]\nsimp only [zero_comp]\n[GOAL]\ncase w.w.hnc\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nj : Subtype p\u1d9c\nk : Subtype p\n\u22a2 \u00ac(p \u2191k)\u1d9c\n[PROOFSTEP]\nexact not_not.mpr k.2\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 \u2200 {Z' : C} (g' : \u2a01 f \u27f6 Z') (eq' : biproduct.fromSubtype f p \u226b g' = 0),\n biproduct.toSubtype f p\u1d9c \u226b (fun {W} g x => biproduct.fromSubtype f p\u1d9c \u226b g) g' eq' = g'\n[PROOFSTEP]\nintro W g' w\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nw : biproduct.fromSubtype f p \u226b g' = 0\n\u22a2 biproduct.toSubtype f p\u1d9c \u226b (fun {W} g x => biproduct.fromSubtype f p\u1d9c \u226b g) g' w = g'\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nw : biproduct.fromSubtype f p \u226b g' = 0\nj : K\n\u22a2 biproduct.\u03b9 f j \u226b biproduct.toSubtype f p\u1d9c \u226b (fun {W} g x => biproduct.fromSubtype f p\u1d9c \u226b g) g' w =\n biproduct.\u03b9 f j \u226b g'\n[PROOFSTEP]\nsimp only [biproduct.toSubtype_fromSubtype_assoc, Pi.compl_apply, biproduct.\u03b9_map_assoc]\n[GOAL]\ncase w\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nw : biproduct.fromSubtype f p \u226b g' = 0\nj : K\n\u22a2 (if (p j)\u1d9c then \ud835\udfd9 (f j) else 0) \u226b biproduct.\u03b9 (fun b => f b) j \u226b g' = biproduct.\u03b9 f j \u226b g'\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nw : biproduct.fromSubtype f p \u226b g' = 0\nj : K\nh : (p j)\u1d9c\n\u22a2 \ud835\udfd9 (f j) \u226b biproduct.\u03b9 (fun b => f b) j \u226b g' = biproduct.\u03b9 f j \u226b g'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nw : biproduct.fromSubtype f p \u226b g' = 0\nj : K\nh : \u00ac(p j)\u1d9c\n\u22a2 0 \u226b biproduct.\u03b9 (fun b => f b) j \u226b g' = biproduct.\u03b9 f j \u226b g'\n[PROOFSTEP]\nreplace w := biproduct.\u03b9 _ (\u27e8j, not_not.mp h\u27e9 : p) \u226b= w\n[GOAL]\ncase neg\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\nW : C\ng' : \u2a01 f \u27f6 W\nj : K\nh : \u00ac(p j)\u1d9c\nw :\n biproduct.\u03b9 (Subtype.restrict p f) { val := j, property := (_ : j \u2208 p) } \u226b biproduct.fromSubtype f p \u226b g' =\n biproduct.\u03b9 (Subtype.restrict p f) { val := j, property := (_ : j \u2208 p) } \u226b 0\n\u22a2 0 \u226b biproduct.\u03b9 (fun b => f b) j \u226b g' = biproduct.\u03b9 f j \u226b g'\n[PROOFSTEP]\nsimpa using w.symm\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nK : Type\ninst\u271d\u00b9 : Fintype K\ninst\u271d : HasFiniteBiproducts C\nf : K \u2192 C\np : Set K\n\u22a2 \u2200 {Z' : C} (g' : \u2a01 f \u27f6 Z') (eq' : biproduct.fromSubtype f p \u226b g' = 0) (m : \u2a01 Subtype.restrict p\u1d9c f \u27f6 Z'),\n biproduct.toSubtype f p\u1d9c \u226b m = g' \u2192 m = (fun {W} g x => biproduct.fromSubtype f p\u1d9c \u226b g) g' eq'\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nk : K\n\u22a2 matrix m \u226b \u03c0 g k = desc fun j => m j k\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nk : K\nj\u271d : J\n\u22a2 \u03b9 (fun j => f j) j\u271d \u226b matrix m \u226b \u03c0 g k = \u03b9 (fun j => f j) j\u271d \u226b desc fun j => m j k\n[PROOFSTEP]\nsimp [biproduct.matrix]\n[GOAL]\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nj : J\n\u22a2 \u03b9 f j \u226b matrix m = lift fun k => m j k\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nj : J\nj\u271d : K\n\u22a2 (\u03b9 f j \u226b matrix m) \u226b \u03c0 (fun k => g k) j\u271d = (lift fun k => m j k) \u226b \u03c0 (fun k => g k) j\u271d\n[PROOFSTEP]\nsimp [biproduct.matrix]\n[GOAL]\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nj : J\nk : K\n\u22a2 components (matrix m) j k = m j k\n[PROOFSTEP]\nsimp [biproduct.components]\n[GOAL]\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : \u2a01 f \u27f6 \u2a01 g\n\u22a2 (matrix fun j k => components m j k) = m\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : \u2a01 f \u27f6 \u2a01 g\nj\u271d\u00b9 : K\nj\u271d : J\n\u22a2 \u03b9 (fun j => f j) j\u271d \u226b (matrix fun j k => components m j k) \u226b \u03c0 (fun k => g k) j\u271d\u00b9 =\n \u03b9 (fun j => f j) j\u271d \u226b m \u226b \u03c0 (fun k => g k) j\u271d\u00b9\n[PROOFSTEP]\nsimp [biproduct.components]\n[GOAL]\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\n\u22a2 components (matrix m) = m\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nJ : Type\ninst\u271d\u2074 : Fintype J\nK : Type\ninst\u271d\u00b3 : Fintype K\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nf : J \u2192 C\ng : K \u2192 C\nm : (j : J) \u2192 (k : K) \u2192 f j \u27f6 g k\nx\u271d\u00b9 : J\nx\u271d : K\n\u22a2 components (matrix m) x\u271d\u00b9 x\u271d = m x\u271d\u00b9 x\u271d\n[PROOFSTEP]\napply biproduct.matrix_components\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nb : Bicone f\nhb : Bicone.IsBilimit b\n\u22a2 (IsLimit.conePointUniqueUpToIso hb.isLimit (isLimit f)).inv = desc b.\u03b9\n[PROOFSTEP]\nrefine' biproduct.hom_ext' _ _ fun j => hb.isLimit.hom_ext fun j' => _\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nb : Bicone f\nhb : Bicone.IsBilimit b\nj : J\nj' : Discrete J\n\u22a2 (\u03b9 f j \u226b (IsLimit.conePointUniqueUpToIso hb.isLimit (isLimit f)).inv) \u226b NatTrans.app (Bicone.toCone b).\u03c0 j' =\n (\u03b9 f j \u226b desc b.\u03b9) \u226b NatTrans.app (Bicone.toCone b).\u03c0 j'\n[PROOFSTEP]\nrw [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp, Bicone.toCone_\u03c0_app, biproduct.bicone_\u03c0, biproduct.\u03b9_desc,\n biproduct.\u03b9_\u03c0, b.toCone_\u03c0_app, b.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nb : Bicone f\nhb : Bicone.IsBilimit b\n\u22a2 lift b.\u03c0 \u226b desc b.\u03b9 = \ud835\udfd9 b.pt\n[PROOFSTEP]\nrw [\u2190 biproduct.conePointUniqueUpToIso_hom f hb, \u2190 biproduct.conePointUniqueUpToIso_inv f hb, Iso.hom_inv_id]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nb : Bicone f\nhb : Bicone.IsBilimit b\n\u22a2 desc b.\u03b9 \u226b lift b.\u03c0 = \ud835\udfd9 (\u2a01 f)\n[PROOFSTEP]\nrw [\u2190 biproduct.conePointUniqueUpToIso_hom f hb, \u2190 biproduct.conePointUniqueUpToIso_inv f hb, Iso.inv_hom_id]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\n\u22a2 HasZeroObject C\n[PROOFSTEP]\nrefine' \u27e8\u27e8biproduct Empty.elim, fun X => \u27e8\u27e8\u27e80\u27e9, _\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, _\u27e9\u27e9\u27e9\u27e9\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\n\u22a2 \u2200 (a : \u2a01 Empty.elim \u27f6 X), a = default\n[PROOFSTEP]\nintro a\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\na : \u2a01 Empty.elim \u27f6 X\n\u22a2 a = default\n[PROOFSTEP]\napply biproduct.hom_ext'\n[GOAL]\ncase refine'_1.w\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\na : \u2a01 Empty.elim \u27f6 X\n\u22a2 \u2200 (j : Empty), biproduct.\u03b9 Empty.elim j \u226b a = biproduct.\u03b9 Empty.elim j \u226b default\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\n\u22a2 \u2200 (a : X \u27f6 \u2a01 Empty.elim), a = default\n[PROOFSTEP]\nintro a\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\na : X \u27f6 \u2a01 Empty.elim\n\u22a2 a = default\n[PROOFSTEP]\napply biproduct.hom_ext\n[GOAL]\ncase refine'_2.w\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasFiniteBiproducts C\nX : C\na : X \u27f6 \u2a01 Empty.elim\n\u22a2 \u2200 (j : Empty), a \u226b biproduct.\u03c0 Empty.elim j = default \u226b biproduct.\u03c0 Empty.elim j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : Unique J\nf : J \u2192 C\nj : J\n\u22a2 f default = f j\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : Unique J\nf : J \u2192 C\nj : J\n\u22a2 default = j\n[PROOFSTEP]\nrw [\u2190 Unique.uniq]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : Unique J\nf : J \u2192 C\nj : J\n\u22a2 f j = f default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : Unique J\nf : J \u2192 C\nj : J\n\u22a2 j = default\n[PROOFSTEP]\nrw [\u2190 Unique.uniq]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nj j' : WalkingPair\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) j \u226b (fun j => WalkingPair.casesOn j b.fst b.snd) j' =\n if h : j = j' then eqToHom (_ : pairFunction X Y j = pairFunction X Y j') else 0\n[PROOFSTEP]\nrcases j with \u27e8\u27e9\n[GOAL]\ncase left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nj' : WalkingPair\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.left \u226b (fun j => WalkingPair.casesOn j b.fst b.snd) j' =\n if h : WalkingPair.left = j' then eqToHom (_ : pairFunction X Y WalkingPair.left = pairFunction X Y j') else 0\n[PROOFSTEP]\nrcases j' with \u27e8\u27e9\n[GOAL]\ncase right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nj' : WalkingPair\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.right \u226b (fun j => WalkingPair.casesOn j b.fst b.snd) j' =\n if h : WalkingPair.right = j' then eqToHom (_ : pairFunction X Y WalkingPair.right = pairFunction X Y j') else 0\n[PROOFSTEP]\nrcases j' with \u27e8\u27e9\n[GOAL]\ncase left.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.left \u226b\n (fun j => WalkingPair.casesOn j b.fst b.snd) WalkingPair.left =\n if h : WalkingPair.left = WalkingPair.left then\n eqToHom (_ : pairFunction X Y WalkingPair.left = pairFunction X Y WalkingPair.left)\n else 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.left \u226b\n (fun j => WalkingPair.casesOn j b.fst b.snd) WalkingPair.right =\n if h : WalkingPair.left = WalkingPair.right then\n eqToHom (_ : pairFunction X Y WalkingPair.left = pairFunction X Y WalkingPair.right)\n else 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.right \u226b\n (fun j => WalkingPair.casesOn j b.fst b.snd) WalkingPair.left =\n if h : WalkingPair.right = WalkingPair.left then\n eqToHom (_ : pairFunction X Y WalkingPair.right = pairFunction X Y WalkingPair.left)\n else 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 (fun j => WalkingPair.casesOn j b.inl b.inr) WalkingPair.right \u226b\n (fun j => WalkingPair.casesOn j b.fst b.snd) WalkingPair.right =\n if h : WalkingPair.right = WalkingPair.right then\n eqToHom (_ : pairFunction X Y WalkingPair.right = pairFunction X Y WalkingPair.right)\n else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nj : Discrete WalkingPair\n\u22a2 NatTrans.app (Bicone.toCone (toBicone b)).\u03c0 j =\n (Iso.refl (Bicone.toCone (toBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 j\n[PROOFSTEP]\ncases' j with as\n[GOAL]\ncase mk\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nas : WalkingPair\n\u22a2 NatTrans.app (Bicone.toCone (toBicone b)).\u03c0 { as := as } =\n (Iso.refl (Bicone.toCone (toBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 { as := as }\n[PROOFSTEP]\ncases as\n[GOAL]\ncase mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 NatTrans.app (Bicone.toCone (toBicone b)).\u03c0 { as := WalkingPair.left } =\n (Iso.refl (Bicone.toCone (toBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 NatTrans.app (Bicone.toCone (toBicone b)).\u03c0 { as := WalkingPair.right } =\n (Iso.refl (Bicone.toCone (toBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nj : Discrete WalkingPair\n\u22a2 NatTrans.app (Bicone.toCocone (toBicone b)).\u03b9 j \u226b (Iso.refl (Bicone.toCocone (toBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 j\n[PROOFSTEP]\ncases' j with as\n[GOAL]\ncase mk\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\nas : WalkingPair\n\u22a2 NatTrans.app (Bicone.toCocone (toBicone b)).\u03b9 { as := as } \u226b (Iso.refl (Bicone.toCocone (toBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 { as := as }\n[PROOFSTEP]\ncases as\n[GOAL]\ncase mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 NatTrans.app (Bicone.toCocone (toBicone b)).\u03b9 { as := WalkingPair.left } \u226b\n (Iso.refl (Bicone.toCocone (toBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nb : BinaryBicone X Y\n\u22a2 NatTrans.app (Bicone.toCocone (toBicone b)).\u03b9 { as := WalkingPair.right } \u226b\n (Iso.refl (Bicone.toCocone (toBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 \u03b9 b WalkingPair.left \u226b \u03c0 b WalkingPair.left = \ud835\udfd9 X\n[PROOFSTEP]\nsimp [Bicone.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 \u03b9 b WalkingPair.left \u226b \u03c0 b WalkingPair.right = 0\n[PROOFSTEP]\nsimp [Bicone.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 \u03b9 b WalkingPair.right \u226b \u03c0 b WalkingPair.left = 0\n[PROOFSTEP]\nsimp [Bicone.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 \u03b9 b WalkingPair.right \u226b \u03c0 b WalkingPair.right = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp [Bicone.\u03b9_\u03c0]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nj : Discrete WalkingPair\n\u22a2 NatTrans.app (BinaryBicone.toCone (toBinaryBicone b)).\u03c0 j =\n (Iso.refl (BinaryBicone.toCone (toBinaryBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 NatTrans.app (BinaryBicone.toCone (toBinaryBicone b)).\u03c0 { as := WalkingPair.left } =\n (Iso.refl (BinaryBicone.toCone (toBinaryBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 NatTrans.app (BinaryBicone.toCone (toBinaryBicone b)).\u03c0 { as := WalkingPair.right } =\n (Iso.refl (BinaryBicone.toCone (toBinaryBicone b)).pt).hom \u226b NatTrans.app (toCone b).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nj : Discrete WalkingPair\n\u22a2 NatTrans.app (BinaryBicone.toCocone (toBinaryBicone b)).\u03b9 j \u226b\n (Iso.refl (BinaryBicone.toCocone (toBinaryBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 NatTrans.app (BinaryBicone.toCocone (toBinaryBicone b)).\u03b9 { as := WalkingPair.left } \u226b\n (Iso.refl (BinaryBicone.toCocone (toBinaryBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\n\u22a2 NatTrans.app (BinaryBicone.toCocone (toBinaryBicone b)).\u03b9 { as := WalkingPair.right } \u226b\n (Iso.refl (BinaryBicone.toCocone (toBinaryBicone b)).pt).hom =\n NatTrans.app (toCocone b).\u03b9 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : BinaryBicone X Y\nx\u271d : Bicone.IsBilimit (toBicone b)\nh : IsLimit (Bicone.toCone (toBicone b))\nh' : IsColimit (Bicone.toCocone (toBicone b))\n\u22a2 (fun h => { isLimit := \u2191(toBiconeIsLimit b).symm h.isLimit, isColimit := \u2191(toBiconeIsColimit b).symm h.isColimit })\n ((fun h => { isLimit := \u2191(toBiconeIsLimit b) h.isLimit, isColimit := \u2191(toBiconeIsColimit b) h.isColimit })\n { isLimit := h, isColimit := h' }) =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : BinaryBicone X Y\nx\u271d : Bicone.IsBilimit (toBicone b)\nh : IsLimit (Bicone.toCone (toBicone b))\nh' : IsColimit (Bicone.toCocone (toBicone b))\n\u22a2 { isLimit := \u2191(toBiconeIsLimit b).symm (\u2191(toBiconeIsLimit b) h),\n isColimit := \u2191(toBiconeIsColimit b).symm (\u2191(toBiconeIsColimit b) h') } =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : BinaryBicone X Y\nx\u271d : IsBilimit b\nh : IsLimit (toCone b)\nh' : IsColimit (toCocone b)\n\u22a2 (fun h => { isLimit := \u2191(toBiconeIsLimit b) h.isLimit, isColimit := \u2191(toBiconeIsColimit b) h.isColimit })\n ((fun h =>\n { isLimit := \u2191(toBiconeIsLimit b).symm h.isLimit, isColimit := \u2191(toBiconeIsColimit b).symm h.isColimit })\n { isLimit := h, isColimit := h' }) =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : BinaryBicone X Y\nx\u271d : IsBilimit b\nh : IsLimit (toCone b)\nh' : IsColimit (toCocone b)\n\u22a2 { isLimit := \u2191(toBiconeIsLimit b) (\u2191(toBiconeIsLimit b).symm h),\n isColimit := \u2191(toBiconeIsColimit b) (\u2191(toBiconeIsColimit b).symm h') } =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nx\u271d : BinaryBicone.IsBilimit (toBinaryBicone b)\nh : IsLimit (BinaryBicone.toCone (toBinaryBicone b))\nh' : IsColimit (BinaryBicone.toCocone (toBinaryBicone b))\n\u22a2 (fun h =>\n { isLimit := \u2191(toBinaryBiconeIsLimit b).symm h.isLimit,\n isColimit := \u2191(toBinaryBiconeIsColimit b).symm h.isColimit })\n ((fun h =>\n { isLimit := \u2191(toBinaryBiconeIsLimit b) h.isLimit, isColimit := \u2191(toBinaryBiconeIsColimit b) h.isColimit })\n { isLimit := h, isColimit := h' }) =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nx\u271d : BinaryBicone.IsBilimit (toBinaryBicone b)\nh : IsLimit (BinaryBicone.toCone (toBinaryBicone b))\nh' : IsColimit (BinaryBicone.toCocone (toBinaryBicone b))\n\u22a2 { isLimit := \u2191(toBinaryBiconeIsLimit b).symm (\u2191(toBinaryBiconeIsLimit b) h),\n isColimit := \u2191(toBinaryBiconeIsColimit b).symm (\u2191(toBinaryBiconeIsColimit b) h') } =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nx\u271d : IsBilimit b\nh : IsLimit (toCone b)\nh' : IsColimit (toCocone b)\n\u22a2 (fun h => { isLimit := \u2191(toBinaryBiconeIsLimit b) h.isLimit, isColimit := \u2191(toBinaryBiconeIsColimit b) h.isColimit })\n ((fun h =>\n { isLimit := \u2191(toBinaryBiconeIsLimit b).symm h.isLimit,\n isColimit := \u2191(toBinaryBiconeIsColimit b).symm h.isColimit })\n { isLimit := h, isColimit := h' }) =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nb : Bicone (pairFunction X Y)\nx\u271d : IsBilimit b\nh : IsLimit (toCone b)\nh' : IsColimit (toCocone b)\n\u22a2 { isLimit := \u2191(toBinaryBiconeIsLimit b) (\u2191(toBinaryBiconeIsLimit b).symm h),\n isColimit := \u2191(toBinaryBiconeIsColimit b) (\u2191(toBinaryBiconeIsColimit b).symm h') } =\n { isLimit := h, isColimit := h' }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoProd X Y).hom = prod.lift fst snd\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 inl \u226b (isoProd X Y).hom \u226b prod.fst = inl \u226b prod.lift fst snd \u226b prod.fst\n[PROOFSTEP]\nsimp [biprod.isoProd]\n[GOAL]\ncase h\u2081.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 inr \u226b (isoProd X Y).hom \u226b prod.fst = inr \u226b prod.lift fst snd \u226b prod.fst\n[PROOFSTEP]\nsimp [biprod.isoProd]\n[GOAL]\ncase h\u2082.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 inl \u226b (isoProd X Y).hom \u226b prod.snd = inl \u226b prod.lift fst snd \u226b prod.snd\n[PROOFSTEP]\nsimp [biprod.isoProd]\n[GOAL]\ncase h\u2082.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 inr \u226b (isoProd X Y).hom \u226b prod.snd = inr \u226b prod.lift fst snd \u226b prod.snd\n[PROOFSTEP]\nsimp [biprod.isoProd]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoProd X Y).inv = lift prod.fst prod.snd\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoProd X Y).inv \u226b fst = lift prod.fst prod.snd \u226b fst\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoProd X Y).inv \u226b snd = lift prod.fst prod.snd \u226b snd\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (isoCoprod X Y).inv = coprod.desc inl inr\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (coprod.inl \u226b (isoCoprod X Y).inv) \u226b fst = (coprod.inl \u226b coprod.desc inl inr) \u226b fst\n[PROOFSTEP]\nsimp [biprod.isoCoprod]\n[GOAL]\ncase h\u2081.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (coprod.inl \u226b (isoCoprod X Y).inv) \u226b snd = (coprod.inl \u226b coprod.desc inl inr) \u226b snd\n[PROOFSTEP]\nsimp [biprod.isoCoprod]\n[GOAL]\ncase h\u2082.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (coprod.inr \u226b (isoCoprod X Y).inv) \u226b fst = (coprod.inr \u226b coprod.desc inl inr) \u226b fst\n[PROOFSTEP]\nsimp [biprod.isoCoprod]\n[GOAL]\ncase h\u2082.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (coprod.inr \u226b (isoCoprod X Y).inv) \u226b snd = (coprod.inr \u226b coprod.desc inl inr) \u226b snd\n[PROOFSTEP]\nsimp [biprod.isoCoprod]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 (biprod.isoCoprod X Y).hom = biprod.desc coprod.inl coprod.inr\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 biprod.inl \u226b (biprod.isoCoprod X Y).hom = biprod.inl \u226b biprod.desc coprod.inl coprod.inr\n[PROOFSTEP]\nsimp [\u2190 Iso.eq_comp_inv]\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 biprod.inr \u226b (biprod.isoCoprod X Y).hom = biprod.inr \u226b biprod.desc coprod.inl coprod.inr\n[PROOFSTEP]\nsimp [\u2190 Iso.eq_comp_inv]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 map f g = map' f g\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (inl \u226b map f g) \u226b fst = (inl \u226b map' f g) \u226b fst\n[PROOFSTEP]\nsimp only [mapPair_left, IsColimit.\u03b9_map, IsLimit.map_\u03c0, biprod.inl_fst_assoc, Category.assoc, \u2190\n BinaryBicone.toCone_\u03c0_app_left, \u2190 BinaryBiproduct.bicone_fst, \u2190 BinaryBicone.toCocone_\u03b9_app_left, \u2190\n BinaryBiproduct.bicone_inl]\n[GOAL]\ncase h\u2080.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone W X)).\u03b9 { as := WalkingPair.left } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone W X)).\u03c0 { as := WalkingPair.left } \u226b f =\n f \u226b\n NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone Y Z)).\u03b9 { as := WalkingPair.left } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone Y Z)).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 inl \u226b fst \u226b f = f \u226b inl \u226b fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2080.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (inl \u226b map f g) \u226b snd = (inl \u226b map' f g) \u226b snd\n[PROOFSTEP]\nsimp only [mapPair_left, IsColimit.\u03b9_map, IsLimit.map_\u03c0, zero_comp, biprod.inl_snd_assoc, Category.assoc, \u2190\n BinaryBicone.toCone_\u03c0_app_right, \u2190 BinaryBiproduct.bicone_snd, \u2190 BinaryBicone.toCocone_\u03b9_app_left, \u2190\n BinaryBiproduct.bicone_inl]\n[GOAL]\ncase h\u2080.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone W X)).\u03b9 { as := WalkingPair.left } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone W X)).\u03c0 { as := WalkingPair.right } \u226b\n NatTrans.app (mapPair f g) { as := WalkingPair.right } =\n f \u226b\n NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone Y Z)).\u03b9 { as := WalkingPair.left } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone Y Z)).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (inr \u226b map f g) \u226b fst = (inr \u226b map' f g) \u226b fst\n[PROOFSTEP]\nsimp only [mapPair_right, biprod.inr_fst_assoc, IsColimit.\u03b9_map, IsLimit.map_\u03c0, zero_comp, Category.assoc, \u2190\n BinaryBicone.toCone_\u03c0_app_left, \u2190 BinaryBiproduct.bicone_fst, \u2190 BinaryBicone.toCocone_\u03b9_app_right, \u2190\n BinaryBiproduct.bicone_inr]\n[GOAL]\ncase h\u2081.h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone W X)).\u03b9 { as := WalkingPair.right } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone W X)).\u03c0 { as := WalkingPair.left } \u226b\n NatTrans.app (mapPair f g) { as := WalkingPair.left } =\n g \u226b\n NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone Y Z)).\u03b9 { as := WalkingPair.right } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone Y Z)).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (inr \u226b map f g) \u226b snd = (inr \u226b map' f g) \u226b snd\n[PROOFSTEP]\nsimp only [mapPair_right, IsColimit.\u03b9_map, IsLimit.map_\u03c0, biprod.inr_snd_assoc, Category.assoc, \u2190\n BinaryBicone.toCone_\u03c0_app_right, \u2190 BinaryBiproduct.bicone_snd, \u2190 BinaryBicone.toCocone_\u03b9_app_right, \u2190\n BinaryBiproduct.bicone_inr]\n[GOAL]\ncase h\u2081.h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone W X)).\u03b9 { as := WalkingPair.right } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone W X)).\u03c0 { as := WalkingPair.right } \u226b g =\n g \u226b\n NatTrans.app (BinaryBicone.toCocone (BinaryBiproduct.bicone Y Z)).\u03b9 { as := WalkingPair.right } \u226b\n NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone Y Z)).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 inl \u226b map f g = f \u226b inl\n[PROOFSTEP]\nrw [biprod.map_eq_map']\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 inl \u226b map' f g = f \u226b inl\n[PROOFSTEP]\nexact IsColimit.\u03b9_map (BinaryBiproduct.isColimit W X) _ _ \u27e8WalkingPair.left\u27e9\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 inr \u226b map f g = g \u226b inr\n[PROOFSTEP]\nrw [biprod.map_eq_map']\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nP Q W X Y Z : C\ninst\u271d\u00b9 : HasBinaryBiproduct W X\ninst\u271d : HasBinaryBiproduct Y Z\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 inr \u226b map' f g = g \u226b inr\n[PROOFSTEP]\nexact IsColimit.\u03b9_map (BinaryBiproduct.isColimit W X) _ _ \u27e8WalkingPair.right\u27e9\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 (IsLimit.conePointUniqueUpToIso hb.isLimit (BinaryBiproduct.isLimit X Y)).inv = desc b.inl b.inr\n[PROOFSTEP]\nrefine' biprod.hom_ext' _ _ (hb.isLimit.hom_ext fun j => _) (hb.isLimit.hom_ext fun j => _)\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 (inl \u226b (IsLimit.conePointUniqueUpToIso hb.isLimit (BinaryBiproduct.isLimit X Y)).inv) \u226b\n NatTrans.app (BinaryBicone.toCone b).\u03c0 j =\n (inl \u226b desc b.inl b.inr) \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 (inr \u226b (IsLimit.conePointUniqueUpToIso hb.isLimit (BinaryBiproduct.isLimit X Y)).inv) \u226b\n NatTrans.app (BinaryBicone.toCone b).\u03c0 j =\n (inr \u226b desc b.inl b.inr) \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\n[PROOFSTEP]\nall_goals\n simp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]\n rcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 (inl \u226b (IsLimit.conePointUniqueUpToIso hb.isLimit (BinaryBiproduct.isLimit X Y)).inv) \u226b\n NatTrans.app (BinaryBicone.toCone b).\u03c0 j =\n (inl \u226b desc b.inl b.inr) \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\n[PROOFSTEP]\nsimp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]\n[GOAL]\ncase refine'_1\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 inl \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 j =\n inl \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 (inr \u226b (IsLimit.conePointUniqueUpToIso hb.isLimit (BinaryBiproduct.isLimit X Y)).inv) \u226b\n NatTrans.app (BinaryBicone.toCone b).\u03c0 j =\n (inr \u226b desc b.inl b.inr) \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\n[PROOFSTEP]\nsimp only [Category.assoc, IsLimit.conePointUniqueUpToIso_inv_comp]\n[GOAL]\ncase refine'_2\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 inr \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 j =\n inr \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase refine'_1.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inl \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.left } =\n inl \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.left }\ncase refine'_1.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inl \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.right } =\n inl \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.right }\ncase refine'_2.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inr \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.left } =\n inr \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.left }\ncase refine'_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inr \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.right } =\n inr \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nall_goals simp\n[GOAL]\ncase refine'_1.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inl \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.left } =\n inl \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inl \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.right } =\n inl \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.mk.left\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inr \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.left } =\n inr \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.mk.right\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 inr \u226b NatTrans.app (BinaryBicone.toCone (BinaryBiproduct.bicone X Y)).\u03c0 { as := WalkingPair.right } =\n inr \u226b desc b.inl b.inr \u226b NatTrans.app (BinaryBicone.toCone b).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 lift b.fst b.snd \u226b desc b.inl b.inr = \ud835\udfd9 b.pt\n[PROOFSTEP]\nrw [\u2190 biprod.conePointUniqueUpToIso_hom X Y hb, \u2190 biprod.conePointUniqueUpToIso_inv X Y hb, Iso.hom_inv_id]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 desc b.inl b.inr \u226b lift b.fst b.snd = \ud835\udfd9 (X \u229e Y)\n[PROOFSTEP]\nrw [\u2190 biprod.conePointUniqueUpToIso_hom X Y hb, \u2190 biprod.conePointUniqueUpToIso_inv X Y hb, Iso.inv_hom_id]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 IsIso inl \u2194 \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 IsIso inl \u2192 \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nh : IsIso inl\n\u22a2 \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n[PROOFSTEP]\nhave := (cancel_epi (inv biprod.inl : X \u229e Y \u27f6 X)).2 <| @biprod.inl_fst _ _ _ X Y _\n[GOAL]\ncase mp\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nh : IsIso inl\nthis : inv inl \u226b inl \u226b fst = inv inl \u226b \ud835\udfd9 X\n\u22a2 \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n[PROOFSTEP]\nrw [IsIso.inv_hom_id_assoc, Category.comp_id] at this \n[GOAL]\ncase mp\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nh : IsIso inl\nthis : fst = inv inl\n\u22a2 \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n[PROOFSTEP]\nrw [this, IsIso.inv_hom_id]\n[GOAL]\ncase mpr\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\n\u22a2 \ud835\udfd9 (X \u229e Y) = fst \u226b inl \u2192 IsIso inl\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nh : \ud835\udfd9 (X \u229e Y) = fst \u226b inl\n\u22a2 IsIso inl\n[PROOFSTEP]\nexact \u27e8\u27e8biprod.fst, biprod.inl_fst, h.symm\u27e9\u27e9\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.fst 0\n\u22a2 (Fork.\u03b9 s \u226b c.snd) \u226b Fork.\u03b9 (fstKernelFork c) = Fork.\u03b9 s\n[PROOFSTEP]\napply BinaryFan.IsLimit.hom_ext i\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.fst 0\n\u22a2 ((Fork.\u03b9 s \u226b c.snd) \u226b Fork.\u03b9 (fstKernelFork c)) \u226b BinaryFan.fst (toCone c) = Fork.\u03b9 s \u226b BinaryFan.fst (toCone c)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.fst 0\n\u22a2 ((Fork.\u03b9 s \u226b c.snd) \u226b Fork.\u03b9 (fstKernelFork c)) \u226b BinaryFan.snd (toCone c) = Fork.\u03b9 s \u226b BinaryFan.snd (toCone c)\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.fst 0\nm\u271d :\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((const WalkingParallelPair).obj (fstKernelFork c).pt).obj WalkingParallelPair.zero\nhm : m\u271d \u226b Fork.\u03b9 (fstKernelFork c) = Fork.\u03b9 s\n\u22a2 m\u271d = Fork.\u03b9 s \u226b c.snd\n[PROOFSTEP]\nsimp [\u2190 hm]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.snd 0\n\u22a2 (Fork.\u03b9 s \u226b c.fst) \u226b Fork.\u03b9 (sndKernelFork c) = Fork.\u03b9 s\n[PROOFSTEP]\napply BinaryFan.IsLimit.hom_ext i\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.snd 0\n\u22a2 ((Fork.\u03b9 s \u226b c.fst) \u226b Fork.\u03b9 (sndKernelFork c)) \u226b BinaryFan.fst (toCone c) = Fork.\u03b9 s \u226b BinaryFan.fst (toCone c)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.snd 0\n\u22a2 ((Fork.\u03b9 s \u226b c.fst) \u226b Fork.\u03b9 (sndKernelFork c)) \u226b BinaryFan.snd (toCone c) = Fork.\u03b9 s \u226b BinaryFan.snd (toCone c)\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsLimit (toCone c)\ns : Fork c.snd 0\nm\u271d :\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((const WalkingParallelPair).obj (sndKernelFork c).pt).obj WalkingParallelPair.zero\nhm : m\u271d \u226b Fork.\u03b9 (sndKernelFork c) = Fork.\u03b9 s\n\u22a2 m\u271d = Fork.\u03b9 s \u226b c.fst\n[PROOFSTEP]\nsimp [\u2190 hm]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inl 0\n\u22a2 Cofork.\u03c0 (inlCokernelCofork c) \u226b c.inr \u226b Cofork.\u03c0 s = Cofork.\u03c0 s\n[PROOFSTEP]\napply BinaryCofan.IsColimit.hom_ext i\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inl 0\n\u22a2 BinaryCofan.inl (toCocone c) \u226b Cofork.\u03c0 (inlCokernelCofork c) \u226b c.inr \u226b Cofork.\u03c0 s =\n BinaryCofan.inl (toCocone c) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inl 0\n\u22a2 BinaryCofan.inr (toCocone c) \u226b Cofork.\u03c0 (inlCokernelCofork c) \u226b c.inr \u226b Cofork.\u03c0 s =\n BinaryCofan.inr (toCocone c) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inl 0\nm\u271d :\n ((const WalkingParallelPair).obj (inlCokernelCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nhm : Cofork.\u03c0 (inlCokernelCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = c.inr \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp [\u2190 hm]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inr 0\n\u22a2 Cofork.\u03c0 (inrCokernelCofork c) \u226b c.inl \u226b Cofork.\u03c0 s = Cofork.\u03c0 s\n[PROOFSTEP]\napply BinaryCofan.IsColimit.hom_ext i\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inr 0\n\u22a2 BinaryCofan.inl (toCocone c) \u226b Cofork.\u03c0 (inrCokernelCofork c) \u226b c.inl \u226b Cofork.\u03c0 s =\n BinaryCofan.inl (toCocone c) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inr 0\n\u22a2 BinaryCofan.inr (toCocone c) \u226b Cofork.\u03c0 (inrCokernelCofork c) \u226b c.inl \u226b Cofork.\u03c0 s =\n BinaryCofan.inr (toCocone c) \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasZeroMorphisms C\nP Q X Y : C\nc : BinaryBicone X Y\ni : IsColimit (toCocone c)\ns : Cofork c.inr 0\nm\u271d :\n ((const WalkingParallelPair).obj (inrCokernelCofork c).pt).obj WalkingParallelPair.one \u27f6\n ((const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nhm : Cofork.\u03c0 (inrCokernelCofork c) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = c.inl \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp [\u2190 hm]\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero Y\n\u22a2 biprod.fst \u226b biprod.inl = \ud835\udfd9 (X \u229e Y)\n[PROOFSTEP]\napply CategoryTheory.Limits.biprod.hom_ext\n[GOAL]\ncase h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero Y\n\u22a2 (biprod.fst \u226b biprod.inl) \u226b biprod.fst = \ud835\udfd9 (X \u229e Y) \u226b biprod.fst\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.inl_fst, Category.comp_id, Category.id_comp, biprod.inl_snd, comp_zero]\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero Y\n\u22a2 (biprod.fst \u226b biprod.inl) \u226b biprod.snd = \ud835\udfd9 (X \u229e Y) \u226b biprod.snd\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.inl_fst, Category.comp_id, Category.id_comp, biprod.inl_snd, comp_zero]\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero Y\n\u22a2 0 = biprod.snd\n[PROOFSTEP]\napply hY.eq_of_tgt\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero X\n\u22a2 biprod.snd \u226b biprod.inr = \ud835\udfd9 (X \u229e Y)\n[PROOFSTEP]\napply CategoryTheory.Limits.biprod.hom_ext\n[GOAL]\ncase h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero X\n\u22a2 (biprod.snd \u226b biprod.inr) \u226b biprod.fst = \ud835\udfd9 (X \u229e Y) \u226b biprod.fst\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.inr_snd, Category.comp_id, Category.id_comp, biprod.inr_fst, comp_zero]\n[GOAL]\ncase h\u2081\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero X\n\u22a2 (biprod.snd \u226b biprod.inr) \u226b biprod.snd = \ud835\udfd9 (X \u229e Y) \u226b biprod.snd\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.inr_snd, Category.comp_id, Category.id_comp, biprod.inr_fst, comp_zero]\n[GOAL]\ncase h\u2080\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q X Y : C\ninst\u271d : HasBinaryBiproduct X Y\nhY : IsZero X\n\u22a2 0 = biprod.fst\n[PROOFSTEP]\napply hY.eq_of_tgt\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP\u271d Q\u271d : C\ninst\u271d : HasBinaryBiproducts C\nP Q : C\n\u22a2 braiding' P Q = braiding P Q\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q : C\ninst\u271d : HasBinaryBiproducts C\nW X Y Z : C\nf : X \u27f6 Y\ng : Z \u27f6 W\n\u22a2 map f g \u226b (braiding Y W).hom = (braiding X Z).hom \u226b map g f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP Q : C\ninst\u271d : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\n\u22a2 (braiding X W).hom \u226b map f g \u226b (braiding Y Z).hom = map g f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP\u271d Q\u271d : C\ninst\u271d : HasBinaryBiproducts C\nP Q : C\n\u22a2 lift snd fst \u226b lift snd fst = \ud835\udfd9 (P \u229e Q)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroMorphisms C\nP\u271d Q\u271d : C\ninst\u271d : HasBinaryBiproducts C\nP Q : C\n\u22a2 (braiding P Q).hom \u226b (braiding Q P).hom = \ud835\udfd9 (P \u229e Q)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\n\u22a2 f \u226b biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst = \ud835\udfd9 W\n[PROOFSTEP]\nhave t := congrArg (fun p : W \u229e X \u27f6 W \u229e X => biprod.inl \u226b p \u226b biprod.fst) (IsIso.hom_inv_id (biprod.map f g))\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\nt :\n (fun p => biprod.inl \u226b p \u226b biprod.fst) (biprod.map f g \u226b inv (biprod.map f g)) =\n (fun p => biprod.inl \u226b p \u226b biprod.fst) (\ud835\udfd9 (W \u229e X))\n\u22a2 f \u226b biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst = \ud835\udfd9 W\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.assoc, biprod.inl_map_assoc] at t \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\nt : f \u226b biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst = biprod.inl \u226b biprod.fst\n\u22a2 f \u226b biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst = \ud835\udfd9 W\n[PROOFSTEP]\nsimp [t]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\n\u22a2 (biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst) \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\nhave t := congrArg (fun p : Y \u229e Z \u27f6 Y \u229e Z => biprod.inl \u226b p \u226b biprod.fst) (IsIso.inv_hom_id (biprod.map f g))\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\nt :\n (fun p => biprod.inl \u226b p \u226b biprod.fst) (inv (biprod.map f g) \u226b biprod.map f g) =\n (fun p => biprod.inl \u226b p \u226b biprod.fst) (\ud835\udfd9 (Y \u229e Z))\n\u22a2 (biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst) \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.assoc, biprod.map_fst] at t \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\nt : biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst \u226b f = biprod.inl \u226b biprod.fst\n\u22a2 (biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst) \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\nt : biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst \u226b f = biprod.inl \u226b biprod.fst\n\u22a2 biprod.inl \u226b inv (biprod.map f g) \u226b biprod.fst \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\nsimp [t]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\n\u22a2 IsIso (biprod.map g f)\n[PROOFSTEP]\nrw [\u2190 biprod.braiding_map_braiding]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBinaryBiproducts C\nW X Y Z : C\nf : W \u27f6 Y\ng : X \u27f6 Z\ninst\u271d : IsIso (biprod.map f g)\n\u22a2 IsIso ((biprod.braiding X W).hom \u226b biprod.map f g \u226b (biprod.braiding Y Z).hom)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Biproducts", "llama_tokens": 61692, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.2903443617412255}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nH : Nodup l\ni : Fin (length l)\n\u22a2 (fun x => { val := indexOf (\u2191x) l, isLt := (_ : indexOf (\u2191x) l < length l) })\n ((fun i => { val := get l i, property := (_ : get l { val := \u2191i, isLt := (_ : \u2191i < length l) } \u2208 l) }) i) =\n i\n[PROOFSTEP]\nsimp only [List.get_indexOf, eq_self_iff_true, Fin.eta, Subtype.coe_mk, H]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nH : Nodup l\nx : { x // x \u2208 l }\n\u22a2 (fun i => { val := get l i, property := (_ : get l { val := \u2191i, isLt := (_ : \u2191i < length l) } \u2208 l) })\n ((fun x => { val := indexOf (\u2191x) l, isLt := (_ : indexOf (\u2191x) l < length l) }) x) =\n x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nnd : Nodup l\nh : \u2200 (x : \u03b1), x \u2208 l\ni : Fin (length l)\n\u22a2 (fun a => { val := indexOf a l, isLt := (_ : indexOf a l < length l) }) ((fun i => get l i) i) = i\n[PROOFSTEP]\nsimp [List.get_indexOf, nd]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nnd : Nodup l\nh : \u2200 (x : \u03b1), x \u2208 l\na : \u03b1\n\u22a2 (fun i => get l i) ((fun a => { val := indexOf a l, isLt := (_ : indexOf a l < length l) }) a) = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n\u22a2 l <+ l'\n[PROOFSTEP]\ninduction' l with hd tl IH generalizing l' f\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? [] ix = get? l' (\u2191f ix)\n\u22a2 [] <+ l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nhave : some hd = _ := hf 0\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nthis : some hd = get? l' (\u2191f 0)\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nrw [eq_comm, List.get?_eq_some] at this \n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nthis : \u2203 h, get l' { val := \u2191f 0, isLt := h } = hd\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nobtain \u27e8w, h\u27e9 := this\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nlet f' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => f (i + 1) - (f 0 + 1)) fun a b =>\n by\n dsimp only\n rw [tsub_le_tsub_iff_right, OrderEmbedding.le_iff_le, Nat.succ_le_succ_iff]\n rw [Nat.succ_le_iff, OrderEmbedding.lt_iff_lt]\n exact b.succ_pos\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\na b : \u2115\n\u22a2 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\na b : \u2115\n\u22a2 \u2191f (a + 1) - (\u2191f 0 + 1) \u2264 \u2191f (b + 1) - (\u2191f 0 + 1) \u2194 a \u2264 b\n[PROOFSTEP]\nrw [tsub_le_tsub_iff_right, OrderEmbedding.le_iff_le, Nat.succ_le_succ_iff]\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\na b : \u2115\n\u22a2 \u2191f 0 + 1 \u2264 \u2191f (b + 1)\n[PROOFSTEP]\nrw [Nat.succ_le_iff, OrderEmbedding.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\na b : \u2115\n\u22a2 0 < b + 1\n[PROOFSTEP]\nexact b.succ_pos\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nhave : \u2200 ix, tl.get? ix = (l'.drop (f 0 + 1)).get? (f' ix) :=\n by\n intro ix\n rw [List.get?_drop, OrderEmbedding.coe_ofMapLEIff, add_tsub_cancel_of_le, \u2190 hf, List.get?]\n rw [Nat.succ_le_iff, OrderEmbedding.lt_iff_lt]\n exact ix.succ_pos\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\n\u22a2 \u2200 (ix : \u2115), get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n[PROOFSTEP]\nintro ix\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nix : \u2115\n\u22a2 get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n[PROOFSTEP]\nrw [List.get?_drop, OrderEmbedding.coe_ofMapLEIff, add_tsub_cancel_of_le, \u2190 hf, List.get?]\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nix : \u2115\n\u22a2 \u2191f 0 + 1 \u2264 \u2191f (ix + 1)\n[PROOFSTEP]\nrw [Nat.succ_le_iff, OrderEmbedding.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nix : \u2115\n\u22a2 0 < ix + 1\n[PROOFSTEP]\nexact ix.succ_pos\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nthis : \u2200 (ix : \u2115), get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n\u22a2 hd :: tl <+ l'\n[PROOFSTEP]\nrw [\u2190 List.take_append_drop (f 0 + 1) l', \u2190 List.singleton_append]\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nthis : \u2200 (ix : \u2115), get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n\u22a2 [hd] ++ tl <+ take (\u2191f 0 + 1) l' ++ drop (\u2191f 0 + 1) l'\n[PROOFSTEP]\napply List.Sublist.append _ (IH _ this)\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nthis : \u2200 (ix : \u2115), get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n\u22a2 [hd] <+ take (\u2191f 0 + 1) l'\n[PROOFSTEP]\nrw [List.singleton_sublist, \u2190 h, l'.get_take _ (Nat.lt_succ_self _)]\n[GOAL]\n\u03b1 : Type u_1\nl l'\u271d : List \u03b1\nf\u271d : \u2115 \u21aao \u2115\nhf\u271d : \u2200 (ix : \u2115), get? l ix = get? l'\u271d (\u2191f\u271d ix)\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l' : List \u03b1} (f : \u2115 \u21aao \u2115), (\u2200 (ix : \u2115), get? tl ix = get? l' (\u2191f ix)) \u2192 tl <+ l'\nl' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? (hd :: tl) ix = get? l' (\u2191f ix)\nw : \u2191f 0 < length l'\nh : get l' { val := \u2191f 0, isLt := w } = hd\nf' : \u2115 \u21aao \u2115 :=\n OrderEmbedding.ofMapLEIff (fun i => \u2191f (i + 1) - (\u2191f 0 + 1))\n (_ : \u2200 (a b : \u2115), (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) a \u2264 (fun i => \u2191f (i + 1) - (\u2191f 0 + 1)) b \u2194 a \u2264 b)\nthis : \u2200 (ix : \u2115), get? tl ix = get? (drop (\u2191f 0 + 1) l') (\u2191f' ix)\n\u22a2 get (take (Nat.succ (\u2191f 0)) l') { val := \u2191f 0, isLt := (_ : \u2191f 0 < length (take (Nat.succ (\u2191f 0)) l')) } \u2208\n take (\u2191f 0 + 1) l'\n[PROOFSTEP]\napply List.get_mem\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 l <+ l' \u2194 \u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 l <+ l' \u2192 \u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nl l' : List \u03b1\nH : l <+ l'\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n[PROOFSTEP]\ninduction' H with xs ys y _H IH xs ys x _H IH\n[GOAL]\ncase mp.slnil\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? [] ix = get? [] (\u2191f ix)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.cons\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\ny : \u03b1\n_H : xs <+ ys\nIH : \u2203 f, \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? xs ix = get? (y :: ys) (\u2191f ix)\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := IH\n[GOAL]\ncase mp.cons.intro\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\ny : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? xs ix = get? (y :: ys) (\u2191f ix)\n[PROOFSTEP]\nrefine' \u27e8f.trans (OrderEmbedding.ofStrictMono (\u00b7 + 1) fun _ => by simp), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\ny : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\nx\u271d : \u2115\n\u22a2 \u2200 \u2983b : \u2115\u2984, x\u271d < b \u2192 (fun x => x + 1) x\u271d < (fun x => x + 1) b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.cons.intro\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\ny : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2200 (ix : \u2115),\n get? xs ix =\n get? (y :: ys)\n (\u2191(RelEmbedding.trans f (OrderEmbedding.ofStrictMono (fun x => x + 1) (_ : \u2200 (x a : \u2115), x < a \u2192 x + 1 < a + 1)))\n ix)\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\ncase mp.cons\u2082\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nIH : \u2203 f, \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? (x :: xs) ix = get? (x :: ys) (\u2191f ix)\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := IH\n[GOAL]\ncase mp.cons\u2082.intro\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? (x :: xs) ix = get? (x :: ys) (\u2191f ix)\n[PROOFSTEP]\nrefine' \u27e8OrderEmbedding.ofMapLEIff (fun ix : \u2115 => if ix = 0 then 0 else (f ix.pred).succ) _, _\u27e9\n[GOAL]\ncase mp.cons\u2082.intro.refine'_1\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2200 (a b : \u2115),\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) a \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) b \u2194\n a \u2264 b\n[PROOFSTEP]\nrintro \u27e8_ | a\u27e9 \u27e8_ | b\u27e9\n[GOAL]\ncase mp.cons\u2082.intro.refine'_1.zero.zero\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) Nat.zero \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) Nat.zero \u2194\n Nat.zero \u2264 Nat.zero\n[PROOFSTEP]\nsimp [Nat.succ_le_succ_iff]\n[GOAL]\ncase mp.cons\u2082.intro.refine'_1.zero.succ\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\nn\u271d : \u2115\n\u22a2 (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) Nat.zero \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) (Nat.succ n\u271d) \u2194\n Nat.zero \u2264 Nat.succ n\u271d\n[PROOFSTEP]\nsimp [Nat.succ_le_succ_iff]\n[GOAL]\ncase mp.cons\u2082.intro.refine'_1.succ.zero\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\nn\u271d : \u2115\n\u22a2 (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) (Nat.succ n\u271d) \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) Nat.zero \u2194\n Nat.succ n\u271d \u2264 Nat.zero\n[PROOFSTEP]\nsimp [Nat.succ_le_succ_iff]\n[GOAL]\ncase mp.cons\u2082.intro.refine'_1.succ.succ\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\nn\u271d\u00b9 n\u271d : \u2115\n\u22a2 (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) (Nat.succ n\u271d\u00b9) \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) (Nat.succ n\u271d) \u2194\n Nat.succ n\u271d\u00b9 \u2264 Nat.succ n\u271d\n[PROOFSTEP]\nsimp [Nat.succ_le_succ_iff]\n[GOAL]\ncase mp.cons\u2082.intro.refine'_2\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 \u2200 (ix : \u2115),\n get? (x :: xs) ix =\n get? (x :: ys)\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix)))\n (_ :\n \u2200 (a b : \u2115),\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) a \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) b \u2194\n a \u2264 b))\n ix)\n[PROOFSTEP]\nrintro \u27e8_ | i\u27e9\n[GOAL]\ncase mp.cons\u2082.intro.refine'_2.zero\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\n\u22a2 get? (x :: xs) Nat.zero =\n get? (x :: ys)\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix)))\n (_ :\n \u2200 (a b : \u2115),\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) a \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) b \u2194\n a \u2264 b))\n Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.cons\u2082.intro.refine'_2.succ\n\u03b1 : Type u_1\nl l' xs ys : List \u03b1\nx : \u03b1\n_H : xs <+ ys\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? xs ix = get? ys (\u2191f ix)\nn\u271d : \u2115\n\u22a2 get? (x :: xs) (Nat.succ n\u271d) =\n get? (x :: ys)\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix)))\n (_ :\n \u2200 (a b : \u2115),\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) a \u2264\n (fun ix => if ix = 0 then 0 else Nat.succ (\u2191f (Nat.pred ix))) b \u2194\n a \u2264 b))\n (Nat.succ n\u271d))\n[PROOFSTEP]\nsimpa using hf _\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)) \u2192 l <+ l'\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n\u22a2 l <+ l'\n[PROOFSTEP]\nexact sublist_of_orderEmbedding_get?_eq f hf\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 l <+ l' \u2194 \u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n[PROOFSTEP]\nrw [sublist_iff_exists_orderEmbedding_get?_eq]\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)) \u2194 \u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)) \u2192 \u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n[PROOFSTEP]\nhave h : \u2200 {i : \u2115} (_ : i < l.length), f i < l'.length :=\n by\n intro i hi\n specialize hf i\n rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf \n obtain \u27e8h, -\u27e9 := hf\n exact h\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n\u22a2 \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\ni : \u2115\nhi : i < length l\n\u22a2 \u2191f i < length l'\n[PROOFSTEP]\nspecialize hf i\n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\ni : \u2115\nhi : i < length l\nhf : get? l i = get? l' (\u2191f i)\n\u22a2 \u2191f i < length l'\n[PROOFSTEP]\nrw [get?_eq_get hi, eq_comm, get?_eq_some] at hf \n[GOAL]\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\ni : \u2115\nhi : i < length l\nhf : \u2203 h, get l' { val := \u2191f i, isLt := h } = get l { val := i, isLt := hi }\n\u22a2 \u2191f i < length l'\n[PROOFSTEP]\nobtain \u27e8h, -\u27e9 := hf\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\ni : \u2115\nhi : i < length l\nh : \u2191f i < length l'\n\u22a2 \u2191f i < length l'\n[PROOFSTEP]\nexact h\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\nh : \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\n\u22a2 \u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n[PROOFSTEP]\nrefine' \u27e8OrderEmbedding.ofMapLEIff (fun ix => \u27e8f ix, h ix.is_lt\u27e9) _, _\u27e9\n[GOAL]\ncase mp.intro.refine'_1\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\nh : \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\n\u22a2 \u2200 (a b : Fin (length l)),\n (fun ix => { val := \u2191f \u2191ix, isLt := (_ : \u2191f \u2191ix < length l') }) a \u2264\n (fun ix => { val := \u2191f \u2191ix, isLt := (_ : \u2191f \u2191ix < length l') }) b \u2194\n a \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro.refine'_2\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\nh : \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\n\u22a2 \u2200 (ix : Fin (length l)),\n get l ix =\n get l'\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => { val := \u2191f \u2191ix, isLt := (_ : \u2191f \u2191ix < length l') })\n (_ :\n \u2200 (a a_1 : Fin (length l)),\n { val := \u2191f \u2191a, isLt := (_ : \u2191f \u2191a < length l') } \u2264\n { val := \u2191f \u2191a_1, isLt := (_ : \u2191f \u2191a_1 < length l') } \u2194\n a \u2264 a_1))\n ix)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mp.intro.refine'_2\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\nh : \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\ni : Fin (length l)\n\u22a2 get l i =\n get l'\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => { val := \u2191f \u2191ix, isLt := (_ : \u2191f \u2191ix < length l') })\n (_ :\n \u2200 (a a_1 : Fin (length l)),\n { val := \u2191f \u2191a, isLt := (_ : \u2191f \u2191a < length l') } \u2264\n { val := \u2191f \u2191a_1, isLt := (_ : \u2191f \u2191a_1 < length l') } \u2194\n a \u2264 a_1))\n i)\n[PROOFSTEP]\napply Option.some_injective\n[GOAL]\ncase mp.intro.refine'_2.a\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : \u2115 \u21aao \u2115\nhf : \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\nh : \u2200 {i : \u2115}, i < length l \u2192 \u2191f i < length l'\ni : Fin (length l)\n\u22a2 some (get l i) =\n some\n (get l'\n (\u2191(OrderEmbedding.ofMapLEIff (fun ix => { val := \u2191f \u2191ix, isLt := (_ : \u2191f \u2191ix < length l') })\n (_ :\n \u2200 (a a_1 : Fin (length l)),\n { val := \u2191f \u2191a, isLt := (_ : \u2191f \u2191a < length l') } \u2264\n { val := \u2191f \u2191a_1, isLt := (_ : \u2191f \u2191a_1 < length l') } \u2194\n a \u2264 a_1))\n i))\n[PROOFSTEP]\nsimpa [get?_eq_get i.2, get?_eq_get (h i.2)] using hf i\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)) \u2192 \u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n\u22a2 \u2203 f, \u2200 (ix : \u2115), get? l ix = get? l' (\u2191f ix)\n[PROOFSTEP]\nrefine' \u27e8OrderEmbedding.ofStrictMono (fun i => if hi : i < l.length then f \u27e8i, hi\u27e9 else i + l'.length) _, _\u27e9\n[GOAL]\ncase mpr.intro.refine'_1\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n\u22a2 StrictMono fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l'\n[PROOFSTEP]\nintro i j h\n[GOAL]\ncase mpr.intro.refine'_1\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\n\u22a2 (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') j\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mpr.intro.refine'_1\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\n\u22a2 (if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') <\n if hi : j < length l then \u2191(\u2191f { val := j, isLt := hi }) else j + length l'\n[PROOFSTEP]\nsplit_ifs with hi hj hj\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\nhi : i < length l\nhj : j < length l\n\u22a2 \u2191(\u2191f { val := i, isLt := hi }) < \u2191(\u2191f { val := j, isLt := hj })\n[PROOFSTEP]\nrwa [Fin.val_fin_lt, f.lt_iff_lt]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\nhi : i < length l\nhj : \u00acj < length l\n\u22a2 \u2191(\u2191f { val := i, isLt := hi }) < j + length l'\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\nhi : i < length l\nhj : \u00acj < length l\n\u22a2 \u2191(\u2191f { val := i, isLt := hi }) < length l' + j\n[PROOFSTEP]\nexact lt_add_of_lt_of_pos (Fin.is_lt _) (i.zero_le.trans_lt h)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\nhi : \u00aci < length l\nhj : j < length l\n\u22a2 i + length l' < \u2191(\u2191f { val := j, isLt := hj })\n[PROOFSTEP]\nexact absurd (h.trans hj) hi\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni j : \u2115\nh : i < j\nhi : \u00aci < length l\nhj : \u00acj < length l\n\u22a2 i + length l' < j + length l'\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr.intro.refine'_2\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\n\u22a2 \u2200 (ix : \u2115),\n get? l ix =\n get? l'\n (\u2191(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l')\n (_ :\n \u2200 \u2983i j : \u2115\u2984,\n i < j \u2192\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') j))\n ix)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mpr.intro.refine'_2\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\n\u22a2 get? l i =\n get? l'\n (\u2191(OrderEmbedding.ofStrictMono\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l')\n (_ :\n \u2200 \u2983i j : \u2115\u2984,\n i < j \u2192\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') i <\n (fun i => if hi : i < length l then \u2191(\u2191f { val := i, isLt := hi }) else i + length l') j))\n i)\n[PROOFSTEP]\nsimp only [OrderEmbedding.coe_ofStrictMono]\n[GOAL]\ncase mpr.intro.refine'_2\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\n\u22a2 get? l i = get? l' (if h : i < length l then \u2191(\u2191f { val := i, isLt := (_ : i < length l) }) else i + length l')\n[PROOFSTEP]\nsplit_ifs with hi\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\nhi : i < length l\n\u22a2 get? l i = get? l' \u2191(\u2191f { val := i, isLt := (_ : i < length l) })\n[PROOFSTEP]\nrw [get?_eq_get hi, get?_eq_get, \u2190 hf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\nhi : \u00aci < length l\n\u22a2 get? l i = get? l' (i + length l')\n[PROOFSTEP]\nrw [get?_eq_none.mpr, get?_eq_none.mpr]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\nhi : \u00aci < length l\n\u22a2 length l' \u2264 i + length l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nl l' : List \u03b1\nf : Fin (length l) \u21aao Fin (length l')\nhf : \u2200 (ix : Fin (length l)), get l ix = get l' (\u2191f ix)\ni : \u2115\nhi : \u00aci < length l\n\u22a2 length l \u2264 i\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 Duplicate x l \u2194 \u2203 n m x_1, x = get l n \u2227 x = get l m\n[PROOFSTEP]\nclassical\nrw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist, sublist_iff_exists_fin_orderEmbedding_get_eq]\nconstructor\n\u00b7 rintro \u27e8f, hf\u27e9\n refine' \u27e8f \u27e80, by simp\u27e9, f \u27e81, by simp\u27e9, f.lt_iff_lt.2 (show (0 : \u2115) < 1 from zero_lt_one), _\u27e9\n \u00b7 rw [\u2190 hf, \u2190 hf]; simp\n\u00b7 rintro \u27e8n, m, hnm, h, h'\u27e9\n refine' \u27e8OrderEmbedding.ofStrictMono (fun i => if (i : \u2115) = 0 then n else m) _, _\u27e9\n \u00b7 rintro \u27e8\u27e8_ | i\u27e9, hi\u27e9 \u27e8\u27e8_ | j\u27e9, hj\u27e9\n \u00b7 simp\n \u00b7 simp [hnm]\n \u00b7 simp\n \u00b7 simp only [Nat.lt_succ_iff, Nat.succ_le_succ_iff, replicate, length, nonpos_iff_eq_zero] at hi hj \n simp [hi, hj]\n \u00b7 rintro \u27e8\u27e8_ | i\u27e9, hi\u27e9\n \u00b7 simpa using h\n \u00b7 simpa using h'\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 Duplicate x l \u2194 \u2203 n m x_1, x = get l n \u2227 x = get l m\n[PROOFSTEP]\nrw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist, sublist_iff_exists_fin_orderEmbedding_get_eq]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)) \u2194\n \u2203 n m x_1, x = get l n \u2227 x = get l m\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 (\u2203 f, \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)) \u2192\n \u2203 n m x_1, x = get l n \u2227 x = get l m\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nf : Fin (length (replicate 2 x)) \u21aao Fin (length l)\nhf : \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n\u22a2 \u2203 n m x_1, x = get l n \u2227 x = get l m\n[PROOFSTEP]\nrefine' \u27e8f \u27e80, by simp\u27e9, f \u27e81, by simp\u27e9, f.lt_iff_lt.2 (show (0 : \u2115) < 1 from zero_lt_one), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nf : Fin (length (replicate 2 x)) \u21aao Fin (length l)\nhf : \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n\u22a2 0 < length (replicate 2 x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nf : Fin (length (replicate 2 x)) \u21aao Fin (length l)\nhf : \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n\u22a2 1 < length (replicate 2 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nf : Fin (length (replicate 2 x)) \u21aao Fin (length l)\nhf : \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n\u22a2 x = get l (\u2191f { val := 0, isLt := (_ : 0 < Nat.succ 1) }) \u2227 x = get l (\u2191f { val := 1, isLt := (_ : 1 < Nat.succ 1) })\n[PROOFSTEP]\nrw [\u2190 hf, \u2190 hf]\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nf : Fin (length (replicate 2 x)) \u21aao Fin (length l)\nhf : \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n\u22a2 x = get (replicate 2 x) { val := 0, isLt := (_ : 0 < Nat.succ 1) } \u2227\n x = get (replicate 2 x) { val := 1, isLt := (_ : 1 < Nat.succ 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\n\u22a2 (\u2203 n m x_1, x = get l n \u2227 x = get l m) \u2192\n \u2203 f, \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n[PROOFSTEP]\nrintro \u27e8n, m, hnm, h, h'\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\n\u22a2 \u2203 f, \u2200 (ix : Fin (length (replicate 2 x))), get (replicate 2 x) ix = get l (\u2191f ix)\n[PROOFSTEP]\nrefine' \u27e8OrderEmbedding.ofStrictMono (fun i => if (i : \u2115) = 0 then n else m) _, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\n\u22a2 StrictMono fun i => if \u2191i = 0 then n else m\n[PROOFSTEP]\nrintro \u27e8\u27e8_ | i\u27e9, hi\u27e9 \u27e8\u27e8_ | j\u27e9, hj\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1.mk.zero.mk.zero\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nhi hj : Nat.zero < length (replicate 2 x)\n\u22a2 { val := Nat.zero, isLt := hi } < { val := Nat.zero, isLt := hj } \u2192\n (fun i => if \u2191i = 0 then n else m) { val := Nat.zero, isLt := hi } <\n (fun i => if \u2191i = 0 then n else m) { val := Nat.zero, isLt := hj }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1.mk.zero.mk.succ\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nhi : Nat.zero < length (replicate 2 x)\nn\u271d : \u2115\nhj : Nat.succ n\u271d < length (replicate 2 x)\n\u22a2 { val := Nat.zero, isLt := hi } < { val := Nat.succ n\u271d, isLt := hj } \u2192\n (fun i => if \u2191i = 0 then n else m) { val := Nat.zero, isLt := hi } <\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d, isLt := hj }\n[PROOFSTEP]\nsimp [hnm]\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1.mk.succ.mk.zero\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nn\u271d : \u2115\nhi : Nat.succ n\u271d < length (replicate 2 x)\nhj : Nat.zero < length (replicate 2 x)\n\u22a2 { val := Nat.succ n\u271d, isLt := hi } < { val := Nat.zero, isLt := hj } \u2192\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d, isLt := hi } <\n (fun i => if \u2191i = 0 then n else m) { val := Nat.zero, isLt := hj }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1.mk.succ.mk.succ\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nn\u271d\u00b9 : \u2115\nhi : Nat.succ n\u271d\u00b9 < length (replicate 2 x)\nn\u271d : \u2115\nhj : Nat.succ n\u271d < length (replicate 2 x)\n\u22a2 { val := Nat.succ n\u271d\u00b9, isLt := hi } < { val := Nat.succ n\u271d, isLt := hj } \u2192\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d\u00b9, isLt := hi } <\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d, isLt := hj }\n[PROOFSTEP]\nsimp only [Nat.lt_succ_iff, Nat.succ_le_succ_iff, replicate, length, nonpos_iff_eq_zero] at hi hj \n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_1.mk.succ.mk.succ\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nn\u271d\u00b9 : \u2115\nhi\u271d : Nat.succ n\u271d\u00b9 < length (replicate 2 x)\nn\u271d : \u2115\nhj\u271d : Nat.succ n\u271d < length (replicate 2 x)\nhi : n\u271d\u00b9 = 0\nhj : n\u271d = 0\n\u22a2 { val := Nat.succ n\u271d\u00b9, isLt := hi\u271d } < { val := Nat.succ n\u271d, isLt := hj\u271d } \u2192\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d\u00b9, isLt := hi\u271d } <\n (fun i => if \u2191i = 0 then n else m) { val := Nat.succ n\u271d, isLt := hj\u271d }\n[PROOFSTEP]\nsimp [hi, hj]\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\n\u22a2 \u2200 (ix : Fin (length (replicate 2 x))),\n get (replicate 2 x) ix =\n get l\n (\u2191(OrderEmbedding.ofStrictMono (fun i => if \u2191i = 0 then n else m)\n (_ :\n \u2200 \u2983a b : Fin (length (replicate 2 x))\u2984,\n a < b \u2192 (fun i => if \u2191i = 0 then n else m) a < (fun i => if \u2191i = 0 then n else m) b))\n ix)\n[PROOFSTEP]\nrintro \u27e8\u27e8_ | i\u27e9, hi\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_2.mk.zero\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nhi : Nat.zero < length (replicate 2 x)\n\u22a2 get (replicate 2 x) { val := Nat.zero, isLt := hi } =\n get l\n (\u2191(OrderEmbedding.ofStrictMono (fun i => if \u2191i = 0 then n else m)\n (_ :\n \u2200 \u2983a b : Fin (length (replicate 2 x))\u2984,\n a < b \u2192 (fun i => if \u2191i = 0 then n else m) a < (fun i => if \u2191i = 0 then n else m) b))\n { val := Nat.zero, isLt := hi })\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr.intro.intro.intro.intro.refine'_2.mk.succ\n\u03b1 : Type u_1\nl : List \u03b1\nx : \u03b1\nn m : Fin (length l)\nhnm : n < m\nh : x = get l n\nh' : x = get l m\nn\u271d : \u2115\nhi : Nat.succ n\u271d < length (replicate 2 x)\n\u22a2 get (replicate 2 x) { val := Nat.succ n\u271d, isLt := hi } =\n get l\n (\u2191(OrderEmbedding.ofStrictMono (fun i => if \u2191i = 0 then n else m)\n (_ :\n \u2200 \u2983a b : Fin (length (replicate 2 x))\u2984,\n a < b \u2192 (fun i => if \u2191i = 0 then n else m) a < (fun i => if \u2191i = 0 then n else m) b))\n { val := Nat.succ n\u271d, isLt := hi })\n[PROOFSTEP]\nsimpa using h'\n", "meta": {"mathlib_filename": "Mathlib.Data.List.NodupEquivFin", "llama_tokens": 18850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.2902650722165359}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 E \u2192\u2097[\u211d] E \u2192\u2097[\u211d] \u211d\n[PROOFSTEP]\nlet z : AlternatingMap \u211d E \u211d (Fin 0) \u2243\u2097[\u211d] \u211d := AlternatingMap.constLinearEquivOfIsEmpty.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz : AlternatingMap \u211d E \u211d (Fin 0) \u2243\u2097[\u211d] \u211d := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n\u22a2 E \u2192\u2097[\u211d] E \u2192\u2097[\u211d] \u211d\n[PROOFSTEP]\nlet y : AlternatingMap \u211d E \u211d (Fin 1) \u2192\u2097[\u211d] E \u2192\u2097[\u211d] \u211d :=\n LinearMap.llcomp \u211d E (AlternatingMap \u211d E \u211d (Fin 0)) \u211d z \u2218\u2097 AlternatingMap.curryLeftLinearMap\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz : AlternatingMap \u211d E \u211d (Fin 0) \u2243\u2097[\u211d] \u211d := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\ny : AlternatingMap \u211d E \u211d (Fin 1) \u2192\u2097[\u211d] E \u2192\u2097[\u211d] \u211d :=\n LinearMap.comp (\u2191(LinearMap.llcomp \u211d E (AlternatingMap \u211d E \u211d (Fin 0)) \u211d) \u2191z) AlternatingMap.curryLeftLinearMap\n\u22a2 E \u2192\u2097[\u211d] E \u2192\u2097[\u211d] \u211d\n[PROOFSTEP]\nexact y \u2218\u2097 AlternatingMap.curryLeftLinearMap (R' := \u211d) o.volumeForm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = \u2191(volumeForm o) ![x, y]\n[PROOFSTEP]\nsimp [areaForm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(\u2191(areaForm o) x) x = 0\n[PROOFSTEP]\nrw [areaForm_to_volumeForm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(volumeForm o) ![x, x] = 0\n[PROOFSTEP]\nrefine' o.volumeForm.map_eq_zero_of_eq ![x, x] _ (_ : (0 : Fin 2) \u2260 1)\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 Matrix.vecCons x ![x] 0 = Matrix.vecCons x ![x] 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = -\u2191(\u2191(areaForm o) y) x\n[PROOFSTEP]\nsimp only [areaForm_to_volumeForm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(volumeForm o) ![x, y] = -\u2191(volumeForm o) ![y, x]\n[PROOFSTEP]\nconvert o.volumeForm.map_swap ![y, x] (_ : (0 : Fin 2) \u2260 1)\n[GOAL]\ncase h.e'_2.h.e'_6\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 ![x, y] = ![y, x] \u2218 \u2191(Equiv.swap 0 1)\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_2.h.e'_6.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\ni : Fin (Nat.succ 1)\n\u22a2 Matrix.vecCons x ![y] i = (![y, x] \u2218 \u2191(Equiv.swap 0 1)) i\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.e'_2.h.e'_6.h.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < Nat.succ 1) } =\n (![y, x] \u2218 \u2191(Equiv.swap 0 1)) { val := 0, isLt := (_ : 0 < Nat.succ 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.e'_6.h.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < Nat.succ 1) 1) } =\n (![y, x] \u2218 \u2191(Equiv.swap 0 1)) { val := 1, isLt := (_ : (fun a => a < Nat.succ 1) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 areaForm (-o) = -areaForm o\n[PROOFSTEP]\next x y\n[GOAL]\ncase h.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm (-o)) x) y = \u2191(\u2191(-areaForm o) x) y\n[PROOFSTEP]\nsimp [areaForm_to_volumeForm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 |\u2191(\u2191(areaForm o) x) y| \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.volumeForm_apply_le ![x, y]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 |\u2191(\u2191(areaForm o) x) y| = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 (Finset.prod Finset.univ fun i => \u2016Matrix.vecCons x ![y] i\u2016) = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp [Fin.prod_univ_succ]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 Pairwise fun i j => inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0\n[PROOFSTEP]\nintro i j hij\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\ni j : Fin 2\nhij : i \u2260 j\n\u22a2 inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nj : Fin 2\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 j\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) (Matrix.vecCons x ![y] j) = 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nj : Fin 2\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 j\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (Matrix.vecCons x ![y] j) = 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase tail.head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n 0\n[PROOFSTEP]\nsimpa [real_inner_comm] using h\n[GOAL]\ncase tail.head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 \u2191(\u2191(areaForm (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n \u2191(\u2191(areaForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y)\n[PROOFSTEP]\nhave : \u03c6.symm \u2218 ![x, y] = ![\u03c6.symm x, \u03c6.symm y] := by\n ext i\n fin_cases i <;> rfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 \u2191(LinearIsometryEquiv.symm \u03c6) \u2218 ![x, y] = ![\u2191(LinearIsometryEquiv.symm \u03c6) x, \u2191(LinearIsometryEquiv.symm \u03c6) y]\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\ni : Fin (Nat.succ (Nat.succ 0))\n\u22a2 (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 ![x, y]) i =\n Matrix.vecCons (\u2191(LinearIsometryEquiv.symm \u03c6) x) ![\u2191(LinearIsometryEquiv.symm \u03c6) y] i\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.head\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 ![x, y]) { val := 0, isLt := (_ : 0 < Nat.succ (Nat.succ 0)) } =\n Matrix.vecCons (\u2191(LinearIsometryEquiv.symm \u03c6) x) ![\u2191(LinearIsometryEquiv.symm \u03c6) y]\n { val := 0, isLt := (_ : 0 < Nat.succ (Nat.succ 0)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.tail.head\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 (\u2191(LinearIsometryEquiv.symm \u03c6) \u2218 ![x, y]) { val := 1, isLt := (_ : (fun a => a < Nat.succ (Nat.succ 0)) 1) } =\n Matrix.vecCons (\u2191(LinearIsometryEquiv.symm \u03c6) x) ![\u2191(LinearIsometryEquiv.symm \u03c6) y]\n { val := 1, isLt := (_ : (fun a => a < Nat.succ (Nat.succ 0)) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\nthis : \u2191(LinearIsometryEquiv.symm \u03c6) \u2218 ![x, y] = ![\u2191(LinearIsometryEquiv.symm \u03c6) x, \u2191(LinearIsometryEquiv.symm \u03c6) y]\n\u22a2 \u2191(\u2191(areaForm (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n \u2191(\u2191(areaForm o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y)\n[PROOFSTEP]\nsimp [areaForm_to_volumeForm, volumeForm_map, this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) (\u2191\u03c6 x)) (\u2191\u03c6 y) = \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nconvert o.areaForm_map \u03c6 (\u03c6 x) (\u03c6 y)\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 o = \u2191(map (Fin 2) \u03c6.toLinearEquiv) o\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 \u2191(map (Fin 2) \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'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 Fintype.card (Fin 2) = finrank \u211d E\n[PROOFSTEP]\nrw [@Fact.out (finrank \u211d E = 2), Fintype.card_fin]\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_6\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 x = \u2191(LinearIsometryEquiv.symm \u03c6) (\u2191\u03c6 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.e'_6\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 y = \u2191(LinearIsometryEquiv.symm \u03c6) (\u2191\u03c6 y)\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner (\u2191(rightAngleRotationAux\u2081 o) x) y = \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nsimp only [rightAngleRotationAux\u2081, LinearEquiv.trans_symm, LinearIsometryEquiv.toLinearEquiv_symm, LinearMap.coe_comp,\n LinearEquiv.coe_coe, Function.comp_apply, LinearEquiv.trans_apply, LinearIsometryEquiv.coe_toLinearEquiv]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner\n (\u2191(LinearIsometryEquiv.symm (InnerProductSpace.toDual \u211d E))\n (\u2191(LinearEquiv.symm (LinearEquiv.symm LinearMap.toContinuousLinearMap)) (\u2191(areaForm o) x)))\n y =\n \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nrw [InnerProductSpace.toDual_symm_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(LinearEquiv.symm (LinearEquiv.symm LinearMap.toContinuousLinearMap)) (\u2191(areaForm o) x)) y = \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner x (\u2191(rightAngleRotationAux\u2081 o) y) = -\u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nrw [real_inner_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner (\u2191(rightAngleRotationAux\u2081 o) y) x = -\u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nsimp [o.areaForm_swap y x]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\n\u22a2 \u2016\u2191{ toAddHom := src\u271d.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : \u211d) (x : E),\n AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id \u211d) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n x\u2016 =\n \u2016x\u2016\n[PROOFSTEP]\ndsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\n\u22a2 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\n\u22a2 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\ncases' eq_or_lt_of_le (norm_nonneg (o.rightAngleRotationAux\u2081 x)) with h h\n[GOAL]\ncase refine'_1.inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nh : 0 = \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n\u22a2 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase refine'_1.inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nh : 0 = \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n\u22a2 0 \u2264 \u2016x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase refine'_1.inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nh : 0 < \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n\u22a2 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_right _ h\n[GOAL]\ncase refine'_1.inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nh : 0 < \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n\u22a2 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 * \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 \u2264 \u2016x\u2016 * \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nrw [\u2190 real_inner_self_eq_norm_mul_norm, o.inner_rightAngleRotationAux\u2081_left]\n[GOAL]\ncase refine'_1.inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nh : 0 < \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n\u22a2 \u2191(\u2191(areaForm o) x) (\u2191(rightAngleRotationAux\u2081 o) x) \u2264 \u2016x\u2016 * \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nexact o.areaForm_le x (o.rightAngleRotationAux\u2081 x)\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nlet K : Submodule \u211d E := \u211d \u2219 x\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nhave : Nontrivial K\u15ee := by\n apply @FiniteDimensional.nontrivial_of_finrank_pos \u211d\n have : finrank \u211d K \u2264 Finset.card { x } := by\n rw [\u2190 Set.toFinset_singleton]\n exact finrank_span_le_card ({ x } : Set E)\n have : Finset.card { x } = 1 := Finset.card_singleton x\n have : finrank \u211d K + finrank \u211d K\u15ee = finrank \u211d E := K.finrank_add_finrank_orthogonal\n have : finrank \u211d E = 2 := Fact.out\n linarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\n\u22a2 Nontrivial { x // x \u2208 K\u15ee }\n[PROOFSTEP]\napply @FiniteDimensional.nontrivial_of_finrank_pos \u211d\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\n\u22a2 0 < finrank \u211d { x // x \u2208 K\u15ee }\n[PROOFSTEP]\nhave : finrank \u211d K \u2264 Finset.card { x } := by\n rw [\u2190 Set.toFinset_singleton]\n exact finrank_span_le_card ({ x } : Set E)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\n\u22a2 finrank \u211d { x // x \u2208 K } \u2264 Finset.card {x}\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_singleton]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\n\u22a2 finrank \u211d { x // x \u2208 K } \u2264 Finset.card (Set.toFinset {x})\n[PROOFSTEP]\nexact finrank_span_le_card ({ x } : Set E)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : finrank \u211d { x // x \u2208 K } \u2264 Finset.card {x}\n\u22a2 0 < finrank \u211d { x // x \u2208 K\u15ee }\n[PROOFSTEP]\nhave : Finset.card { x } = 1 := Finset.card_singleton x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis\u271d : finrank \u211d { x // x \u2208 K } \u2264 Finset.card {x}\nthis : Finset.card {x} = 1\n\u22a2 0 < finrank \u211d { x // x \u2208 K\u15ee }\n[PROOFSTEP]\nhave : finrank \u211d K + finrank \u211d K\u15ee = finrank \u211d E := K.finrank_add_finrank_orthogonal\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis\u271d\u00b9 : finrank \u211d { x // x \u2208 K } \u2264 Finset.card {x}\nthis\u271d : Finset.card {x} = 1\nthis : finrank \u211d { x // x \u2208 K } + finrank \u211d { x // x \u2208 K\u15ee } = finrank \u211d E\n\u22a2 0 < finrank \u211d { x // x \u2208 K\u15ee }\n[PROOFSTEP]\nhave : finrank \u211d E = 2 := Fact.out\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis\u271d\u00b2 : finrank \u211d { x // x \u2208 K } \u2264 Finset.card {x}\nthis\u271d\u00b9 : Finset.card {x} = 1\nthis\u271d : finrank \u211d { x // x \u2208 K } + finrank \u211d { x // x \u2208 K\u15ee } = finrank \u211d E\nthis : finrank \u211d E = 2\n\u22a2 0 < finrank \u211d { x // x \u2208 K\u15ee }\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nobtain \u27e8w, hw\u2080\u27e9 : \u2203 w : K\u15ee, w \u2260 0 := exists_ne 0\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nhave hw' : \u27eax, (w : E)\u27eb = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nhave hw : (w : E) \u2260 0 := fun h => hw\u2080 (Submodule.coe_eq_zero.mp h)\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\nhw : \u2191w \u2260 0\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_right _ (by rwa [norm_pos_iff] : 0 < \u2016(w : E)\u2016)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\nhw : \u2191w \u2260 0\n\u22a2 0 < \u2016\u2191w\u2016\n[PROOFSTEP]\nrwa [norm_pos_iff]\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\nhw : \u2191w \u2260 0\n\u22a2 \u2016x\u2016 * \u2016\u2191w\u2016 \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 * \u2016\u2191w\u2016\n[PROOFSTEP]\nrw [\u2190 o.abs_areaForm_of_orthogonal hw']\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\nhw : \u2191w \u2260 0\n\u22a2 |\u2191(\u2191(areaForm o) x) \u2191w| \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 * \u2016\u2191w\u2016\n[PROOFSTEP]\nrw [\u2190 o.inner_rightAngleRotationAux\u2081_left x w]\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nsrc\u271d : E \u2192\u2097[\u211d] E := rightAngleRotationAux\u2081 o\nx : E\nK : Submodule \u211d E := Submodule.span \u211d {x}\nthis : Nontrivial { x // x \u2208 K\u15ee }\nw : { x // x \u2208 K\u15ee }\nhw\u2080 : w \u2260 0\nhw' : inner x \u2191w = 0\nhw : \u2191w \u2260 0\n\u22a2 |inner (\u2191(rightAngleRotationAux\u2081 o) x) \u2191w| \u2264 \u2016\u2191(rightAngleRotationAux\u2081 o) x\u2016 * \u2016\u2191w\u2016\n[PROOFSTEP]\nexact abs_real_inner_le_norm (o.rightAngleRotationAux\u2081 x) w\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x) = -x\n[PROOFSTEP]\napply ext_inner_left \u211d\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2200 (v : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)),\n inner v (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner v (-x)\n[PROOFSTEP]\nintro y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)\n[PROOFSTEP]\nhave : \u27eao.rightAngleRotationAux\u2081 y, o.rightAngleRotationAux\u2081 x\u27eb = \u27eay, x\u27eb :=\n LinearIsometry.inner_map_map o.rightAngleRotationAux\u2082 y x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\nthis : inner (\u2191(rightAngleRotationAux\u2081 o) y) (\u2191(rightAngleRotationAux\u2081 o) x) = inner y x\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)\n[PROOFSTEP]\nrw [o.inner_rightAngleRotationAux\u2081_right, \u2190 o.inner_rightAngleRotationAux\u2081_left, this, inner_neg_right]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx\u271d : E\n\u22a2 \u2191(LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o)) x\u271d = \u2191LinearMap.id x\u271d\n[PROOFSTEP]\nsimp [rightAngleRotationAux\u2082]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx\u271d : E\n\u22a2 \u2191(LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap) x\u271d = \u2191LinearMap.id x\u271d\n[PROOFSTEP]\nsimp [rightAngleRotationAux\u2082]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner (\u2191(rightAngleRotation o) x) y = \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nrw [rightAngleRotation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner\n (\u2191(LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id))\n x)\n y =\n \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nexact o.inner_rightAngleRotationAux\u2081_left x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner x (\u2191(rightAngleRotation o) y) = -\u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nrw [rightAngleRotation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner x\n (\u2191(LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id))\n y) =\n -\u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nexact o.inner_rightAngleRotationAux\u2081_right x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(rightAngleRotation o) (\u2191(rightAngleRotation o) x) = -x\n[PROOFSTEP]\nrw [rightAngleRotation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id))\n (\u2191(LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id))\n x) =\n -x\n[PROOFSTEP]\nexact o.rightAngleRotationAux\u2081_rightAngleRotationAux\u2081 x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 LinearIsometryEquiv.symm (rightAngleRotation o) =\n LinearIsometryEquiv.trans (rightAngleRotation o) (LinearIsometryEquiv.neg \u211d)\n[PROOFSTEP]\nrw [rightAngleRotation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 LinearIsometryEquiv.symm\n (LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id)) =\n LinearIsometryEquiv.trans\n (LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o)\n (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id)\n (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id))\n (LinearIsometryEquiv.neg \u211d)\n[PROOFSTEP]\nexact LinearIsometryEquiv.toLinearIsometry_injective rfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 inner (\u2191(rightAngleRotation o) x) x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner x (\u2191(rightAngleRotation o) y) = -inner (\u2191(rightAngleRotation o) x) y\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner (\u2191(rightAngleRotation o) x) y = -inner x (\u2191(rightAngleRotation o) y)\n[PROOFSTEP]\nsimp [o.inner_rightAngleRotation_swap x y]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) (\u2191(rightAngleRotation o) x)) y = -inner x y\n[PROOFSTEP]\nrw [\u2190 o.inner_comp_rightAngleRotation, o.inner_rightAngleRotation_right, neg_neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) (\u2191(rightAngleRotation o) y) = inner x y\n[PROOFSTEP]\nrw [\u2190 o.inner_rightAngleRotation_left, o.inner_comp_rightAngleRotation]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) (\u2191(rightAngleRotation o) x)) (\u2191(rightAngleRotation o) y) = \u2191(\u2191(areaForm o) x) y\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u22a2 LinearIsometryEquiv.trans (rightAngleRotation o) (rightAngleRotation o) = LinearIsometryEquiv.neg \u211d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx\u271d : E\n\u22a2 \u2191(LinearIsometryEquiv.trans (rightAngleRotation o) (rightAngleRotation o)) x\u271d = \u2191(LinearIsometryEquiv.neg \u211d) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(rightAngleRotation (-o)) x = -\u2191(rightAngleRotation o) x\n[PROOFSTEP]\napply ext_inner_right \u211d\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2200 (v : E), inner (\u2191(rightAngleRotation (-o)) x) v = inner (-\u2191(rightAngleRotation o) x) v\n[PROOFSTEP]\nintro y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 inner (\u2191(rightAngleRotation (-o)) x) y = inner (-\u2191(rightAngleRotation o) x) y\n[PROOFSTEP]\nrw [inner_rightAngleRotation_left]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(areaForm (-o)) x) y = inner (-\u2191(rightAngleRotation o) x) y\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx : F\n\u22a2 \u2191(rightAngleRotation (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x =\n \u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))\n[PROOFSTEP]\napply ext_inner_right \u211d\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx : F\n\u22a2 \u2200 (v : F),\n inner (\u2191(rightAngleRotation (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) v =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) v\n[PROOFSTEP]\nintro y\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 inner (\u2191(rightAngleRotation (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) y\n[PROOFSTEP]\nrw [inner_rightAngleRotation_left]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 \u2191(\u2191(areaForm (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) y\n[PROOFSTEP]\ntrans \u27eaJ (\u03c6.symm x), \u03c6.symm y\u27eb\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 \u2191(\u2191(areaForm (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n inner (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y)\n[PROOFSTEP]\nsimp [o.areaForm_map]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 inner (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y) =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) y\n[PROOFSTEP]\ntrans \u27ea\u03c6 (J (\u03c6.symm x)), \u03c6 (\u03c6.symm y)\u27eb\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 inner (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y) =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) (\u2191\u03c6 (\u2191(LinearIsometryEquiv.symm \u03c6) y))\n[PROOFSTEP]\nrw [\u03c6.inner_map_map]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) (\u2191\u03c6 (\u2191(LinearIsometryEquiv.symm \u03c6) y)) =\n inner (\u2191\u03c6 (\u2191(rightAngleRotation o) (\u2191(LinearIsometryEquiv.symm \u03c6) x))) y\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : E\n\u22a2 \u2191\u03c6 (\u2191(rightAngleRotation o) x) = \u2191(rightAngleRotation o) (\u2191\u03c6 x)\n[PROOFSTEP]\nconvert (o.rightAngleRotation_map \u03c6 (\u03c6 x)).symm\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_6\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : E\n\u22a2 x = \u2191(LinearIsometryEquiv.symm \u03c6) (\u2191\u03c6 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : E\n\u22a2 o = \u2191(map (Fin 2) \u03c6.toLinearEquiv) o\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : E\n\u22a2 \u2191(map (Fin 2) \u03c6.toLinearEquiv) o = o\n[PROOFSTEP]\nrwa [\u2190 o.map_eq_iff_det_pos \u03c6.toLinearEquiv] at h\u03c6 \n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_5\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx : E\n\u22a2 Fintype.card (Fin 2) = finrank \u211d E\n[PROOFSTEP]\nrw [@Fact.out (finrank \u211d E = 2), Fintype.card_fin]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\ni : Fin 2\n\u22a2 Matrix.vecCons x ![\u2191(rightAngleRotation o) x] i \u2260 0\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\n\u22a2 Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) } \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\n\u22a2 Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\n\u22a2 \u2200 (i j : Fin 2),\n i \u2260 j \u2192\n inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] i) (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] j) = 0\n[PROOFSTEP]\nintro i j hij\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\ni j : Fin 2\nhij : i \u2260 j\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] i) (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] j) = 0\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nj : Fin 2\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 j\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] j) =\n 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nj : Fin 2\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 j\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] j) =\n 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase tail.head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 0, isLt := (_ : 0 < 2) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase tail.head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : x \u2260 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n (Matrix.vecCons x ![\u2191(rightAngleRotation o) x] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\n\u22a2 inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a = \u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : a = 0\n\u22a2 inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a = \u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a = \u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x\n[PROOFSTEP]\napply (o.basisRightAngleRotation a ha).ext\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2200 (i : Fin 2),\n \u2191(inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a) (\u2191(basisRightAngleRotation o a ha) i) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x) (\u2191(basisRightAngleRotation o a ha) i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\ni : Fin 2\n\u22a2 \u2191(inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a) (\u2191(basisRightAngleRotation o a ha) i) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x) (\u2191(basisRightAngleRotation o a ha) i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase neg.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a)\n (\u2191(basisRightAngleRotation o a ha) { val := 0, isLt := (_ : 0 < 2) }) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x) (\u2191(basisRightAngleRotation o a ha) { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nsimp only [Fin.mk_zero, coe_basisRightAngleRotation, Matrix.cons_val_zero, LinearMap.add_apply, LinearMap.smul_apply,\n inner\u209b\u2097_apply, real_inner_self_eq_norm_sq, smul_eq_mul, areaForm_apply_self, mul_zero, add_zero, Real.rpow_two,\n real_inner_comm]\n[GOAL]\ncase neg.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 inner a x * \u2016a\u2016 ^ 2 = \u2016a\u2016 ^ 2 * inner a x\n[PROOFSTEP]\nring\n[GOAL]\ncase neg.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(inner a x \u2022 \u2191(inner\u209b\u2097 \u211d) a + \u2191(\u2191(areaForm o) a) x \u2022 \u2191(areaForm o) a)\n (\u2191(basisRightAngleRotation o a ha) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(inner\u209b\u2097 \u211d) x) (\u2191(basisRightAngleRotation o a ha) { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nsimp only [Fin.mk_one, coe_basisRightAngleRotation, Matrix.cons_val_one, Matrix.head_cons, LinearMap.add_apply,\n LinearMap.smul_apply, inner\u209b\u2097_apply, inner_rightAngleRotation_right, areaForm_apply_self, neg_zero, smul_eq_mul,\n mul_zero, areaForm_rightAngleRotation_right, real_inner_self_eq_norm_sq, zero_add, Real.rpow_two, mul_neg]\n[GOAL]\ncase neg.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(\u2191(areaForm o) a) x * \u2016a\u2016 ^ 2 = -(\u2016a\u2016 ^ 2 * \u2191(\u2191(areaForm o) x) a)\n[PROOFSTEP]\nrw [o.areaForm_swap]\n[GOAL]\ncase neg.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 -\u2191(\u2191(areaForm o) x) a * \u2016a\u2016 ^ 2 = -(\u2016a\u2016 ^ 2 * \u2191(\u2191(areaForm o) x) a)\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na b : E\n\u22a2 inner a b ^ 2 + \u2191(\u2191(areaForm o) a) b ^ 2 = \u2016a\u2016 ^ 2 * \u2016b\u2016 ^ 2\n[PROOFSTEP]\nsimpa [sq, real_inner_self_eq_norm_sq] using o.inner_mul_inner_add_areaForm_mul_areaForm a b b\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\n\u22a2 inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a = \u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : a = 0\n\u22a2 inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a = \u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a = \u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x\n[PROOFSTEP]\napply (o.basisRightAngleRotation a ha).ext\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2200 (i : Fin 2),\n \u2191(inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a) (\u2191(basisRightAngleRotation o a ha) i) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x) (\u2191(basisRightAngleRotation o a ha) i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\ni : Fin 2\n\u22a2 \u2191(inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a) (\u2191(basisRightAngleRotation o a ha) i) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x) (\u2191(basisRightAngleRotation o a ha) i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase neg.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a)\n (\u2191(basisRightAngleRotation o a ha) { val := 0, isLt := (_ : 0 < 2) }) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x) (\u2191(basisRightAngleRotation o a ha) { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nsimp only [o.areaForm_swap a x, neg_smul, sub_neg_eq_add, Fin.mk_zero, coe_basisRightAngleRotation,\n Matrix.cons_val_zero, LinearMap.add_apply, LinearMap.smul_apply, areaForm_apply_self, smul_eq_mul, mul_zero,\n inner\u209b\u2097_apply, real_inner_self_eq_norm_sq, zero_add, Real.rpow_two]\n[GOAL]\ncase neg.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(\u2191(areaForm o) x) a * \u2016a\u2016 ^ 2 = \u2016a\u2016 ^ 2 * \u2191(\u2191(areaForm o) x) a\n[PROOFSTEP]\nring\n[GOAL]\ncase neg.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 \u2191(inner a x \u2022 \u2191(areaForm o) a - \u2191(\u2191(areaForm o) a) x \u2022 \u2191(inner\u209b\u2097 \u211d) a)\n (\u2191(basisRightAngleRotation o a ha) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n \u2191(\u2016a\u2016 ^ 2 \u2022 \u2191(areaForm o) x) (\u2191(basisRightAngleRotation o a ha) { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nsimp only [Fin.mk_one, coe_basisRightAngleRotation, Matrix.cons_val_one, Matrix.head_cons, LinearMap.sub_apply,\n LinearMap.smul_apply, areaForm_rightAngleRotation_right, real_inner_self_eq_norm_sq, smul_eq_mul, inner\u209b\u2097_apply,\n inner_rightAngleRotation_right, areaForm_apply_self, neg_zero, mul_zero, sub_zero, Real.rpow_two, real_inner_comm]\n[GOAL]\ncase neg.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x : E\nha : \u00aca = 0\n\u22a2 inner a x * \u2016a\u2016 ^ 2 = \u2016a\u2016 ^ 2 * inner a x\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2194 SameRay \u211d x y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : x = 0\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2194 SameRay \u211d x y\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2194 SameRay \u211d x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.mp\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2192 SameRay \u211d x y\n[PROOFSTEP]\nlet a : \u211d := (o.basisRightAngleRotation x hx).repr y 0\n[GOAL]\ncase neg.mp\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2192 SameRay \u211d x y\n[PROOFSTEP]\nlet b : \u211d := (o.basisRightAngleRotation x hx).repr y 1\n[GOAL]\ncase neg.mp\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2192 SameRay \u211d x y\n[PROOFSTEP]\nsuffices \u21910 \u2264 a * \u2016x\u2016 ^ 2 \u2227 b * \u2016x\u2016 ^ 2 = 0 \u2192 SameRay \u211d x (a \u2022 x + b \u2022 J x)\n by\n rw [\u2190 (o.basisRightAngleRotation x hx).sum_repr y]\n simp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero, Fin.succ_zero_eq_one',\n Fintype.univ_of_isEmpty, Finset.sum_empty, areaForm_apply_self, map_smul, map_add, real_inner_smul_right,\n inner_add_right, Matrix.cons_val_one, Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,\n mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right, real_inner_self_eq_norm_sq]\n exact this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nthis : 0 \u2264 a * \u2016x\u2016 ^ 2 \u2227 b * \u2016x\u2016 ^ 2 = 0 \u2192 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0 \u2192 SameRay \u211d x y\n[PROOFSTEP]\nrw [\u2190 (o.basisRightAngleRotation x hx).sum_repr y]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nthis : 0 \u2264 a * \u2016x\u2016 ^ 2 \u2227 b * \u2016x\u2016 ^ 2 = 0 \u2192 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n\u22a2 0 \u2264\n inner x\n (Finset.sum Finset.univ fun i =>\n \u2191(\u2191(basisRightAngleRotation o x hx).repr y) i \u2022 \u2191(basisRightAngleRotation o x hx) i) \u2227\n \u2191(\u2191(areaForm o) x)\n (Finset.sum Finset.univ fun i =>\n \u2191(\u2191(basisRightAngleRotation o x hx).repr y) i \u2022 \u2191(basisRightAngleRotation o x hx) i) =\n 0 \u2192\n SameRay \u211d x\n (Finset.sum Finset.univ fun i =>\n \u2191(\u2191(basisRightAngleRotation o x hx).repr y) i \u2022 \u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nsimp only [Fin.sum_univ_succ, coe_basisRightAngleRotation, Matrix.cons_val_zero, Fin.succ_zero_eq_one',\n Fintype.univ_of_isEmpty, Finset.sum_empty, areaForm_apply_self, map_smul, map_add, real_inner_smul_right,\n inner_add_right, Matrix.cons_val_one, Matrix.head_cons, Algebra.id.smul_eq_mul, areaForm_rightAngleRotation_right,\n mul_zero, add_zero, zero_add, neg_zero, inner_rightAngleRotation_right, real_inner_self_eq_norm_sq]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nthis : 0 \u2264 a * \u2016x\u2016 ^ 2 \u2227 b * \u2016x\u2016 ^ 2 = 0 \u2192 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n\u22a2 0 \u2264 \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0 * \u2016x\u2016 ^ 2 \u2227\n \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1 * \u2016x\u2016 ^ 2 = 0 \u2192\n SameRay \u211d x\n (\u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0 \u2022 x +\n \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1 \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase neg.mp\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\n\u22a2 0 \u2264 a * \u2016x\u2016 ^ 2 \u2227 b * \u2016x\u2016 ^ 2 = 0 \u2192 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nrintro \u27e8ha, hb\u27e9\n[GOAL]\ncase neg.mp.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\n\u22a2 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nhave hx' : 0 < \u2016x\u2016 := by simpa using hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\n\u22a2 0 < \u2016x\u2016\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase neg.mp.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\nhx' : 0 < \u2016x\u2016\n\u22a2 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nhave ha' : 0 \u2264 a := nonneg_of_mul_nonneg_left ha (by positivity)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\nhx' : 0 < \u2016x\u2016\n\u22a2 0 < \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase neg.mp.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\nhx' : 0 < \u2016x\u2016\nha' : 0 \u2264 a\n\u22a2 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nhave hb' : b = 0 := eq_zero_of_ne_zero_of_mul_right_eq_zero (pow_ne_zero 2 hx'.ne') hb\n[GOAL]\ncase neg.mp.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\na : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 0\nb : \u211d := \u2191(\u2191(basisRightAngleRotation o x hx).repr y) 1\nha : 0 \u2264 a * \u2016x\u2016 ^ 2\nhb : b * \u2016x\u2016 ^ 2 = 0\nhx' : 0 < \u2016x\u2016\nha' : 0 \u2264 a\nhb' : b = 0\n\u22a2 SameRay \u211d x (a \u2022 x + b \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nsimpa [hb'] using SameRay.sameRay_nonneg_smul_right x ha'\n[GOAL]\ncase neg.mpr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\n\u22a2 SameRay \u211d x y \u2192 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg.mpr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u00acx = 0\nh : SameRay \u211d x y\n\u22a2 0 \u2264 inner x y \u2227 \u2191(\u2191(areaForm o) x) y = 0\n[PROOFSTEP]\nobtain \u27e8r, hr, rfl\u27e9 := h.exists_nonneg_left hx\n[GOAL]\ncase neg.mpr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : \u00acx = 0\nr : \u211d\nhr : 0 \u2264 r\nh : SameRay \u211d x (r \u2022 x)\n\u22a2 0 \u2264 inner x (r \u2022 x) \u2227 \u2191(\u2191(areaForm o) x) (r \u2022 x) = 0\n[PROOFSTEP]\nsimp only [inner_smul_right, real_inner_self_eq_norm_sq, LinearMap.map_smul\u209b\u2097, areaForm_apply_self,\n Algebra.id.smul_eq_mul, mul_zero, eq_self_iff_true, and_true_iff]\n[GOAL]\ncase neg.mpr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\nhx : \u00acx = 0\nr : \u211d\nhr : 0 \u2264 r\nh : SameRay \u211d x (r \u2022 x)\n\u22a2 0 \u2264 r * \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(starRingEnd ((fun x => \u2102) x)) (\u2191(\u2191(kahler o) y) x)\n[PROOFSTEP]\nhave : \u2200 r : \u211d, Complex.ofReal' r = @IsROrC.ofReal \u2102 _ r := fun r => rfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(starRingEnd ((fun x => \u2102) x)) (\u2191(\u2191(kahler o) y) x)\n[PROOFSTEP]\nsimp only [kahler_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u2191(inner x y) + \u2191(\u2191(areaForm o) x) y \u2022 Complex.I = \u2191(starRingEnd \u2102) (\u2191(inner y x) + \u2191(\u2191(areaForm o) y) x \u2022 Complex.I)\n[PROOFSTEP]\nrw [real_inner_comm, areaForm_swap]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u2191(inner y x) + -\u2191(\u2191(areaForm o) y) x \u2022 Complex.I = \u2191(starRingEnd \u2102) (\u2191(inner y x) + \u2191(\u2191(areaForm o) y) x \u2022 Complex.I)\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(\u2191(kahler o) x) x = \u2191(\u2016x\u2016 ^ 2)\n[PROOFSTEP]\nsimp [kahler_apply_apply, real_inner_self_eq_norm_sq]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) (\u2191(rightAngleRotation o) x)) y = -Complex.I * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [o.areaForm_rightAngleRotation_left, o.inner_rightAngleRotation_left, o.kahler_apply_apply,\n Complex.ofReal_neg, Complex.real_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(\u2191(areaForm o) x) y) + -\u2191(inner x y) * Complex.I =\n -Complex.I * (\u2191(inner x y) + \u2191(\u2191(\u2191(areaForm o) x) y) * Complex.I)\n[PROOFSTEP]\nlinear_combination \u03c9 x y * Complex.I_sq\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) (\u2191(rightAngleRotation o) y) = Complex.I * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [o.areaForm_rightAngleRotation_right, o.inner_rightAngleRotation_right, o.kahler_apply_apply,\n Complex.ofReal_neg, Complex.real_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 -\u2191(\u2191(\u2191(areaForm o) x) y) + \u2191(inner x y) * Complex.I = Complex.I * (\u2191(inner x y) + \u2191(\u2191(\u2191(areaForm o) x) y) * Complex.I)\n[PROOFSTEP]\nlinear_combination -\u03c9 x y * Complex.I_sq\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) (\u2191(rightAngleRotation o) x)) (\u2191(rightAngleRotation o) y) = \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [kahler_rightAngleRotation_left, kahler_rightAngleRotation_right]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 Complex.I * (-Complex.I * \u2191(\u2191(kahler o) x) y) = \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nlinear_combination -o.kahler x y * Complex.I_sq\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 -(Complex.I * (Complex.I * \u2191(\u2191(kahler o) x) y)) = \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nlinear_combination -o.kahler x y * Complex.I_sq\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler (-o)) x) y = \u2191(starRingEnd ((fun x => \u2102) y)) (\u2191(\u2191(kahler o) x) y)\n[PROOFSTEP]\nhave : \u2200 r : \u211d, Complex.ofReal' r = @IsROrC.ofReal \u2102 _ r := fun r => rfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nthis : \u2200 (r : \u211d), \u2191r = \u2191r\n\u22a2 \u2191(\u2191(kahler (-o)) x) y = \u2191(starRingEnd ((fun x => \u2102) y)) (\u2191(\u2191(kahler o) x) y)\n[PROOFSTEP]\nsimp [kahler_apply_apply, this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 \u2191(\u2191(kahler o) x) a * \u2191(\u2191(kahler o) a) y = \u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\ntrans (\u2191(\u2016a\u2016 ^ 2) : \u2102) * o.kahler x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 \u2191(\u2191(kahler o) x) a * \u2191(\u2191(kahler o) a) y = \u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (\u2191(\u2191(kahler o) x) a * \u2191(\u2191(kahler o) a) y).re = (\u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y).re\n[PROOFSTEP]\nsimp only [o.kahler_apply_apply, Complex.add_im, Complex.add_re, Complex.I_im, Complex.I_re, Complex.mul_im,\n Complex.mul_re, Complex.ofReal_im, Complex.ofReal_re, Complex.real_smul]\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (inner x a + (\u2191(\u2191(areaForm o) x) a * 0 - 0 * 1)) * (inner a y + (\u2191(\u2191(areaForm o) a) y * 0 - 0 * 1)) -\n (0 + (\u2191(\u2191(areaForm o) x) a * 1 + 0 * 0)) * (0 + (\u2191(\u2191(areaForm o) a) y * 1 + 0 * 0)) =\n \u2016a\u2016 ^ 2 * (inner x y + (\u2191(\u2191(areaForm o) x) y * 0 - 0 * 1)) - 0 * (0 + (\u2191(\u2191(areaForm o) x) y * 1 + 0 * 0))\n[PROOFSTEP]\nrw [real_inner_comm a x, o.areaForm_swap x a]\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (inner a x + (-\u2191(\u2191(areaForm o) a) x * 0 - 0 * 1)) * (inner a y + (\u2191(\u2191(areaForm o) a) y * 0 - 0 * 1)) -\n (0 + (-\u2191(\u2191(areaForm o) a) x * 1 + 0 * 0)) * (0 + (\u2191(\u2191(areaForm o) a) y * 1 + 0 * 0)) =\n \u2016a\u2016 ^ 2 * (inner x y + (\u2191(\u2191(areaForm o) x) y * 0 - 0 * 1)) - 0 * (0 + (\u2191(\u2191(areaForm o) x) y * 1 + 0 * 0))\n[PROOFSTEP]\nlinear_combination o.inner_mul_inner_add_areaForm_mul_areaForm a x y\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (\u2191(\u2191(kahler o) x) a * \u2191(\u2191(kahler o) a) y).im = (\u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y).im\n[PROOFSTEP]\nsimp only [o.kahler_apply_apply, Complex.add_im, Complex.add_re, Complex.I_im, Complex.I_re, Complex.mul_im,\n Complex.mul_re, Complex.ofReal_im, Complex.ofReal_re, Complex.real_smul]\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (inner x a + (\u2191(\u2191(areaForm o) x) a * 0 - 0 * 1)) * (0 + (\u2191(\u2191(areaForm o) a) y * 1 + 0 * 0)) +\n (0 + (\u2191(\u2191(areaForm o) x) a * 1 + 0 * 0)) * (inner a y + (\u2191(\u2191(areaForm o) a) y * 0 - 0 * 1)) =\n \u2016a\u2016 ^ 2 * (0 + (\u2191(\u2191(areaForm o) x) y * 1 + 0 * 0)) + 0 * (inner x y + (\u2191(\u2191(areaForm o) x) y * 0 - 0 * 1))\n[PROOFSTEP]\nrw [real_inner_comm a x, o.areaForm_swap x a]\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 (inner a x + (-\u2191(\u2191(areaForm o) a) x * 0 - 0 * 1)) * (0 + (\u2191(\u2191(areaForm o) a) y * 1 + 0 * 0)) +\n (0 + (-\u2191(\u2191(areaForm o) a) x * 1 + 0 * 0)) * (inner a y + (\u2191(\u2191(areaForm o) a) y * 0 - 0 * 1)) =\n \u2016a\u2016 ^ 2 * (0 + (\u2191(\u2191(areaForm o) x) y * 1 + 0 * 0)) + 0 * (inner x y + (\u2191(\u2191(areaForm o) x) y * 0 - 0 * 1))\n[PROOFSTEP]\nlinear_combination o.inner_mul_areaForm_sub a x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\na x y : E\n\u22a2 \u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y = \u2191(\u2016a\u2016 ^ 2) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191Complex.normSq (\u2191(\u2191(kahler o) x) y) = \u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2\n[PROOFSTEP]\nsimpa [kahler_apply_apply, Complex.normSq, sq] using o.inner_sq_add_areaForm_sq x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191Complex.abs (\u2191(\u2191(kahler o) x) y) = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq_eq_sq, Complex.sq_abs]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191Complex.normSq (\u2191(\u2191(kahler o) x) y) = (\u2016x\u2016 * \u2016y\u2016) ^ 2\n[PROOFSTEP]\nlinear_combination o.normSq_kahler x y\n[GOAL]\ncase ha\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 0 \u2264 \u2191Complex.abs (\u2191(\u2191(kahler o) x) y)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase hb\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 0 \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2016\u2191(\u2191(kahler o) x) y\u2016 = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimpa using o.abs_kahler x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nhave : \u2016x\u2016 * \u2016y\u2016 = 0 := by simpa [hx] using (o.norm_kahler x y).symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\n\u22a2 \u2016x\u2016 * \u2016y\u2016 = 0\n[PROOFSTEP]\nsimpa [hx] using (o.norm_kahler x y).symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\nthis : \u2016x\u2016 * \u2016y\u2016 = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\ncases' eq_zero_or_eq_zero_of_mul_eq_zero this with h h\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\nthis : \u2016x\u2016 * \u2016y\u2016 = 0\nh : \u2016x\u2016 = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\nthis : \u2016x\u2016 * \u2016y\u2016 = 0\nh : \u2016x\u2016 = 0\n\u22a2 x = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\nthis : \u2016x\u2016 * \u2016y\u2016 = 0\nh : \u2016y\u2016 = 0\n\u22a2 x = 0 \u2228 y = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : \u2191(\u2191(kahler o) x) y = 0\nthis : \u2016x\u2016 * \u2016y\u2016 = 0\nh : \u2016y\u2016 = 0\n\u22a2 y = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = 0 \u2194 x = 0 \u2228 y = 0\n[PROOFSTEP]\nrefine' \u27e8o.eq_zero_or_eq_zero_of_kahler_eq_zero, _\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 x = 0 \u2228 y = 0 \u2192 \u2191(\u2191(kahler o) x) y = 0\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\ny : E\n\u22a2 \u2191(\u2191(kahler o) 0) y = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(\u2191(kahler o) x) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2191(\u2191(kahler o) x) y \u2260 0\n[PROOFSTEP]\napply mt o.eq_zero_or_eq_zero_of_kahler_eq_zero\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u00ac(x = 0 \u2228 y = 0)\n[PROOFSTEP]\ntauto\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y \u2260 0 \u2194 x \u2260 0 \u2227 y \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, fun h => o.kahler_ne_zero h.1 h.2\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y \u2260 0 \u2192 x \u2260 0 \u2227 y \u2260 0\n[PROOFSTEP]\ncontrapose\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u00ac(x \u2260 0 \u2227 y \u2260 0) \u2192 \u00ac\u2191(\u2191(kahler o) x) y \u2260 0\n[PROOFSTEP]\nsimp only [not_and_or, Classical.not_not, kahler_apply_apply, Complex.real_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 x = 0 \u2228 y = 0 \u2192 \u2191(inner x y) + \u2191(\u2191(\u2191(areaForm o) x) y) * Complex.I = 0\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\ny : E\n\u22a2 \u2191(inner 0 y) + \u2191(\u2191(\u2191(areaForm o) 0) y) * Complex.I = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(inner x 0) + \u2191(\u2191(\u2191(areaForm o) x) 0) * Complex.I = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \u211d E\ninst\u271d\u00b2 : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nhF : Fact (finrank \u211d F = 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] F\nx y : F\n\u22a2 \u2191(\u2191(kahler (\u2191(map (Fin 2) \u03c6.toLinearEquiv) o)) x) y =\n \u2191(\u2191(kahler o) (\u2191(LinearIsometryEquiv.symm \u03c6) x)) (\u2191(LinearIsometryEquiv.symm \u03c6) y)\n[PROOFSTEP]\nsimp [kahler_apply_apply, areaForm_map]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\n\u03c6 : E \u2243\u2097\u1d62[\u211d] E\nh\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv\nx y : E\n\u22a2 \u2191(\u2191(kahler o) (\u2191\u03c6 x)) (\u2191\u03c6 y) = \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp [kahler_apply_apply, o.areaForm_comp_linearIsometryEquiv \u03c6 h\u03c6]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nw z : \u2102\n\u22a2 \u2191(\u2191(Orientation.areaForm Complex.orientation) w) z = (\u2191(starRingEnd \u2102) w * z).im\n[PROOFSTEP]\nlet o := Complex.orientation\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no\u271d : Orientation \u211d E (Fin 2)\nw z : \u2102\no : Orientation \u211d \u2102 (Fin 2) := Complex.orientation\n\u22a2 \u2191(\u2191(Orientation.areaForm Complex.orientation) w) z = (\u2191(starRingEnd \u2102) w * z).im\n[PROOFSTEP]\nsimp only [o.areaForm_to_volumeForm, o.volumeForm_robust Complex.orthonormalBasisOneI rfl, (Basis.det_apply),\n Matrix.det_fin_two, (Basis.toMatrix_apply), toBasis_orthonormalBasisOneI, Matrix.cons_val_zero, coe_basisOneI_repr,\n Matrix.cons_val_one, Matrix.head_cons, mul_im, conj_re, conj_im]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no\u271d : Orientation \u211d E (Fin 2)\nw z : \u2102\no : Orientation \u211d \u2102 (Fin 2) := Complex.orientation\n\u22a2 w.re * z.im - z.re * w.im = w.re * z.im + -w.im * z.re\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz : \u2102\n\u22a2 \u2191(Orientation.rightAngleRotation Complex.orientation) z = I * z\n[PROOFSTEP]\napply ext_inner_right \u211d\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz : \u2102\n\u22a2 \u2200 (v : \u2102), inner (\u2191(Orientation.rightAngleRotation Complex.orientation) z) v = inner (I * z) v\n[PROOFSTEP]\nintro w\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz w : \u2102\n\u22a2 inner (\u2191(Orientation.rightAngleRotation Complex.orientation) z) w = inner (I * z) w\n[PROOFSTEP]\nrw [Orientation.inner_rightAngleRotation_left]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz w : \u2102\n\u22a2 \u2191(\u2191(Orientation.areaForm Complex.orientation) z) w = inner (I * z) w\n[PROOFSTEP]\nsimp only [Complex.areaForm, Complex.inner, mul_re, mul_im, conj_re, conj_im, map_mul, conj_I, neg_re, neg_im, I_re,\n I_im]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nz w : \u2102\n\u22a2 z.re * w.im + -z.im * w.re = (-0 * z.re - -1 * -z.im) * w.re - (-0 * -z.im + -1 * z.re) * w.im\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nw z : \u2102\n\u22a2 \u2191(\u2191(Orientation.kahler Complex.orientation) w) z = \u2191(starRingEnd \u2102) w * z\n[PROOFSTEP]\nrw [Orientation.kahler_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nw z : \u2102\n\u22a2 \u2191(inner w z) + \u2191(\u2191(Orientation.areaForm Complex.orientation) w) z \u2022 I = \u2191(starRingEnd \u2102) w * z\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nw z : \u2102\n\u22a2 (\u2191(inner w z) + \u2191(\u2191(Orientation.areaForm Complex.orientation) w) z \u2022 I).re = (\u2191(starRingEnd \u2102) w * z).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nw z : \u2102\n\u22a2 (\u2191(inner w z) + \u2191(\u2191(Orientation.areaForm Complex.orientation) w) z \u2022 I).im = (\u2191(starRingEnd \u2102) w * z).im\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = (\u2191(starRingEnd \u2102) (\u2191f x) * \u2191f y).im\n[PROOFSTEP]\nrw [\u2190 Complex.areaForm, \u2190 hf, areaForm_map (hF := _)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = \u2191(\u2191(areaForm o) (\u2191(LinearIsometryEquiv.symm f) (\u2191f x))) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\niterate 2 rw [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = \u2191(\u2191(areaForm o) (\u2191(LinearIsometryEquiv.symm f) (\u2191f x))) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(areaForm o) x) y = \u2191(\u2191(areaForm o) x) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx : E\n\u22a2 \u2191f (\u2191(rightAngleRotation o) x) = I * \u2191f x\n[PROOFSTEP]\nrw [\u2190 Complex.rightAngleRotation, \u2190 hf, rightAngleRotation_map (hF := _), LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(starRingEnd \u2102) (\u2191f x) * \u2191f y\n[PROOFSTEP]\nrw [\u2190 Complex.kahler, \u2190 hf, kahler_map (hF := _)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(\u2191(kahler o) (\u2191(LinearIsometryEquiv.symm f) (\u2191f x))) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\niterate 2 rw [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(\u2191(kahler o) (\u2191(LinearIsometryEquiv.symm f) (\u2191f x))) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nf : E \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx y : E\n\u22a2 \u2191(\u2191(kahler o) x) y = \u2191(\u2191(kahler o) x) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.symm_apply_apply]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.TwoDim", "llama_tokens": 41954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.2902211242444727}} {"text": "[GOAL]\nX : Type u_1\nx : X\n\u22a2 AddMonoidHom.comp toFreeAbelianGroup (singleAddHom x) =\n \u2191(AddMonoidHom.flip (smulAddHom \u2124 (FreeAbelianGroup X))) (of x)\n[PROOFSTEP]\next\n[GOAL]\ncase h1\nX : Type u_1\nx : X\n\u22a2 \u2191(AddMonoidHom.comp toFreeAbelianGroup (singleAddHom x)) 1 =\n \u2191(\u2191(AddMonoidHom.flip (smulAddHom \u2124 (FreeAbelianGroup X))) (of x)) 1\n[PROOFSTEP]\nsimp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul, toFreeAbelianGroup,\n Finsupp.liftAddHom_apply_single]\n[GOAL]\nX : Type u_1\n\u22a2 AddMonoidHom.comp toFinsupp toFreeAbelianGroup = AddMonoidHom.id (X \u2192\u2080 \u2124)\n[PROOFSTEP]\next x y\n[GOAL]\ncase H.h1.h\nX : Type u_1\nx y : X\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.id (X \u2192\u2080 \u2124)) (singleAddHom x)) 1) y\n[PROOFSTEP]\nsimp only [AddMonoidHom.id_comp]\n[GOAL]\ncase H.h1.h\nX : Type u_1\nx y : X\n\u22a2 \u2191(\u2191(AddMonoidHom.comp (AddMonoidHom.comp toFinsupp toFreeAbelianGroup) (singleAddHom x)) 1) y =\n \u2191(\u2191(singleAddHom x) 1) y\n[PROOFSTEP]\nrw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]\n[GOAL]\ncase H.h1.h\nX : Type u_1\nx y : X\n\u22a2 \u2191(\u2191(AddMonoidHom.comp toFinsupp (\u2191(AddMonoidHom.flip (smulAddHom \u2124 (FreeAbelianGroup X))) (of x))) 1) y =\n \u2191(\u2191(singleAddHom x) 1) y\n[PROOFSTEP]\nsimp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul, lift.of,\n AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]\n[GOAL]\nX : Type u_1\n\u22a2 AddMonoidHom.comp toFreeAbelianGroup toFinsupp = AddMonoidHom.id (FreeAbelianGroup X)\n[PROOFSTEP]\next\n[GOAL]\ncase H\nX : Type u_1\nx\u271d : X\n\u22a2 \u2191(AddMonoidHom.comp toFreeAbelianGroup toFinsupp) (of x\u271d) = \u2191(AddMonoidHom.id (FreeAbelianGroup X)) (of x\u271d)\n[PROOFSTEP]\nrw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of, liftAddHom_apply_single, AddMonoidHom.flip_apply,\n smulAddHom_apply, one_smul, AddMonoidHom.id_apply]\n[GOAL]\nX\u271d : Type u_1\nX : Type u_2\nx : FreeAbelianGroup X\n\u22a2 \u2191toFreeAbelianGroup (\u2191toFinsupp x) = x\n[PROOFSTEP]\nrw [\u2190 AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]\n[GOAL]\nX : Type u_1\nx : X\n\u22a2 \u2191toFinsupp (of x) = single x 1\n[PROOFSTEP]\nsimp only [toFinsupp, lift.of]\n[GOAL]\nX : Type u_1\nf : X \u2192\u2080 \u2124\n\u22a2 \u2191toFinsupp (\u2191toFreeAbelianGroup f) = f\n[PROOFSTEP]\nrw [\u2190 AddMonoidHom.comp_apply, toFinsupp_comp_toFreeAbelianGroup, AddMonoidHom.id_apply]\n[GOAL]\nX : Type u_1\nx : X\na : FreeAbelianGroup X\n\u22a2 x \u2208 support a \u2194 \u2191(coeff x) a \u2260 0\n[PROOFSTEP]\nrw [support, Finsupp.mem_support_iff]\n[GOAL]\nX : Type u_1\nx : X\na : FreeAbelianGroup X\n\u22a2 \u2191(\u2191toFinsupp a) x \u2260 0 \u2194 \u2191(coeff x) a \u2260 0\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nX : Type u_1\nx : X\na : FreeAbelianGroup X\n\u22a2 \u00acx \u2208 support a \u2194 \u2191(coeff x) a = 0\n[PROOFSTEP]\nrw [support, Finsupp.not_mem_support_iff]\n[GOAL]\nX : Type u_1\nx : X\na : FreeAbelianGroup X\n\u22a2 \u2191(\u2191toFinsupp a) x = 0 \u2194 \u2191(coeff x) a = 0\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nX : Type u_1\n\u22a2 support 0 = \u2205\n[PROOFSTEP]\nsimp only [support, Finsupp.support_zero, AddMonoidHom.map_zero]\n[GOAL]\nX : Type u_1\nx : X\n\u22a2 support (of x) = {x}\n[PROOFSTEP]\nrw [support, toFinsupp_of, Finsupp.support_single_ne_zero _ one_ne_zero]\n[GOAL]\nX : Type u_1\na : FreeAbelianGroup X\n\u22a2 support (-a) = support a\n[PROOFSTEP]\nsimp only [support, AddMonoidHom.map_neg, Finsupp.support_neg]\n[GOAL]\nX : Type u_1\nk : \u2124\nh : k \u2260 0\na : FreeAbelianGroup X\n\u22a2 support (k \u2022 a) = support a\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nX : Type u_1\nk : \u2124\nh : k \u2260 0\na : FreeAbelianGroup X\nx : X\n\u22a2 x \u2208 support (k \u2022 a) \u2194 x \u2208 support a\n[PROOFSTEP]\nsimp only [mem_support_iff, AddMonoidHom.map_zsmul]\n[GOAL]\ncase a\nX : Type u_1\nk : \u2124\nh : k \u2260 0\na : FreeAbelianGroup X\nx : X\n\u22a2 k \u2022 \u2191(coeff x) a \u2260 0 \u2194 \u2191(coeff x) a \u2260 0\n[PROOFSTEP]\nsimp only [h, zsmul_int_int, false_or_iff, Ne.def, mul_eq_zero]\n[GOAL]\nX : Type u_1\nk : \u2115\nh : k \u2260 0\na : FreeAbelianGroup X\n\u22a2 support (k \u2022 a) = support a\n[PROOFSTEP]\napply support_zsmul k _ a\n[GOAL]\nX : Type u_1\nk : \u2115\nh : k \u2260 0\na : FreeAbelianGroup X\n\u22a2 \u2191k \u2260 0\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\nX : Type u_1\na b : FreeAbelianGroup X\n\u22a2 support (a + b) \u2286 support a \u222a support b\n[PROOFSTEP]\nsimp only [support, AddMonoidHom.map_add]\n[GOAL]\nX : Type u_1\na b : FreeAbelianGroup X\n\u22a2 (\u2191toFinsupp a + \u2191toFinsupp b).support \u2286 (\u2191toFinsupp a).support \u222a (\u2191toFinsupp b).support\n[PROOFSTEP]\napply Finsupp.support_add\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.FreeAbelianGroupFinsupp", "llama_tokens": 2091, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.2900627781786525}} {"text": "[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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj j' : J\n\u22a2 (fun j => F.map (Bicone.\u03b9 b j)) j \u226b (fun j => F.map (Bicone.\u03c0 b j)) j' =\n if h : j = j' then eqToHom (_ : (F.obj \u2218 f) j = (F.obj \u2218 f) j') else 0\n[PROOFSTEP]\nrw [\u2190 F.map_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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj j' : J\n\u22a2 F.map (Bicone.\u03b9 b j \u226b Bicone.\u03c0 b j') = if h : j = j' then eqToHom (_ : (F.obj \u2218 f) j = (F.obj \u2218 f) j') else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj j' : J\nh : j = j'\n\u22a2 F.map (Bicone.\u03b9 b j \u226b Bicone.\u03c0 b j') = eqToHom (_ : (F.obj \u2218 f) j = (F.obj \u2218 f) j')\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj : J\n\u22a2 F.map (Bicone.\u03b9 b j \u226b Bicone.\u03c0 b j) = eqToHom (_ : (F.obj \u2218 f) j = (F.obj \u2218 f) j)\n[PROOFSTEP]\nsimp only [bicone_\u03b9_\u03c0_self, CategoryTheory.Functor.map_id, eqToHom_refl]\n[GOAL]\ncase pos\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj : J\n\u22a2 \ud835\udfd9 (F.obj (f j)) = \ud835\udfd9 ((F.obj \u2218 f) j)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\nb : Bicone f\nj j' : J\nh : \u00acj = j'\n\u22a2 F.map (Bicone.\u03b9 b j \u226b Bicone.\u03c0 b j') = 0\n[PROOFSTEP]\nrw [bicone_\u03b9_\u03c0_ne _ h, F.map_zero]\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nX Y : C\nb : BinaryBicone X Y\n\u22a2 F.map b.inl \u226b F.map b.fst = \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, b.inl_fst, F.map_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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nX Y : C\nb : BinaryBicone X Y\n\u22a2 F.map b.inl \u226b F.map b.snd = 0\n[PROOFSTEP]\nrw [\u2190 F.map_comp, b.inl_snd, F.map_zero]\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nX Y : C\nb : BinaryBicone X Y\n\u22a2 F.map b.inr \u226b F.map b.fst = 0\n[PROOFSTEP]\nrw [\u2190 F.map_comp, b.inr_fst, F.map_zero]\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\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d : PreservesZeroMorphisms F\nX Y : C\nb : BinaryBicone X Y\n\u22a2 F.map b.inr \u226b F.map b.snd = \ud835\udfd9 (F.obj Y)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, b.inr_snd, F.map_id]\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\ninst\u271d : PreservesBiproducts F\nJ : Type\nx\u271d : Fintype J\n\u22a2 PreservesBiproductsOfShape J F\n[PROOFSTEP]\nletI := preservesBiproductsShrink.{0} F\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\ninst\u271d : PreservesBiproducts F\nJ : Type\nx\u271d : Fintype J\nthis : PreservesBiproducts F := preservesBiproductsShrink F\n\u22a2 PreservesBiproductsOfShape J F\n[PROOFSTEP]\ninfer_instance\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 NatTrans.app\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).\u03c0\n j =\n (Iso.refl\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom \u226b\n NatTrans.app (BinaryBicone.toCone (mapBinaryBicone F b)).\u03c0 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 NatTrans.app\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).\u03c0\n { as := WalkingPair.left } =\n (Iso.refl\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom \u226b\n NatTrans.app (BinaryBicone.toCone (mapBinaryBicone F b)).\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 NatTrans.app\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).\u03c0\n { as := WalkingPair.right } =\n (Iso.refl\n ((Cones.postcompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).hom).obj\n (Bicone.toCone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom \u226b\n NatTrans.app (BinaryBicone.toCone (mapBinaryBicone F b)).\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\nj : Discrete WalkingPair\n\u22a2 NatTrans.app\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).\u03b9\n j \u226b\n (Iso.refl\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom =\n NatTrans.app (BinaryBicone.toCocone (mapBinaryBicone F b)).\u03b9 j\n[PROOFSTEP]\nrcases j with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.left\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 NatTrans.app\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).\u03b9\n { as := WalkingPair.left } \u226b\n (Iso.refl\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom =\n NatTrans.app (BinaryBicone.toCocone (mapBinaryBicone F b)).\u03b9 { as := WalkingPair.left }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right\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\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b9 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d : PreservesBiproduct (pairFunction X Y) F\nb : BinaryBicone X Y\nhb : BinaryBicone.IsBilimit b\n\u22a2 NatTrans.app\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).\u03b9\n { as := WalkingPair.right } \u226b\n (Iso.refl\n ((Cocones.precompose (diagramIsoPair (Discrete.functor (F.obj \u2218 pairFunction X Y))).inv).obj\n (Bicone.toCocone (mapBicone F (BinaryBicone.toBicone b)))).pt).hom =\n NatTrans.app (BinaryBicone.toCocone (mapBinaryBicone F b)).\u03b9 { as := WalkingPair.right }\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nJ : Type w\u2081\nF : C \u2964 D\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\ninst\u271d\u00b9 : HasBiproduct (F.obj \u2218 f)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biproductComparison' F f \u226b biproductComparison F f = \ud835\udfd9 (\u2a01 F.obj \u2218 f)\n[PROOFSTEP]\nclassical\next\nsimp [biproduct.\u03b9_\u03c0, \u2190 Functor.map_comp, eqToHom_map]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nJ : Type w\u2081\nF : C \u2964 D\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\ninst\u271d\u00b9 : HasBiproduct (F.obj \u2218 f)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biproductComparison' F f \u226b biproductComparison F f = \ud835\udfd9 (\u2a01 F.obj \u2218 f)\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nJ : Type w\u2081\nF : C \u2964 D\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\ninst\u271d\u00b9 : HasBiproduct (F.obj \u2218 f)\ninst\u271d : PreservesZeroMorphisms F\nj\u271d\u00b9 j\u271d : J\n\u22a2 biproduct.\u03b9 (F.obj \u2218 f) j\u271d \u226b (biproductComparison' F f \u226b biproductComparison F f) \u226b biproduct.\u03c0 (F.obj \u2218 f) j\u271d\u00b9 =\n biproduct.\u03b9 (F.obj \u2218 f) j\u271d \u226b \ud835\udfd9 (\u2a01 F.obj \u2218 f) \u226b biproduct.\u03c0 (F.obj \u2218 f) j\u271d\u00b9\n[PROOFSTEP]\nsimp [biproduct.\u03b9_\u03c0, \u2190 Functor.map_comp, eqToHom_map]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nJ : Type w\u2081\nF : C \u2964 D\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\ninst\u271d\u00b9 : HasBiproduct (F.obj \u2218 f)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biproductComparison' F f \u226b biproductComparison F f = \ud835\udfd9 (\u2a01 F.obj \u2218 f)\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nJ : Type w\u2081\nF : C \u2964 D\nf : J \u2192 C\ninst\u271d\u00b2 : HasBiproduct f\ninst\u271d\u00b9 : HasBiproduct (F.obj \u2218 f)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biproductComparison' F f \u226b biproductComparison F f = \ud835\udfd9 (\u2a01 F.obj \u2218 f)\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biprodComparison' F X Y \u226b biprodComparison F X Y = \ud835\udfd9 (F.obj X \u229e F.obj Y)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080.h\u2080\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 (biprod.inl \u226b biprodComparison' F X Y \u226b biprodComparison F X Y) \u226b biprod.fst =\n (biprod.inl \u226b \ud835\udfd9 (F.obj X \u229e F.obj Y)) \u226b biprod.fst\n[PROOFSTEP]\nsimp [\u2190 Functor.map_comp]\n[GOAL]\ncase h\u2080.h\u2081\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 (biprod.inl \u226b biprodComparison' F X Y \u226b biprodComparison F X Y) \u226b biprod.snd =\n (biprod.inl \u226b \ud835\udfd9 (F.obj X \u229e F.obj Y)) \u226b biprod.snd\n[PROOFSTEP]\nsimp [\u2190 Functor.map_comp]\n[GOAL]\ncase h\u2081.h\u2080\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 (biprod.inr \u226b biprodComparison' F X Y \u226b biprodComparison F X Y) \u226b biprod.fst =\n (biprod.inr \u226b \ud835\udfd9 (F.obj X \u229e F.obj Y)) \u226b biprod.fst\n[PROOFSTEP]\nsimp [\u2190 Functor.map_comp]\n[GOAL]\ncase h\u2081.h\u2081\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 (biprod.inr \u226b biprodComparison' F X Y \u226b biprodComparison F X Y) \u226b biprod.snd =\n (biprod.inr \u226b \ud835\udfd9 (F.obj X \u229e F.obj Y)) \u226b biprod.snd\n[PROOFSTEP]\nsimp [\u2190 Functor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biprodComparison' F X Y \u226b biprodComparison F X Y = \ud835\udfd9 (F.obj X \u229e F.obj Y)\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\nX Y : C\ninst\u271d\u00b2 : HasBinaryBiproduct X Y\ninst\u271d\u00b9 : HasBinaryBiproduct (F.obj X) (F.obj Y)\ninst\u271d : PreservesZeroMorphisms F\n\u22a2 biprodComparison' F X Y \u226b biprodComparison F X Y = \ud835\udfd9 (F.obj X \u229e F.obj Y)\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 W \u27f6 f j\n\u22a2 F.map (lift g) \u226b (mapBiproduct F f).hom = lift fun j => F.map (g j)\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 W \u27f6 f j\nj : J\n\u22a2 (F.map (lift g) \u226b (mapBiproduct F f).hom) \u226b \u03c0 (F.obj \u2218 f) j = (lift fun j => F.map (g j)) \u226b \u03c0 (F.obj \u2218 f) j\n[PROOFSTEP]\ndsimp only [Function.comp]\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 W \u27f6 f j\nj : J\n\u22a2 (F.map (lift g) \u226b (mapBiproduct F f).hom) \u226b \u03c0 (fun x => F.obj (f x)) j =\n (lift fun j => F.map (g j)) \u226b \u03c0 (fun x => F.obj (f x)) j\n[PROOFSTEP]\nhaveI : HasBiproduct fun j => F.obj (f j) := hasBiproduct_of_preserves F f\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 W \u27f6 f j\nj : J\nthis : HasBiproduct fun j => F.obj (f j)\n\u22a2 (F.map (lift g) \u226b (mapBiproduct F f).hom) \u226b \u03c0 (fun x => F.obj (f x)) j =\n (lift fun j => F.map (g j)) \u226b \u03c0 (fun x => F.obj (f x)) j\n[PROOFSTEP]\nsimp only [mapBiproduct_hom, Category.assoc, biproduct.lift_\u03c0, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 f j \u27f6 W\n\u22a2 (mapBiproduct F f).inv \u226b F.map (desc g) = desc fun j => F.map (g j)\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 f j \u27f6 W\nj : J\n\u22a2 \u03b9 (F.obj \u2218 f) j \u226b (mapBiproduct F f).inv \u226b F.map (desc g) = \u03b9 (F.obj \u2218 f) j \u226b desc fun j => F.map (g j)\n[PROOFSTEP]\ndsimp only [Function.comp]\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 f j \u27f6 W\nj : J\n\u22a2 \u03b9 (fun x => F.obj (f x)) j \u226b (mapBiproduct F f).inv \u226b F.map (desc g) =\n \u03b9 (fun x => F.obj (f x)) j \u226b desc fun j => F.map (g j)\n[PROOFSTEP]\nhaveI : HasBiproduct fun j => F.obj (f j) := hasBiproduct_of_preserves F f\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 f j \u27f6 W\nj : J\nthis : HasBiproduct fun j => F.obj (f j)\n\u22a2 \u03b9 (fun x => F.obj (f x)) j \u226b (mapBiproduct F f).inv \u226b F.map (desc g) =\n \u03b9 (fun x => F.obj (f x)) j \u226b desc fun j => F.map (g j)\n[PROOFSTEP]\nsimp only [mapBiproduct_inv, \u2190 Category.assoc, biproduct.\u03b9_desc, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nJ : Type w\u2081\nf : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : PreservesBiproduct f F\nW : C\ng : (j : J) \u2192 f j \u27f6 W\n\u22a2 ((mapBiproduct F f).hom \u226b desc fun j => F.map (g j)) = F.map (desc g)\n[PROOFSTEP]\nrw [\u2190 biproduct.mapBiproduct_inv_map_desc, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : W \u27f6 X\ng : W \u27f6 Y\n\u22a2 F.map (lift f g) \u226b (mapBiprod F X Y).hom = lift (F.map f) (F.map g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : W \u27f6 X\ng : W \u27f6 Y\n\u22a2 (F.map (lift f g) \u226b (mapBiprod F X Y).hom) \u226b fst = lift (F.map f) (F.map g) \u226b fst\n[PROOFSTEP]\nsimp [mapBiprod, \u2190 F.map_comp]\n[GOAL]\ncase h\u2081\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : W \u27f6 X\ng : W \u27f6 Y\n\u22a2 (F.map (lift f g) \u226b (mapBiprod F X Y).hom) \u226b snd = lift (F.map f) (F.map g) \u226b snd\n[PROOFSTEP]\nsimp [mapBiprod, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : W \u27f6 X\ng : W \u27f6 Y\n\u22a2 lift (F.map f) (F.map g) \u226b (mapBiprod F X Y).inv = F.map (lift f g)\n[PROOFSTEP]\nrw [\u2190 biprod.map_lift_mapBiprod, Category.assoc, Iso.hom_inv_id, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : X \u27f6 W\ng : Y \u27f6 W\n\u22a2 (mapBiprod F X Y).inv \u226b F.map (desc f g) = desc (F.map f) (F.map g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : X \u27f6 W\ng : Y \u27f6 W\n\u22a2 inl \u226b (mapBiprod F X Y).inv \u226b F.map (desc f g) = inl \u226b desc (F.map f) (F.map g)\n[PROOFSTEP]\nsimp [mapBiprod, \u2190 F.map_comp]\n[GOAL]\ncase h\u2081\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : X \u27f6 W\ng : Y \u27f6 W\n\u22a2 inr \u226b (mapBiprod F X Y).inv \u226b F.map (desc f g) = inr \u226b desc (F.map f) (F.map g)\n[PROOFSTEP]\nsimp [mapBiprod, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : HasZeroMorphisms C\ninst\u271d\u00b3 : HasZeroMorphisms D\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesZeroMorphisms F\nX Y : C\ninst\u271d\u00b9 : HasBinaryBiproduct X Y\ninst\u271d : PreservesBinaryBiproduct X Y F\nW : C\nf : X \u27f6 W\ng : Y \u27f6 W\n\u22a2 (mapBiprod F X Y).hom \u226b desc (F.map f) (F.map g) = F.map (desc f g)\n[PROOFSTEP]\nrw [\u2190 biprod.mapBiprod_inv_map_desc, Iso.hom_inv_id_assoc]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts", "llama_tokens": 10936, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.42250463481418826, "lm_q1q2_score": 0.2898168239892278}} {"text": "[GOAL]\nJ : Type u\u2081\ninst\u271d : Fintype J\nj : Bicone J\n\u22a2 j \u2208 List.toFinset [Bicone.left, Bicone.right] \u222a Finset.image Bicone.diagram Fintype.elems\n[PROOFSTEP]\ncases j\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Fintype J\n\u22a2 Bicone.left \u2208 List.toFinset [Bicone.left, Bicone.right] \u222a Finset.image Bicone.diagram Fintype.elems\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Fintype J\n\u22a2 Bicone.right \u2208 List.toFinset [Bicone.left, Bicone.right] \u222a Finset.image Bicone.diagram Fintype.elems\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Fintype J\nval\u271d : J\n\u22a2 Bicone.diagram val\u271d \u2208 List.toFinset [Bicone.left, Bicone.right] \u222a Finset.image Bicone.diagram Fintype.elems\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Fintype J\nval\u271d : J\n\u22a2 val\u271d \u2208 Fintype.elems\n[PROOFSTEP]\napply Fintype.complete\n[GOAL]\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj k : Bicone J\nf g : BiconeHom J j k\n\u22a2 Decidable (f = g)\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\ng : BiconeHom J Bicone.left Bicone.left\n\u22a2 Decidable (left_id = g)\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\ng : BiconeHom J Bicone.right Bicone.right\n\u22a2 Decidable (right_id = g)\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\ng : BiconeHom J Bicone.left (Bicone.diagram j\u271d)\n\u22a2 Decidable (left j\u271d = g)\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\ng : BiconeHom J Bicone.right (Bicone.diagram j\u271d)\n\u22a2 Decidable (right j\u271d = g)\n[PROOFSTEP]\ncases g\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\ng : BiconeHom J (Bicone.diagram j\u271d) (Bicone.diagram k\u271d)\n\u22a2 Decidable (diagram f\u271d = g)\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left_id.left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 Decidable (left_id = left_id)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id.right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 Decidable (right_id = right_id)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left.left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 Decidable (left j\u271d = left j\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 Decidable (right j\u271d = right j\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d\u00b9 f\u271d : j\u271d \u27f6 k\u271d\n\u22a2 Decidable (diagram f\u271d\u00b9 = diagram f\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left_id.left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 Decidable True\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase right_id.right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 Decidable True\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase left.left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 Decidable True\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase right.right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 Decidable True\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d\u00b9 f\u271d : j\u271d \u27f6 k\u271d\n\u22a2 Decidable (f\u271d\u00b9 = f\u271d)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nX\u271d Y\u271d Z\u271d : Bicone J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 X\u271d \u27f6 Z\u271d\n[PROOFSTEP]\nrcases f with (_ | _ | _ | _ | f)\n[GOAL]\ncase left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\ng : Bicone.left \u27f6 Z\u271d\n\u22a2 Bicone.left \u27f6 Z\u271d\n[PROOFSTEP]\nexact g\n[GOAL]\ncase right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\ng : Bicone.right \u27f6 Z\u271d\n\u22a2 Bicone.right \u27f6 Z\u271d\n[PROOFSTEP]\nexact g\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 Bicone.left \u27f6 Z\u271d\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 Bicone.left \u27f6 Bicone.diagram k\u271d\n[PROOFSTEP]\napply BiconeHom.left\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 Bicone.right \u27f6 Z\u271d\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 Bicone.right \u27f6 Bicone.diagram k\u271d\n[PROOFSTEP]\napply BiconeHom.right\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d k\u271d : J\nf : j\u271d \u27f6 k\u271d\ng : Bicone.diagram k\u271d \u27f6 Z\u271d\n\u22a2 Bicone.diagram j\u271d \u27f6 Z\u271d\n[PROOFSTEP]\nrcases g with (_ | _ | _ | _ | g)\n[GOAL]\ncase diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d\u00b9 : J\nf : j\u271d \u27f6 k\u271d\u00b9\nk\u271d : J\ng : k\u271d\u00b9 \u27f6 k\u271d\n\u22a2 Bicone.diagram j\u271d \u27f6 Bicone.diagram k\u271d\n[PROOFSTEP]\nexact BiconeHom.diagram (f \u226b g)\n[GOAL]\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nX\u271d Y\u271d : Bicone J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 \ud835\udfd9 X\u271d \u226b f = f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 \ud835\udfd9 Bicone.left \u226b BiconeHom.left_id = BiconeHom.left_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 \ud835\udfd9 Bicone.right \u226b BiconeHom.right_id = BiconeHom.right_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 \ud835\udfd9 Bicone.left \u226b BiconeHom.left j\u271d = BiconeHom.left j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 \ud835\udfd9 Bicone.right \u226b BiconeHom.right j\u271d = BiconeHom.right j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 \ud835\udfd9 (Bicone.diagram j\u271d) \u226b BiconeHom.diagram f\u271d = BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nX\u271d Y\u271d : Bicone J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f \u226b \ud835\udfd9 Y\u271d = f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 BiconeHom.left_id \u226b \ud835\udfd9 Bicone.left = BiconeHom.left_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 BiconeHom.right_id \u226b \ud835\udfd9 Bicone.right = BiconeHom.right_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 BiconeHom.left j\u271d \u226b \ud835\udfd9 (Bicone.diagram j\u271d) = BiconeHom.left j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 BiconeHom.right j\u271d \u226b \ud835\udfd9 (Bicone.diagram j\u271d) = BiconeHom.right j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 BiconeHom.diagram f\u271d \u226b \ud835\udfd9 (Bicone.diagram k\u271d) = BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nW\u271d X\u271d Y\u271d Z\u271d : Bicone 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 left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nY\u271d Z\u271d : Bicone J\nh : Y\u271d \u27f6 Z\u271d\ng : Bicone.left \u27f6 Y\u271d\n\u22a2 (BiconeHom.left_id \u226b g) \u226b h = BiconeHom.left_id \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nY\u271d Z\u271d : Bicone J\nh : Y\u271d \u27f6 Z\u271d\ng : Bicone.right \u27f6 Y\u271d\n\u22a2 (BiconeHom.right_id \u226b g) \u226b h = BiconeHom.right_id \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nY\u271d Z\u271d : Bicone J\nh : Y\u271d \u27f6 Z\u271d\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Y\u271d\n\u22a2 (BiconeHom.left j\u271d \u226b g) \u226b h = BiconeHom.left j\u271d \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nY\u271d Z\u271d : Bicone J\nh : Y\u271d \u27f6 Z\u271d\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Y\u271d\n\u22a2 (BiconeHom.right j\u271d \u226b g) \u226b h = BiconeHom.right j\u271d \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nY\u271d Z\u271d : Bicone J\nh : Y\u271d \u27f6 Z\u271d\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\ng : Bicone.diagram k\u271d \u27f6 Y\u271d\n\u22a2 (BiconeHom.diagram f\u271d \u226b g) \u226b h = BiconeHom.diagram f\u271d \u226b g \u226b h\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left_id.left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nh : Bicone.left \u27f6 Z\u271d\n\u22a2 (BiconeHom.left_id \u226b BiconeHom.left_id) \u226b h = BiconeHom.left_id \u226b BiconeHom.left_id \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase left_id.left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d : J\nh : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 (BiconeHom.left_id \u226b BiconeHom.left j\u271d) \u226b h = BiconeHom.left_id \u226b BiconeHom.left j\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase right_id.right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nh : Bicone.right \u27f6 Z\u271d\n\u22a2 (BiconeHom.right_id \u226b BiconeHom.right_id) \u226b h = BiconeHom.right_id \u226b BiconeHom.right_id \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase right_id.right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d : J\nh : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 (BiconeHom.right_id \u226b BiconeHom.right j\u271d) \u226b h = BiconeHom.right_id \u226b BiconeHom.right j\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase left.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\nh : Bicone.diagram k\u271d \u27f6 Z\u271d\n\u22a2 (BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d) \u226b h = BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase right.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\nh : Bicone.diagram k\u271d \u27f6 Z\u271d\n\u22a2 (BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d) \u226b h = BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nZ\u271d : Bicone J\nj\u271d k\u271d\u00b9 : J\nf\u271d\u00b9 : j\u271d \u27f6 k\u271d\u00b9\nk\u271d : J\nf\u271d : k\u271d\u00b9 \u27f6 k\u271d\nh : Bicone.diagram k\u271d \u27f6 Z\u271d\n\u22a2 (BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d) \u226b h = BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d \u226b h\n[PROOFSTEP]\ncases h\n[GOAL]\ncase left_id.left_id.left_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 (BiconeHom.left_id \u226b BiconeHom.left_id) \u226b BiconeHom.left_id =\n BiconeHom.left_id \u226b BiconeHom.left_id \u226b BiconeHom.left_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left_id.left_id.left\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 (BiconeHom.left_id \u226b BiconeHom.left_id) \u226b BiconeHom.left j\u271d =\n BiconeHom.left_id \u226b BiconeHom.left_id \u226b BiconeHom.left j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left_id.left.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 (BiconeHom.left_id \u226b BiconeHom.left j\u271d) \u226b BiconeHom.diagram f\u271d =\n BiconeHom.left_id \u226b BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id.right_id.right_id\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\n\u22a2 (BiconeHom.right_id \u226b BiconeHom.right_id) \u226b BiconeHom.right_id =\n BiconeHom.right_id \u226b BiconeHom.right_id \u226b BiconeHom.right_id\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id.right_id.right\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d : J\n\u22a2 (BiconeHom.right_id \u226b BiconeHom.right_id) \u226b BiconeHom.right j\u271d =\n BiconeHom.right_id \u226b BiconeHom.right_id \u226b BiconeHom.right j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right_id.right.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 (BiconeHom.right_id \u226b BiconeHom.right j\u271d) \u226b BiconeHom.diagram f\u271d =\n BiconeHom.right_id \u226b BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left.diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d\u00b9 : J\nf\u271d\u00b9 : j\u271d \u27f6 k\u271d\u00b9\nk\u271d : J\nf\u271d : k\u271d\u00b9 \u27f6 k\u271d\n\u22a2 (BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d\u00b9) \u226b BiconeHom.diagram f\u271d =\n BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d\u00b9 : J\nf\u271d\u00b9 : j\u271d \u27f6 k\u271d\u00b9\nk\u271d : J\nf\u271d : k\u271d\u00b9 \u27f6 k\u271d\n\u22a2 (BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d\u00b9) \u226b BiconeHom.diagram f\u271d =\n BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram.diagram.diagram\nJ : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} J\nj\u271d k\u271d\u00b2 : J\nf\u271d\u00b2 : j\u271d \u27f6 k\u271d\u00b2\nk\u271d\u00b9 : J\nf\u271d\u00b9 : k\u271d\u00b2 \u27f6 k\u271d\u00b9\nk\u271d : J\nf\u271d : k\u271d\u00b9 \u27f6 k\u271d\n\u22a2 (BiconeHom.diagram f\u271d\u00b2 \u226b BiconeHom.diagram f\u271d\u00b9) \u226b BiconeHom.diagram f\u271d =\n BiconeHom.diagram f\u271d\u00b2 \u226b BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nX\u271d Y\u271d : Bicone J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X\u271d \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y\u271d\n[PROOFSTEP]\nrcases f with (_ | _ | _ | _ | f)\n[GOAL]\ncase left_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\ncase right_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\ncase left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d : J\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j\u271d)\n[PROOFSTEP]\nexact c\u2081.\u03c0.app _\n[GOAL]\ncase right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d : J\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j\u271d)\n[PROOFSTEP]\nexact c\u2082.\u03c0.app _\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d k\u271d : J\nf : j\u271d \u27f6 k\u271d\n\u22a2 (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j\u271d) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram k\u271d)\n[PROOFSTEP]\nexact F.map f\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nX : Bicone J\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({ obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.obj\n X)\n[PROOFSTEP]\ncases X\n[GOAL]\ncase left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (\ud835\udfd9 Bicone.left) =\n \ud835\udfd9\n ({ obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.obj\n Bicone.left)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (\ud835\udfd9 Bicone.right) =\n \ud835\udfd9\n ({ obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.obj\n Bicone.right)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nval\u271d : J\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (\ud835\udfd9 (Bicone.diagram val\u271d)) =\n \ud835\udfd9\n ({ obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.obj\n (Bicone.diagram val\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nX\u271d Y\u271d Z\u271d : Bicone J\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (f \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n f \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\nrcases f with (_ | _ | _ | _ | _)\n[GOAL]\ncase left_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nZ\u271d : Bicone J\ng : Bicone.left \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.left_id \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n BiconeHom.left_id \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\ncase right_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nZ\u271d : Bicone J\ng : Bicone.right \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.right_id \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n BiconeHom.right_id \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\nexact (Category.id_comp _).symm\n[GOAL]\ncase left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nZ\u271d : Bicone J\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.left j\u271d \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.left j\u271d) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase left.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.left j\u271d \u226b BiconeHom.diagram f\u271d) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.left j\u271d) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d)\n[PROOFSTEP]\nexact (Category.id_comp _).symm.trans (c\u2081.\u03c0.naturality _)\n[GOAL]\ncase right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nZ\u271d : Bicone J\nj\u271d : J\ng : Bicone.diagram j\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.right j\u271d \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.right j\u271d) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase right.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.right j\u271d \u226b BiconeHom.diagram f\u271d) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.right j\u271d) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d)\n[PROOFSTEP]\nexact (Category.id_comp _).symm.trans (c\u2082.\u03c0.naturality _)\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nZ\u271d : Bicone J\nj\u271d k\u271d : J\nf\u271d : j\u271d \u27f6 k\u271d\ng : Bicone.diagram k\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d \u226b g) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase diagram.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : J \u2964 C\nc\u2081 c\u2082 : Cone F\nj\u271d k\u271d\u00b9 : J\nf\u271d\u00b9 : j\u271d \u27f6 k\u271d\u00b9\nk\u271d : J\nf\u271d : k\u271d\u00b9 \u27f6 k\u271d\n\u22a2 { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d\u00b9 \u226b BiconeHom.diagram f\u271d) =\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d\u00b9) \u226b\n { obj := fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j,\n map := fun {X Y} f =>\n BiconeHom.casesOn (motive := fun a a_1 x =>\n X = a \u2192\n Y = a_1 \u2192\n HEq f x \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n f\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.left \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f BiconeHom.left_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.left_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left))\n (_ : Bicone.left = Y) f)\n (_ : Bicone.left = X) f)\n (fun h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.right \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f BiconeHom.right_id \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n (_ : BiconeHom.right_id = f) \u25b8\n \ud835\udfd9 ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right))\n (_ : Bicone.right = Y) f)\n (_ : Bicone.right = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.left \u27f6 Y) \u2192\n HEq f (BiconeHom.left j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.left \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.left j = f) \u25b8 NatTrans.app c\u2081.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.left = X) f)\n (fun j h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram j \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.right \u27f6 Y) \u2192\n HEq f (BiconeHom.right j) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Bicone.right \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.right j = f) \u25b8 NatTrans.app c\u2082.\u03c0 j) (_ : Bicone.diagram j = Y) f)\n (_ : Bicone.right = X) f)\n (fun {j k} f_1 h =>\n Eq.ndrec (motive := fun {X} =>\n (f : X \u27f6 Y) \u2192\n Y = Bicone.diagram k \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) X \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h =>\n Eq.ndrec (motive := fun {Y} =>\n (f : Bicone.diagram j \u27f6 Y) \u2192\n HEq f (BiconeHom.diagram f_1) \u2192\n ((fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) (Bicone.diagram j) \u27f6\n (fun X => Bicone.casesOn X c\u2081.pt c\u2082.pt fun j => F.obj j) Y))\n (fun f h => (_ : BiconeHom.diagram f_1 = f) \u25b8 F.map f_1) (_ : Bicone.diagram k = Y) f)\n (_ : Bicone.diagram j = X) f)\n (_ : X = X) (_ : Y = Y) (_ : HEq f f) }.map\n (BiconeHom.diagram f\u271d)\n[PROOFSTEP]\napply F.map_comp\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nj k : Bicone J\n\u22a2 Fintype (j \u27f6 k)\n[PROOFSTEP]\ncases j\n[GOAL]\ncase left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nk : Bicone J\n\u22a2 Fintype (Bicone.left \u27f6 k)\n[PROOFSTEP]\ncases k\n[GOAL]\ncase right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nk : Bicone J\n\u22a2 Fintype (Bicone.right \u27f6 k)\n[PROOFSTEP]\ncases k\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nk : Bicone J\nval\u271d : J\n\u22a2 Fintype (Bicone.diagram val\u271d \u27f6 k)\n[PROOFSTEP]\ncases k\n[GOAL]\ncase left.left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 Fintype (Bicone.left \u27f6 Bicone.left)\n[PROOFSTEP]\nexact\n { elems := { BiconeHom.left_id }\n complete := fun f => by cases f; simp }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nf : Bicone.left \u27f6 Bicone.left\n\u22a2 f \u2208 {BiconeHom.left_id}\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 BiconeHom.left_id \u2208 {BiconeHom.left_id}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left.right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 Fintype (Bicone.left \u27f6 Bicone.right)\n[PROOFSTEP]\nexact\n { elems := \u2205\n complete := fun f => by cases f }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nf : Bicone.left \u27f6 Bicone.right\n\u22a2 f \u2208 \u2205\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 Fintype (Bicone.left \u27f6 Bicone.diagram val\u271d)\n[PROOFSTEP]\nexact\n { elems := {BiconeHom.left _}\n complete := fun f => by cases f; simp }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\nf : Bicone.left \u27f6 Bicone.diagram val\u271d\n\u22a2 f \u2208 {BiconeHom.left val\u271d}\n[PROOFSTEP]\ncases f\n[GOAL]\ncase left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 BiconeHom.left val\u271d \u2208 {BiconeHom.left val\u271d}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 Fintype (Bicone.right \u27f6 Bicone.left)\n[PROOFSTEP]\nexact\n { elems := \u2205\n complete := fun f => by cases f }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nf : Bicone.right \u27f6 Bicone.left\n\u22a2 f \u2208 \u2205\n[PROOFSTEP]\ncases f\n[GOAL]\ncase right.right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 Fintype (Bicone.right \u27f6 Bicone.right)\n[PROOFSTEP]\nexact\n { elems := { BiconeHom.right_id }\n complete := fun f => by cases f; simp }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nf : Bicone.right \u27f6 Bicone.right\n\u22a2 f \u2208 {BiconeHom.right_id}\n[PROOFSTEP]\ncases f\n[GOAL]\ncase right_id\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 BiconeHom.right_id \u2208 {BiconeHom.right_id}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 Fintype (Bicone.right \u27f6 Bicone.diagram val\u271d)\n[PROOFSTEP]\nexact\n { elems := {BiconeHom.right _}\n complete := fun f => by cases f; simp }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\nf : Bicone.right \u27f6 Bicone.diagram val\u271d\n\u22a2 f \u2208 {BiconeHom.right val\u271d}\n[PROOFSTEP]\ncases f\n[GOAL]\ncase right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 BiconeHom.right val\u271d \u2208 {BiconeHom.right val\u271d}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase diagram.left\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 Fintype (Bicone.diagram val\u271d \u27f6 Bicone.left)\n[PROOFSTEP]\nexact\n { elems := \u2205\n complete := fun f => by cases f }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\nf : Bicone.diagram val\u271d \u27f6 Bicone.left\n\u22a2 f \u2208 \u2205\n[PROOFSTEP]\ncases f\n[GOAL]\ncase diagram.right\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\n\u22a2 Fintype (Bicone.diagram val\u271d \u27f6 Bicone.right)\n[PROOFSTEP]\nexact\n { elems := \u2205\n complete := fun f => by cases f }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d : J\nf : Bicone.diagram val\u271d \u27f6 Bicone.right\n\u22a2 f \u2208 \u2205\n[PROOFSTEP]\ncases f\n[GOAL]\ncase diagram.diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d\u00b9 val\u271d : J\n\u22a2 Fintype (Bicone.diagram val\u271d\u00b9 \u27f6 Bicone.diagram val\u271d)\n[PROOFSTEP]\nexact\n { elems := Finset.image BiconeHom.diagram Fintype.elems\n complete := fun f => by\n rcases f with (_ | _ | _ | _ | f)\n simp only [Finset.mem_image]\n use f\n simpa using Fintype.complete _ }\n[GOAL]\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d\u00b9 val\u271d : J\nf : Bicone.diagram val\u271d\u00b9 \u27f6 Bicone.diagram val\u271d\n\u22a2 f \u2208 Finset.image BiconeHom.diagram Fintype.elems\n[PROOFSTEP]\nrcases f with (_ | _ | _ | _ | f)\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d\u00b9 val\u271d : J\nf : val\u271d\u00b9 \u27f6 val\u271d\n\u22a2 BiconeHom.diagram f \u2208 Finset.image BiconeHom.diagram Fintype.elems\n[PROOFSTEP]\nsimp only [Finset.mem_image]\n[GOAL]\ncase diagram\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d\u00b9 val\u271d : J\nf : val\u271d\u00b9 \u27f6 val\u271d\n\u22a2 \u2203 a, a \u2208 Fintype.elems \u2227 BiconeHom.diagram a = BiconeHom.diagram f\n[PROOFSTEP]\nuse f\n[GOAL]\ncase h\nJ : Type v\u2081\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\nval\u271d\u00b9 val\u271d : J\nf : val\u271d\u00b9 \u27f6 val\u271d\n\u22a2 f \u2208 Fintype.elems \u2227 BiconeHom.diagram f = BiconeHom.diagram f\n[PROOFSTEP]\nsimpa using Fintype.complete _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Bicones", "llama_tokens": 75860, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.28964107553793816}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Primcodable \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1), p a \u2194 p (id a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 r ((g \u2218 f) a)\n[PROOFSTEP]\nerw [\u2190 h\u2082, \u2190 h\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 p a\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : r ((g \u2218 f) a)\n\u22a2 p a\n[PROOFSTEP]\nrwa [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Primcodable \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1), p a \u2194 p (id a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 r ((g \u2218 f) a)\n[PROOFSTEP]\nerw [\u2190 h\u2082, \u2190 h\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 p a\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : r ((g \u2218 f) a)\n\u22a2 p a\n[PROOFSTEP]\nrwa [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\ne : \u03b1 \u2243 \u03b2\nq : \u03b2 \u2192 Prop\nh : Computable \u2191e.symm\n\u22a2 q \u2264\u2081 (q \u2218 \u2191e)\n[PROOFSTEP]\nconvert OneOneReducible.of_equiv _ h\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\ne : \u03b1 \u2243 \u03b2\nq : \u03b2 \u2192 Prop\nh : Computable \u2191e.symm\n\u22a2 q = (q \u2218 \u2191e) \u2218 \u2191e.symm\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\ne : \u03b1 \u2243 \u03b2\nq : \u03b2 \u2192 Prop\nh : Computable \u2191e.symm\nx\u271d : \u03b2\n\u22a2 q x\u271d = ((q \u2218 \u2191e) \u2218 \u2191e.symm) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nh\u2081 : p \u2264\u2080 q\nh\u2082 : ComputablePred q\n\u22a2 ComputablePred p\n[PROOFSTEP]\nrcases h\u2081 with \u27e8f, c, hf\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nh\u2082 : ComputablePred q\nf : \u03b1 \u2192 \u03b2\nc : Computable f\nhf : \u2200 (a : \u03b1), p a \u2194 q (f a)\n\u22a2 ComputablePred p\n[PROOFSTEP]\nrw [show p = fun a => q (f a) from Set.ext hf]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nh\u2082 : ComputablePred q\nf : \u03b1 \u2192 \u03b2\nc : Computable f\nhf : \u2200 (a : \u03b1), p a \u2194 q (f a)\n\u22a2 ComputablePred fun a => q (f a)\n[PROOFSTEP]\nrcases computable_iff.1 h\u2082 with \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc : Computable f\ng : \u03b2 \u2192 Bool\nhg : Computable g\nh\u2082 : ComputablePred fun a => g a = true\nhf : \u2200 (a : \u03b1), p a \u2194 (fun a => g a = true) (f a)\n\u22a2 ComputablePred fun a => (fun a => g a = true) (f a)\n[PROOFSTEP]\nexact \u27e8by infer_instance, by simpa using hg.comp c\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc : Computable f\ng : \u03b2 \u2192 Bool\nhg : Computable g\nh\u2082 : ComputablePred fun a => g a = true\nhf : \u2200 (a : \u03b1), p a \u2194 (fun a => g a = true) (f a)\n\u22a2 DecidablePred fun a => (fun a => g a = true) (f a)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03c3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03c3\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc : Computable f\ng : \u03b2 \u2192 Bool\nhg : Computable g\nh\u2082 : ComputablePred fun a => g a = true\nhf : \u2200 (a : \u03b1), p a \u2194 (fun a => g a = true) (f a)\n\u22a2 Computable fun a => decide ((fun a => (fun a => g a = true) (f a)) a)\n[PROOFSTEP]\nsimpa using hg.comp c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b3\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 r (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\nx : \u03b1 \u2295 \u03b2\n\u22a2 (p \u2295' q) x \u2194 r ((f \u2295' g) x)\n[PROOFSTEP]\ncases x <;> [apply h\u2081; apply h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b3\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 r (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\nx : \u03b1 \u2295 \u03b2\n\u22a2 (p \u2295' q) x \u2194 r ((f \u2295' g) x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b3\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 r (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\nval\u271d : \u03b1\n\u22a2 (p \u2295' q) (Sum.inl val\u271d) \u2194 r ((f \u2295' g) (Sum.inl val\u271d))\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b3\nc\u2081 : Computable f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 r (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\nval\u271d : \u03b2\n\u22a2 (p \u2295' q) (Sum.inr val\u271d) \u2194 r ((f \u2295' g) (Sum.inr val\u271d))\n[PROOFSTEP]\napply h\u2082\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : Set \u03b1\n\u22a2 \u2200 (a : \u03b1), p a \u2194 toNat p (Encodable.encode a)\n[PROOFSTEP]\nsimp [toNat, setOf]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : Set \u03b1\n\u22a2 ManyOneEquiv (toNat p) p\n[PROOFSTEP]\nsimp [ManyOneEquiv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : Set \u03b1\nq : Set \u03b2\n\u22a2 ManyOneEquiv (toNat p) (toNat q) \u2194 ManyOneEquiv p q\n[PROOFSTEP]\nsimp [ManyOneEquiv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 d\u2082 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np : Set \u2115\nq\u2081 q\u2082 : \u2115 \u2192 Prop\nhq : ManyOneEquiv q\u2081 q\u2082\n\u22a2 ManyOneEquiv p p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 d\u2082 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\n\u22a2 \u2200 (p q : \u2115 \u2192 Prop),\n ManyOneEquiv p q \u2192\n (fun p => ManyOneDegree.liftOn d\u2082 (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) p =\n (fun p => ManyOneDegree.liftOn d\u2082 (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) q\n[PROOFSTEP]\nintro p\u2081 p\u2082 hp\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 d\u2082 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np\u2081 p\u2082 : \u2115 \u2192 Prop\nhp : ManyOneEquiv p\u2081 p\u2082\n\u22a2 (fun p => ManyOneDegree.liftOn d\u2082 (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) p\u2081 =\n (fun p => ManyOneDegree.liftOn d\u2082 (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) p\u2082\n[PROOFSTEP]\ninduction d\u2082 using ManyOneDegree.ind_on\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np\u2081 p\u2082 : \u2115 \u2192 Prop\nhp : ManyOneEquiv p\u2081 p\u2082\np\u271d : Set \u2115\n\u22a2 (fun p => ManyOneDegree.liftOn (of p\u271d) (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) p\u2081 =\n (fun p => ManyOneDegree.liftOn (of p\u271d) (f p) (_ : \u2200 (q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv q\u2081 q\u2082 \u2192 f p q\u2081 = f p q\u2082)) p\u2082\n[PROOFSTEP]\napply h\n[GOAL]\ncase h.a\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np\u2081 p\u2082 : \u2115 \u2192 Prop\nhp : ManyOneEquiv p\u2081 p\u2082\np\u271d : Set \u2115\n\u22a2 ManyOneEquiv p\u2081 p\u2082\ncase h.a\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np\u2081 p\u2082 : \u2115 \u2192 Prop\nhp : ManyOneEquiv p\u2081 p\u2082\np\u271d : Set \u2115\n\u22a2 ManyOneEquiv (toNat p\u271d) (toNat p\u271d)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h.a\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\n\u03c6 : Sort ?u.72115\nd\u2081 : ManyOneDegree\nf : Set \u2115 \u2192 Set \u2115 \u2192 \u03c6\nh : \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop), ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 f p\u2081 q\u2081 = f p\u2082 q\u2082\np\u2081 p\u2082 : \u2115 \u2192 Prop\nhp : ManyOneEquiv p\u2081 p\u2082\np\u271d : Set \u2115\n\u22a2 ManyOneEquiv (toNat p\u271d) (toNat p\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\n\u22a2 of p = of q \u2194 ManyOneEquiv p q\n[PROOFSTEP]\nrw [of, of, Quotient.eq'']\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\n\u22a2 Setoid.r (toNat p) (toNat q) \u2194 ManyOneEquiv p q\n[PROOFSTEP]\nunfold Setoid.r\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\n\u22a2 { r := ManyOneEquiv, iseqv := proof_1 }.1 (toNat p) (toNat q) \u2194 ManyOneEquiv p q\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd : ManyOneDegree\n\u22a2 d \u2264 d\n[PROOFSTEP]\ninduction d using ManyOneDegree.ind_on\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d : Set \u2115\n\u22a2 of p\u271d \u2264 of p\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d : Set \u2115\n\u22a2 p\u271d \u2264\u2080 p\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\n\u22a2 d\u2081 \u2264 d\u2082 \u2192 d\u2082 \u2264 d\u2081 \u2192 d\u2081 = d\u2082\n[PROOFSTEP]\ninduction d\u2081 using ManyOneDegree.ind_on\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d \u2264 d\u2082 \u2192 d\u2082 \u2264 of p\u271d \u2192 of p\u271d = d\u2082\n[PROOFSTEP]\ninduction d\u2082 using ManyOneDegree.ind_on\n[GOAL]\ncase h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d \u2264 of p\u271d\u00b9 \u2192 of p\u271d\u00b9 = of p\u271d\n[PROOFSTEP]\nintro hp hq\n[GOAL]\ncase h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b9 p\u271d : Set \u2115\nhp : of p\u271d\u00b9 \u2264 of p\u271d\nhq : of p\u271d \u2264 of p\u271d\u00b9\n\u22a2 of p\u271d\u00b9 = of p\u271d\n[PROOFSTEP]\nsimp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and_iff]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 d\u2083 : ManyOneDegree\n\u22a2 d\u2081 \u2264 d\u2082 \u2192 d\u2082 \u2264 d\u2083 \u2192 d\u2081 \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2081 using ManyOneDegree.ind_on\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 d\u2083 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d \u2264 d\u2082 \u2192 d\u2082 \u2264 d\u2083 \u2192 of p\u271d \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2082 using ManyOneDegree.ind_on\n[GOAL]\ncase h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2083 : ManyOneDegree\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d \u2264 d\u2083 \u2192 of p\u271d\u00b9 \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2083 using ManyOneDegree.ind_on\n[GOAL]\ncase h.h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b2 p\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b2 \u2264 of p\u271d\u00b9 \u2192 of p\u271d\u00b9 \u2264 of p\u271d \u2192 of p\u271d\u00b2 \u2264 of p\u271d\n[PROOFSTEP]\napply ManyOneReducible.trans\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\n\u22a2 \u2200 (p\u2081 p\u2082 q\u2081 q\u2082 : \u2115 \u2192 Prop),\n ManyOneEquiv p\u2081 p\u2082 \u2192 ManyOneEquiv q\u2081 q\u2082 \u2192 (fun a b => of (a \u2295' b)) p\u2081 q\u2081 = (fun a b => of (a \u2295' b)) p\u2082 q\u2082\n[PROOFSTEP]\nrintro a b c d \u27e8hl\u2081, hr\u2081\u27e9 \u27e8hl\u2082, hr\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\na b c d : \u2115 \u2192 Prop\nhl\u2081 : a \u2264\u2080 b\nhr\u2081 : b \u2264\u2080 a\nhl\u2082 : c \u2264\u2080 d\nhr\u2082 : d \u2264\u2080 c\n\u22a2 (fun a b => of (a \u2295' b)) a c = (fun a b => of (a \u2295' b)) b d\n[PROOFSTEP]\nrw [of_eq_of]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\na b c d : \u2115 \u2192 Prop\nhl\u2081 : a \u2264\u2080 b\nhr\u2081 : b \u2264\u2080 a\nhl\u2082 : c \u2264\u2080 d\nhr\u2082 : d \u2264\u2080 c\n\u22a2 ManyOneEquiv (a \u2295' c) (b \u2295' d)\n[PROOFSTEP]\nexact\n \u27e8disjoin_manyOneReducible (hl\u2081.trans OneOneReducible.disjoin_left.to_many_one)\n (hl\u2082.trans OneOneReducible.disjoin_right.to_many_one),\n disjoin_manyOneReducible (hr\u2081.trans OneOneReducible.disjoin_left.to_many_one)\n (hr\u2082.trans OneOneReducible.disjoin_right.to_many_one)\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 d\u2083 : ManyOneDegree\n\u22a2 d\u2081 + d\u2082 \u2264 d\u2083 \u2194 d\u2081 \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2081 using ManyOneDegree.ind_on\n[GOAL]\ncase h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2082 d\u2083 : ManyOneDegree\np\u271d : Set \u2115\n\u22a2 of p\u271d + d\u2082 \u2264 d\u2083 \u2194 of p\u271d \u2264 d\u2083 \u2227 d\u2082 \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2082 using ManyOneDegree.ind_on\n[GOAL]\ncase h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2083 : ManyOneDegree\np\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b9 + of p\u271d \u2264 d\u2083 \u2194 of p\u271d\u00b9 \u2264 d\u2083 \u2227 of p\u271d \u2264 d\u2083\n[PROOFSTEP]\ninduction d\u2083 using ManyOneDegree.ind_on\n[GOAL]\ncase h.h.h\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\np\u271d\u00b2 p\u271d\u00b9 p\u271d : Set \u2115\n\u22a2 of p\u271d\u00b2 + of p\u271d\u00b9 \u2264 of p\u271d \u2194 of p\u271d\u00b2 \u2264 of p\u271d \u2227 of p\u271d\u00b9 \u2264 of p\u271d\n[PROOFSTEP]\nsimpa only [\u2190 add_of, of_le_of] using disjoin_le\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\n\u22a2 d\u2081 + ?m.117648 d\u2081 d\u2082 \u2264 d\u2081 + d\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u2075 : Primcodable \u03b1\ninst\u271d\u2074 : Inhabited \u03b1\n\u03b2 : Type v\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Inhabited \u03b2\n\u03b3 : Type w\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Inhabited \u03b3\nd\u2081 d\u2082 : ManyOneDegree\n\u22a2 ?m.117829 d\u2081 d\u2082 + d\u2082 \u2264 d\u2081 + d\u2082\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Computability.Reduce", "llama_tokens": 9456, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.28959989467088487}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03c4 : \u211d\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : MetricSpace \u03b1\ni : Fin (0 + 1)\nhi : i < last 0\n\u22a2 (fun x => 1) i \u2264 dist (default i) (default (last 0)) \u2227 (fun x => 1) (last 0) \u2264 \u03c4 * (fun x => 1) i\n[PROOFSTEP]\nrw [Subsingleton.elim (\u03b1 := Fin 1) i (last 0)] at hi \n[GOAL]\n\u03b1 : Type u_1\n\u03c4 : \u211d\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : MetricSpace \u03b1\ni : Fin (0 + 1)\nhi : last 0 < last 0\n\u22a2 (fun x => 1) i \u2264 dist (default i) (default (last 0)) \u2227 (fun x => 1) (last 0) \u2264 \u03c4 * (fun x => 1) i\n[PROOFSTEP]\nexact (lt_irrefl _ hi).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03c4 : \u211d\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : MetricSpace \u03b1\ni : Fin (0 + 1)\nhi : i < last 0\n\u22a2 dist (default i) (default (last 0)) \u2264 (fun x => 1) i + (fun x => 1) (last 0)\n[PROOFSTEP]\nrw [Subsingleton.elim (\u03b1 := Fin 1) i (last 0)] at hi \n[GOAL]\n\u03b1 : Type u_1\n\u03c4 : \u211d\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : MetricSpace \u03b1\ni : Fin (0 + 1)\nhi : last 0 < last 0\n\u22a2 dist (default i) (default (last 0)) \u2264 (fun x => 1) i + (fun x => 1) (last 0)\n[PROOFSTEP]\nexact (lt_irrefl _ hi).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\n\u22a2 dist (c a i) (c a (last N)) \u2264 r a i + r a (last N)\n[PROOFSTEP]\nrcases lt_or_le i (last N) with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nH : i < last N\n\u22a2 dist (c a i) (c a (last N)) \u2264 r a i + r a (last N)\n[PROOFSTEP]\nexact a.inter i H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nH : last N \u2264 i\n\u22a2 dist (c a i) (c a (last N)) \u2264 r a i + r a (last N)\n[PROOFSTEP]\nhave I : i = last N := top_le_iff.1 H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nH : last N \u2264 i\nI : i = last N\n\u22a2 dist (c a i) (c a (last N)) \u2264 r a i + r a (last N)\n[PROOFSTEP]\nhave := (a.rpos (last N)).le\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nH : last N \u2264 i\nI : i = last N\nthis : 0 \u2264 r a (last N)\n\u22a2 dist (c a i) (c a (last N)) \u2264 r a i + r a (last N)\n[PROOFSTEP]\nsimp only [I, add_nonneg this this, dist_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nh : 1 \u2264 \u03c4\n\u22a2 r a (last N) \u2264 \u03c4 * r a i\n[PROOFSTEP]\nrcases lt_or_le i (last N) with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nh : 1 \u2264 \u03c4\nH : i < last N\n\u22a2 r a (last N) \u2264 \u03c4 * r a i\n[PROOFSTEP]\nexact (a.hlast i H).2\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nh : 1 \u2264 \u03c4\nH : last N \u2264 i\n\u22a2 r a (last N) \u2264 \u03c4 * r a i\n[PROOFSTEP]\nhave : i = last N := top_le_iff.1 H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nh : 1 \u2264 \u03c4\nH : last N \u2264 i\nthis : i = last N\n\u22a2 r a (last N) \u2264 \u03c4 * r a i\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\nN : \u2115\n\u03c4 : \u211d\na : SatelliteConfig \u03b1 N \u03c4\ni : Fin (Nat.succ N)\nh : 1 \u2264 \u03c4\nH : last N \u2264 i\nthis : i = last N\n\u22a2 r a (last N) \u2264 \u03c4 * r a (last N)\n[PROOFSTEP]\nexact le_mul_of_one_le_left (a.rpos _).le h\n[GOAL]\na\u271d : Ordinal.{u}\ni : Ordinal.{u} := a\u271d\nj : { j // j < i }\n\u22a2 (invImage (fun a => a) Ordinal.wellFoundedRelation).1 (\u2191j) a\u271d\n[PROOFSTEP]\nexact j.2\n[GOAL]\na\u271d : Ordinal.{u}\ni : Ordinal.{u} := a\u271d\nj : { j // j < i }\n\u22a2 (invImage (fun a => a) Ordinal.wellFoundedRelation).1 (\u2191j) a\u271d\n[PROOFSTEP]\nexact j.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\n\u22a2 Monotone (iUnionUpTo p)\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni j : Ordinal.{u}\nhij : i \u2264 j\n\u22a2 iUnionUpTo p i \u2264 iUnionUpTo p j\n[PROOFSTEP]\nsimp only [iUnionUpTo]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni j : Ordinal.{u}\nhij : i \u2264 j\n\u22a2 \u22c3 (j : { j // j < i }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2264\n \u22c3 (j_1 : { j_1 // j_1 < j }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j_1)) (BallPackage.r p.toBallPackage (index p \u2191j_1))\n[PROOFSTEP]\nexact iUnion_mono' fun r => \u27e8\u27e8r, r.2.trans_le hij\u27e9, Subset.rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\na\u271d : Ordinal.{u}\ni : Ordinal.{u} := a\u271d\nj : { j // j < i }\nx\u271d :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))\n\u22a2 (invImage (fun a => a) Ordinal.wellFoundedRelation).1 (\u2191j) a\u271d\n[PROOFSTEP]\nexact j.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\n\u22a2 Set.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\n\u22a2 False\n[PROOFSTEP]\nsuffices H : Function.Injective p.index\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nH : Function.Injective (index p)\n\u22a2 False\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\n\u22a2 Function.Injective (index p)\n[PROOFSTEP]\nexact not_injective_of_ordinal p.index H\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\n\u22a2 Function.Injective (index p)\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nx y : Ordinal.{u}\nhxy : index p x = index p y\n\u22a2 x = y\n[PROOFSTEP]\nwlog x_le_y : x \u2264 y generalizing x y\n[GOAL]\ncase H.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nx y : Ordinal.{u}\nhxy : index p x = index p y\nthis : \u2200 \u2983x y : Ordinal.{u}\u2984, index p x = index p y \u2192 x \u2264 y \u2192 x = y\nx_le_y : \u00acx \u2264 y\n\u22a2 x = y\n[PROOFSTEP]\nexact (this hxy.symm (le_of_not_le x_le_y)).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\n\u22a2 x = y\n[PROOFSTEP]\nrcases eq_or_lt_of_le x_le_y with (rfl | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nx : Ordinal.{u}\nhxy : index p x = index p x\nx_le_y : x \u2264 x\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nh :\n \u00acSet.Nonempty\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [nonempty_def, not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh :\n \u2200 (x : Ordinal.{u}),\n \u2203 x_1, \u00acBallPackage.c p.toBallPackage x_1 \u2208 iUnionUpTo p x \u2227 R p x \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x_1\n\u22a2 x = y\n[PROOFSTEP]\nspecialize h y\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\n\u22a2 x = y\n[PROOFSTEP]\nhave A : p.c (p.index y) \u2209 p.iUnionUpTo y :=\n by\n have : p.index y = Classical.epsilon fun b : \u03b2 => p.c b \u2209 p.iUnionUpTo y \u2227 p.R y \u2264 p.\u03c4 * p.r b := by\n rw [TauPackage.index]; rfl\n rw [this]\n exact (Classical.epsilon_spec h).1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p y) \u2208 iUnionUpTo p y\n[PROOFSTEP]\nhave : p.index y = Classical.epsilon fun b : \u03b2 => p.c b \u2209 p.iUnionUpTo y \u2227 p.R y \u2264 p.\u03c4 * p.r b := by\n rw [TauPackage.index]; rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\n\u22a2 index p y =\n Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b\n[PROOFSTEP]\nrw [TauPackage.index]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\n\u22a2 (Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208\n \u22c3 (j : { j // j < y }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2227\n \u2a06 (b :\n { b //\n \u00acBallPackage.c p.toBallPackage b \u2208\n \u22c3 (j : { j // j < y }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) }),\n BallPackage.r p.toBallPackage \u2191b \u2264\n p.\u03c4 * BallPackage.r p.toBallPackage b) =\n Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nthis :\n index p y =\n Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p y) \u2208 iUnionUpTo p y\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nthis :\n index p y =\n Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b\n\u22a2 \u00acBallPackage.c p.toBallPackage\n (Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b) \u2208\n iUnionUpTo p y\n[PROOFSTEP]\nexact (Classical.epsilon_spec h).1\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nA : \u00acBallPackage.c p.toBallPackage (index p y) \u2208 iUnionUpTo p y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists,\n Subtype.coe_mk] at A \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nA :\n \u2200 (x : Ordinal.{u}),\n x < y \u2192\n \u00acBallPackage.c p.toBallPackage (index p y) \u2208\n ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))\n\u22a2 x = y\n[PROOFSTEP]\nspecialize A x H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nA :\n \u00acBallPackage.c p.toBallPackage (index p y) \u2208\n ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))\n\u22a2 x = y\n[PROOFSTEP]\nsimp [hxy] at A \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx y : Ordinal.{u}\nhxy : index p x = index p y\nx_le_y : x \u2264 y\nH : x < y\nh : \u2203 x, \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p y \u2227 R p y \u2264 p.\u03c4 * BallPackage.r p.toBallPackage x\nA : BallPackage.r p.toBallPackage (index p y) \u2264 0\n\u22a2 x = y\n[PROOFSTEP]\nexact (lt_irrefl _ ((p.rpos (p.index y)).trans_le A)).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\n\u22a2 BallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\n[PROOFSTEP]\nhave A : \u2200 z : \u03b2, p.c z \u2208 p.iUnionUpTo p.lastStep \u2228 p.\u03c4 * p.r z < p.R p.lastStep :=\n by\n have : p.lastStep \u2208 {i | \u00ac\u2203 b : \u03b2, p.c b \u2209 p.iUnionUpTo i \u2227 p.R i \u2264 p.\u03c4 * p.r b} := csInf_mem p.lastStep_nonempty\n simpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\n\u22a2 \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\n[PROOFSTEP]\nhave : p.lastStep \u2208 {i | \u00ac\u2203 b : \u03b2, p.c b \u2209 p.iUnionUpTo i \u2227 p.R i \u2264 p.\u03c4 * p.r b} := csInf_mem p.lastStep_nonempty\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nthis :\n lastStep p \u2208\n {i | \u00ac\u2203 b, \u00acBallPackage.c p.toBallPackage b \u2208 iUnionUpTo p i \u2227 R p i \u2264 p.\u03c4 * BallPackage.r p.toBallPackage b}\n\u22a2 \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\n[PROOFSTEP]\nsimpa only [not_exists, mem_setOf_eq, not_and_or, not_le, not_not_mem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\n\u22a2 BallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nrcases A x with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : BallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nexact h H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nhave Rpos : 0 < p.R p.lastStep := by apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\n\u22a2 0 < R p (lastStep p)\n[PROOFSTEP]\napply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nhave B : p.\u03c4\u207b\u00b9 * p.R p.lastStep < p.R p.lastStep :=\n by\n conv_rhs => rw [\u2190 one_mul (p.R p.lastStep)]\n exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n\u22a2 p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 one_mul (p.R p.lastStep)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n| R p (lastStep p)\n[PROOFSTEP]\nrw [\u2190 one_mul (p.R p.lastStep)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n| R p (lastStep p)\n[PROOFSTEP]\nrw [\u2190 one_mul (p.R p.lastStep)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n| R p (lastStep p)\n[PROOFSTEP]\nrw [\u2190 one_mul (p.R p.lastStep)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\n\u22a2 p.\u03c4\u207b\u00b9 * R p (lastStep p) < 1 * R p (lastStep p)\n[PROOFSTEP]\nexact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, hy1, hy2\u27e9 : \u2203 y : \u03b2, p.c y \u2209 p.iUnionUpTo p.lastStep \u2227 p.\u03c4\u207b\u00b9 * p.R p.lastStep < p.r y :=\n by\n have := exists_lt_of_lt_csSup ?_ B\n \u00b7 simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists, Subtype.coe_mk]\n rw [\u2190 image_univ, nonempty_image_iff]\n exact \u27e8\u27e8_, h\u27e9, mem_univ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\n\u22a2 \u2203 y,\n \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p) \u2227\n p.\u03c4\u207b\u00b9 * R p (lastStep p) < BallPackage.r p.toBallPackage y\n[PROOFSTEP]\nhave := exists_lt_of_lt_csSup ?_ B\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\nthis : \u2203 a, (a \u2208 range fun b => BallPackage.r p.toBallPackage \u2191b) \u2227 p.\u03c4\u207b\u00b9 * R p (lastStep p) < a\n\u22a2 \u2203 y,\n \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p) \u2227\n p.\u03c4\u207b\u00b9 * R p (lastStep p) < BallPackage.r p.toBallPackage y\n[PROOFSTEP]\nsimpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists, Subtype.coe_mk]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\n\u22a2 Set.Nonempty (range fun b => BallPackage.r p.toBallPackage \u2191b)\n[PROOFSTEP]\nrw [\u2190 image_univ, nonempty_image_iff]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\n\u22a2 Set.Nonempty univ\n[PROOFSTEP]\nexact \u27e8\u27e8_, h\u27e9, mem_univ _\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\ny : \u03b2\nhy1 : \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\nhy2 : p.\u03c4\u207b\u00b9 * R p (lastStep p) < BallPackage.r p.toBallPackage y\n\u22a2 False\n[PROOFSTEP]\nrcases A y with (Hy | Hy)\n[GOAL]\ncase inr.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\ny : \u03b2\nhy1 : \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\nhy2 : p.\u03c4\u207b\u00b9 * R p (lastStep p) < BallPackage.r p.toBallPackage y\nHy : BallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nexact hy1 Hy\n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\ny : \u03b2\nhy1 : \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\nhy2 : p.\u03c4\u207b\u00b9 * R p (lastStep p) < BallPackage.r p.toBallPackage y\nHy : p.\u03c4 * BallPackage.r p.toBallPackage y < R p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 div_eq_inv_mul] at hy2 \n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\ny : \u03b2\nhy1 : \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\nhy2 : R p (lastStep p) / p.\u03c4 < BallPackage.r p.toBallPackage y\nHy : p.\u03c4 * BallPackage.r p.toBallPackage y < R p (lastStep p)\n\u22a2 False\n[PROOFSTEP]\nhave := (div_le_iff' (_root_.zero_lt_one.trans p.one_lt_tau)).1 hy2.le\n[GOAL]\ncase inr.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\nx : \u03b2\nA :\n \u2200 (z : \u03b2),\n BallPackage.c p.toBallPackage z \u2208 iUnionUpTo p (lastStep p) \u2228\n p.\u03c4 * BallPackage.r p.toBallPackage z < R p (lastStep p)\nh : \u00acBallPackage.c p.toBallPackage x \u2208 iUnionUpTo p (lastStep p)\nH : p.\u03c4 * BallPackage.r p.toBallPackage x < R p (lastStep p)\nRpos : 0 < R p (lastStep p)\nB : p.\u03c4\u207b\u00b9 * R p (lastStep p) < R p (lastStep p)\ny : \u03b2\nhy1 : \u00acBallPackage.c p.toBallPackage y \u2208 iUnionUpTo p (lastStep p)\nhy2 : R p (lastStep p) / p.\u03c4 < BallPackage.r p.toBallPackage y\nHy : p.\u03c4 * BallPackage.r p.toBallPackage y < R p (lastStep p)\nthis : R p (lastStep p) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage y\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ (Hy.trans_le this)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni : Ordinal.{u}\nhi : i < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\n\u22a2 color p i < N\n[PROOFSTEP]\ninduction' i using Ordinal.induction with i IH\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\n\u22a2 color p i < N\n[PROOFSTEP]\nlet A : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229 closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),\n {p.color j}\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\n\u22a2 color p i < N\n[PROOFSTEP]\nhave color_i : p.color i = sInf (univ \\ A) := by rw [color]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\n\u22a2 color p i = sInf (univ \\ A)\n[PROOFSTEP]\nrw [color]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 color p i < N\n[PROOFSTEP]\nrw [color_i]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) < N\n[PROOFSTEP]\nhave N_mem : N \u2208 univ \\ A :=\n by\n simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ,\n mem_diff, Subtype.exists, Subtype.coe_mk]\n intro j ji _\n exact (IH j ji (ji.trans hi)).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 N \u2208 univ \\ A\n[PROOFSTEP]\nsimp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, not_and, mem_univ,\n mem_diff, Subtype.exists, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal.{u}),\n x < i \u2192\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2192\n \u00acN = color p x\n[PROOFSTEP]\nintro j ji _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nj : Ordinal.{u}\nji : j < i\na\u271d :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))\n\u22a2 \u00acN = color p j\n[PROOFSTEP]\nexact (IH j ji (ji.trans hi)).ne'\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\n\u22a2 sInf (univ \\ A) < N\n[PROOFSTEP]\nsuffices sInf (univ \\ A) \u2260 N\n by\n rcases(csInf_le (OrderBot.bddBelow (univ \\ A)) N_mem).lt_or_eq with (H | H)\n \u00b7 exact H\n \u00b7 exact (this H).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\n\u22a2 sInf (univ \\ A) < N\n[PROOFSTEP]\nrcases(csInf_le (OrderBot.bddBelow (univ \\ A)) N_mem).lt_or_eq with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) < N\n\u22a2 sInf (univ \\ A) < N\n[PROOFSTEP]\nexact H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) = N\n\u22a2 sInf (univ \\ A) < N\n[PROOFSTEP]\nexact (this H).elim\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\n\u22a2 sInf (univ \\ A) \u2260 N\n[PROOFSTEP]\nintro Inf_eq_N\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\n\u22a2 False\n[PROOFSTEP]\nhave :\n \u2200 k,\n k < N \u2192\n \u2203 j,\n j < i \u2227\n (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229 closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty \u2227\n k = p.color j :=\n by\n intro k hk\n rw [\u2190 Inf_eq_N] at hk \n have : k \u2208 A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk\n simp [and_assoc, -exists_and_left] at this \n simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\n\u22a2 \u2200 (k : \u2115),\n k < N \u2192\n \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n[PROOFSTEP]\nintro k hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < N\n\u22a2 \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n[PROOFSTEP]\nrw [\u2190 Inf_eq_N] at hk \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\n\u22a2 \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n[PROOFSTEP]\nhave : k \u2208 A := by simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\n\u22a2 k \u2208 A\n[PROOFSTEP]\nsimpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using Nat.not_mem_of_lt_sInf hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis : k \u2208 A\n\u22a2 \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n[PROOFSTEP]\nsimp [and_assoc, -exists_and_left] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis :\n \u2203 a,\n a < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p a)) (BallPackage.r p.toBallPackage (index p a)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p a\n\u22a2 \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n[PROOFSTEP]\nsimpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists, Subtype.coe_mk]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nthis :\n \u2200 (k : \u2115),\n k < N \u2192\n \u2203 j,\n j < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p j\n\u22a2 False\n[PROOFSTEP]\nchoose! g hg using this\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\n\u22a2 False\n[PROOFSTEP]\nlet G : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\n\u22a2 False\n[PROOFSTEP]\nhave color_G : \u2200 n, n \u2264 N \u2192 p.color (G n) = n := by\n intro n hn\n rcases hn.eq_or_lt with (rfl | H)\n \u00b7 simp only; simp only [color_i, Inf_eq_N, if_true, eq_self_iff_true]\n \u00b7 simp only; simp only [H.ne, (hg n H).right.right.symm, if_false]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\n\u22a2 \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\n[PROOFSTEP]\nintro n hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\n\u22a2 color p (G n) = n\n[PROOFSTEP]\nrcases hn.eq_or_lt with (rfl | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\ni : Ordinal.{u}\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal.{u}\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n \u2200 (k : \u2115),\n k < n \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (G n) = n\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\ni : Ordinal.{u}\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal.{u}\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n \u2200 (k : \u2115),\n k < n \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (if True then i else g n) = n\n[PROOFSTEP]\nsimp only [color_i, Inf_eq_N, if_true, eq_self_iff_true]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (G n) = n\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (if n = N then i else g n) = n\n[PROOFSTEP]\nsimp only [H.ne, (hg n H).right.right.symm, if_false]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\n\u22a2 False\n[PROOFSTEP]\nhave G_lt_last : \u2200 n, n \u2264 N \u2192 G n < p.lastStep := by\n intro n hn\n rcases hn.eq_or_lt with (rfl | H)\n \u00b7 simp only; simp only [hi, if_true, eq_self_iff_true]\n \u00b7 simp only; simp only [H.ne, (hg n H).left.trans hi, if_false]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\n\u22a2 \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\n[PROOFSTEP]\nintro n hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\n\u22a2 G n < lastStep p\n[PROOFSTEP]\nrcases hn.eq_or_lt with (rfl | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\ni : Ordinal.{u}\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal.{u}\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n \u2200 (k : \u2115),\n k < n \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 G n < lastStep p\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\ni : Ordinal.{u}\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal.{u}\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n \u2200 (k : \u2115),\n k < n \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 (if True then i else g n) < lastStep p\n[PROOFSTEP]\nsimp only [hi, if_true, eq_self_iff_true]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 G n < lastStep p\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 (if n = N then i else g n) < lastStep p\n[PROOFSTEP]\nsimp only [H.ne, (hg n H).left.trans hi, if_false]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\n\u22a2 False\n[PROOFSTEP]\nhave fGn : \u2200 n, n \u2264 N \u2192 p.c (p.index (G n)) \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r (p.index (G n)) :=\n by\n intro n hn\n have : p.index (G n) = Classical.epsilon fun t => p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by\n rw [index]; rfl\n rw [this]\n have : \u2203 t, p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by\n simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)\n exact Classical.epsilon_spec this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\n\u22a2 \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n[PROOFSTEP]\nintro n hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n[PROOFSTEP]\nhave : p.index (G n) = Classical.epsilon fun t => p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by rw [index];\n rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 index p (G n) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n[PROOFSTEP]\nrw [index]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 (Classical.epsilon fun b =>\n \u00acBallPackage.c p.toBallPackage b \u2208\n \u22c3 (j : { j // j < G n }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2227\n \u2a06 (b :\n { b //\n \u00acBallPackage.c p.toBallPackage b \u2208\n \u22c3 (j : { j // j < G n }),\n ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) }),\n BallPackage.r p.toBallPackage \u2191b \u2264\n p.\u03c4 * BallPackage.r p.toBallPackage b) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n index p (G n) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n index p (G n) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n (Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n iUnionUpTo p (G n) \u2227\n R p (G n) \u2264\n p.\u03c4 *\n BallPackage.r p.toBallPackage\n (Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)\n[PROOFSTEP]\nhave : \u2203 t, p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by\n simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n index p (G n) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u2203 t, \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n[PROOFSTEP]\nsimpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis\u271d :\n index p (G n) =\n Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\nthis : \u2203 t, \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n (Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n iUnionUpTo p (G n) \u2227\n R p (G n) \u2264\n p.\u03c4 *\n BallPackage.r p.toBallPackage\n (Classical.epsilon fun t =>\n \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)\n[PROOFSTEP]\nexact Classical.epsilon_spec this\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n\u22a2 False\n[PROOFSTEP]\nhave Gab :\n \u2200 a b : Fin (Nat.succ N),\n G a < G b \u2192\n p.r (p.index (G a)) \u2264 dist (p.c (p.index (G a))) (p.c (p.index (G b))) \u2227\n p.r (p.index (G b)) \u2264 p.\u03c4 * p.r (p.index (G a)) :=\n by\n intro a b G_lt\n have ha : (a : \u2115) \u2264 N := Nat.lt_succ_iff.1 a.2\n have hb : (b : \u2115) \u2264 N := Nat.lt_succ_iff.1 b.2\n constructor\n \u00b7 have := (fGn b hb).1\n simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists,\n Subtype.coe_mk] at this \n simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt\n \u00b7 apply le_trans _ (fGn a ha).2\n have B : p.c (p.index (G b)) \u2209 p.iUnionUpTo (G a) := by intro H;\n exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)\n let b' : { t // p.c t \u2209 p.iUnionUpTo (G a) } := \u27e8p.index (G b), B\u27e9\n apply @le_ciSup _ _ _ (fun t : { t // p.c t \u2209 p.iUnionUpTo (G a) } => p.r t) _ b'\n refine' \u27e8p.r_bound, fun t ht => _\u27e9\n simp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht \n rcases ht with \u27e8u, hu\u27e9\n rw [\u2190 hu.2]\n exact\n p.r_le\n _\n -- therefore, one may use them to construct a satellite configuration with `N+1` points\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n\u22a2 \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n[PROOFSTEP]\nintro a b G_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n[PROOFSTEP]\nhave ha : (a : \u2115) \u2264 N := Nat.lt_succ_iff.1 a.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n[PROOFSTEP]\nhave hb : (b : \u2115) \u2264 N := Nat.lt_succ_iff.1 b.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))\n[PROOFSTEP]\nhave := (fGn b hb).1\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191b)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))\n[PROOFSTEP]\nsimp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, Subtype.exists,\n Subtype.coe_mk] at this \n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis :\n \u2200 (x : Ordinal.{u}),\n (x < if \u2191b = N then i else g \u2191b) \u2192\n \u00acBallPackage.c p.toBallPackage (index p (if \u2191b = N then i else g \u2191b)) \u2208\n ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))\n[PROOFSTEP]\nsimpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n[PROOFSTEP]\napply le_trans _ (fGn a ha).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)\n[PROOFSTEP]\nhave B : p.c (p.index (G b)) \u2209 p.iUnionUpTo (G a) := by intro H; exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nH : BallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 False\n[PROOFSTEP]\nexact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)\n[PROOFSTEP]\nlet b' : { t // p.c t \u2209 p.iUnionUpTo (G a) } := \u27e8p.index (G b), B\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)\n[PROOFSTEP]\napply @le_ciSup _ _ _ (fun t : { t // p.c t \u2209 p.iUnionUpTo (G a) } => p.r t) _ b'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BddAbove (range fun t => BallPackage.r p.toBallPackage \u2191t)\n[PROOFSTEP]\nrefine' \u27e8p.r_bound, fun t ht => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht : t \u2208 range fun t => BallPackage.r p.toBallPackage \u2191t\n\u22a2 t \u2264 p.r_bound\n[PROOFSTEP]\nsimp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht :\n \u2203 a_1,\n \u00acBallPackage.c p.toBallPackage a_1 \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227\n BallPackage.r p.toBallPackage a_1 = t\n\u22a2 t \u2264 p.r_bound\n[PROOFSTEP]\nrcases ht with \u27e8u, hu\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 t \u2264 p.r_bound\n[PROOFSTEP]\nrw [\u2190 hu.2]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 BallPackage.r p.toBallPackage u \u2264 p.r_bound\n[PROOFSTEP]\nexact\n p.r_le\n _\n -- therefore, one may use them to construct a satellite configuration with `N+1` points\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 False\n[PROOFSTEP]\nlet sc : SatelliteConfig \u03b1 N p.\u03c4 :=\n { c := fun k => p.c (p.index (G k))\n r := fun k => p.r (p.index (G k))\n rpos := fun k => p.rpos (p.index (G k))\n h := by\n intro a b a_ne_b\n wlog G_le : G a \u2264 G b generalizing a b\n \u00b7 exact (this b a a_ne_b.symm (le_of_not_le G_le)).symm\n have G_lt : G a < G b := by\n rcases G_le.lt_or_eq with (H | H); \u00b7 exact H\n have A : (a : \u2115) \u2260 b := Fin.val_injective.ne a_ne_b\n rw [\u2190 color_G a (Nat.lt_succ_iff.1 a.2), \u2190 color_G b (Nat.lt_succ_iff.1 b.2), H] at A \n exact (A rfl).elim\n exact Or.inl (Gab a b G_lt)\n hlast := by\n intro a ha\n have I : (a : \u2115) < N := ha\n have : G a < G (Fin.last N) := by dsimp; simp [I.ne, (hg a I).1]\n exact Gab _ _ this\n inter := by\n intro a ha\n have I : (a : \u2115) < N := ha\n have J : G (Fin.last N) = i := by dsimp; simp only [if_true, eq_self_iff_true]\n have K : G a = g a := by dsimp; simp [I.ne, (hg a I).1]\n convert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1 }\n -- this is a contradiction\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i j : Fin (Nat.succ N)),\n i \u2260 j \u2192\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) j) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) j)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j\n[PROOFSTEP]\nintro a b a_ne_b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\n[PROOFSTEP]\nwlog G_le : G a \u2264 G b generalizing a b\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nthis :\n \u2200 (a b : Fin (Nat.succ N)),\n a \u2260 b \u2192\n G \u2191a \u2264 G \u2191b \u2192\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\nG_le : \u00acG \u2191a \u2264 G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\n[PROOFSTEP]\nexact (this b a a_ne_b.symm (le_of_not_le G_le)).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\n[PROOFSTEP]\nhave G_lt : G a < G b := by\n rcases G_le.lt_or_eq with (H | H); \u00b7 exact H\n have A : (a : \u2115) \u2260 b := Fin.val_injective.ne a_ne_b\n rw [\u2190 color_G a (Nat.lt_succ_iff.1 a.2), \u2190 color_G b (Nat.lt_succ_iff.1 b.2), H] at A \n exact (A rfl).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\n\u22a2 G \u2191a < G \u2191b\n[PROOFSTEP]\nrcases G_le.lt_or_eq with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a < G \u2191b\n\u22a2 G \u2191a < G \u2191b\n[PROOFSTEP]\nexact H\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\n\u22a2 G \u2191a < G \u2191b\n[PROOFSTEP]\nhave A : (a : \u2115) \u2260 b := Fin.val_injective.ne a_ne_b\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : \u2191a \u2260 \u2191b\n\u22a2 G \u2191a < G \u2191b\n[PROOFSTEP]\nrw [\u2190 color_G a (Nat.lt_succ_iff.1 a.2), \u2190 color_G b (Nat.lt_succ_iff.1 b.2), H] at A \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : color p (G \u2191b) \u2260 color p (G \u2191b)\n\u22a2 G \u2191a < G \u2191b\n[PROOFSTEP]\nexact (A rfl).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nG_lt : G \u2191a < G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\n[PROOFSTEP]\nexact Or.inl (Gab a b G_lt)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i : Fin (N + 1)),\n i < last N \u2192\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a\n[PROOFSTEP]\nhave I : (a : \u2115) < N := ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a\n[PROOFSTEP]\nhave : G a < G (Fin.last N) := by dsimp; simp [I.ne, (hg a I).1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 G \u2191a < G \u2191(last N)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if \u2191a = N then i else g \u2191a) < if N = N then i else g N\n[PROOFSTEP]\nsimp [I.ne, (hg a I).1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nthis : G \u2191a < G \u2191(last N)\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a\n[PROOFSTEP]\nexact Gab _ _ this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i : Fin (N + 1)),\n i < last N \u2192\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)\n[PROOFSTEP]\nhave I : (a : \u2115) < N := ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)\n[PROOFSTEP]\nhave J : G (Fin.last N) = i := by dsimp; simp only [if_true, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 G \u2191(last N) = i\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if N = N then i else g N) = i\n[PROOFSTEP]\nsimp only [if_true, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)\n[PROOFSTEP]\nhave K : G a = g a := by dsimp; simp [I.ne, (hg a I).1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 G \u2191a = g \u2191a\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 (if \u2191a = N then i else g \u2191a) = g \u2191a\n[PROOFSTEP]\nsimp [I.ne, (hg a I).1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\nK : G \u2191a = g \u2191a\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)\n[PROOFSTEP]\nconvert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal.{u}\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal.{u}\nIH : \u2200 (k : Ordinal.{u}), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n \u22c3 (j : { j // j < i }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal.{u}\nhg :\n \u2200 (k : \u2115),\n k < N \u2192\n g k < i \u2227\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n k = color p (g k)\nG : \u2115 \u2192 Ordinal.{u} := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n \u2200 (n : \u2115),\n n \u2264 N \u2192\n \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n \u2200 (a b : Fin (Nat.succ N)),\n G \u2191a < G \u2191b \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\nsc : SatelliteConfig \u03b1 N p.\u03c4 :=\n { c := fun k => BallPackage.c p.toBallPackage (index p (G \u2191k)),\n r := fun k => BallPackage.r p.toBallPackage (index p (G \u2191k)),\n rpos := (_ : \u2200 (k : Fin (Nat.succ N)), 0 < BallPackage.r p.toBallPackage (index p (G \u2191k))),\n h :=\n (_ :\n \u2200 (a b : Fin (Nat.succ N)),\n a \u2260 b \u2192\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b),\n hlast :=\n (_ :\n \u2200 (a : Fin (N + 1)),\n a < last N \u2192\n BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n dist (BallPackage.c p.toBallPackage (index p (G \u2191a)))\n (BallPackage.c p.toBallPackage (index p (G \u2191(last N)))) \u2227\n BallPackage.r p.toBallPackage (index p (G \u2191(last N))) \u2264\n p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))),\n inter :=\n (_ :\n \u2200 (a : Fin (N + 1)),\n a < last N \u2192\n dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)) }\n\u22a2 False\n[PROOFSTEP]\nexact hN.false sc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\n\u22a2 \u2203 s,\n (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 \u2203 s,\n (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nrefine' \u27e8fun _ => \u2205, fun _ => pairwiseDisjoint_empty, _\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 (fun x => \u2205) i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nrw [\u2190 image_univ, eq_empty_of_isEmpty (univ : Set \u03b2)]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : IsEmpty \u03b2\n\u22a2 q.c '' \u2205 \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 (fun x => \u2205) i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nsimp\n -- Now, assume `\u03b2` is nonempty.\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\n\u22a2 \u2203 s,\n (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nlet p : TauPackage \u03b2 \u03b1 :=\n { q with\n \u03c4\n one_lt_tau := h\u03c4 }\n -- we use for `s i` the balls of color `i`.\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n\u22a2 \u2203 s,\n (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nlet s := fun i : Fin N => \u22c3 (k : Ordinal.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : Set \u03b2)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\n\u22a2 \u2203 s,\n (\u2200 (i : Fin N), PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) \u2227\n range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nrefine' \u27e8s, fun i => _, _\u27e9\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\n\u22a2 PairwiseDisjoint (s i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nintro x hx y hy x_ne_y\n[GOAL]\ncase inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\nx : \u03b2\nhx : x \u2208 s i\ny : \u03b2\nhy : y \u2208 s i\nx_ne_y : x \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) x y\n[PROOFSTEP]\nobtain \u27e8jx, jx_lt, jxi, rfl\u27e9 : \u2203 jx : Ordinal, jx < p.lastStep \u2227 p.color jx = i \u2227 x = p.index jx := by\n simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\nx : \u03b2\nhx : x \u2208 s i\ny : \u03b2\nhy : y \u2208 s i\nx_ne_y : x \u2260 y\n\u22a2 \u2203 jx, jx < lastStep p \u2227 color p jx = \u2191i \u2227 x = index p jx\n[PROOFSTEP]\nsimpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\ny : \u03b2\nhy : y \u2208 s i\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 y\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) y\n[PROOFSTEP]\nobtain \u27e8jy, jy_lt, jyi, rfl\u27e9 : \u2203 jy : Ordinal, jy < p.lastStep \u2227 p.color jy = i \u2227 y = p.index jy := by\n simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\ny : \u03b2\nhy : y \u2208 s i\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 y\n\u22a2 \u2203 jy, jy < lastStep p \u2227 color p jy = \u2191i \u2227 y = index p jy\n[PROOFSTEP]\nsimpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy\n[GOAL]\ncase inr.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nwlog jxy : jx \u2264 jy generalizing jx jy\n[GOAL]\ncase inr.refine'_1.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\nthis :\n \u2200 (jx : Ordinal.{u}),\n jx < lastStep p \u2192\n color p jx = \u2191i \u2192\n index p jx \u2208 s i \u2192\n \u2200 (jy : Ordinal.{u}),\n jy < lastStep p \u2192\n color p jy = \u2191i \u2192\n index p jy \u2208 s i \u2192\n index p jx \u2260 index p jy \u2192\n jx \u2264 jy \u2192\n (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx)\n (index p jy)\njxy : \u00acjx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nexact (this jy jy_lt jyi hy jx jx_lt jxi hx x_ne_y.symm (le_of_not_le jxy)).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nreplace jxy : jx < jy\n[GOAL]\ncase jxy\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\n\u22a2 jx < jy\n[PROOFSTEP]\nrcases lt_or_eq_of_le jxy with (H | rfl)\n[GOAL]\ncase jxy.inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\nH : jx < jy\n\u22a2 jx < jy\n[PROOFSTEP]\n{exact H\n}\n[GOAL]\ncase jxy.inl\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx \u2264 jy\nH : jx < jy\n\u22a2 jx < jy\n[PROOFSTEP]\nexact H\n[GOAL]\ncase jxy.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy_lt : jx < lastStep p\njyi : color p jx = \u2191i\nhy : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 index p jx\njxy : jx \u2264 jx\n\u22a2 jx < jx\n[PROOFSTEP]\n{exact (x_ne_y rfl).elim\n}\n[GOAL]\ncase jxy.inr\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy_lt : jx < lastStep p\njyi : color p jx = \u2191i\nhy : index p jx \u2208 s i\nx_ne_y : index p jx \u2260 index p jx\njxy : jx \u2264 jx\n\u22a2 jx < jx\n[PROOFSTEP]\nexact (x_ne_y rfl).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nlet A : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229 closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),\n {p.color j}\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nhave color_j : p.color jy = sInf (univ \\ A) := by rw [TauPackage.color]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\n\u22a2 color p jy = sInf (univ \\ A)\n[PROOFSTEP]\nrw [TauPackage.color]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nhave h : p.color jy \u2208 univ \\ A := by\n rw [color_j]\n apply csInf_mem\n refine' \u27e8N, _\u27e9\n simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and, mem_univ, mem_diff,\n Subtype.exists, Subtype.coe_mk]\n intro k hk _\n exact (p.color_lt (hk.trans jy_lt) hN).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 color p jy \u2208 univ \\ A\n[PROOFSTEP]\nrw [color_j]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) \u2208 univ \\ A\n[PROOFSTEP]\napply csInf_mem\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 Set.Nonempty (univ \\ A)\n[PROOFSTEP]\nrefine' \u27e8N, _\u27e9\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 N \u2208 univ \\ A\n[PROOFSTEP]\nsimp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and, mem_univ, mem_diff,\n Subtype.exists, Subtype.coe_mk]\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal.{u}),\n x < jy \u2192\n Set.Nonempty\n (closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x)) \u2229\n closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))) \u2192\n \u00acN =\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x\n[PROOFSTEP]\nintro k hk _\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nk : Ordinal.{u}\nhk : k < jy\na\u271d :\n Set.Nonempty\n (closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n k))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n k)) \u2229\n closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy)))\n\u22a2 \u00acN =\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n k\n[PROOFSTEP]\nexact (p.color_lt (hk.trans jy_lt) hN).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh : color p jy \u2208 univ \\ A\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nsimp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and, mem_univ, mem_diff,\n Subtype.exists, Subtype.coe_mk] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n \u2200 (x : Ordinal.{u}),\n x < jy \u2192\n Set.Nonempty\n (closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x)) \u2229\n closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))) \u2192\n \u00accolor\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy =\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n x\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\nspecialize h jx jxy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n Set.Nonempty\n (closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx)) \u2229\n closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))) \u2192\n \u00accolor\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy =\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx\n\u22a2 (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\ni : Fin N\njx : Ordinal.{u}\njx_lt : jx < lastStep p\njxi : color p jx = \u2191i\nhx : index p jx \u2208 s i\njy : Ordinal.{u}\njy_lt : jy < lastStep p\njyi : color p jy = \u2191i\nhy : index p jy \u2208 s i\nx_ne_y : index p jx \u2260 index p jy\njxy : jx < jy\nA : Set \u2115 :=\n \u22c3 (j : { j // j < jy }) (_ :\n Set.Nonempty\n (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n closedBall (BallPackage.c p.toBallPackage (index p jy)) (BallPackage.r p.toBallPackage (index p jy)))),\n {color p \u2191j}\ncolor_j : color p jy = sInf (univ \\ A)\nh :\n \u00ac(Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j))\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx)\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy)\n\u22a2 Set.Nonempty\n (closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx)) \u2229\n closedBall\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))\n (BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy))) \u2227\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jy =\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n jx\n[PROOFSTEP]\nsimpa only [jxi, jyi, and_true_iff, eq_self_iff_true, \u2190 not_disjoint_iff_nonempty_inter] using h\n[GOAL]\ncase inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\n\u22a2 range q.c \u2286 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nrefine' range_subset_iff.2 fun b => _\n[GOAL]\ncase inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\nb : \u03b2\n\u22a2 BallPackage.c q b \u2208 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a : Ordinal, a < p.lastStep \u2227 dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by\n simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists, Subtype.coe_mk] using\n p.mem_iUnionUpTo_lastStep b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\nb : \u03b2\n\u22a2 \u2203 a,\n a < lastStep p \u2227\n dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n BallPackage.r p.toBallPackage (index p a)\n[PROOFSTEP]\nsimpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists, Subtype.coe_mk] using\n p.mem_iUnionUpTo_lastStep b\n[GOAL]\ncase inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n a < lastStep p \u2227\n dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n BallPackage.r p.toBallPackage (index p a)\n\u22a2 BallPackage.c q b \u2208 \u22c3 (i : Fin N) (j : \u03b2) (_ : j \u2208 s i), ball (BallPackage.c q j) (BallPackage.r q j)\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, mem_ball, mem_singleton_iff, biUnion_and', exists_eq_left, iUnion_exists,\n exists_and_left]\n[GOAL]\ncase inr.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d : MetricSpace \u03b1\n\u03b2 : Type u\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nq : BallPackage \u03b2 \u03b1\nh\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1 :=\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\ns : Fin N \u2192 Set \u03b2 := fun i => \u22c3 (k : Ordinal.{u}) (_ : k < lastStep p) (_ : color p k = \u2191i), {index p k}\nb : \u03b2\na : Ordinal.{u}\nha :\n a < lastStep p \u2227\n dist (BallPackage.c p.toBallPackage b) (BallPackage.c p.toBallPackage (index p a)) <\n BallPackage.r p.toBallPackage (index p a)\n\u22a2 \u2203 i i_1,\n color\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n i_1 =\n \u2191i \u2227\n i_1 <\n lastStep\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 } \u2227\n dist (BallPackage.c q b)\n (BallPackage.c q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n i_1)) <\n BallPackage.r q\n (index\n {\n toBallPackage :=\n { c := q.c, r := q.r, rpos := (_ : \u2200 (b : \u03b2), 0 < BallPackage.r q b), r_bound := q.r_bound,\n r_le := (_ : \u2200 (b : \u03b2), BallPackage.r q b \u2264 q.r_bound) },\n \u03c4 := \u03c4, one_lt_tau := h\u03c4 }\n i_1)\n[PROOFSTEP]\nexact \u27e8\u27e8p.color a, p.color_lt ha.1 hN\u27e9, a, rfl, ha\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nrcases le_or_lt (\u03bc s) 0 with (h\u03bcs | h\u03bcs)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave : \u03bc s = 0 := le_bot_iff.1 h\u03bcs\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nrefine' \u27e8\u2205, by simp only [Finset.coe_empty, empty_subset], _, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2205 \u2286 s\n[PROOFSTEP]\nsimp only [Finset.coe_empty, empty_subset]\n[GOAL]\ncase inl.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 \u2205), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty, nonpos_iff_eq_zero, mul_zero]\n[GOAL]\ncase inl.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : \u2191\u2191\u03bc s \u2264 0\nthis : \u2191\u2191\u03bc s = 0\n\u22a2 PairwiseDisjoint \u2191\u2205 fun x => closedBall x (r x)\n[PROOFSTEP]\nsimp only [Finset.coe_empty, pairwiseDisjoint_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b1\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nsimp only [eq_empty_of_isEmpty s, measure_empty] at h\u03bcs \n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u271d : IsEmpty \u03b1\nh\u03bcs : 0 < 0\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nexact (lt_irrefl _ h\u03bcs).elim\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave Npos : N \u2260 0 := by\n rintro rfl\n inhabit \u03b1\n exact not_isEmpty_of_nonempty _ hN\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\n\u22a2 N \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\n\u22a2 False\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nhN : IsEmpty (SatelliteConfig \u03b1 0 \u03c4)\ninhabited_h : Inhabited \u03b1\n\u22a2 False\n[PROOFSTEP]\nexact not_isEmpty_of_nonempty _ hN\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nobtain \u27e8o, so, omeas, \u03bco\u27e9 : \u2203 o : Set \u03b1, s \u2286 o \u2227 MeasurableSet o \u2227 \u03bc o = \u03bc s := exists_measurable_superset \u03bc s\n[GOAL]\ncase inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nlet a : BallPackage s \u03b1 :=\n { c := fun x => x\n r := fun x => r x\n rpos := fun x => rpos x x.2\n r_bound := 1\n r_le := fun x => rle x x.2 }\n[GOAL]\ncase inr.inr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nrcases exist_disjoint_covering_families h\u03c4 hN a with \u27e8u, hu, hu'\u27e9\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave u_count : \u2200 i, (u i).Countable := by\n intro i\n refine' (hu i).countable_of_nonempty_interior fun j _ => _\n have : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)\n exact this.mono ball_subset_interior_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\n\u22a2 \u2200 (i : Fin N), Set.Countable (u i)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\n\u22a2 Set.Countable (u i)\n[PROOFSTEP]\nrefine' (hu i).countable_of_nonempty_interior fun j _ => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))\n[PROOFSTEP]\nhave : (ball (j : \u03b1) (r j)).Nonempty := nonempty_ball.2 (a.rpos _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\ni : Fin N\nj : \u2191s\nx\u271d : j \u2208 u i\nthis : Set.Nonempty (ball (\u2191j) (r \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c a j) (BallPackage.r a j)))\n[PROOFSTEP]\nexact this.mono ball_subset_interior_closedBall\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nlet v : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : s) (_ : x \u2208 u i), closedBall x (r x)\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave A : s = \u22c3 i : Fin N, s \u2229 v i :=\n by\n refine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)\n intro x hx\n obtain \u27e8i, y, hxy, h'\u27e9 : \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) :=\n by\n have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n simpa only [mem_iUnion, bex_def] using hu' this\n refine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9\n simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]\n exact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\n\u22a2 s = \u22c3 (i : Fin N), s \u2229 v i\n[PROOFSTEP]\nrefine' Subset.antisymm _ (iUnion_subset fun i => inter_subset_left _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\n\u22a2 s \u2286 \u22c3 (i : Fin N), s \u2229 v i\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (i : Fin N), s \u2229 v i\n[PROOFSTEP]\nobtain \u27e8i, y, hxy, h'\u27e9 : \u2203 (i : Fin N) (i_1 : \u21a5s), i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1) :=\n by\n have : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n simpa only [mem_iUnion, bex_def] using hu' this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1)\n[PROOFSTEP]\nhave : x \u2208 range a.c := by simpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 range a.c\n[PROOFSTEP]\nsimpa only [Subtype.range_coe_subtype, setOf_mem_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\nthis : x \u2208 range a.c\n\u22a2 \u2203 i i_1, i_1 \u2208 u i \u2227 x \u2208 ball (\u2191i_1) (r \u2191i_1)\n[PROOFSTEP]\nsimpa only [mem_iUnion, bex_def] using hu' this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 \u22c3 (i : Fin N), s \u2229 v i\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8i, \u27e8hx, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 x \u2208 v i\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 s) } \u2208 u i) \u2227 x \u2208 closedBall x_1 (r x_1)\n[PROOFSTEP]\nexact \u27e8y, \u27e8y.2, by simpa only [Subtype.coe_eta]\u27e9, ball_subset_closedBall h'\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nx : \u03b1\nhx : x \u2208 s\ni : Fin N\ny : \u2191s\nhxy : y \u2208 u i\nh' : x \u2208 ball (\u2191y) (r \u2191y)\n\u22a2 { val := \u2191y, property := (_ : \u2191y \u2208 s) } \u2208 u i\n[PROOFSTEP]\nsimpa only [Subtype.coe_eta]\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave S : \u2211 _i : Fin N, \u03bc s / N \u2264 \u2211 i, \u03bc (s \u2229 v i) :=\n calc\n \u2211 _i : Fin N, \u03bc s / N = \u03bc s :=\n by\n simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul]\n rw [ENNReal.mul_div_cancel']\n \u00b7 simp only [Npos, Ne.def, Nat.cast_eq_zero, not_false_iff]\n \u00b7 exact ENNReal.nat_ne_top _\n _ \u2264 \u2211 i, \u03bc (s \u2229 v i) := by\n conv_lhs => rw [A]\n apply measure_iUnion_fintype_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N = \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2191N * (\u2191\u2191\u03bc s / \u2191N) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [ENNReal.mul_div_cancel']\n[GOAL]\ncase h0\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2191N \u2260 0\n[PROOFSTEP]\nsimp only [Npos, Ne.def, Nat.cast_eq_zero, not_false_iff]\n[GOAL]\ncase hI\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2191N \u2260 \u22a4\n[PROOFSTEP]\nexact ENNReal.nat_ne_top _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n[PROOFSTEP]\nconv_lhs => rw [A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [A]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : Fin N), s \u2229 v i) \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n[PROOFSTEP]\napply measure_iUnion_fintype_le\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nobtain \u27e8i, -, hi\u27e9 : \u2203 (i : Fin N), i \u2208 Finset.univ \u2227 \u03bc s / N \u2264 \u03bc (s \u2229 v i) :=\n by\n apply ENNReal.exists_le_of_sum_le _ S\n exact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 i, i \u2208 Finset.univ \u2227 \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n[PROOFSTEP]\napply ENNReal.exists_le_of_sum_le _ S\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 Finset.Nonempty Finset.univ\n[PROOFSTEP]\nexact \u27e8\u27e80, bot_lt_iff_ne_bot.2 Npos\u27e9, Finset.mem_univ _\u27e9\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nreplace hi : \u03bc s / (N + 1) < \u03bc (s \u2229 v i)\n[GOAL]\ncase hi\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n[PROOFSTEP]\napply lt_of_lt_of_le _ hi\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc s / \u2191N\n[PROOFSTEP]\napply (ENNReal.mul_lt_mul_left h\u03bcs.ne' (measure_lt_top \u03bc s).ne).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 (\u2191N + 1)\u207b\u00b9 < (\u2191N)\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.inv_lt_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N < \u2191N + 1\n[PROOFSTEP]\nconv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / \u2191N \u2264 \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191N + 0 < \u2191N + 1\n[PROOFSTEP]\nexact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave B : \u03bc (o \u2229 v i) = \u2211' x : u i, \u03bc (o \u2229 closedBall x (r x)) :=\n by\n have : o \u2229 v i = \u22c3 (x : s) (_ : x \u2208 u i), o \u2229 closedBall x (r x) := by simp only [inter_iUnion]\n rw [this, measure_biUnion (u_count i)]\n \u00b7 exact (hu i).mono fun k => inter_subset_right _ _\n \u00b7 exact fun b _ => omeas.inter measurableSet_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nhave : o \u2229 v i = \u22c3 (x : s) (_ : x \u2208 u i), o \u2229 closedBall x (r x) := by simp only [inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\n\u22a2 o \u2229 v i = \u22c3 (x : \u2191s) (_ : x \u2208 u i), o \u2229 closedBall (\u2191x) (r \u2191x)\n[PROOFSTEP]\nsimp only [inter_iUnion]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 (x : \u2191s) (_ : x \u2208 u i), o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nrw [this, measure_biUnion (u_count i)]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 (x : \u2191s) (_ : x \u2208 u i), o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 PairwiseDisjoint (u i) fun x => o \u2229 closedBall (\u2191x) (r \u2191x)\n[PROOFSTEP]\nexact (hu i).mono fun k => inter_subset_right _ _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nthis : o \u2229 v i = \u22c3 (x : \u2191s) (_ : x \u2208 u i), o \u2229 closedBall (\u2191x) (r \u2191x)\n\u22a2 \u2200 (b : \u2191s), b \u2208 u i \u2192 MeasurableSet (o \u2229 closedBall (\u2191b) (r \u2191b))\n[PROOFSTEP]\nexact fun b _ => omeas.inter measurableSet_closedBall\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nobtain \u27e8w, hw\u27e9 : \u2203 w : Finset (u i), \u03bc s / (N + 1) < \u2211 x : u i in w, \u03bc (o \u2229 closedBall (x : \u03b1) (r (x : \u03b1))) :=\n by\n have C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by rw [B]; exact ENNReal.summable.hasSum\n have : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))\n exact ((tendsto_order.1 C).1 _ this).exists\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nhave C : HasSum (fun x : u i => \u03bc (o \u2229 closedBall x (r x))) (\u03bc (o \u2229 v i)) := by rw [B]; exact ENNReal.summable.hasSum\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\n[PROOFSTEP]\nrw [B]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)))\n[PROOFSTEP]\nexact ENNReal.summable.hasSum\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nhave : \u03bc s / (N + 1) < \u03bc (o \u2229 v i) := hi.trans_le (measure_mono (inter_subset_inter_left _ so))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nC : HasSum (fun x => \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))) (\u2191\u2191\u03bc (o \u2229 v i))\nthis : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (o \u2229 v i)\n\u22a2 \u2203 w, \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nexact ((tendsto_order.1 C).1 _ this).exists\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2203 t,\n \u2191t \u2286 s \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2227\n PairwiseDisjoint \u2191t fun x => closedBall x (r x)\n[PROOFSTEP]\nrefine'\n \u27e8Finset.image (fun x : u i => x) w, _, _, _\u27e9\n -- show that the finset is included in `s`.\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191(Finset.image (fun x => \u2191\u2191x) w) \u2286 s\n[PROOFSTEP]\nsimp only [image_subset_iff, Finset.coe_image]\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191w \u2286 (fun x => \u2191\u2191x) \u207b\u00b9' s\n[PROOFSTEP]\nintro y _\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\ny : \u2191(u i)\na\u271d : y \u2208 \u2191w\n\u22a2 y \u2208 (fun x => \u2191\u2191x) \u207b\u00b9' s\n[PROOFSTEP]\nsimp only [Subtype.coe_prop, mem_preimage]\n -- show that it covers a large enough proportion of `s`. For measure computations, we do not\n -- use `s` (which might not be measurable), but its measurable superset `o`. Since their measures\n -- are the same, this does not spoil the estimates\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 Finset.image (fun x => \u2191\u2191x) w), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsuffices H : \u03bc (o \\ \u22c3 x \u2208 w, closedBall (\u2191x) (r \u2191x)) \u2264 N / (N + 1) * \u03bc s\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 Finset.image (fun x => \u2191\u2191x) w), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [Finset.set_biUnion_finset_image]\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nH : \u2191\u2191\u03bc (o \\ \u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (y : \u2191(u i)) (_ : y \u2208 w), closedBall (\u2191\u2191y) (r \u2191\u2191y)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact le_trans (measure_mono (diff_subset_diff so (Subset.refl _))) H\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc (o \\ \u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 diff_inter_self_eq_diff, measure_diff_le_iff_le_add _ (inter_subset_right _ _) (measure_lt_top \u03bc _).ne]\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n[PROOFSTEP]\napply MeasurableSet.inter _ omeas\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 MeasurableSet (\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nhaveI : Encodable (u i) := (u_count i).toEncodable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : Encodable \u2191(u i)\n\u22a2 MeasurableSet (\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nexact MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => measurableSet_closedBall\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o \u2264 \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\ncalc\n \u03bc o = 1 / (N + 1) * \u03bc s + N / (N + 1) * \u03bc s := by\n rw [\u03bco, \u2190 add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul] <;> simp\n _ \u2264 \u03bc ((\u22c3 x \u2208 w, closedBall (\u2191x) (r \u2191x)) \u2229 o) + N / (N + 1) * \u03bc s :=\n by\n refine' add_le_add _ le_rfl\n rw [div_eq_mul_inv, one_mul, mul_comm, \u2190 div_eq_mul_inv]\n apply hw.le.trans (le_of_eq _)\n rw [\u2190 Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion\u2082, Finset.set_biUnion_coe, measure_biUnion_finset]\n \u00b7 have : (w : Set (u i)).PairwiseDisjoint fun b : u i => closedBall (b : \u03b1) (r (b : \u03b1)) := by intro k _ l _ hkl;\n exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)\n exact this.mono fun k => inter_subset_right _ _\n \u00b7 intro b _\n apply omeas.inter measurableSet_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc o = 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u03bco, \u2190 add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul]\n[GOAL]\ncase h0\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191N + 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hI\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191N + 1 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264\n \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o) + \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrefine' add_le_add _ le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 1 / (\u2191N + 1) * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n[PROOFSTEP]\nrw [div_eq_mul_inv, one_mul, mul_comm, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2191\u2191\u03bc s / (\u2191N + 1) \u2264 \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n[PROOFSTEP]\napply hw.le.trans (le_of_eq _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc ((\u22c3 (x : \u2191(u i)) (_ : x \u2208 w), closedBall (\u2191\u2191x) (r \u2191\u2191x)) \u2229 o)\n[PROOFSTEP]\nrw [\u2190 Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion\u2082, Finset.set_biUnion_coe, measure_biUnion_finset]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)\n[PROOFSTEP]\nhave : (w : Set (u i)).PairwiseDisjoint fun b : u i => closedBall (b : \u03b1) (r (b : \u03b1)) := by intro k _ l _ hkl;\n exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)\n[PROOFSTEP]\nintro k _ l _ hkl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u2191(u i)\na\u271d\u00b9 : k \u2208 \u2191w\nl : \u2191(u i)\na\u271d : l \u2208 \u2191w\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)) k l\n[PROOFSTEP]\nexact hu i k.2 l.2 (Subtype.val_injective.ne hkl)\n[GOAL]\ncase hd\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nthis : PairwiseDisjoint \u2191w fun b => closedBall (\u2191\u2191b) (r \u2191\u2191b)\n\u22a2 PairwiseDisjoint \u2191w fun x => o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x)\n[PROOFSTEP]\nexact this.mono fun k => inter_subset_right _ _\n[GOAL]\ncase hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 \u2200 (b : \u2191(u i)), b \u2208 w \u2192 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))\n[PROOFSTEP]\nintro b _\n[GOAL]\ncase hm\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nb : \u2191(u i)\na\u271d : b \u2208 w\n\u22a2 MeasurableSet (o \u2229 closedBall (\u2191\u2191b) (r \u2191\u2191b))\n[PROOFSTEP]\napply omeas.inter measurableSet_closedBall\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\n\u22a2 PairwiseDisjoint \u2191(Finset.image (fun x => \u2191\u2191x) w) fun x => closedBall x (r x)\n[PROOFSTEP]\nintro k hk l hl hkl\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u03b1\nhk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : k \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) k l\n[PROOFSTEP]\nobtain \u27e8k', _, rfl\u27e9 : \u2203 k' : u i, k' \u2208 w \u2227 \u2191k' = k := by\n simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk : \u03b1\nhk : k \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : k \u2260 l\n\u22a2 \u2203 k', k' \u2208 w \u2227 \u2191\u2191k' = k\n[PROOFSTEP]\nsimpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hk\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nk' : \u2191(u i)\nleft\u271d : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 l\n\u22a2 (Disjoint on fun x => closedBall x (r x)) (\u2191\u2191k') l\n[PROOFSTEP]\nobtain \u27e8l', _, rfl\u27e9 : \u2203 l' : u i, l' \u2208 w \u2227 \u2191l' = l := by\n simpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nl : \u03b1\nhl : l \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nk' : \u2191(u i)\nleft\u271d : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 l\n\u22a2 \u2203 l', l' \u2208 w \u2227 \u2191\u2191l' = l\n[PROOFSTEP]\nsimpa only [mem_image, Finset.mem_coe, Finset.coe_image] using hl\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'\n[PROOFSTEP]\nhave k'nel' : (k' : s) \u2260 l' := by intro h; rw [h] at hkl ; exact hkl rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\n\u22a2 \u2191k' \u2260 \u2191l'\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hkl \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191l' \u2260 \u2191\u2191l'\nh : \u2191k' = \u2191l'\n\u22a2 False\n[PROOFSTEP]\nexact hkl rfl\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\ns : Set \u03b1\nr : \u03b1 \u2192 \u211d\nrpos : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < r x\nrle : \u2200 (x : \u03b1), x \u2208 s \u2192 r x \u2264 1\nh\u03bcs : 0 < \u2191\u2191\u03bc s\nh\u271d : Nonempty \u03b1\nNpos : N \u2260 0\no : Set \u03b1\nso : s \u2286 o\nomeas : MeasurableSet o\n\u03bco : \u2191\u2191\u03bc o = \u2191\u2191\u03bc s\na : BallPackage (\u2191s) \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r \u2191x, rpos := (_ : \u2200 (x : \u2191s), 0 < r \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s), r \u2191x \u2264 1) }\nu : Fin N \u2192 Set \u2191s\nhu : \u2200 (i : Fin N), PairwiseDisjoint (u i) fun j => closedBall (BallPackage.c a j) (BallPackage.r a j)\nhu' : range a.c \u2286 \u22c3 (i : Fin N) (j : \u2191s) (_ : j \u2208 u i), ball (BallPackage.c a j) (BallPackage.r a j)\nu_count : \u2200 (i : Fin N), Set.Countable (u i)\nv : Fin N \u2192 Set \u03b1 := fun i => \u22c3 (x : \u2191s) (_ : x \u2208 u i), closedBall (\u2191x) (r \u2191x)\nA : s = \u22c3 (i : Fin N), s \u2229 v i\nS : \u2211 _i : Fin N, \u2191\u2191\u03bc s / \u2191N \u2264 \u2211 i : Fin N, \u2191\u2191\u03bc (s \u2229 v i)\ni : Fin N\nhi : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2191\u2191\u03bc (s \u2229 v i)\nB : \u2191\u2191\u03bc (o \u2229 v i) = \u2211' (x : \u2191(u i)), \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nw : Finset \u2191(u i)\nhw : \u2191\u2191\u03bc s / (\u2191N + 1) < \u2211 x in w, \u2191\u2191\u03bc (o \u2229 closedBall (\u2191\u2191x) (r \u2191\u2191x))\nk' : \u2191(u i)\nleft\u271d\u00b9 : k' \u2208 w\nhk : \u2191\u2191k' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nl' : \u2191(u i)\nleft\u271d : l' \u2208 w\nhl : \u2191\u2191l' \u2208 \u2191(Finset.image (fun x => \u2191\u2191x) w)\nhkl : \u2191\u2191k' \u2260 \u2191\u2191l'\nk'nel' : \u2191k' \u2260 \u2191l'\n\u22a2 (Disjoint on fun x => closedBall x (r x)) \u2191\u2191k' \u2191\u2191l'\n[PROOFSTEP]\nexact hu i k'.2 l'.2 k'nel'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nrcases HasBesicovitchCovering.no_satelliteConfig (\u03b1 := \u03b1) with\n \u27e8N, \u03c4, h\u03c4, hN\u27e9\n /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a\n subset of `s` by admissible balls. -/\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nlet P : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop := fun t =>\n ((t : Set (\u03b1 \u00d7 \u211d)).PairwiseDisjoint fun p => closedBall p.1 p.2) \u2227\n (\u2200 p : \u03b1 \u00d7 \u211d, p \u2208 t \u2192 p.1 \u2208 s) \u2227\n \u2200 p : \u03b1 \u00d7 \u211d,\n p \u2208 t \u2192\n p.2 \u2208\n f\n p.1\n /- Given a finite good covering of a subset `s`, one can find a larger finite good covering,\n covering additionally a proportion at least `1/(N+1)` of leftover points. This follows from\n `exist_finset_disjoint_balls_large_measure` applied to balls not intersecting the initial\n covering. -/\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nhave :\n \u2200 t : Finset (\u03b1 \u00d7 \u211d),\n P t \u2192\n \u2203 u : Finset (\u03b1 \u00d7 \u211d),\n t \u2286 u \u2227\n P u \u2227\n \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.1 p.2) \u2264\n N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2) :=\n by\n intro t ht\n set B := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2 with hB\n have B_closed : IsClosed B := isClosed_biUnion (Finset.finite_toSet _) fun i _ => isClosed_ball\n set s' := s \\ B\n have : \u2200 x \u2208 s', \u2203 r \u2208 f x \u2229 Ioo 0 1, Disjoint B (closedBall x r) :=\n by\n intro x hx\n have xs : x \u2208 s := ((mem_diff x).1 hx).1\n rcases eq_empty_or_nonempty B with (hB | hB)\n \u00b7 rcases hf x xs 1 zero_lt_one with \u27e8r, hr, h'r\u27e9\n exact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, empty_disjoint]\u27e9\n \u00b7 let r := infDist x B\n have : 0 < min r 1 := lt_min ((B_closed.not_mem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one\n rcases hf x xs _ this with \u27e8r, hr, h'r\u27e9\n refine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.trans_le (min_le_right _ _)\u27e9\u27e9, _\u27e9\n rw [disjoint_comm]\n exact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _))\n choose! r hr using this\n obtain \u27e8v, vs', h\u03bcv, hv\u27e9 :\n \u2203 v : Finset \u03b1,\n \u2191v \u2286 s' \u2227\n \u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 N / (N + 1) * \u03bc s' \u2227\n (v : Set \u03b1).PairwiseDisjoint fun x : \u03b1 => closedBall x (r x) :=\n haveI rI : \u2200 x \u2208 s', r x \u2208 Ioo (0 : \u211d) 1 := fun x hx => (hr x hx).1.2\n exist_finset_disjoint_balls_large_measure \u03bc h\u03c4 hN s' r (fun x hx => (rI x hx).1) fun x hx => (rI x hx).2.le\n refine' \u27e8t \u222a Finset.image (fun x => (x, r x)) v, Finset.subset_union_left _ _, \u27e8_, _, _\u27e9, _\u27e9\n \u00b7 simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and_iff, Finset.coe_image]\n constructor\n \u00b7 intro p hp q hq hpq\n rcases(mem_image _ _ _).1 hp with \u27e8p', p'v, rfl\u27e9\n rcases(mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9\n refine' hv p'v q'v fun hp'q' => _\n rw [hp'q'] at hpq \n exact hpq rfl\n \u00b7 intro p hp q hq hpq\n rcases(mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9\n apply disjoint_of_subset_left _ (hr q' (vs' q'v)).2\n rw [hB, \u2190 Finset.set_biUnion_coe]\n exact subset_biUnion_of_mem (u := fun x : \u03b1 \u00d7 \u211d => closedBall x.1 x.2) hp\n \u00b7 intro p hp\n rcases Finset.mem_union.1 hp with (h'p | h'p)\n \u00b7 exact ht.2.1 p h'p\n \u00b7 rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9\n exact ((mem_diff _).1 (vs' (Finset.mem_coe.2 p'v))).1\n \u00b7 intro p hp\n rcases Finset.mem_union.1 hp with (h'p | h'p)\n \u00b7 exact ht.2.2 p h'p\n \u00b7 rcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9\n exact (hr p' (vs' p'v)).1.1\n \u00b7 convert h\u03bcv using 2\n rw [Finset.set_biUnion_union, \u2190 diff_diff, Finset.set_biUnion_finset_image]\n /- Define `F` associating to a finite good covering the above enlarged good covering, covering\n a proportion `1/(N+1)` of leftover points. Iterating `F`, one will get larger and larger good\n coverings, missing in the end only a measure-zero set. -/\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\n\u22a2 \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n \u2203 u,\n t \u2286 u \u2227\n P u \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\n\u22a2 \u2203 u,\n t \u2286 u \u2227\n P u \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\n[PROOFSTEP]\nset B := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2 with hB\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)\n[PROOFSTEP]\nhave B_closed : IsClosed B := isClosed_biUnion (Finset.finite_toSet _) fun i _ => isClosed_ball\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ B)\n[PROOFSTEP]\nset s' := s \\ B\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\n[PROOFSTEP]\nhave : \u2200 x \u2208 s', \u2203 r \u2208 f x \u2229 Ioo 0 1, Disjoint B (closedBall x r) :=\n by\n intro x hx\n have xs : x \u2208 s := ((mem_diff x).1 hx).1\n rcases eq_empty_or_nonempty B with (hB | hB)\n \u00b7 rcases hf x xs 1 zero_lt_one with \u27e8r, hr, h'r\u27e9\n exact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, empty_disjoint]\u27e9\n \u00b7 let r := infDist x B\n have : 0 < min r 1 := lt_min ((B_closed.not_mem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one\n rcases hf x xs _ this with \u27e8r, hr, h'r\u27e9\n refine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.trans_le (min_le_right _ _)\u27e9\u27e9, _\u27e9\n rw [disjoint_comm]\n exact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\n\u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nhave xs : x \u2208 s := ((mem_diff x).1 hx).1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty B with (hB | hB)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nrcases hf x xs 1 zero_lt_one with \u27e8r, hr, h'r\u27e9\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nexact \u27e8r, \u27e8hr, h'r\u27e9, by simp only [hB, empty_disjoint]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : B = \u2205\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 1\n\u22a2 Disjoint B (closedBall x r)\n[PROOFSTEP]\nsimp only [hB, empty_disjoint]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nlet r := infDist x B\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nhave : 0 < min r 1 := lt_min ((B_closed.not_mem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr : \u211d := infDist x B\nthis : 0 < min r 1\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nrcases hf x xs _ this with \u27e8r, hr, h'r\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n[PROOFSTEP]\nrefine' \u27e8r, \u27e8hr, \u27e8h'r.1, h'r.2.trans_le (min_le_right _ _)\u27e9\u27e9, _\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint B (closedBall x r)\n[PROOFSTEP]\nrw [disjoint_comm]\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB\u271d : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nx : \u03b1\nhx : x \u2208 s'\nxs : x \u2208 s\nhB : Set.Nonempty B\nr\u271d : \u211d := infDist x B\nthis : 0 < min r\u271d 1\nr : \u211d\nhr : r \u2208 f x\nh'r : r \u2208 Ioo 0 (min r\u271d 1)\n\u22a2 Disjoint (closedBall x r) B\n[PROOFSTEP]\nexact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nthis : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r, r \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x r)\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\n[PROOFSTEP]\nchoose! r hr using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\n[PROOFSTEP]\nobtain \u27e8v, vs', h\u03bcv, hv\u27e9 :\n \u2203 v : Finset \u03b1,\n \u2191v \u2286 s' \u2227\n \u03bc (s' \\ \u22c3 x \u2208 v, closedBall x (r x)) \u2264 N / (N + 1) * \u03bc s' \u2227\n (v : Set \u03b1).PairwiseDisjoint fun x : \u03b1 => closedBall x (r x) :=\n haveI rI : \u2200 x \u2208 s', r x \u2208 Ioo (0 : \u211d) 1 := fun x hx => (hr x hx).1.2\n exist_finset_disjoint_balls_large_measure \u03bc h\u03c4 hN s' r (fun x hx => (rI x hx).1) fun x hx => (rI x hx).2.le\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2203 u, t \u2286 u \u2227 P u \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\n[PROOFSTEP]\nrefine' \u27e8t \u222a Finset.image (fun x => (x, r x)) v, Finset.subset_union_left _ _, \u27e8_, _, _\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint \u2191(t \u222a Finset.image (fun x => (x, r x)) v) fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nsimp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and_iff, Finset.coe_image]\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 (PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.fst p.snd) \u2227\n \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n i \u2208 \u2191t \u2192\n \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984,\n j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.fst i.snd) (closedBall j.fst j.snd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 PairwiseDisjoint ((fun x => (x, r x)) '' \u2191v) fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nintro p hp q hq hpq\n[GOAL]\ncase intro.intro.intro.refine'_1.left\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 (fun x => (x, r x)) '' \u2191v\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.fst p.snd) p q\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hp with \u27e8p', p'v, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1.left.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 q\n\u22a2 (Disjoint on fun p => closedBall p.fst p.snd) (p', r p') q\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\n\u22a2 (Disjoint on fun p => closedBall p.fst p.snd) (p', r p') (q', r q')\n[PROOFSTEP]\nrefine' hv p'v q'v fun hp'q' => _\n[GOAL]\ncase intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : (p', r p') \u2260 (q', r q')\nhp'q' : p' = (q', r q').fst\n\u22a2 False\n[PROOFSTEP]\nrw [hp'q'] at hpq \n[GOAL]\ncase intro.intro.intro.refine'_1.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 \u2191v\nhp : (p', r p') \u2208 (fun x => (x, r x)) '' \u2191v\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : ((q', r q').fst, r (q', r q').fst) \u2260 (q', r q')\nhp'q' : p' = (q', r q').fst\n\u22a2 False\n[PROOFSTEP]\nexact hpq rfl\n[GOAL]\ncase intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 \u2983i : \u03b1 \u00d7 \u211d\u2984,\n i \u2208 \u2191t \u2192\n \u2200 \u2983j : \u03b1 \u00d7 \u211d\u2984, j \u2208 (fun x => (x, r x)) '' \u2191v \u2192 i \u2260 j \u2192 Disjoint (closedBall i.fst i.snd) (closedBall j.fst j.snd)\n[PROOFSTEP]\nintro p hp q hq hpq\n[GOAL]\ncase intro.intro.intro.refine'_1.right\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq : \u03b1 \u00d7 \u211d\nhq : q \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 q\n\u22a2 Disjoint (closedBall p.fst p.snd) (closedBall q.fst q.snd)\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hq with \u27e8q', q'v, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1.right.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 Disjoint (closedBall p.fst p.snd) (closedBall (q', r q').fst (q', r q').snd)\n[PROOFSTEP]\napply disjoint_of_subset_left _ (hr q' (vs' q'v)).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.fst p.snd \u2286 B\n[PROOFSTEP]\nrw [hB, \u2190 Finset.set_biUnion_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u2191t\nq' : \u03b1\nq'v : q' \u2208 \u2191v\nhq : (q', r q') \u2208 (fun x => (x, r x)) '' \u2191v\nhpq : p \u2260 (q', r q')\n\u22a2 closedBall p.fst p.snd \u2286 \u22c3 (x : \u03b1 \u00d7 \u211d) (_ : x \u2208 \u2191t), closedBall x.fst x.snd\n[PROOFSTEP]\nexact subset_biUnion_of_mem (u := fun x : \u03b1 \u00d7 \u211d => closedBall x.1 x.2) hp\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.fst \u2208 s\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.fst \u2208 s\n[PROOFSTEP]\nrcases Finset.mem_union.1 hp with (h'p | h'p)\n[GOAL]\ncase intro.intro.intro.refine'_2.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.fst \u2208 s\n[PROOFSTEP]\nexact ht.2.1 p h'p\n[GOAL]\ncase intro.intro.intro.refine'_2.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.fst \u2208 s\n[PROOFSTEP]\nrcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_2.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').fst \u2208 s\n[PROOFSTEP]\nexact ((mem_diff _).1 (vs' (Finset.mem_coe.2 p'v))).1\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u222a Finset.image (fun x => (x, r x)) v \u2192 p.snd \u2208 f p.fst\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\n\u22a2 p.snd \u2208 f p.fst\n[PROOFSTEP]\nrcases Finset.mem_union.1 hp with (h'p | h'p)\n[GOAL]\ncase intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 t\n\u22a2 p.snd \u2208 f p.fst\n[PROOFSTEP]\nexact ht.2.2 p h'p\n[GOAL]\ncase intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : p \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 p.snd \u2208 f p.fst\n[PROOFSTEP]\nrcases Finset.mem_image.1 h'p with \u27e8p', p'v, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_3.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\np' : \u03b1\np'v : p' \u2208 v\nhp : (p', r p') \u2208 t \u222a Finset.image (fun x => (x, r x)) v\nh'p : (p', r p') \u2208 Finset.image (fun x => (x, r x)) v\n\u22a2 (p', r p').snd \u2208 f (p', r p').fst\n[PROOFSTEP]\nexact (hr p' (vs' p'v)).1.1\n[GOAL]\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\n[PROOFSTEP]\nconvert h\u03bcv using 2\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt : Finset (\u03b1 \u00d7 \u211d)\nht : P t\nB : Set \u03b1 := \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nhB : B = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd\nB_closed : IsClosed B\ns' : Set \u03b1 := s \\ B\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 s' \u2192 r x \u2208 f x \u2229 Ioo 0 1 \u2227 Disjoint B (closedBall x (r x))\nv : Finset \u03b1\nvs' : \u2191v \u2286 s'\nh\u03bcv : \u2191\u2191\u03bc (s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)) \u2264 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc s'\nhv : PairwiseDisjoint \u2191v fun x => closedBall x (r x)\n\u22a2 s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t \u222a Finset.image (fun x => (x, r x)) v), closedBall p.fst p.snd =\n s' \\ \u22c3 (x : \u03b1) (_ : x \u2208 v), closedBall x (r x)\n[PROOFSTEP]\nrw [Finset.set_biUnion_union, \u2190 diff_diff, Finset.set_biUnion_finset_image]\n /- Define `F` associating to a finite good covering the above enlarged good covering, covering\n a proportion `1/(N+1)` of leftover points. Iterating `F`, one will get larger and larger good\n coverings, missing in the end only a measure-zero set. -/\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nthis :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n \u2203 u,\n t \u2286 u \u2227\n P u \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nchoose! F hF using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nlet u n := F^[n] \u2205\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nhave u_succ : \u2200 n : \u2115, u n.succ = F (u n) := fun n => by simp only [Function.comp_apply, Function.iterate_succ']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nn : \u2115\n\u22a2 u (Nat.succ n) = F (u n)\n[PROOFSTEP]\nsimp only [Function.comp_apply, Function.iterate_succ']\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nhave Pu : \u2200 n, P (u n) := by\n intro n\n induction' n with n IH\n \u00b7 simp only [Prod.forall, id.def, Function.iterate_zero, Nat.zero_eq]\n simp only [Finset.not_mem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall\u2082_true_iff, and_self_iff,\n pairwiseDisjoint_empty]\n \u00b7 rw [u_succ]\n exact (hF (u n) IH).2.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 \u2200 (n : \u2115), P (u n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\n\u22a2 P (u n)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 P (u Nat.zero)\n[PROOFSTEP]\nsimp only [Prod.forall, id.def, Function.iterate_zero, Nat.zero_eq]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\n\u22a2 (PairwiseDisjoint \u2191\u2205 fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 a \u2208 s) \u2227 \u2200 (a : \u03b1) (b : \u211d), (a, b) \u2208 \u2205 \u2192 b \u2208 f a\n[PROOFSTEP]\nsimp only [Finset.not_mem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall\u2082_true_iff, and_self_iff,\n pairwiseDisjoint_empty]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (u (Nat.succ n))\n[PROOFSTEP]\nrw [u_succ]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nn : \u2115\nIH : P (u n)\n\u22a2 P (F (u n))\n[PROOFSTEP]\nexact (hF (u n) IH).2.1\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nrefine' \u27e8\u22c3 n, u n, countable_iUnion fun n => (u n).countable_toSet, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 (n : \u2115), \u2191(u n) \u2192 p.fst \u2208 s\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 (n : \u2115), \u2191(u n)\n\u22a2 p.fst \u2208 s\n[PROOFSTEP]\nrcases mem_iUnion.1 hp with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 (n : \u2115), \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.fst \u2208 s\n[PROOFSTEP]\nexact (Pu n).2.1 p (Finset.mem_coe.1 hn)\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 \u22c3 (n : \u2115), \u2191(u n) \u2192 p.snd \u2208 f p.fst\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 (n : \u2115), \u2191(u n)\n\u22a2 p.snd \u2208 f p.fst\n[PROOFSTEP]\nrcases mem_iUnion.1 hp with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_2.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\np : \u03b1 \u00d7 \u211d\nhp : p \u2208 \u22c3 (n : \u2115), \u2191(u n)\nn : \u2115\nhn : p \u2208 \u2191(u n)\n\u22a2 p.snd \u2208 f p.fst\n[PROOFSTEP]\nexact (Pu n).2.2 p (Finset.mem_coe.1 hn)\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\nhave A :\n \u2200 n,\n \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 n : \u2115, (u n : Set (\u03b1 \u00d7 \u211d))), closedBall p.fst p.snd) \u2264\n \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) :=\n by\n intro n\n apply measure_mono\n apply diff_subset_diff (Subset.refl _)\n exact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd \u2286\n s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd\n[PROOFSTEP]\napply diff_subset_diff (Subset.refl _)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd \u2286\n \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd\n[PROOFSTEP]\nexact biUnion_subset_biUnion_left (subset_iUnion (fun i => (u i : Set (\u03b1 \u00d7 \u211d))) n)\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\nhave B : \u2200 n, \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n * \u03bc s :=\n by\n intro n\n induction' n with n IH\n \u00b7\n simp only [le_refl, diff_empty, one_mul, iUnion_false, iUnion_empty, pow_zero, Nat.zero_eq, Function.iterate_zero,\n id.def, Finset.not_mem_empty]\n calc\n \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n.succ), closedBall p.fst p.snd) \u2264\n N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) :=\n by rw [u_succ]; exact (hF (u n) (Pu n)).2.2\n _ \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n.succ * \u03bc s := by rw [pow_succ, mul_assoc]; exact mul_le_mul_left' IH _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n\u22a2 \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u Nat.zero), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.zero * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [le_refl, diff_empty, one_mul, iUnion_false, iUnion_empty, pow_zero, Nat.zero_eq, Function.iterate_zero,\n id.def, Finset.not_mem_empty]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u (Nat.succ n)), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s\n[PROOFSTEP]\ncalc\n \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n.succ), closedBall p.fst p.snd) \u2264\n N / (N + 1) * \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) :=\n by rw [u_succ]; exact (hF (u n) (Pu n)).2.2\n _ \u2264 (N / (N + 1) : \u211d\u22650\u221e) ^ n.succ * \u03bc s := by rw [pow_succ, mul_assoc]; exact mul_le_mul_left' IH _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u (Nat.succ n)), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n[PROOFSTEP]\nrw [u_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F (u n)), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\n[PROOFSTEP]\nexact (hF (u n) (Pu n)).2.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ Nat.succ n * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [pow_succ, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nn : \u2115\nIH : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * ((\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s)\n[PROOFSTEP]\nexact mul_le_mul_left' IH _\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\nhave C : Tendsto (fun n : \u2115 => ((N : \u211d\u22650\u221e) / (N + 1)) ^ n * \u03bc s) atTop (\ud835\udcdd (0 * \u03bc s)) :=\n by\n apply ENNReal.Tendsto.mul_const _ (Or.inr (measure_lt_top \u03bc s).ne)\n apply ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1\n rw [ENNReal.div_lt_iff, one_mul]\n \u00b7 conv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n exact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one\n \u00b7 simp only [true_or_iff, add_eq_zero_iff, Ne.def, not_false_iff, one_ne_zero, and_false_iff]\n \u00b7 simp only [ENNReal.nat_ne_top, Ne.def, not_false_iff, or_true_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\napply ENNReal.Tendsto.mul_const _ (Or.inr (measure_lt_top \u03bc s).ne)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 Tendsto (fun x => (\u2191N / (\u2191N + 1)) ^ x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1\n[GOAL]\ncase hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N / (\u2191N + 1) < 1\n[PROOFSTEP]\nrw [ENNReal.div_lt_iff, one_mul]\n[GOAL]\ncase hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N < \u2191N + 1\n[PROOFSTEP]\nconv_lhs => rw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n| \u2191N\n[PROOFSTEP]\nrw [\u2190 add_zero (N : \u211d\u22650\u221e)]\n[GOAL]\ncase hr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 0 < \u2191N + 1\n[PROOFSTEP]\nexact ENNReal.add_lt_add_left (ENNReal.nat_ne_top N) zero_lt_one\n[GOAL]\ncase hr.h0\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 0 \u2228 \u2191N \u2260 0\n[PROOFSTEP]\nsimp only [true_or_iff, add_eq_zero_iff, Ne.def, not_false_iff, one_ne_zero, and_false_iff]\n[GOAL]\ncase hr.ht\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\n\u22a2 \u2191N + 1 \u2260 \u22a4 \u2228 \u2191N \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.nat_ne_top, Ne.def, not_false_iff, or_true_iff]\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd (0 * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\nrw [zero_mul] at C \n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\napply le_bot_iff.1\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nA :\n \u2200 (n : \u2115),\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd)\nB : \u2200 (n : \u2115), \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 u n), closedBall p.fst p.snd) \u2264 (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s\nC : Tendsto (fun n => (\u2191N / (\u2191N + 1)) ^ n * \u2191\u2191\u03bc s) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 \u22c3 (n : \u2115), \u2191(u n)), closedBall p.fst p.snd) \u2264 \u22a5\n[PROOFSTEP]\nexact le_of_tendsto_of_tendsto' tendsto_const_nhds C fun n => (A n).trans (B n)\n[GOAL]\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 PairwiseDisjoint (\u22c3 (n : \u2115), \u2191(u n)) fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nrefine' (pairwiseDisjoint_iUnion _).2 fun n => (Pu n).1\n[GOAL]\ncase intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => \u2191(u n)\n[PROOFSTEP]\napply (monotone_nat_of_le_succ fun n => ?_).directed_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(u (n + 1))\n[PROOFSTEP]\nrw [\u2190 Nat.succ_eq_add_one, u_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nhN : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nP : Finset (\u03b1 \u00d7 \u211d) \u2192 Prop :=\n fun t =>\n (PairwiseDisjoint \u2191t fun p => closedBall p.fst p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227 \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nF : Finset (\u03b1 \u00d7 \u211d) \u2192 Finset (\u03b1 \u00d7 \u211d)\nhF :\n \u2200 (t : Finset (\u03b1 \u00d7 \u211d)),\n P t \u2192\n t \u2286 F t \u2227\n P (F t) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 F t), closedBall p.fst p.snd) \u2264\n \u2191N / (\u2191N + 1) * \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd)\nu : \u2115 \u2192 Finset (\u03b1 \u00d7 \u211d) := fun n => F^[n] \u2205\nu_succ : \u2200 (n : \u2115), u (Nat.succ n) = F (u n)\nPu : \u2200 (n : \u2115), P (u n)\nn : \u2115\n\u22a2 \u2191(u n) \u2264 \u2191(F (u n))\n[PROOFSTEP]\nexact (hF (u n) (Pu n)).1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nrcases exists_absolutelyContinuous_isFiniteMeasure \u03bc with \u27e8\u03bd, h\u03bd, h\u03bc\u03bd\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nrcases exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux \u03bd f s hf with \u27e8t, t_count, ts, tr, t\u03bd, tdisj\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u03bd : Measure \u03b1\nh\u03bd : IsFiniteMeasure \u03bd\nh\u03bc\u03bd : \u03bc \u226a \u03bd\nt : Set (\u03b1 \u00d7 \u211d)\nt_count : Set.Countable t\nts : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s\ntr : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst\nt\u03bd : \u2191\u2191\u03bd (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0\ntdisj : PairwiseDisjoint t fun p => closedBall p.fst p.snd\n\u22a2 \u2203 t,\n Set.Countable t \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (\u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.fst p.snd) = 0 \u2227\n PairwiseDisjoint t fun p => closedBall p.fst p.snd\n[PROOFSTEP]\nexact \u27e8t, t_count, ts, tr, h\u03bc\u03bd t\u03bd, tdisj\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nlet g x := f x \u2229 Ioo 0 (R x)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave hg : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 Ioo 0 \u03b4).Nonempty :=\n by\n intro x hx \u03b4 \u03b4pos\n rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n exact \u27e8r, \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9, \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nintro x hx \u03b4 \u03b4pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nrcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nhr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x))\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nexact \u27e8r, \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9, \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nrcases exists_disjoint_closedBall_covering_ae_aux \u03bc g s hg with \u27e8v, v_count, vs, vg, \u03bcv, v_disj\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nlet t := Prod.fst '' v\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave : \u2200 x \u2208 t, \u2203 r : \u211d, (x, r) \u2208 v := by\n intro x hx\n rcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n exact \u27e8q, hp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 r, (x, r) \u2208 v\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 \u2203 r, ((p, q).fst, r) \u2208 v\n[PROOFSTEP]\nexact \u27e8q, hp\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nthis : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nchoose! r hr using this\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave im_t : (fun x => (x, r x)) '' t = v :=\n by\n have I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).2.1.le\n apply Subset.antisymm\n \u00b7 simp only [image_subset_iff]\n rintro \u27e8x, p\u27e9 hxp\n simp only [mem_preimage]\n exact hr _ (mem_image_of_mem _ hxp)\n \u00b7 rintro \u27e8x, p\u27e9 hxp\n have hxrx : (x, r x) \u2208 v := hr _ (mem_image_of_mem _ hxp)\n have : p = r x := by\n by_contra h\n have A : (x, p) \u2260 (x, r x) := by simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n have H := v_disj hxp hxrx A\n contrapose H\n rw [not_disjoint_iff_nonempty_inter]\n refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9\n rw [this]\n apply mem_image_of_mem\n exact mem_image_of_mem _ hxp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\n\u22a2 (fun x => (x, r x)) '' t = v\n[PROOFSTEP]\nhave I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).2.1.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t = v\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t \u2286 v\n[PROOFSTEP]\nsimp only [image_subset_iff]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)\n[PROOFSTEP]\nrintro \u27e8x, p\u27e9 hxp\n[GOAL]\ncase h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)\n[PROOFSTEP]\nsimp only [mem_preimage]\n[GOAL]\ncase h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, r x) \u2208 v\n[PROOFSTEP]\nexact hr _ (mem_image_of_mem _ hxp)\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 (fun x => (x, r x)) '' t\n[PROOFSTEP]\nrintro \u27e8x, p\u27e9 hxp\n[GOAL]\ncase h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t\n[PROOFSTEP]\nhave hxrx : (x, r x) \u2208 v := hr _ (mem_image_of_mem _ hxp)\n[GOAL]\ncase h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t\n[PROOFSTEP]\nhave : p = r x := by\n by_contra h\n have A : (x, p) \u2260 (x, r x) := by simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n have H := v_disj hxp hxrx A\n contrapose H\n rw [not_disjoint_iff_nonempty_inter]\n refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 p = r x\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 False\n[PROOFSTEP]\nhave A : (x, p) \u2260 (x, r x) := by simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 (x, p) \u2260 (x, r x)\n[PROOFSTEP]\nsimpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\n\u22a2 False\n[PROOFSTEP]\nhave H := v_disj hxp hxrx A\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : (Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)\n\u22a2 False\n[PROOFSTEP]\ncontrapose H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 \u00ac(Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)\n[PROOFSTEP]\nrw [not_disjoint_iff_nonempty_inter]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 Set.Nonempty ((fun p => closedBall p.fst p.snd) (x, p) \u2229 (fun p => closedBall p.fst p.snd) (x, r x))\n[PROOFSTEP]\nrefine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 x \u2208 (fun p => closedBall p.fst p.snd) (x, p) \u2229 (fun p => closedBall p.fst p.snd) (x, r x)\n[PROOFSTEP]\nsimp (config := { proj := false }) [I _ hxp, I _ hxrx]\n[GOAL]\ncase h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t\n[PROOFSTEP]\napply mem_image_of_mem\n[GOAL]\ncase h\u2082.mk.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact mem_image_of_mem _ hxp\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nrefine' \u27e8t, r, v_count.image _, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 t \u2286 s\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 (p, q).fst \u2208 s\n[PROOFSTEP]\nexact vs _ hp\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 r x \u2208 f x \u2229 Ioo 0 (R x)\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nleft\u271d : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 r (p, q).fst \u2208 f (p, q).fst \u2229 Ioo 0 (R (p, q).fst)\n[PROOFSTEP]\nexact vg _ (hr _ hx)\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\n[PROOFSTEP]\nhave :\n \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.1 p.2 := by\n conv_rhs => rw [biUnion_image]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n[PROOFSTEP]\nconv_rhs => rw [biUnion_image]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n| \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n[PROOFSTEP]\nrw [biUnion_image]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n| \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n[PROOFSTEP]\nrw [biUnion_image]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n| \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n[PROOFSTEP]\nrw [biUnion_image]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\n[PROOFSTEP]\nrw [this, im_t]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\n[PROOFSTEP]\nexact \u03bcv\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nhave A : InjOn (fun x : \u03b1 => (x, r x)) t := by\n simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 InjOn (fun x => (x, r x)) t\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff, eq_self_iff_true]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nA : InjOn (fun x => (x, r x)) t\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)\n[PROOFSTEP]\nrwa [\u2190 im_t, A.pairwiseDisjoint_image] at v_disj \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nobtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 U, U \u2287 s \u2227 IsOpen U \u2227 \u03bc U \u2264 \u03bc s + \u03b5 / 2 :=\n Set.exists_isOpen_le_add _ _\n (by simpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\n\u22a2 \u03b5 / 2 \u2260 0\n[PROOFSTEP]\nsimpa only [or_false_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top] using h\u03b5\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave : \u2200 x \u2208 s, \u2203 R > 0, ball x R \u2286 u := fun x hx => Metric.mem_nhds_iff.1 (u_open.mem_nhds (su hx))\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 R, R > 0 \u2227 ball x R \u2286 u\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nchoose! R hR using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nobtain \u27e8t0, r0, t0_count, t0s, hr0, \u03bct0, t0_disj\u27e9 :\n \u2203 (t0 : Set \u03b1) (r0 : \u03b1 \u2192 \u211d),\n t0.Countable \u2227\n t0 \u2286 s \u2227\n (\u2200 x \u2208 t0, r0 x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n \u03bc (s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)) = 0 \u2227 t0.PairwiseDisjoint fun x => closedBall x (r0 x) :=\n exists_disjoint_closedBall_covering_ae \u03bc f s hf R fun x hx =>\n (hR x hx).1\n -- we have constructed an almost everywhere covering of `s` by disjoint balls. Let `s'` be the\n -- remaining set.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nlet s' := s \\ \u22c3 x \u2208 t0, closedBall x (r0 x)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave s's : s' \u2286 s := diff_subset _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nobtain \u27e8N, \u03c4, h\u03c4, H\u27e9 : \u2203 N \u03c4, 1 < \u03c4 \u2227 IsEmpty (Besicovitch.SatelliteConfig \u03b1 N \u03c4) :=\n HasBesicovitchCovering.no_satelliteConfig\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nobtain \u27e8v, s'v, v_open, \u03bcv\u27e9 : \u2203 v, v \u2287 s' \u2227 IsOpen v \u2227 \u03bc v \u2264 \u03bc s' + \u03b5 / 2 / N :=\n Set.exists_isOpen_le_add _ _\n (by\n simp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top,\n not_false_iff, and_false_iff, false_and_iff, or_self_iff])\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\n\u22a2 \u03b5 / 2 / \u2191N \u2260 0\n[PROOFSTEP]\nsimp only [h\u03b5, ENNReal.nat_ne_top, WithTop.mul_eq_top_iff, Ne.def, ENNReal.div_eq_zero_iff, ENNReal.one_ne_top,\n not_false_iff, and_false_iff, false_and_iff, or_self_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave : \u2200 x \u2208 s', \u2203 r1 \u2208 f x \u2229 Ioo (0 : \u211d) 1, closedBall x r1 \u2286 v :=\n by\n intro x hx\n rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9\n rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9\n exact\n \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\n\u22a2 \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with \u27e8r, rpos, hr\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n[PROOFSTEP]\nrcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with \u27e8R', hR'\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nx : \u03b1\nhx : x \u2208 s'\nr : \u211d\nrpos : r > 0\nhr : ball x r \u2286 v\nR' : \u211d\nhR' : R' \u2208 f x \u2229 Ioo 0 (min r 1)\n\u22a2 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n[PROOFSTEP]\nexact\n \u27e8R', \u27e8hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)\u27e9,\n Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nthis : \u2200 (x : \u03b1), x \u2208 s' \u2192 \u2203 r1, r1 \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x r1 \u2286 v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nchoose! r1 hr1 using this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nlet q : BallPackage s' \u03b1 :=\n { c := fun x => x\n r := fun x => r1 x\n rpos := fun x => (hr1 x.1 x.2).1.2.1\n r_bound := 1\n r_le := fun x => (hr1 x.1 x.2).1.2.2.le }\n -- by Besicovitch, we cover `s'` with at most `N` families of disjoint balls, all included in\n -- a suitable neighborhood `v` of `s'`.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nobtain \u27e8S, S_disj, hS\u27e9 :\n \u2203 S : Fin N \u2192 Set s',\n (\u2200 i : Fin N, (S i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) \u2227\n range q.c \u2286 \u22c3 i : Fin N, \u22c3 j \u2208 S i, ball (q.c j) (q.r j) :=\n exist_disjoint_covering_families h\u03c4 H q\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave S_count : \u2200 i, (S i).Countable := by\n intro i\n apply (S_disj i).countable_of_nonempty_interior fun j _ => ?_\n have : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _)\n exact this.mono ball_subset_interior_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\n\u22a2 \u2200 (i : Fin N), Set.Countable (S i)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\n\u22a2 Set.Countable (S i)\n[PROOFSTEP]\napply (S_disj i).countable_of_nonempty_interior fun j _ => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))\n[PROOFSTEP]\nhave : (ball (j : \u03b1) (r1 j)).Nonempty := nonempty_ball.2 (q.rpos _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\ni : Fin N\nj : \u2191s'\nx\u271d : j \u2208 S i\nthis : Set.Nonempty (ball (\u2191j) (r1 \u2191j))\n\u22a2 Set.Nonempty (interior (closedBall (BallPackage.c q j) (BallPackage.r q j)))\n[PROOFSTEP]\nexact this.mono ball_subset_interior_closedBall\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nlet r x := if x \u2208 s' then r1 x else r0 x\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave r_t0 : \u2200 x \u2208 t0, r x = r0 x := by\n intro x hx\n have : \u00acx \u2208 s' :=\n by\n simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le, mem_diff, not_forall]\n intro _\n refine' \u27e8x, hx, _\u27e9\n rw [dist_self]\n exact (hr0 x hx).2.1.le\n simp only [if_neg this]\n -- the desired covering set is given by the union of the families constructed in the first and\n -- second steps.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x = r0 x\n[PROOFSTEP]\nhave : \u00acx \u2208 s' :=\n by\n simp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le, mem_diff, not_forall]\n intro _\n refine' \u27e8x, hx, _\u27e9\n rw [dist_self]\n exact (hr0 x hx).2.1.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 \u00acx \u2208 s'\n[PROOFSTEP]\nsimp only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_lt, not_le, mem_diff, not_forall]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 x \u2208 s \u2192 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1\n[PROOFSTEP]\nintro _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 \u2203 x_1, x_1 \u2208 t0 \u2227 dist x x_1 \u2264 r0 x_1\n[PROOFSTEP]\nrefine' \u27e8x, hx, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 dist x x \u2264 r0 x\n[PROOFSTEP]\nrw [dist_self]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\na\u271d : x \u2208 s\n\u22a2 0 \u2264 r0 x\n[PROOFSTEP]\nexact (hr0 x hx).2.1.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nx : \u03b1\nhx : x \u2208 t0\nthis : \u00acx \u2208 s'\n\u22a2 r x = r0 x\n[PROOFSTEP]\nsimp only [if_neg this]\n -- the desired covering set is given by the union of the families constructed in the first and\n -- second steps.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2203 t r,\n Set.Countable t \u2227\n t \u2286 s \u2227\n (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x) \u2227\n s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x) \u2227 \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nrefine'\n \u27e8t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, r, _, _, _, _, _\u27e9\n -- it remains to check that they have the desired properties\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 Set.Countable (t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i)\n[PROOFSTEP]\nexact t0_count.union (countable_iUnion fun i => (S_count i).image _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i \u2286 s\n[PROOFSTEP]\nsimp only [t0s, true_and_iff, union_subset_iff, image_subset_iff, iUnion_subset_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (i : Fin N), S i \u2286 (fun a => \u2191a) \u207b\u00b9' s\n[PROOFSTEP]\nintro i x _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\nx : { x // x \u2208 s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x) }\na\u271d : x \u2208 S i\n\u22a2 x \u2208 (fun a => \u2191a) \u207b\u00b9' s\n[PROOFSTEP]\nexact s's x.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i \u2192 r x \u2208 f x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n rw [r_t0 x hx]\n exact (hr0 _ hx).1\n|\n inr hx =>\n have h'x : x \u2208 s' := by\n simp only [mem_iUnion, mem_image] at hx \n rcases hx with \u27e8i, y, _, rfl\u27e9\n exact y.2\n simp only [if_pos h'x, (hr1 x h'x).1.1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\ncases hx with\n| inl hx =>\n rw [r_t0 x hx]\n exact (hr0 _ hx).1\n|\n inr hx =>\n have h'x : x \u2208 s' := by\n simp only [mem_iUnion, mem_image] at hx \n rcases hx with \u27e8i, y, _, rfl\u27e9\n exact y.2\n simp only [if_pos h'x, (hr1 x h'x).1.1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\n\n| inl hx =>\n rw [r_t0 x hx]\n exact (hr0 _ hx).1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\nrw [r_t0 x hx]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 r0 x \u2208 f x\n[PROOFSTEP]\nexact (hr0 _ hx).1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 (i : Fin N), Subtype.val '' S i\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\n\n|\n inr hx =>\n have h'x : x \u2208 s' := by\n simp only [mem_iUnion, mem_image] at hx \n rcases hx with \u27e8i, y, _, rfl\u27e9\n exact y.2\n simp only [if_pos h'x, (hr1 x h'x).1.1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 (i : Fin N), Subtype.val '' S i\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\nhave h'x : x \u2208 s' := by\n simp only [mem_iUnion, mem_image] at hx \n rcases hx with \u27e8i, y, _, rfl\u27e9\n exact y.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 (i : Fin N), Subtype.val '' S i\n\u22a2 x \u2208 s'\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_image] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : \u2203 i x_1, x_1 \u2208 S i \u2227 \u2191x_1 = x\n\u22a2 x \u2208 s'\n[PROOFSTEP]\nrcases hx with \u27e8i, y, _, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\ni : Fin N\ny : { x // x \u2208 s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x) }\nleft\u271d : y \u2208 S i\n\u22a2 \u2191y \u2208 s'\n[PROOFSTEP]\nexact y.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 \u22c3 (i : Fin N), Subtype.val '' S i\nh'x : x \u2208 s'\n\u22a2 r x \u2208 f x\n[PROOFSTEP]\nsimp only [if_pos h'x, (hr1 x h'x).1.1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 s \u2286 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nby_cases h'x : x \u2208 s'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nobtain \u27e8i, y, ySi, xy\u27e9 : \u2203 (i : Fin N) (y : \u21a5s'), y \u2208 S i \u2227 x \u2208 ball (y : \u03b1) (r1 y) :=\n by\n have A : x \u2208 range q.c := by\n simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, mem_setOf_eq,\n Subtype.range_coe_subtype, mem_diff] using h'x\n simpa only [mem_iUnion, mem_image, bex_def] using hS A\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y)\n[PROOFSTEP]\nhave A : x \u2208 range q.c := by\n simpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, mem_setOf_eq,\n Subtype.range_coe_subtype, mem_diff] using h'x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\n\u22a2 x \u2208 range q.c\n[PROOFSTEP]\nsimpa only [not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le, mem_setOf_eq,\n Subtype.range_coe_subtype, mem_diff] using h'x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\nA : x \u2208 range q.c\n\u22a2 \u2203 i y, y \u2208 S i \u2227 x \u2208 ball (\u2191y) (r1 \u2191y)\n[PROOFSTEP]\nsimpa only [mem_iUnion, mem_image, bex_def] using hS A\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nrefine' mem_iUnion\u2082.2 \u27e8y, Or.inr _, _\u27e9\n[GOAL]\ncase pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2191y \u2208 \u22c3 (i : Fin N), Subtype.val '' S i\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_image]\n[GOAL]\ncase pos.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 \u2203 i x, x \u2208 S i \u2227 \u2191x = \u2191y\n[PROOFSTEP]\nexact \u27e8i, y, ySi, rfl\u27e9\n[GOAL]\ncase pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)\n[PROOFSTEP]\nhave : (y : \u03b1) \u2208 s' := y.2\n[GOAL]\ncase pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r \u2191y)\n[PROOFSTEP]\nsimp only [if_pos this]\n[GOAL]\ncase pos.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : x \u2208 s'\ni : Fin N\ny : \u2191s'\nySi : y \u2208 S i\nxy : x \u2208 ball (\u2191y) (r1 \u2191y)\nthis : \u2191y \u2208 s'\n\u22a2 x \u2208 closedBall (\u2191y) (r1 \u2191y)\n[PROOFSTEP]\nexact ball_subset_closedBall xy\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nobtain \u27e8y, yt0, hxy\u27e9 : \u2203 y : \u03b1, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y) := by simpa [hx, -mem_closedBall] using h'x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\n\u22a2 \u2203 y, y \u2208 t0 \u2227 x \u2208 closedBall y (r0 y)\n[PROOFSTEP]\nsimpa [hx, -mem_closedBall] using h'x\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i), closedBall x (r x)\n[PROOFSTEP]\nrefine' mem_iUnion\u2082.2 \u27e8y, Or.inl yt0, _\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 s\nh'x : \u00acx \u2208 s'\ny : \u03b1\nyt0 : y \u2208 t0\nhxy : x \u2208 closedBall y (r0 y)\n\u22a2 x \u2208 closedBall y (r y)\n[PROOFSTEP]\nrwa [r_t0 _ yt0]\n -- the only nontrivial property is the measure control, which we check now\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave A : (\u2211' x : t0, \u03bc (closedBall x (r x))) \u2264 \u03bc s + \u03b5 / 2 :=\n calc\n (\u2211' x : t0, \u03bc (closedBall x (r x))) = \u2211' x : t0, \u03bc (closedBall x (r0 x)) := by congr 1; ext x; rw [r_t0 x x.2]\n _ = \u03bc (\u22c3 x : t0, closedBall x (r0 x)) :=\n by\n haveI : Encodable t0 := t0_count.toEncodable\n rw [measure_iUnion]\n \u00b7 exact (pairwise_subtype_iff_pairwise_set _ _).2 t0_disj\n \u00b7 exact fun i => measurableSet_closedBall\n _ \u2264 \u03bc u := by\n apply measure_mono\n simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n intro x hx\n apply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2\n _ \u2264 \u03bc s + \u03b5 / 2 := \u03bcu\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\next x\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u2191t0\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\nrw [r_t0 x x.2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 (x : \u2191t0), closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\nhaveI : Encodable t0 := t0_count.toEncodable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r0 \u2191x)) = \u2191\u2191\u03bc (\u22c3 (x : \u2191t0), closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191x) (r0 \u2191x))\n[PROOFSTEP]\nexact (pairwise_subtype_iff_pairwise_set _ _).2 t0_disj\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nthis : Encodable \u2191t0\n\u22a2 \u2200 (i : \u2191t0), MeasurableSet (closedBall (\u2191i) (r0 \u2191i))\n[PROOFSTEP]\nexact fun i => measurableSet_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2191\u2191\u03bc (\u22c3 (x : \u2191t0), closedBall (\u2191x) (r0 \u2191x)) \u2264 \u2191\u2191\u03bc u\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u22c3 (x : \u2191t0), closedBall (\u2191x) (r0 \u2191x) \u2286 u\n[PROOFSTEP]\nsimp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\n\u22a2 \u2200 (x : \u03b1), x \u2208 t0 \u2192 closedBall x (r0 x) \u2286 u\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nx : \u03b1\nhx : x \u2208 t0\n\u22a2 closedBall x (r0 x) \u2286 u\n[PROOFSTEP]\napply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\n\u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nhave B : \u2200 i : Fin N, (\u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x))) \u2264 \u03b5 / 2 / N := fun i =>\n calc\n (\u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x))) = \u2211' x : S i, \u03bc (closedBall x (r x)) :=\n by\n have : InjOn ((\u2191) : s' \u2192 \u03b1) (S i) := Subtype.val_injective.injOn _\n let F : S i \u2243 ((\u2191) : s' \u2192 \u03b1) '' S i := this.bijOn_image.equiv _\n exact (F.tsum_eq fun x => \u03bc (closedBall x (r x))).symm\n _ = \u2211' x : S i, \u03bc (closedBall x (r1 x)) := by congr 1; ext x; have : (x : \u03b1) \u2208 s' := x.1.2; simp only [if_pos this]\n _ = \u03bc (\u22c3 x : S i, closedBall x (r1 x)) :=\n by\n haveI : Encodable (S i) := (S_count i).toEncodable\n rw [measure_iUnion]\n \u00b7 exact (pairwise_subtype_iff_pairwise_set _ _).2 (S_disj i)\n \u00b7 exact fun i => measurableSet_closedBall\n _ \u2264 \u03bc v := by\n apply measure_mono\n simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n intro x xs' _\n exact (hr1 x xs').2\n _ \u2264 \u03b5 / 2 / N := by have : \u03bc s' = 0 := \u03bct0; rwa [this, zero_add] at \u03bcv \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nhave : InjOn ((\u2191) : s' \u2192 \u03b1) (S i) := Subtype.val_injective.injOn _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nlet F : S i \u2243 ((\u2191) : s' \u2192 \u03b1) '' S i := this.bijOn_image.equiv _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : InjOn Subtype.val (S i)\nF : \u2191(S i) \u2243 \u2191(Subtype.val '' S i) := BijOn.equiv Subtype.val (_ : BijOn Subtype.val (S i) (Subtype.val '' S i))\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))\n[PROOFSTEP]\nexact (F.tsum_eq fun x => \u03bc (closedBall x (r x))).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 (fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x))) = fun x => \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\next x\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\nhave : (x : \u03b1) \u2208 s' := x.1.2\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u2191(S i)\nthis : \u2191\u2191x \u2208 s'\n\u22a2 \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r \u2191\u2191x)) = \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\nsimp only [if_pos this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 (x : \u2191(S i)), closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\nhaveI : Encodable (S i) := (S_count i).toEncodable\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2211' (x : \u2191(S i)), \u2191\u2191\u03bc (closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) = \u2191\u2191\u03bc (\u22c3 (x : \u2191(S i)), closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\ncase hn\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 Pairwise (Disjoint on fun x => closedBall (\u2191\u2191x) (r1 \u2191\u2191x))\n[PROOFSTEP]\nexact (pairwise_subtype_iff_pairwise_set _ _).2 (S_disj i)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : Encodable \u2191(S i)\n\u22a2 \u2200 (i_1 : \u2191(S i)), MeasurableSet (closedBall (\u2191\u2191i_1) (r1 \u2191\u2191i_1))\n[PROOFSTEP]\nexact fun i => measurableSet_closedBall\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2191\u2191\u03bc (\u22c3 (x : \u2191(S i)), closedBall (\u2191\u2191x) (r1 \u2191\u2191x)) \u2264 \u2191\u2191\u03bc v\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u22c3 (x : \u2191(S i)), closedBall (\u2191\u2191x) (r1 \u2191\u2191x) \u2286 v\n[PROOFSTEP]\nsimp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2200 (x : \u03b1) (h : x \u2208 s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)),\n { val := x, property := h } \u2208 S i \u2192 closedBall x (r1 x) \u2286 v\n[PROOFSTEP]\nintro x xs' _\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nx : \u03b1\nxs' : x \u2208 s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\nh\u271d : { val := x, property := xs' } \u2208 S i\n\u22a2 closedBall x (r1 x) \u2286 v\n[PROOFSTEP]\nexact (hr1 x xs').2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\n\u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nhave : \u03bc s' = 0 := \u03bct0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\ni : Fin N\nthis : \u2191\u2191\u03bc s' = 0\n\u22a2 \u2191\u2191\u03bc v \u2264 \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nrwa [this, zero_add] at \u03bcv \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211' (x : \u2191(t0 \u222a \u22c3 (i : Fin N), Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\ncalc\n (\u2211' x : \u21a5(t0 \u222a \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i), \u03bc (closedBall x (r x))) \u2264\n (\u2211' x : t0, \u03bc (closedBall x (r x))) + \u2211' x : \u22c3 i : Fin N, ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n ENNReal.tsum_union_le (fun x => \u03bc (closedBall x (r x))) _ _\n _ \u2264 (\u2211' x : t0, \u03bc (closedBall x (r x))) + \u2211 i : Fin N, \u2211' x : ((\u2191) : s' \u2192 \u03b1) '' S i, \u03bc (closedBall x (r x)) :=\n (add_le_add le_rfl (ENNReal.tsum_iUnion_le (fun x => \u03bc (closedBall x (r x))) _))\n _ \u2264 \u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / N := by\n refine' add_le_add A _\n refine' Finset.sum_le_sum _\n intro i _\n exact B i\n _ \u2264 \u03bc s + \u03b5 / 2 + \u03b5 / 2 := by\n refine' add_le_add le_rfl _\n simp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le]\n _ = \u03bc s + \u03b5 := by rw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) +\n \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264\n \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nrefine' add_le_add A _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2211 i : Fin N, \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nrefine' Finset.sum_le_sum _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2200 (i : Fin N), i \u2208 Finset.univ \u2192 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nintro i _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\ni : Fin N\na\u271d : i \u2208 Finset.univ\n\u22a2 \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n[PROOFSTEP]\nexact B i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\nrefine' add_le_add le_rfl _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2211 i : Fin N, \u03b5 / 2 / \u2191N \u2264 \u03b5 / 2\n[PROOFSTEP]\nsimp only [Finset.card_fin, Finset.sum_const, nsmul_eq_mul, ENNReal.mul_div_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2075 : SecondCountableTopology \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : Measure.OuterRegular \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 R x > 0 \u2227 ball x (R x) \u2286 u\nt0 : Set \u03b1\nr0 : \u03b1 \u2192 \u211d\nt0_count : Set.Countable t0\nt0s : t0 \u2286 s\nhr0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r0 x \u2208 f x \u2229 Ioo 0 (R x)\n\u03bct0 : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)) = 0\nt0_disj : PairwiseDisjoint t0 fun x => closedBall x (r0 x)\ns' : Set \u03b1 := s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t0), closedBall x (r0 x)\ns's : s' \u2286 s\nN : \u2115\n\u03c4 : \u211d\nh\u03c4 : 1 < \u03c4\nH : IsEmpty (SatelliteConfig \u03b1 N \u03c4)\nv : Set \u03b1\ns'v : v \u2287 s'\nv_open : IsOpen v\n\u03bcv : \u2191\u2191\u03bc v \u2264 \u2191\u2191\u03bc s' + \u03b5 / 2 / \u2191N\nr1 : \u03b1 \u2192 \u211d\nhr1 : \u2200 (x : \u03b1), x \u2208 s' \u2192 r1 x \u2208 f x \u2229 Ioo 0 1 \u2227 closedBall x (r1 x) \u2286 v\nq : BallPackage (\u2191s') \u03b1 :=\n { c := fun x => \u2191x, r := fun x => r1 \u2191x, rpos := (_ : \u2200 (x : \u2191s'), 0 < r1 \u2191x), r_bound := 1,\n r_le := (_ : \u2200 (x : \u2191s'), r1 \u2191x \u2264 1) }\nS : Fin N \u2192 Set \u2191s'\nS_disj : \u2200 (i : Fin N), PairwiseDisjoint (S i) fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)\nhS : range q.c \u2286 \u22c3 (i : Fin N) (j : \u2191s') (_ : j \u2208 S i), ball (BallPackage.c q j) (BallPackage.r q j)\nS_count : \u2200 (i : Fin N), Set.Countable (S i)\nr : \u03b1 \u2192 \u211d := fun x => if x \u2208 s' then r1 x else r0 x\nr_t0 : \u2200 (x : \u03b1), x \u2208 t0 \u2192 r x = r0 x\nA : \u2211' (x : \u2191t0), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + \u03b5 / 2\nB : \u2200 (i : Fin N), \u2211' (x : \u2191(Subtype.val '' S i)), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03b5 / 2 / \u2191N\n\u22a2 \u2191\u2191\u03bc s + \u03b5 / 2 + \u03b5 / 2 = \u2191\u2191\u03bc s + \u03b5\n[PROOFSTEP]\nrw [add_assoc, ENNReal.add_halves]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (x : \u03b1) (a : Set \u03b1), a \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x \u2192 MeasurableSet a\n[PROOFSTEP]\nintro x y hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ny : Set \u03b1\nhy : y \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 MeasurableSet y\n[PROOFSTEP]\nobtain \u27e8r, _, rfl\u27e9 : \u2203 r : \u211d, 0 < r \u2227 closedBall x r = y := by simpa only [mem_image, mem_Ioi] using hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ny : Set \u03b1\nhy : y \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 \u2203 r, 0 < r \u2227 closedBall x r = y\n[PROOFSTEP]\nsimpa only [mem_image, mem_Ioi] using hy\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\nr : \u211d\nleft\u271d : 0 < r\nhy : closedBall x r \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 MeasurableSet (closedBall x r)\n[PROOFSTEP]\nexact isClosed_ball.measurableSet\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (x : \u03b1) (y : Set \u03b1), y \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x \u2192 Set.Nonempty (interior y)\n[PROOFSTEP]\nintro x y hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ny : Set \u03b1\nhy : y \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 Set.Nonempty (interior y)\n[PROOFSTEP]\nobtain \u27e8r, rpos, rfl\u27e9 : \u2203 r : \u211d, 0 < r \u2227 closedBall x r = y := by simpa only [mem_image, mem_Ioi] using hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ny : Set \u03b1\nhy : y \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 \u2203 r, 0 < r \u2227 closedBall x r = y\n[PROOFSTEP]\nsimpa only [mem_image, mem_Ioi] using hy\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\nr : \u211d\nrpos : 0 < r\nhy : closedBall x r \u2208 (fun x => (fun r => closedBall x r) '' Ioi 0) x\n\u22a2 Set.Nonempty (interior (closedBall x r))\n[PROOFSTEP]\nsimp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (s : Set \u03b1) (f : \u03b1 \u2192 Set (Set \u03b1)),\n (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x) \u2192\n (\u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5) \u2192\n \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nintro s f fsubset ffine\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\n\u22a2 \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nlet g : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\n\u22a2 \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nhave A : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 Ioo 0 \u03b4).Nonempty :=\n by\n intro x xs \u03b4 \u03b4pos\n obtain \u27e8t, tf, ht\u27e9 : \u2203 (t : Set \u03b1), t \u2208 f x \u2227 t \u2286 closedBall x (\u03b4 / 2) := ffine x xs (\u03b4 / 2) (half_pos \u03b4pos)\n obtain \u27e8r, rpos, rfl\u27e9 : \u2203 r : \u211d, 0 < r \u2227 closedBall x r = t := by simpa using fsubset x xs tf\n rcases le_total r (\u03b4 / 2) with (H | H)\n \u00b7 exact \u27e8r, \u27e8rpos, tf\u27e9, \u27e8rpos, H.trans_lt (half_lt_self \u03b4pos)\u27e9\u27e9\n \u00b7 have : closedBall x r = closedBall x (\u03b4 / 2) := Subset.antisymm ht (closedBall_subset_closedBall H)\n rw [this] at tf \n refine' \u27e8\u03b4 / 2, \u27e8half_pos \u03b4pos, tf\u27e9, \u27e8half_pos \u03b4pos, half_lt_self \u03b4pos\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nintro x xs \u03b4 \u03b4pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nobtain \u27e8t, tf, ht\u27e9 : \u2203 (t : Set \u03b1), t \u2208 f x \u2227 t \u2286 closedBall x (\u03b4 / 2) := ffine x xs (\u03b4 / 2) (half_pos \u03b4pos)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nt : Set \u03b1\ntf : t \u2208 f x\nht : t \u2286 closedBall x (\u03b4 / 2)\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nobtain \u27e8r, rpos, rfl\u27e9 : \u2203 r : \u211d, 0 < r \u2227 closedBall x r = t := by simpa using fsubset x xs tf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nt : Set \u03b1\ntf : t \u2208 f x\nht : t \u2286 closedBall x (\u03b4 / 2)\n\u22a2 \u2203 r, 0 < r \u2227 closedBall x r = t\n[PROOFSTEP]\nsimpa using fsubset x xs tf\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nrpos : 0 < r\ntf : closedBall x r \u2208 f x\nht : closedBall x r \u2286 closedBall x (\u03b4 / 2)\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nrcases le_total r (\u03b4 / 2) with (H | H)\n[GOAL]\ncase intro.intro.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nrpos : 0 < r\ntf : closedBall x r \u2208 f x\nht : closedBall x r \u2286 closedBall x (\u03b4 / 2)\nH : r \u2264 \u03b4 / 2\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nexact \u27e8r, \u27e8rpos, tf\u27e9, \u27e8rpos, H.trans_lt (half_lt_self \u03b4pos)\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nrpos : 0 < r\ntf : closedBall x r \u2208 f x\nht : closedBall x r \u2286 closedBall x (\u03b4 / 2)\nH : \u03b4 / 2 \u2264 r\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nhave : closedBall x r = closedBall x (\u03b4 / 2) := Subset.antisymm ht (closedBall_subset_closedBall H)\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nrpos : 0 < r\ntf : closedBall x r \u2208 f x\nht : closedBall x r \u2286 closedBall x (\u03b4 / 2)\nH : \u03b4 / 2 \u2264 r\nthis : closedBall x r = closedBall x (\u03b4 / 2)\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nrw [this] at tf \n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nx : \u03b1\nxs : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nrpos : 0 < r\ntf : closedBall x (\u03b4 / 2) \u2208 f x\nht : closedBall x r \u2286 closedBall x (\u03b4 / 2)\nH : \u03b4 / 2 \u2264 r\nthis : closedBall x r = closedBall x (\u03b4 / 2)\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n[PROOFSTEP]\nrefine' \u27e8\u03b4 / 2, \u27e8half_pos \u03b4pos, tf\u27e9, \u27e8half_pos \u03b4pos, half_lt_self \u03b4pos\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nobtain \u27e8t, r, _, ts, tg, \u03bct, tdisj\u27e9 :\n \u2203 (t : Set \u03b1) (r : \u03b1 \u2192 \u211d),\n t.Countable \u2227\n t \u2286 s \u2227\n (\u2200 x \u2208 t, r x \u2208 g x \u2229 Ioo 0 1) \u2227\n \u03bc (s \\ \u22c3 x \u2208 t, closedBall x (r x)) = 0 \u2227 t.PairwiseDisjoint fun x => closedBall x (r x) :=\n exists_disjoint_closedBall_covering_ae \u03bc g s A (fun _ => 1) fun _ _ => zero_lt_one\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\n\u22a2 \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nlet F : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n\u22a2 \u2203 t,\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.fst \u2208 s) \u2227\n (PairwiseDisjoint t fun p => p.snd) \u2227\n (\u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 t \u2192 p.snd \u2208 f p.fst) \u2227 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 t), p.snd) = 0\n[PROOFSTEP]\nrefine' \u27e8F '' t, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n\u22a2 \u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 F '' t \u2192 p.fst \u2208 s\n[PROOFSTEP]\nrintro - \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\nx : \u03b1\nhx : x \u2208 t\n\u22a2 (F x).fst \u2208 s\n[PROOFSTEP]\nexact ts hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n\u22a2 PairwiseDisjoint (F '' t) fun p => p.snd\n[PROOFSTEP]\nrintro p \u27e8x, hx, rfl\u27e9 q \u27e8y, hy, rfl\u27e9 hxy\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\nx : \u03b1\nhx : x \u2208 t\ny : \u03b1\nhy : y \u2208 t\nhxy : F x \u2260 F y\n\u22a2 (Disjoint on fun p => p.snd) (F x) (F y)\n[PROOFSTEP]\nexact tdisj hx hy (ne_of_apply_ne F hxy)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n\u22a2 \u2200 (p : \u03b1 \u00d7 Set \u03b1), p \u2208 F '' t \u2192 p.snd \u2208 f p.fst\n[PROOFSTEP]\nrintro - \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\nx : \u03b1\nhx : x \u2208 t\n\u22a2 (F x).snd \u2208 f (F x).fst\n[PROOFSTEP]\nexact (tg x hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nf : \u03b1 \u2192 Set (Set \u03b1)\nfsubset : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2286 (fun x => (fun r => closedBall x r) '' Ioi 0) x\nffine : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 a, a \u2208 f x \u2227 a \u2286 closedBall x \u03b5\ng : \u03b1 \u2192 Set \u211d := fun x => {r | 0 < r \u2227 closedBall x r \u2208 f x}\nA : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nt : Set \u03b1\nr : \u03b1 \u2192 \u211d\nleft\u271d : Set.Countable t\nts : t \u2286 s\ntg : \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 g x \u2229 Ioo 0 1\n\u03bct : \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\ntdisj : PairwiseDisjoint t fun x => closedBall x (r x)\nF : \u03b1 \u2192 \u03b1 \u00d7 Set \u03b1 := fun x => (x, closedBall x (r x))\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 F '' t), p.snd) = 0\n[PROOFSTEP]\nrwa [biUnion_image]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\n\u22a2 Tendsto (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x)\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 s \u2208 map (fun r => closedBall x r) (\ud835\udcdd[Ioi 0] 0)\n[PROOFSTEP]\nsimp only [mem_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 :\n \u2203 (\u03b5 : \u211d), \u03b5 > 0 \u2227 \u2200 a : Set \u03b1, a \u2208 (Besicovitch.vitaliFamily \u03bc).setsAt x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s :=\n (VitaliFamily.mem_filterAt_iff _).1 hs\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0\n[PROOFSTEP]\nhave : Ioc (0 : \u211d) \u03b5 \u2208 \ud835\udcdd[>] (0 : \u211d) := Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, \u03b5pos\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\n\u22a2 (fun r => closedBall x r) \u207b\u00b9' s \u2208 \ud835\udcdd[Ioi 0] 0\n[PROOFSTEP]\nfilter_upwards [this] with _ hr\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 a\u271d \u2208 (fun r => closedBall x r) \u207b\u00b9' s\n[PROOFSTEP]\napply h\u03b5\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x\n[PROOFSTEP]\nexact mem_image_of_mem _ hr.1\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nx : \u03b1\ns : Set (Set \u03b1)\nhs : s \u2208 VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (a : Set \u03b1), a \u2208 VitaliFamily.setsAt (Besicovitch.vitaliFamily \u03bc) x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s\nthis : Ioc 0 \u03b5 \u2208 \ud835\udcdd[Ioi 0] 0\na\u271d : \u211d\nhr : a\u271d \u2208 Ioc 0 \u03b5\n\u22a2 (fun r => closedBall x r) a\u271d \u2286 closedBall x \u03b5\n[PROOFSTEP]\nexact closedBall_subset_closedBall hr.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b9\u2070 : SecondCountableTopology \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : OpensMeasurableSpace \u03b1\ninst\u271d\u2077 : HasBesicovitchCovering \u03b1\ninst\u271d\u2076 : MetricSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : BorelSpace \u03b2\ninst\u271d\u00b3 : SecondCountableTopology \u03b2\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b2\n\u03c1 \u03bc : Measure \u03b2\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsLocallyFiniteMeasure \u03c1\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, Tendsto (fun r => \u2191\u2191\u03c1 (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (Measure.rnDeriv \u03c1 \u03bc x))\n[PROOFSTEP]\nfilter_upwards [VitaliFamily.ae_tendsto_rnDeriv (Besicovitch.vitaliFamily \u03bc) \u03c1] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u00b9\u2070 : SecondCountableTopology \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : OpensMeasurableSpace \u03b1\ninst\u271d\u2077 : HasBesicovitchCovering \u03b1\ninst\u271d\u2076 : MetricSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : BorelSpace \u03b2\ninst\u271d\u00b3 : SecondCountableTopology \u03b2\ninst\u271d\u00b2 : HasBesicovitchCovering \u03b2\n\u03c1 \u03bc : Measure \u03b2\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsLocallyFiniteMeasure \u03c1\nx : \u03b2\nhx : Tendsto (fun a => \u2191\u2191\u03c1 a / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x) (\ud835\udcdd (Measure.rnDeriv \u03c1 \u03bc x))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03c1 (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (Measure.rnDeriv \u03c1 \u03bc x))\n[PROOFSTEP]\nexact hx.comp (tendsto_filterAt \u03bc x)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 x))\n[PROOFSTEP]\nfilter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet (Besicovitch.vitaliFamily \u03bc) hs]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2200 (a : \u03b2),\n Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) a)\n (\ud835\udcdd (indicator s 1 a)) \u2192\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 a))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b2\nhx : Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x) (\ud835\udcdd (indicator s 1 x))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 x))\n[PROOFSTEP]\nexact hx.comp (tendsto_filterAt \u03bc x)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202Measure.restrict \u03bc s,\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nfilter_upwards [VitaliFamily.ae_tendsto_measure_inter_div (Besicovitch.vitaliFamily \u03bc) s] with x hx using\n hx.comp (tendsto_filterAt \u03bc x)\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Covering.Besicovitch", "llama_tokens": 375726, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.28938926673008236}} {"text": "[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : MeasurableSMul M \u03b1\nc : M\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure M \u03b1 \u03bc\n\u22a2 map (fun x => c \u2022 x) \u03bc = \u03bc\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : MeasurableSMul M \u03b1\nc : M\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure M \u03b1 \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map (fun x => c \u2022 x) \u03bc) s = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [map_apply (measurable_const_smul c) hs]\n[GOAL]\ncase h\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : SMul M \u03b1\ninst\u271d\u00b9 : MeasurableSMul M \u03b1\nc : M\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure M \u03b1 \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ((fun x x_1 => x \u2022 x_1) c \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact SMulInvariantMeasure.measure_preimage_smul c hs\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 1 \u2194 2\n[GOAL]\ncase tfae_1_iff_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact \u27e8fun h => h.1, fun h => \u27e8h\u27e9\u27e9\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 1 \u2192 6\n[GOAL]\ncase tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n[PROOFSTEP]\nintro h c\n[GOAL]\ncase tfae_1_to_6\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nh : SMulInvariantMeasure G \u03b1 \u03bc\nc : G\n\u22a2 map (fun x => c \u2022 x) \u03bc = \u03bc\n[PROOFSTEP]\nexact (measurePreserving_smul c \u03bc).map_eq\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 6 \u2192 7\n[GOAL]\ncase tfae_6_to_7\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\n\u22a2 (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n[PROOFSTEP]\nexact fun H c => \u27e8measurable_const_smul c, H c\u27e9\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 7 \u2192 4\n[GOAL]\ncase tfae_7_to_4\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\n\u22a2 (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact fun H c => (H c).measure_preimage_emb (measurableEmbedding_const_smul c)\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 4 \u2192 5\n[GOAL]\ncase tfae_4_to_5\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact fun H c s => by\n rw [\u2190 preimage_smul_inv]\n apply H\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 preimage_smul_inv]\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\n\u22a2 \u2191\u2191\u03bc ((fun x => c\u207b\u00b9 \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\napply H\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 5 \u2192 3\n[GOAL]\ncase tfae_5_to_3\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact fun H c s _ => H c s\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_have 3 \u2192 2\n[GOAL]\ncase tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\n\u22a2 (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nintro H c s hs\n[GOAL]\ncase tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [preimage_smul]\n[GOAL]\ncase tfae_3_to_2\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nH : \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\nc : G\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc (c\u207b\u00b9 \u2022 s) = \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact H c\u207b\u00b9 s hs\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ntfae_1_iff_2 :\n SMulInvariantMeasure G \u03b1 \u03bc \u2194 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_1_to_6 : SMulInvariantMeasure G \u03b1 \u03bc \u2192 \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc\ntfae_6_to_7 : (\u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc) \u2192 \u2200 (c : G), MeasurePreserving fun x => c \u2022 x\ntfae_7_to_4 :\n (\u2200 (c : G), MeasurePreserving fun x => c \u2022 x) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\ntfae_4_to_5 : (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_5_to_3 :\n (\u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192 \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s\ntfae_3_to_2 :\n (\u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s) \u2192\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s\n\u22a2 List.TFAE\n [SMulInvariantMeasure G \u03b1 \u03bc, \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc ((fun x => c \u2022 x) \u207b\u00b9' s) = \u2191\u2191\u03bc s, \u2200 (c : G) (s : Set \u03b1), \u2191\u2191\u03bc (c \u2022 s) = \u2191\u2191\u03bc s,\n \u2200 (c : G), map (fun x => c \u2022 x) \u03bc = \u03bc, \u2200 (c : G), MeasurePreserving fun x => c \u2022 x]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\ns : Set \u03b1\nhs : NullMeasurableSet s\nc : G\n\u22a2 NullMeasurableSet (c \u2022 s)\n[PROOFSTEP]\nsimpa only [\u2190 preimage_smul_inv] using hs.preimage (measurePreserving_smul _ _).quasiMeasurePreserving\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns\u271d : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc\u271d : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\ns : Set \u03b1\nh : \u2191\u2191\u03bc s = 0\nc : G\n\u22a2 \u2191\u2191\u03bc (c \u2022 s) = 0\n[PROOFSTEP]\nrwa [measure_smul]\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2077 : Group G\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : ContinuousConstSMul G \u03b1\ninst\u271d : MulAction.IsMinimal G \u03b1\nK U : Set \u03b1\nhK : IsCompact K\nh\u03bcK : \u2191\u2191\u03bc K \u2260 0\nhU : IsOpen U\nhne : Set.Nonempty U\nt : Finset G\nht : K \u2286 \u22c3 (g : G) (_ : g \u2208 t), g \u2022 U\nh\u03bcU : \u2191\u2191\u03bc U = 0\nx\u271d\u00b9 : G\nx\u271d : x\u271d\u00b9 \u2208 \u2191t\n\u22a2 \u2191\u2191\u03bc (x\u271d\u00b9 \u2022 U) = 0\n[PROOFSTEP]\nrwa [measure_smul]\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2077 : Group G\ninst\u271d\u2076 : MulAction G \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : ContinuousConstSMul G \u03b1\ninst\u271d : MulAction.IsMinimal G \u03b1\nK U : Set \u03b1\nhU : IsOpen U\nhne : Set.Nonempty U\nh\u03bcU : \u2191\u2191\u03bc U \u2260 \u22a4\nx : \u03b1\ng : G\nhg : g \u2022 x \u2208 U\n\u22a2 \u2191\u2191\u03bc ((fun x x_1 => x \u2022 x_1) g \u207b\u00b9' U) \u2260 \u22a4\n[PROOFSTEP]\nrwa [measure_preimage_smul]\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : MulAction G \u03b1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d\u2074 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ContinuousConstSMul G \u03b1\ninst\u271d\u00b9 : MulAction.IsMinimal G \u03b1\nK U : Set \u03b1\ninst\u271d : Regular \u03bc\nh\u03bc : \u03bc \u2260 0\nhU : IsOpen U\n\u22a2 \u2191\u2191\u03bc U = 0 \u2194 U = \u2205\n[PROOFSTEP]\nrw [\u2190 not_iff_not, \u2190 Ne.def, \u2190 pos_iff_ne_zero, measure_pos_iff_nonempty_of_smulInvariant G h\u03bc hU,\n nonempty_iff_ne_empty]\n[GOAL]\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nhy : y \u2208 Subgroup.zpowers x\n\u22a2 y \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Subgroup.mem_zpowers_iff.mp hy\n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nlet e : \u03b1 \u2243 \u03b1 := MulAction.toPermHom G \u03b1 x\n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\ne : \u03b1 \u2243 \u03b1 := \u2191(MulAction.toPermHom G \u03b1) x\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nhave he : QuasiMeasurePreserving e \u03bc \u03bc := (measurePreserving_smul x \u03bc).quasiMeasurePreserving\n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\ne : \u03b1 \u2243 \u03b1 := \u2191(MulAction.toPermHom G \u03b1) x\nhe : QuasiMeasurePreserving \u2191e\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nhave he' : QuasiMeasurePreserving e.symm \u03bc \u03bc := (measurePreserving_smul x\u207b\u00b9 \u03bc).quasiMeasurePreserving\n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\ne : \u03b1 \u2243 \u03b1 := \u2191(MulAction.toPermHom G \u03b1) x\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nhave h := he.image_zpow_ae_eq he' k hs\n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\ne : \u03b1 \u2243 \u03b1 := \u2191(MulAction.toPermHom G \u03b1) x\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nh : \u2191(e ^ k) '' s =\u1da0[ae \u03bc] s\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nsimp only [\u2190 MonoidHom.map_zpow] at h \n[GOAL]\ncase intro\nG : Type u\nM : Type v\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSMul G \u03b1\nc : G\n\u03bc : Measure \u03b1\ninst\u271d : SMulInvariantMeasure G \u03b1 \u03bc\nx : G\nhs : x \u2022 s =\u1da0[ae \u03bc] s\nk : \u2124\nhy : x ^ k \u2208 Subgroup.zpowers x\ne : \u03b1 \u2243 \u03b1 := \u2191(MulAction.toPermHom G \u03b1) x\nhe : QuasiMeasurePreserving \u2191e\nhe' : QuasiMeasurePreserving \u2191e.symm\nh : (fun a => \u2191(\u2191(MulAction.toPermHom G \u03b1) (x ^ k)) a) '' s =\u1da0[ae \u03bc] s\n\u22a2 x ^ k \u2022 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nsimpa only [MulAction.toPermHom_apply, MulAction.toPerm_apply, image_smul] using h\n[GOAL]\nG\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nletI : MeasurableSpace (Multiplicative G) := (inferInstanceAs (MeasurableSpace G))\n[GOAL]\nG\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nletI : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n \u27e8fun g => VAddInvariantMeasure.measure_preimage_vadd (Multiplicative.toAdd g)\u27e9\n[GOAL]\nG\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nletI : MeasurableSMul (Multiplicative G) \u03b1 :=\n { measurable_const_smul := fun g => measurable_const_vadd (Multiplicative.toAdd g)\n measurable_smul_const := fun a =>\n @measurable_vadd_const (Multiplicative G) \u03b1 (inferInstanceAs (VAdd G \u03b1)) _ _\n (inferInstanceAs (MeasurableVAdd G \u03b1)) a }\n[GOAL]\nG\u271d : Type u\nM : Type v\n\u03b1\u271d : Type w\ns\u271d : Set \u03b1\u271d\nm\u271d : MeasurableSpace \u03b1\u271d\ninst\u271d\u2079 : Group G\u271d\ninst\u271d\u2078 : MulAction G\u271d \u03b1\u271d\ninst\u271d\u2077 : MeasurableSpace G\u271d\ninst\u271d\u2076 : MeasurableSMul G\u271d \u03b1\u271d\nc : G\u271d\n\u03bc\u271d : Measure \u03b1\u271d\ninst\u271d\u2075 : SMulInvariantMeasure G\u271d \u03b1\u271d \u03bc\u271d\nG : Type u\n\u03b1 : Type w\ns : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : AddAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableVAdd G \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : VAddInvariantMeasure G \u03b1 \u03bc\nx y : G\nhs : x +\u1d65 s =\u1da0[ae \u03bc] s\nhy : y \u2208 AddSubgroup.zmultiples x\nthis\u271d\u00b9 : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis\u271d : SMulInvariantMeasure (Multiplicative G) \u03b1 \u03bc :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (\u2191Multiplicative.toAdd g) }\nthis : MeasurableSMul (Multiplicative G) \u03b1 :=\n { measurable_const_smul := fun g => measurable_const_vadd (\u2191Multiplicative.toAdd g),\n measurable_smul_const := fun a => measurable_vadd_const a }\n\u22a2 y +\u1d65 s =\u1da0[ae \u03bc] s\n[PROOFSTEP]\nexact @smul_ae_eq_self_of_mem_zpowers (Multiplicative G) \u03b1 _ _ _ _ _ _ _ _ _ _ hs hy\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Group.Action", "llama_tokens": 13964, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.2893823211917706}} {"text": "[GOAL]\ntm : FinTM2\ns : List (FinTM2.\u0393 tm tm.k\u2080)\nk : tm.K\nh : k = tm.k\u2080\n\u22a2 List (FinTM2.\u0393 tm k)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ntm : FinTM2\ns : List (FinTM2.\u0393 tm tm.k\u2080)\nk : tm.K\nh : k = tm.k\u2080\n\u22a2 List (FinTM2.\u0393 tm tm.k\u2080)\n[PROOFSTEP]\nexact s\n[GOAL]\ntm : FinTM2\ns : List (FinTM2.\u0393 tm tm.k\u2081)\nk : tm.K\nh : k = tm.k\u2081\n\u22a2 List (FinTM2.\u0393 tm k)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ntm : FinTM2\ns : List (FinTM2.\u0393 tm tm.k\u2081)\nk : tm.K\nh : k = tm.k\u2081\n\u22a2 List (FinTM2.\u0393 tm tm.k\u2081)\n[PROOFSTEP]\nexact s\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nc : Option \u03c3\nh\u2081 : EvalsTo f a (some b)\nh\u2082 : EvalsTo f b c\n\u22a2 (flip bind f)^[h\u2082.steps + h\u2081.steps] (some a) = c\n[PROOFSTEP]\nrw [Function.iterate_add_apply, h\u2081.evals_in_steps, h\u2082.evals_in_steps]\n[GOAL]\n\u03b1 : Type\nea : FinEncoding \u03b1\nx\u271d : \u03b1\n\u22a2 { steps := 1,\n evals_in_steps :=\n (_ :\n (flip bind\n (FinTM2.step\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.tm))^[1]\n (some\n (initList\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.tm\n (List.map\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.inputAlphabet.invFun\n (Encoding.encode ea.toEncoding x\u271d)))) =\n (flip bind\n (FinTM2.step\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.tm))^[1]\n (some\n (initList\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.tm\n (List.map\n { tm := idComputer ea,\n inputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2080),\n outputAlphabet :=\n Equiv.cast\n (_ :\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081 =\n FinTM2.\u0393 (idComputer ea) (idComputer ea).k\u2081) }.inputAlphabet.invFun\n (Encoding.encode ea.toEncoding x\u271d))))) }.steps \u2264\n Polynomial.eval (List.length (Encoding.encode ea.toEncoding x\u271d)) 1\n[PROOFSTEP]\nsimp only [Polynomial.eval_one]\n", "meta": {"mathlib_filename": "Mathlib.Computability.TMComputable", "llama_tokens": 1532, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.43014734858584286, "lm_q1q2_score": 0.28920169777218446}} {"text": "[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nj j' : J\nf : j \u27f6 j'\n\u22a2 ((Functor.const J).obj (Shrink \u2191(Functor.sections F))).map f \u226b\n (fun j u => \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm u) j) j' =\n (fun j u => \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm u) j) j \u226b F.map f\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nj j' : J\nf : j \u27f6 j'\nx : ((Functor.const J).obj (Shrink \u2191(Functor.sections F))).obj j\n\u22a2 (((Functor.const J).obj (Shrink \u2191(Functor.sections F))).map f \u226b\n (fun j u => \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm u) j) j')\n x =\n ((fun j u => \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm u) j) j \u226b F.map f) x\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx y : (limitCone F).pt\nw : \u2191(equivShrink \u2191(Functor.sections F)).symm x = \u2191(equivShrink \u2191(Functor.sections F)).symm y\n\u22a2 x = y\n[PROOFSTEP]\naesop\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\n\u22a2 x\u271d =\n (fun s v =>\n \u2191(equivShrink \u2191(Functor.sections F))\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9\n[PROOFSTEP]\next x j\n[GOAL]\ncase h.w.a.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\nx : x\u271d\u00b9.pt\nj : J\n\u22a2 \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm (x\u271d x)) j =\n \u2191(\u2191(equivShrink \u2191(Functor.sections F)).symm\n ((fun s v =>\n \u2191(equivShrink \u2191(Functor.sections F))\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property :=\n (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9 x))\n j\n[PROOFSTEP]\nsimpa using congr_fun (w j) x\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nf : j \u27f6 j'\n\u22a2 ((Functor.const J).obj \u2191(Functor.sections F)).map f \u226b (fun j u => \u2191u j) j' = (fun j u => \u2191u j) j \u226b F.map f\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nf : j \u27f6 j'\nx : ((Functor.const J).obj \u2191(Functor.sections F)).obj j\n\u22a2 (((Functor.const J).obj \u2191(Functor.sections F)).map f \u226b (fun j u => \u2191u j) j') x = ((fun j u => \u2191u j) j \u226b F.map f) x\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\n\u22a2 x\u271d =\n (fun s v =>\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\nx : x\u271d\u00b9.pt\n\u22a2 x\u271d x =\n (fun s v =>\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9 x\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase h.a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\nx : x\u271d\u00b9.pt\n\u22a2 \u2191(x\u271d x) =\n \u2191((fun s v =>\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9 x)\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase h.a.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx\u271d\u00b9 : Cone F\nx\u271d : x\u271d\u00b9.pt \u27f6 (limitCone F).pt\nw : \u2200 (j : J), x\u271d \u226b NatTrans.app (limitCone F).\u03c0 j = NatTrans.app x\u271d\u00b9.\u03c0 j\nx : x\u271d\u00b9.pt\nj : J\n\u22a2 \u2191(x\u271d x) j =\n \u2191((fun s v =>\n { val := fun j => NatTrans.app s.\u03c0 j v,\n property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), (NatTrans.app s.\u03c0 j \u226b F.map f) v = NatTrans.app s.\u03c0 j' v) })\n x\u271d\u00b9 x)\n j\n[PROOFSTEP]\nexact congr_fun (w j) x\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nc : Cone F\nt : IsLimit c\nj : J\nx : c.pt\n\u22a2 \u2191(\u2191(isLimitEquivSections t) x) j = NatTrans.app c.\u03c0 j x\n[PROOFSTEP]\nsimp [isLimitEquivSections, IsLimit.conePointUniqueUpToIso]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nc : Cone F\nt : IsLimit c\nx : \u2191(Functor.sections F)\nj : J\n\u22a2 NatTrans.app c.\u03c0 j (\u2191(isLimitEquivSections t).symm x) = \u2191x j\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := (isLimitEquivSections.{v, u} t).surjective x\n[GOAL]\ncase intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nc : Cone F\nt : IsLimit c\nj : J\nx : c.pt\n\u22a2 NatTrans.app c.\u03c0 j (\u2191(isLimitEquivSections t).symm (\u2191(isLimitEquivSections t) x)) = \u2191(\u2191(isLimitEquivSections t) x) j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx : limit F\nj : J\n\u22a2 \u2191(\u2191(limitEquivSections F) x) j = limit.\u03c0 F j x\n[PROOFSTEP]\nsimp [limitEquivSections, isLimitEquivSections, IsLimit.conePointUniqueUpToIso]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx : (j : J) \u2192 F.obj j\nh : \u2200 (j j' : J) (f : j \u27f6 j'), F.map f (x j) = x j'\nj : J\n\u22a2 limit.\u03c0 F j (mk F x h) = x j\n[PROOFSTEP]\ndsimp [Limit.mk]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx : (j : J) \u2192 F.obj j\nh : \u2200 (j j' : J) (f : j \u27f6 j'), F.map f (x j) = x j'\nj : J\n\u22a2 limit.\u03c0 F j\n (\u2191(limitEquivSections F).symm { val := x, property := (_ : \u2200 {j j' : J} (f : j \u27f6 j'), F.map f (x j) = x j') }) =\n x j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx y : limit F\nw : \u2200 (j : J), limit.\u03c0 F j x = limit.\u03c0 F j y\n\u22a2 x = y\n[PROOFSTEP]\napply (limitEquivSections.{v, u} F).injective\n[GOAL]\ncase a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx y : limit F\nw : \u2200 (j : J), limit.\u03c0 F j x = limit.\u03c0 F j y\n\u22a2 \u2191(limitEquivSections F) x = \u2191(limitEquivSections F) y\n[PROOFSTEP]\next j\n[GOAL]\ncase a.a.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : UnivLE.{v, u}\nF : J \u2964 Type u\nx y : limit F\nw : \u2200 (j : J), limit.\u03c0 F j x = limit.\u03c0 F j y\nj : J\n\u22a2 \u2191(\u2191(limitEquivSections F) x) j = \u2191(\u2191(limitEquivSections F) y) j\n[PROOFSTEP]\nsimp [w j]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nx\u271d\u00b2 x\u271d\u00b9 : (j : J) \u00d7 F.obj j\nj : J\nx : F.obj j\nj' : J\nx' : F.obj j'\nx\u271d : Quot.Rel F { fst := j, snd := x } { fst := j', snd := x' }\nf : { fst := j, snd := x }.fst \u27f6 { fst := j', snd := x' }.fst\nhf : { fst := j', snd := x' }.snd = F.map f { fst := j, snd := x }.snd\n\u22a2 (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j, snd := x } =\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j', snd := x' }\n[PROOFSTEP]\ndsimp at hf \n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nx\u271d\u00b2 x\u271d\u00b9 : (j : J) \u00d7 F.obj j\nj : J\nx : F.obj j\nj' : J\nx' : F.obj j'\nx\u271d : Quot.Rel F { fst := j, snd := x } { fst := j', snd := x' }\nf : { fst := j, snd := x }.fst \u27f6 { fst := j', snd := x' }.fst\nhf : x' = F.map f x\n\u22a2 (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j, snd := x } =\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j', snd := x' }\n[PROOFSTEP]\nrw [hf]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nx\u271d\u00b2 x\u271d\u00b9 : (j : J) \u00d7 F.obj j\nj : J\nx : F.obj j\nj' : J\nx' : F.obj j'\nx\u271d : Quot.Rel F { fst := j, snd := x } { fst := j', snd := x' }\nf : { fst := j, snd := x }.fst \u27f6 { fst := j', snd := x' }.fst\nhf : x' = F.map f x\n\u22a2 (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j, snd := x } =\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) { fst := j', snd := F.map f x }\n[PROOFSTEP]\nexact (congr_fun (Cocone.w s f) x).symm\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nhm : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m =\n (fun s =>\n Quot.lift (fun p => NatTrans.app s.\u03b9 p.fst p.snd)\n (_ :\n \u2200 (x x_1 : (j : J) \u00d7 F.obj j),\n Quot.Rel F x x_1 \u2192\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x = (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x_1))\n s\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nhm : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx : (colimitCocone F).pt\n\u22a2 m x =\n (fun s =>\n Quot.lift (fun p => NatTrans.app s.\u03b9 p.fst p.snd)\n (_ :\n \u2200 (x x_1 : (j : J) \u00d7 F.obj j),\n Quot.Rel F x x_1 \u2192\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x = (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x_1))\n s x\n[PROOFSTEP]\ninduction' x using Quot.ind with x\n[GOAL]\ncase h.mk\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nhm : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx : (j : J) \u00d7 F.obj j\n\u22a2 m (Quot.mk (Quot.Rel F) x) =\n (fun s =>\n Quot.lift (fun p => NatTrans.app s.\u03b9 p.fst p.snd)\n (_ :\n \u2200 (x x_1 : (j : J) \u00d7 F.obj j),\n Quot.Rel F x x_1 \u2192\n (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x = (fun p => NatTrans.app s.\u03b9 p.fst p.snd) x_1))\n s (Quot.mk (Quot.Rel F) x)\n[PROOFSTEP]\nexact congr_fun (hm x.1) x.2\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj : J\nx : F.obj j\n\u22a2 \u2191(colimitEquivQuot F) (colimit.\u03b9 F j x) = Quot.mk (Quot.Rel F) { fst := j, snd := x }\n[PROOFSTEP]\napply (colimitEquivQuot F).symm.injective\n[GOAL]\ncase a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj : J\nx : F.obj j\n\u22a2 \u2191(colimitEquivQuot F).symm (\u2191(colimitEquivQuot F) (colimit.\u03b9 F j x)) =\n \u2191(colimitEquivQuot F).symm (Quot.mk (Quot.Rel F) { fst := j, snd := x })\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nf : j \u27f6 j'\nw : F.map f x = x'\n\u22a2 colimit.\u03b9 F j x = colimit.\u03b9 F j' x'\n[PROOFSTEP]\nrw [\u2190 w, Colimit.w_apply.{v, u}]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nj'' : J\nf : j \u27f6 j''\nf' : j' \u27f6 j''\nw : F.map f x = F.map f' x'\n\u22a2 colimit.\u03b9 F j x = colimit.\u03b9 F j' x'\n[PROOFSTEP]\nrw [\u2190 colimit.w _ f, \u2190 colimit.w _ f']\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nj'' : J\nf : j \u27f6 j''\nf' : j' \u27f6 j''\nw : F.map f x = F.map f' x'\n\u22a2 (F.map f \u226b colimit.\u03b9 F j'') x = (F.map f' \u226b colimit.\u03b9 F j'') x'\n[PROOFSTEP]\nrw [types_comp_apply, types_comp_apply, w]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nw : colimit.\u03b9 F j x = colimit.\u03b9 F j' x'\n\u22a2 EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := x' }\n[PROOFSTEP]\napply Quot.eq.1\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nw : colimit.\u03b9 F j x = colimit.\u03b9 F j' x'\n\u22a2 Quot.mk (Quot.Rel F) { fst := j, snd := x } = Quot.mk (Quot.Rel F) { fst := j', snd := x' }\n[PROOFSTEP]\nsimpa using congr_arg (colimitEquivQuot.{v, u} F) w\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\n\u22a2 \u2203 j y, NatTrans.app t.\u03b9 j y = x\n[PROOFSTEP]\nsuffices (fun x : t.pt => ULift.up (\u2203 j y, t.\u03b9.app j y = x)) = fun _ => ULift.up.{max v u} True\n by\n have := congr_fun this x\n simpa using congr_arg ULift.down this\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\nthis : (fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) = fun x => { down := True }\n\u22a2 \u2203 j y, NatTrans.app t.\u03b9 j y = x\n[PROOFSTEP]\nhave := congr_fun this x\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\nthis\u271d : (fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) = fun x => { down := True }\nthis : { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x } = { down := True }\n\u22a2 \u2203 j y, NatTrans.app t.\u03b9 j y = x\n[PROOFSTEP]\nsimpa using congr_arg ULift.down this\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\n\u22a2 (fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) = fun x => { down := True }\n[PROOFSTEP]\nrefine' h.hom_ext _\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\n\u22a2 \u2200 (j : J),\n (NatTrans.app t.\u03b9 j \u226b fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) =\n NatTrans.app t.\u03b9 j \u226b fun x => { down := True }\n[PROOFSTEP]\nintro j\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\nj : J\n\u22a2 (NatTrans.app t.\u03b9 j \u226b fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) =\n NatTrans.app t.\u03b9 j \u226b fun x => { down := True }\n[PROOFSTEP]\nfunext y\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\nj : J\ny : F.obj j\n\u22a2 (NatTrans.app t.\u03b9 j \u226b fun x => { down := \u2203 j y, NatTrans.app t.\u03b9 j y = x }) y =\n (NatTrans.app t.\u03b9 j \u226b fun x => { down := True }) y\n[PROOFSTEP]\nsimp only [Functor.const_obj_obj, types_comp_apply, ULift.up_inj, eq_iff_iff, iff_true]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nh : IsColimit t\nx : t.pt\nj : J\ny : F.obj j\n\u22a2 \u2203 j_1 y_1, NatTrans.app t.\u03b9 j_1 y_1 = NatTrans.app t.\u03b9 j y\n[PROOFSTEP]\nexact \u27e8j, y, rfl\u27e9\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx : colimit F\n\u22a2 \u2203 j y, colimit.\u03b9 F j y = x\n[PROOFSTEP]\nexact jointly_surjective.{v, u} F (colimit.isColimit F) x\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx y : (j : J) \u00d7 F.obj j\nx\u271d : Quot.Rel F x y\nf : x.fst \u27f6 y.fst\nh : y.snd = F.map f x.snd\n\u22a2 F.map f x.snd = F.map (\ud835\udfd9 y.fst) y.snd\n[PROOFSTEP]\nrw [\u2190 h, FunctorToTypes.map_id_apply]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx y : (j : J) \u00d7 F.obj j\nx\u271d : FilteredColimit.Rel F x y\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\n\u22a2 EqvGen (Quot.Rel F) x y\n[PROOFSTEP]\nrefine' EqvGen.trans _ \u27e8k, F.map f x.2\u27e9 _ _ _\n[GOAL]\ncase refine'_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx y : (j : J) \u00d7 F.obj j\nx\u271d : FilteredColimit.Rel F x y\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\n\u22a2 EqvGen (Quot.Rel F) x { fst := k, snd := F.map f x.snd }\n[PROOFSTEP]\nexact (EqvGen.rel _ _ \u27e8f, rfl\u27e9)\n[GOAL]\ncase refine'_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nx y : (j : J) \u00d7 F.obj j\nx\u271d : FilteredColimit.Rel F x y\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\n\u22a2 EqvGen (Quot.Rel F) { fst := k, snd := F.map f x.snd } y\n[PROOFSTEP]\nexact (EqvGen.symm _ _ (EqvGen.rel _ _ \u27e8g, h\u27e9))\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 IsColimit t\n[PROOFSTEP]\napply IsColimit.ofIsoColimit (colimit.isColimit F)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 colimit.cocone F \u2245 t\n[PROOFSTEP]\nrefine' Cocones.ext (Equiv.toIso (Equiv.ofBijective _ _)) _\n[GOAL]\ncase refine'_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 (colimit.cocone F).pt \u2192 t.pt\n[PROOFSTEP]\nexact colimit.desc F t\n[GOAL]\ncase refine'_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 Function.Bijective (colimit.desc F t)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refine'_2.left\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 Function.Injective (colimit.desc F t)\n[PROOFSTEP]\nshow Function.Injective _\n[GOAL]\ncase refine'_2.left\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 Function.Injective (colimit.desc F t)\n[PROOFSTEP]\nintro a b h\n[GOAL]\ncase refine'_2.left\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\na b : (colimit.cocone F).pt\nh : colimit.desc F t a = colimit.desc F t b\n\u22a2 a = b\n[PROOFSTEP]\nrcases jointly_surjective.{v, u} F (colimit.isColimit F) a with \u27e8i, xi, rfl\u27e9\n[GOAL]\ncase refine'_2.left.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\nb : (colimit.cocone F).pt\ni : J\nxi : F.obj i\nh : colimit.desc F t (NatTrans.app (colimit.cocone F).\u03b9 i xi) = colimit.desc F t b\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 i xi = b\n[PROOFSTEP]\nrcases jointly_surjective.{v, u} F (colimit.isColimit F) b with \u27e8j, xj, rfl\u27e9\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh :\n colimit.desc F t (NatTrans.app (colimit.cocone F).\u03b9 i xi) = colimit.desc F t (NatTrans.app (colimit.cocone F).\u03b9 j xj)\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 i xi = NatTrans.app (colimit.cocone F).\u03b9 j xj\n[PROOFSTEP]\nreplace h : (colimit.\u03b9 F i \u226b colimit.desc F t) xi = (colimit.\u03b9 F j \u226b colimit.desc F t) xj := h\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : (colimit.\u03b9 F i \u226b colimit.desc F t) xi = (colimit.\u03b9 F j \u226b colimit.desc F t) xj\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 i xi = NatTrans.app (colimit.cocone F).\u03b9 j xj\n[PROOFSTEP]\nrw [colimit.\u03b9_desc, colimit.\u03b9_desc] at h \n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 i xi = NatTrans.app (colimit.cocone F).\u03b9 j xj\n[PROOFSTEP]\nrcases hinj i j xi xj h with \u27e8k, f, g, h'\u27e9\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj\nk : J\nf : i \u27f6 k\ng : j \u27f6 k\nh' : F.map f xi = F.map g xj\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 i xi = NatTrans.app (colimit.cocone F).\u03b9 j xj\n[PROOFSTEP]\nchange colimit.\u03b9 F i xi = colimit.\u03b9 F j xj\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj\nk : J\nf : i \u27f6 k\ng : j \u27f6 k\nh' : F.map f xi = F.map g xj\n\u22a2 colimit.\u03b9 F i xi = colimit.\u03b9 F j xj\n[PROOFSTEP]\nrw [\u2190 colimit.w F f, \u2190 colimit.w F g]\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj\nk : J\nf : i \u27f6 k\ng : j \u27f6 k\nh' : F.map f xi = F.map g xj\n\u22a2 (F.map f \u226b colimit.\u03b9 F k) xi = (F.map g \u226b colimit.\u03b9 F k) xj\n[PROOFSTEP]\nchange colimit.\u03b9 F k (F.map f xi) = colimit.\u03b9 F k (F.map g xj)\n[GOAL]\ncase refine'_2.left.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\nj : J\nxj : F.obj j\nh : NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj\nk : J\nf : i \u27f6 k\ng : j \u27f6 k\nh' : F.map f xi = F.map g xj\n\u22a2 colimit.\u03b9 F k (F.map f xi) = colimit.\u03b9 F k (F.map g xj)\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase refine'_2.right\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 Function.Surjective (colimit.desc F t)\n[PROOFSTEP]\nshow Function.Surjective _\n[GOAL]\ncase refine'_2.right\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 Function.Surjective (colimit.desc F t)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_2.right\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\nx : t.pt\n\u22a2 \u2203 a, colimit.desc F t a = x\n[PROOFSTEP]\nrcases hsurj x with \u27e8i, xi, rfl\u27e9\n[GOAL]\ncase refine'_2.right.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\n\u22a2 \u2203 a, colimit.desc F t a = NatTrans.app t.\u03b9 i xi\n[PROOFSTEP]\nuse colimit.\u03b9 F i xi\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\ni : J\nxi : F.obj i\n\u22a2 colimit.desc F t (colimit.\u03b9 F i xi) = NatTrans.app t.\u03b9 i xi\n[PROOFSTEP]\napply Colimit.\u03b9_desc_apply.{v, u}\n[GOAL]\ncase refine'_3\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\n\u22a2 \u2200 (j : J),\n NatTrans.app (colimit.cocone F).\u03b9 j \u226b\n (Equiv.toIso\n (Equiv.ofBijective (colimit.desc F t)\n (_ : Function.Injective (colimit.desc F t) \u2227 Function.Surjective (colimit.desc F t)))).hom =\n NatTrans.app t.\u03b9 j\n[PROOFSTEP]\nintro j\n[GOAL]\ncase refine'_3\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 TypeMax\nt : Cocone F\nhsurj : \u2200 (x : t.pt), \u2203 i xi, x = NatTrans.app t.\u03b9 i xi\nhinj :\n \u2200 (i j : J) (xi : F.obj i) (xj : F.obj j),\n NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2192 \u2203 k f g, F.map f xi = F.map g xj\nj : J\n\u22a2 NatTrans.app (colimit.cocone F).\u03b9 j \u226b\n (Equiv.toIso\n (Equiv.ofBijective (colimit.desc F t)\n (_ : Function.Injective (colimit.desc F t) \u2227 Function.Surjective (colimit.desc F t)))).hom =\n NatTrans.app t.\u03b9 j\n[PROOFSTEP]\napply colimit.\u03b9_desc\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map (f \u226b fl \u226b n) x.snd = F.map (fl \u226b n) (F.map f x.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map (fl \u226b n) (F.map f x.snd) = F.map (fl \u226b n) (F.map g y.snd)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map (fl \u226b n) (F.map g y.snd) = F.map ((g \u226b fl) \u226b n) y.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map ((g \u226b fl) \u226b n) y.snd = F.map ((f' \u226b gl) \u226b n) y.snd\n[PROOFSTEP]\nrw [hn]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map ((f' \u226b gl) \u226b n) y.snd = F.map (gl \u226b n) (F.map f' y.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map (gl \u226b n) (F.map f' y.snd) = F.map (gl \u226b n) (F.map g' z.snd)\n[PROOFSTEP]\nrw [h']\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nx y z : (j : J) \u00d7 F.obj j\nx\u271d\u00b9 : FilteredColimit.Rel F x y\nx\u271d : FilteredColimit.Rel F y z\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\nh : F.map f x.snd = F.map g y.snd\nk' : J\nf' : y.fst \u27f6 k'\ng' : z.fst \u27f6 k'\nh' : F.map f' y.snd = F.map g' z.snd\nl : J\nfl : k \u27f6 l\ngl : k' \u27f6 l\nh\u271d : True\nm : J\nn : l \u27f6 m\nhn : (g \u226b fl) \u226b n = (f' \u226b gl) \u226b n\n\u22a2 F.map (gl \u226b n) (F.map g' z.snd) = F.map (g' \u226b gl \u226b n) z.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\n\u22a2 FilteredColimit.Rel F = EqvGen (Quot.Rel F)\n[PROOFSTEP]\next \u27e8j, x\u27e9 \u27e8j', y\u27e9\n[GOAL]\ncase h.mk.h.mk.a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n\u22a2 FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } \u2194\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mk.h.mk.a.mp\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n\u22a2 FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } \u2192\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }\n[PROOFSTEP]\napply eqvGen_quot_rel_of_rel\n[GOAL]\ncase h.mk.h.mk.a.mpr\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n\u22a2 EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } \u2192\n FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y }\n[PROOFSTEP]\nrw [\u2190 (FilteredColimit.rel_equiv F).eqvGen_iff]\n[GOAL]\ncase h.mk.h.mk.a.mpr\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n\u22a2 EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } \u2192\n EqvGen (FilteredColimit.Rel F) { fst := j, snd := x } { fst := j', snd := y }\n[PROOFSTEP]\nexact EqvGen.mono (rel_of_quot_rel F)\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\ni j : J\nxi : F.obj i\nxj : F.obj j\n\u22a2 NatTrans.app (colimitCocone F).\u03b9 i xi = NatTrans.app (colimitCocone F).\u03b9 j xj \u2194\n FilteredColimit.Rel F { fst := i, snd := xi } { fst := j, snd := xj }\n[PROOFSTEP]\nchange Quot.mk _ _ = Quot.mk _ _ \u2194 _\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\ni j : J\nxi : F.obj i\nxj : F.obj j\n\u22a2 Quot.mk (Quot.Rel F) { fst := i, snd := xi } = Quot.mk (Quot.Rel F) { fst := j, snd := xj } \u2194\n FilteredColimit.Rel F { fst := i, snd := xi } { fst := j, snd := xj }\n[PROOFSTEP]\nrw [Quot.eq, FilteredColimit.rel_eq_eqvGen_quot_rel]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\ni j : J\nxi : F.obj i\nxj : F.obj j\n\u22a2 NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2194 \u2203 k f g, F.map f xi = F.map g xj\n[PROOFSTEP]\nlet t' := colimitCocone.{v, u} F\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\ni j : J\nxi : F.obj i\nxj : F.obj j\nt' : Cocone F := colimitCocone F\n\u22a2 NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2194 \u2203 k f g, F.map f xi = F.map g xj\n[PROOFSTEP]\nlet e : t' \u2245 t := IsColimit.uniqueUpToIso (colimitCoconeIsColimit F) ht\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\ni j : J\nxi : F.obj i\nxj : F.obj j\nt' : Cocone F := colimitCocone F\ne : t' \u2245 t := IsColimit.uniqueUpToIso (colimitCoconeIsColimit F) ht\n\u22a2 NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2194 \u2203 k f g, F.map f xi = F.map g xj\n[PROOFSTEP]\nlet e' : t'.pt \u2245 t.pt := (Cocones.forget _).mapIso e\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\ni j : J\nxi : F.obj i\nxj : F.obj j\nt' : Cocone F := colimitCocone F\ne : t' \u2245 t := IsColimit.uniqueUpToIso (colimitCoconeIsColimit F) ht\ne' : t'.pt \u2245 t.pt := (Cocones.forget F).mapIso e\n\u22a2 NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2194 \u2203 k f g, F.map f xi = F.map g xj\n[PROOFSTEP]\nrefine' Iff.trans _ (colimit_eq_iff_aux.{v, u} F)\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 TypeMax\ninst\u271d : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\ni j : J\nxi : F.obj i\nxj : F.obj j\nt' : Cocone F := colimitCocone F\ne : t' \u2245 t := IsColimit.uniqueUpToIso (colimitCoconeIsColimit F) ht\ne' : t'.pt \u2245 t.pt := (Cocones.forget F).mapIso e\n\u22a2 NatTrans.app t.\u03b9 i xi = NatTrans.app t.\u03b9 j xj \u2194\n NatTrans.app (colimitCocone F).\u03b9 i xi = NatTrans.app (colimitCocone F).\u03b9 j xj\n[PROOFSTEP]\nexact @Equiv.apply_eq_iff_eq _ _ e'.toEquiv ((colimitCocone.{v, u} F).\u03b9.app i xi) ((colimitCocone.{v, u} F).\u03b9.app j xj)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u27f6 \u03b2\nF' : MonoFactorisation f\n\u22a2 lift F' \u226b F'.m = \u03b9 f\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u27f6 \u03b2\nF' : MonoFactorisation f\nx : Image f\n\u22a2 (lift F' \u226b F'.m) x = \u03b9 f x\n[PROOFSTEP]\nchange (F'.e \u226b F'.m) _ = _\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u27f6 \u03b2\nF' : MonoFactorisation f\nx : Image f\n\u22a2 (F'.e \u226b F'.m) \u2191(Classical.indefiniteDescription (fun x_1 => f x_1 = \u2191x) (_ : \u2191x \u2208 Set.range f)) = \u03b9 f x\n[PROOFSTEP]\nrw [F'.fac, (Classical.indefiniteDescription _ x.2).2]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u27f6 \u03b2\nF' : MonoFactorisation f\nx : Image f\n\u22a2 \u2191x = \u03b9 f x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u27f6 \u03b2\n\u22a2 \u2200 {X Y : Type u} (f : X \u27f6 Y), HasImage f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf\u271d : \u03b1 \u27f6 \u03b2\nf g : Arrow (Type u)\nst : f \u27f6 g\nx : (monoFactorisation f.hom).I\n\u22a2 Comma.hom g (CommaMorphism.left st (Classical.choose (_ : \u2191x \u2208 Set.range f.hom))) = CommaMorphism.right st \u2191x\n[PROOFSTEP]\nhave p := st.w\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf\u271d : \u03b1 \u27f6 \u03b2\nf g : Arrow (Type u)\nst : f \u27f6 g\nx : (monoFactorisation f.hom).I\np : (\ud835\udfed (Type u)).map st.left \u226b g.hom = f.hom \u226b (\ud835\udfed (Type u)).map st.right\n\u22a2 Comma.hom g (CommaMorphism.left st (Classical.choose (_ : \u2191x \u2208 Set.range f.hom))) = CommaMorphism.right st \u2191x\n[PROOFSTEP]\nreplace p := congr_fun p (Classical.choose x.2)\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf\u271d : \u03b1 \u27f6 \u03b2\nf g : Arrow (Type u)\nst : f \u27f6 g\nx : (monoFactorisation f.hom).I\np :\n ((\ud835\udfed (Type u)).map st.left \u226b g.hom) (Classical.choose (_ : \u2191x \u2208 Set.range f.hom)) =\n (f.hom \u226b (\ud835\udfed (Type u)).map st.right) (Classical.choose (_ : \u2191x \u2208 Set.range f.hom))\n\u22a2 Comma.hom g (CommaMorphism.left st (Classical.choose (_ : \u2191x \u2208 Set.range f.hom))) = CommaMorphism.right st \u2191x\n[PROOFSTEP]\nsimp only [Functor.id_obj, Functor.id_map, types_comp_apply] at p \n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\n\u03b1 \u03b2 : Type u\nf\u271d : \u03b1 \u27f6 \u03b2\nf g : Arrow (Type u)\nst : f \u27f6 g\nx : (monoFactorisation f.hom).I\np :\n Comma.hom g (CommaMorphism.left st (Classical.choose (_ : \u2203 x_1, (fun x_2 => Comma.hom f x_2 = \u2191x) x_1))) =\n CommaMorphism.right st (Comma.hom f (Classical.choose (_ : \u2203 x_1, (fun x_2 => Comma.hom f x_2 = \u2191x) x_1)))\n\u22a2 Comma.hom g (CommaMorphism.left st (Classical.choose (_ : \u2191x \u2208 Set.range f.hom))) = CommaMorphism.right st \u2191x\n[PROOFSTEP]\nerw [p, Classical.choose_spec x.2]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Types", "llama_tokens": 17897, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.28890034460258746}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nx\u271d\u00b9 x\u271d : HomogeneousIdeal \ud835\udc9c\nx : Submodule A A\nhx : Ideal.IsHomogeneous \ud835\udc9c x\ny : Submodule A A\nhy : Ideal.IsHomogeneous \ud835\udc9c y\nh : x = y\n\u22a2 { toSubmodule := x, is_homogeneous' := hx } = { toSubmodule := y, is_homogeneous' := hy }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2081 : Homogeneous \ud835\udc9c x\nhx\u2082 : x \u2208 I\nj : \u03b9\n\u22a2 \u2191(GradedRing.proj \ud835\udc9c j) (r * x) \u2208 I\n[PROOFSTEP]\nclassical\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r, Finset.sum_mul, map_sum]\napply Ideal.sum_mem\nintro k _\nobtain \u27e8i, hi\u27e9 := hx\u2081\nhave mem\u2081 : (DirectSum.decompose \ud835\udc9c r k : A) * x \u2208 \ud835\udc9c (k + i) := GradedMul.mul_mem (SetLike.coe_mem _) hi\nerw [GradedRing.proj_apply, DirectSum.decompose_of_mem \ud835\udc9c mem\u2081, coe_of_apply]\nsplit_ifs\n\u00b7 exact I.mul_mem_left _ hx\u2082\n\u00b7 exact I.zero_mem\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2081 : Homogeneous \ud835\udc9c x\nhx\u2082 : x \u2208 I\nj : \u03b9\n\u22a2 \u2191(GradedRing.proj \ud835\udc9c j) (r * x) \u2208 I\n[PROOFSTEP]\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r, Finset.sum_mul, map_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2081 : Homogeneous \ud835\udc9c x\nhx\u2082 : x \u2208 I\nj : \u03b9\n\u22a2 \u2211 x_1 in DFinsupp.support (\u2191(decompose \ud835\udc9c) r), \u2191(GradedRing.proj \ud835\udc9c j) (\u2191(\u2191(\u2191(decompose \ud835\udc9c) r) x_1) * x) \u2208 I\n[PROOFSTEP]\napply Ideal.sum_mem\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2081 : Homogeneous \ud835\udc9c x\nhx\u2082 : x \u2208 I\nj : \u03b9\n\u22a2 \u2200 (c : \u03b9), c \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r) \u2192 \u2191(GradedRing.proj \ud835\udc9c j) (\u2191(\u2191(\u2191(decompose \ud835\udc9c) r) c) * x) \u2208 I\n[PROOFSTEP]\nintro k _\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2081 : Homogeneous \ud835\udc9c x\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\n\u22a2 \u2191(GradedRing.proj \ud835\udc9c j) (\u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x) \u2208 I\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := hx\u2081\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\ni : \u03b9\nhi : x \u2208 \ud835\udc9c i\n\u22a2 \u2191(GradedRing.proj \ud835\udc9c j) (\u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x) \u2208 I\n[PROOFSTEP]\nhave mem\u2081 : (DirectSum.decompose \ud835\udc9c r k : A) * x \u2208 \ud835\udc9c (k + i) := GradedMul.mul_mem (SetLike.coe_mem _) hi\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\ni : \u03b9\nhi : x \u2208 \ud835\udc9c i\nmem\u2081 : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x \u2208 \ud835\udc9c (k + i)\n\u22a2 \u2191(GradedRing.proj \ud835\udc9c j) (\u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x) \u2208 I\n[PROOFSTEP]\nerw [GradedRing.proj_apply, DirectSum.decompose_of_mem \ud835\udc9c mem\u2081, coe_of_apply]\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\ni : \u03b9\nhi : x \u2208 \ud835\udc9c i\nmem\u2081 : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x \u2208 \ud835\udc9c (k + i)\n\u22a2 \u2191(if k + i = j then { val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x, property := mem\u2081 } else 0) \u2208 I\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\ni : \u03b9\nhi : x \u2208 \ud835\udc9c i\nmem\u2081 : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x \u2208 \ud835\udc9c (k + i)\nh\u271d : k + i = j\n\u22a2 \u2191{ val := \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x, property := mem\u2081 } \u2208 I\n[PROOFSTEP]\nexact I.mul_mem_left _ hx\u2082\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I : Ideal A\nr x : A\nhx\u2082 : x \u2208 I\nj k : \u03b9\na\u271d : k \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\ni : \u03b9\nhi : x \u2208 \ud835\udc9c i\nmem\u2081 : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) k) * x \u2208 \ud835\udc9c (k + i)\nh\u271d : \u00ack + i = j\n\u22a2 \u21910 \u2208 I\n[PROOFSTEP]\nexact I.zero_mem\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns : Set A\nh : \u2200 (x : A), x \u2208 s \u2192 Homogeneous \ud835\udc9c x\n\u22a2 IsHomogeneous \ud835\udc9c (span s)\n[PROOFSTEP]\nrintro i r hr\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns : Set A\nh : \u2200 (x : A), x \u2208 s \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\nr : A\nhr : r \u2208 span s\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 span s\n[PROOFSTEP]\nrw [Ideal.span, Finsupp.span_eq_range_total] at hr \n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns : Set A\nh : \u2200 (x : A), x \u2208 s \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\nr : A\nhr : r \u2208 LinearMap.range (Finsupp.total (\u2191s) A A Subtype.val)\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 span s\n[PROOFSTEP]\nrw [LinearMap.mem_range] at hr \n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns : Set A\nh : \u2200 (x : A), x \u2208 s \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\nr : A\nhr : \u2203 y, \u2191(Finsupp.total (\u2191s) A A Subtype.val) y = r\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 span s\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := hr\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) (\u2191(Finsupp.total (\u2191s\u271d) A A Subtype.val) s)) i) \u2208 span s\u271d\n[PROOFSTEP]\nrw [Finsupp.total_apply, Finsupp.sum, decompose_sum, DFinsupp.finset_sum_apply, AddSubmonoidClass.coe_finset_sum]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\n\u22a2 \u2211 i_1 in s.support, \u2191(\u2191(\u2191(decompose \ud835\udc9c) (\u2191s i_1 \u2022 \u2191i_1)) i) \u2208 span s\u271d\n[PROOFSTEP]\nrefine' Ideal.sum_mem _ _\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\n\u22a2 \u2200 (c : \u2191s\u271d), c \u2208 s.support \u2192 \u2191(\u2191(\u2191(decompose \ud835\udc9c) (\u2191s c \u2022 \u2191c)) i) \u2208 span s\u271d\n[PROOFSTEP]\nrintro z hz1\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\nz : \u2191s\u271d\nhz1 : z \u2208 s.support\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) (\u2191s z \u2022 \u2191z)) i) \u2208 span s\u271d\n[PROOFSTEP]\nrw [smul_eq_mul]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\nz : \u2191s\u271d\nhz1 : z \u2208 s.support\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) (\u2191s z * \u2191z)) i) \u2208 span s\u271d\n[PROOFSTEP]\nrefine' Ideal.mul_homogeneous_element_mem_of_mem \ud835\udc9c (s z) z _ _ i\n[GOAL]\ncase intro.refine'_1\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\nz : \u2191s\u271d\nhz1 : z \u2208 s.support\n\u22a2 Homogeneous \ud835\udc9c \u2191z\n[PROOFSTEP]\nrcases z with \u27e8z, hz2\u27e9\n[GOAL]\ncase intro.refine'_1.mk\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\nz : A\nhz2 : z \u2208 s\u271d\nhz1 : { val := z, property := hz2 } \u2208 s.support\n\u22a2 Homogeneous \ud835\udc9c \u2191{ val := z, property := hz2 }\n[PROOFSTEP]\napply h _ hz2\n[GOAL]\ncase intro.refine'_2\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u271d : Set A\nh : \u2200 (x : A), x \u2208 s\u271d \u2192 Homogeneous \ud835\udc9c x\ni : \u03b9\ns : \u2191s\u271d \u2192\u2080 A\nz : \u2191s\u271d\nhz1 : z \u2208 s.support\n\u22a2 \u2191z \u2208 span s\u271d\n[PROOFSTEP]\nexact Ideal.subset_span z.2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\n\u22a2 HomogeneousIdeal.toIdeal (homogeneousCore \ud835\udc9c I) = I\n[PROOFSTEP]\napply le_antisymm (I.homogeneousCore'_le \ud835\udc9c) _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\n\u22a2 I \u2264 homogeneousCore' \ud835\udc9c I\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\nx : A\nhx : x \u2208 I\n\u22a2 x \u2208 homogeneousCore' \ud835\udc9c I\n[PROOFSTEP]\nclassical\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c x]\nexact Ideal.sum_mem _ fun j _ => Ideal.subset_span \u27e8\u27e8_, homogeneous_coe _\u27e9, h _ hx, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\nx : A\nhx : x \u2208 I\n\u22a2 x \u2208 homogeneousCore' \ud835\udc9c I\n[PROOFSTEP]\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c x]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\nx : A\nhx : x \u2208 I\n\u22a2 \u2211 i in DFinsupp.support (\u2191(decompose \ud835\udc9c) x), \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) i) \u2208 homogeneousCore' \ud835\udc9c I\n[PROOFSTEP]\nexact Ideal.sum_mem _ fun j _ => Ideal.subset_span \u27e8\u27e8_, homogeneous_coe _\u27e9, h _ hx, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d : Ideal A\nI : HomogeneousIdeal \ud835\udc9c\n\u22a2 Ideal.homogeneousCore \ud835\udc9c (toIdeal I) = I\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d : Ideal A\nI : HomogeneousIdeal \ud835\udc9c\n\u22a2 toIdeal (Ideal.homogeneousCore \ud835\udc9c (toIdeal I)) = toIdeal I\n[PROOFSTEP]\nconvert Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self I.isHomogeneous\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 IsHomogeneous \ud835\udc9c I \u2194 \u2203 S, I = span (Subtype.val '' S)\n[PROOFSTEP]\nrw [Ideal.IsHomogeneous.iff_eq, eq_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : SetLike \u03c3 A\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 I = HomogeneousIdeal.toIdeal (homogeneousCore \ud835\udc9c I) \u2194 \u2203 S, I = span (Subtype.val '' S)\n[PROOFSTEP]\nexact ((Set.image_preimage.compose (Submodule.gi _ _).gc).exists_eq_l _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : r \u2208 \u22a5\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 \u22a5\n[PROOFSTEP]\nsimp only [Ideal.mem_bot] at hr \n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : r = 0\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 \u22a5\n[PROOFSTEP]\nrw [hr, decompose_zero, zero_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : r = 0\n\u22a2 \u21910 \u2208 \u22a5\n[PROOFSTEP]\napply Ideal.zero_mem\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nx\u271d : r \u2208 \u22a4\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 \u22a4\n[PROOFSTEP]\nsimp only [Submodule.mem_top]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI J : Ideal A\nHI : IsHomogeneous \ud835\udc9c I\nHJ : IsHomogeneous \ud835\udc9c J\n\u22a2 IsHomogeneous \ud835\udc9c (I \u2294 J)\n[PROOFSTEP]\nrw [iff_exists] at HI HJ \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI J : Ideal A\nHI : \u2203 S, I = span (Subtype.val '' S)\nHJ : \u2203 S, J = span (Subtype.val '' S)\n\u22a2 \u2203 S, I \u2294 J = span (Subtype.val '' S)\n[PROOFSTEP]\nobtain \u27e8\u27e8s\u2081, rfl\u27e9, \u27e8s\u2082, rfl\u27e9\u27e9 := HI, HJ\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081) \u2294 span (Subtype.val '' s\u2082) = span (Subtype.val '' S)\n[PROOFSTEP]\nrefine' \u27e8s\u2081 \u222a s\u2082, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 span (Subtype.val '' s\u2081) \u2294 span (Subtype.val '' s\u2082) = span (Subtype.val '' (s\u2081 \u222a s\u2082))\n[PROOFSTEP]\nrw [Set.image_union]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 span (Subtype.val '' s\u2081) \u2294 span (Subtype.val '' s\u2082) = span (Subtype.val '' s\u2081 \u222a Subtype.val '' s\u2082)\n[PROOFSTEP]\nexact (Submodule.span_union _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\nh : \u2200 (i : \u03ba), IsHomogeneous \ud835\udc9c (f i)\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a06 (i : \u03ba), f i)\n[PROOFSTEP]\nsimp_rw [iff_exists] at h \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\nh : \u2200 (i : \u03ba), \u2203 S, f i = span (Subtype.val '' S)\n\u22a2 \u2203 S, \u2a06 (i : \u03ba), f i = span (Subtype.val '' S)\n[PROOFSTEP]\nchoose s hs using h\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\ns : \u03ba \u2192 Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\nhs : \u2200 (i : \u03ba), f i = span (Subtype.val '' s i)\n\u22a2 \u2203 S, \u2a06 (i : \u03ba), f i = span (Subtype.val '' S)\n[PROOFSTEP]\nrefine' \u27e8\u22c3 i, s i, _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\ns : \u03ba \u2192 Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\nhs : \u2200 (i : \u03ba), f i = span (Subtype.val '' s i)\n\u22a2 \u2a06 (i : \u03ba), f i = span (Subtype.val '' \u22c3 (i : \u03ba), s i)\n[PROOFSTEP]\nsimp_rw [Set.image_iUnion, Ideal.span_iUnion]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\ns : \u03ba \u2192 Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\nhs : \u2200 (i : \u03ba), f i = span (Subtype.val '' s i)\n\u22a2 \u2a06 (i : \u03ba), f i = \u2a06 (i : \u03ba), span (Subtype.val '' s i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\ns : \u03ba \u2192 Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\nhs : \u2200 (i : \u03ba), f i = span (Subtype.val '' s i)\n\u22a2 (fun i => f i) = fun i => span (Subtype.val '' s i)\n[PROOFSTEP]\nexact funext hs\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\nh : \u2200 (i : \u03ba), IsHomogeneous \ud835\udc9c (f i)\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a05 (i : \u03ba), f i)\n[PROOFSTEP]\nintro i x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\nh : \u2200 (i : \u03ba), IsHomogeneous \ud835\udc9c (f i)\ni : \u03b9\nx : A\nhx : x \u2208 \u2a05 (i : \u03ba), f i\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) i) \u2208 \u2a05 (i : \u03ba), f i\n[PROOFSTEP]\nsimp only [Ideal.mem_iInf] at hx \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\nf : \u03ba \u2192 Ideal A\nh : \u2200 (i : \u03ba), IsHomogeneous \ud835\udc9c (f i)\ni : \u03b9\nx : A\nhx : \u2200 (i : \u03ba), x \u2208 f i\n\u22a2 \u2200 (i_1 : \u03ba), \u2191(\u2191(\u2191(decompose \ud835\udc9c) x) i) \u2208 f i_1\n[PROOFSTEP]\nexact fun j => h _ _ (hx j)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 (I : Ideal A), I \u2208 \u2110 \u2192 IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (SupSet.sSup \u2110)\n[PROOFSTEP]\nrw [sSup_eq_iSup]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 (I : Ideal A), I \u2208 \u2110 \u2192 IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a06 (a : Ideal A) (_ : a \u2208 \u2110), a)\n[PROOFSTEP]\nexact iSup\u2082 h\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 (I : Ideal A), I \u2208 \u2110 \u2192 IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (InfSet.sInf \u2110)\n[PROOFSTEP]\nrw [sInf_eq_iInf]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u2110 : Set (Ideal A)\nh : \u2200 (I : Ideal A), I \u2208 \u2110 \u2192 IsHomogeneous \ud835\udc9c I\n\u22a2 IsHomogeneous \ud835\udc9c (\u2a05 (a : Ideal A) (_ : a \u2208 \u2110), a)\n[PROOFSTEP]\nexact iInf\u2082 h\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\ns : \u03ba \u2192 HomogeneousIdeal \ud835\udc9c\n\u22a2 toIdeal (\u2a06 (i : \u03ba), s i) = \u2a06 (i : \u03ba), toIdeal (s i)\n[PROOFSTEP]\nrw [iSup, toIdeal_sSup, iSup_range]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\ns : \u03ba \u2192 HomogeneousIdeal \ud835\udc9c\n\u22a2 toIdeal (\u2a05 (i : \u03ba), s i) = \u2a05 (i : \u03ba), toIdeal (s i)\n[PROOFSTEP]\nrw [iInf, toIdeal_sInf, iInf_range]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\n\u03ba' : \u03ba \u2192 Sort u_6\ns : (i : \u03ba) \u2192 \u03ba' i \u2192 HomogeneousIdeal \ud835\udc9c\n\u22a2 toIdeal (\u2a06 (i : \u03ba) (j : \u03ba' i), s i j) = \u2a06 (i : \u03ba) (j : \u03ba' i), toIdeal (s i j)\n[PROOFSTEP]\nsimp_rw [toIdeal_iSup]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\n\u03ba : Sort u_5\n\u03ba' : \u03ba \u2192 Sort u_6\ns : (i : \u03ba) \u2192 \u03ba' i \u2192 HomogeneousIdeal \ud835\udc9c\n\u22a2 toIdeal (\u2a05 (i : \u03ba) (j : \u03ba' i), s i j) = \u2a05 (i : \u03ba) (j : \u03ba' i), toIdeal (s i j)\n[PROOFSTEP]\nsimp_rw [toIdeal_iInf]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nHI : IsHomogeneous \ud835\udc9c I\nHJ : IsHomogeneous \ud835\udc9c J\n\u22a2 IsHomogeneous \ud835\udc9c (I * J)\n[PROOFSTEP]\nrw [Ideal.IsHomogeneous.iff_exists] at HI HJ \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nHI : \u2203 S, I = span (Subtype.val '' S)\nHJ : \u2203 S, J = span (Subtype.val '' S)\n\u22a2 \u2203 S, I * J = span (Subtype.val '' S)\n[PROOFSTEP]\nobtain \u27e8\u27e8s\u2081, rfl\u27e9, \u27e8s\u2082, rfl\u27e9\u27e9 := HI, HJ\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081) * span (Subtype.val '' s\u2082) = span (Subtype.val '' S)\n[PROOFSTEP]\nrw [Ideal.span_mul_span']\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081 * Subtype.val '' s\u2082) = span (Subtype.val '' S)\n[PROOFSTEP]\nexact \u27e8s\u2081 * s\u2082, congr_arg _ <| (Set.image_mul (homogeneousSubmonoid \ud835\udc9c).subtype).symm\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 homogeneousCore' \ud835\udc9c I = sSup {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I}\n[PROOFSTEP]\nrefine' (IsLUB.sSup_eq _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 IsLUB {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I} (homogeneousCore' \ud835\udc9c I)\n[PROOFSTEP]\napply IsGreatest.isLUB\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 IsGreatest {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I} (homogeneousCore' \ud835\udc9c I)\n[PROOFSTEP]\nhave coe_mono : Monotone (toIdeal : HomogeneousIdeal \ud835\udc9c \u2192 Ideal A) := fun x y => id\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\n\u22a2 IsGreatest {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I} (homogeneousCore' \ud835\udc9c I)\n[PROOFSTEP]\nconvert coe_mono.map_isGreatest (Ideal.homogeneousCore.gc \ud835\udc9c).isGreatest_u using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\n\u22a2 {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I} = toIdeal '' {a | toIdeal a \u2264 I}\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : Ideal A\n\u22a2 x \u2208 {J | IsHomogeneous \ud835\udc9c J \u2227 J \u2264 I} \u2194 x \u2208 toIdeal '' {a | toIdeal a \u2264 I}\n[PROOFSTEP]\nrw [mem_image, mem_setOf_eq]\n[GOAL]\ncase h.e'_3.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : Ideal A\n\u22a2 IsHomogeneous \ud835\udc9c x \u2227 x \u2264 I \u2194 \u2203 x_1, x_1 \u2208 {a | toIdeal a \u2264 I} \u2227 toIdeal x_1 = x\n[PROOFSTEP]\nrefine' \u27e8fun hI => \u27e8\u27e8x, hI.1\u27e9, \u27e8hI.2, rfl\u27e9\u27e9, _\u27e9\n[GOAL]\ncase h.e'_3.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : Ideal A\n\u22a2 (\u2203 x_1, x_1 \u2208 {a | toIdeal a \u2264 I} \u2227 toIdeal x_1 = x) \u2192 IsHomogeneous \ud835\udc9c x \u2227 x \u2264 I\n[PROOFSTEP]\nrintro \u27e8x, \u27e8hx, rfl\u27e9\u27e9\n[GOAL]\ncase h.e'_3.h.intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ncoe_mono : Monotone toIdeal\nx : HomogeneousIdeal \ud835\udc9c\nhx : x \u2208 {a | toIdeal a \u2264 I}\n\u22a2 IsHomogeneous \ud835\udc9c (toIdeal x) \u2227 toIdeal x \u2264 I\n[PROOFSTEP]\nexact \u27e8x.isHomogeneous, hx\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 IsHomogeneous \ud835\udc9c (span {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r})\n[PROOFSTEP]\nrefine' Ideal.homogeneous_span _ _ fun x hx => _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nx : A\nhx : x \u2208 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r}\n\u22a2 Homogeneous \ud835\udc9c x\n[PROOFSTEP]\nobtain \u27e8i, x, rfl\u27e9 := hx\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ni : \u03b9\nx : { x // x \u2208 I }\n\u22a2 Homogeneous \ud835\udc9c \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i)\n[PROOFSTEP]\napply SetLike.homogeneous_coe\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 I \u2264 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\nintro r hr\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 r \u2208 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\nclassical\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r]\nrefine' Ideal.sum_mem _ _\nintro j _\napply Ideal.subset_span\nuse j\nuse\u27e8r, hr\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 r \u2208 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\nrw [\u2190 DirectSum.sum_support_decompose \ud835\udc9c r]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 \u2211 i in DFinsupp.support (\u2191(decompose \ud835\udc9c) r), \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\nrefine' Ideal.sum_mem _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\n\u22a2 \u2200 (c : \u03b9), c \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r) \u2192 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) c) \u2208 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\nintro j _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) j) \u2208 toIdeal (homogeneousHull \ud835\udc9c I)\n[PROOFSTEP]\napply Ideal.subset_span\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) j) \u2208 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r}\n[PROOFSTEP]\nuse j\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nr : A\nhr : r \u2208 I\nj : \u03b9\na\u271d : j \u2208 DFinsupp.support (\u2191(decompose \ud835\udc9c) r)\n\u22a2 \u2203 x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) j) = \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) j)\n[PROOFSTEP]\nuse\u27e8r, hr\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nI_le_J : I \u2264 J\n\u22a2 homogeneousHull \ud835\udc9c I \u2264 homogeneousHull \ud835\udc9c J\n[PROOFSTEP]\napply Ideal.span_mono\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nI_le_J : I \u2264 J\n\u22a2 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r} \u2286 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r}\n[PROOFSTEP]\nrintro r \u27e8hr1, \u27e8x, hx\u27e9, rfl\u27e9\n[GOAL]\ncase a.intro.intro.mk\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nI_le_J : I \u2264 J\nhr1 : \u03b9\nx : A\nhx : x \u2208 I\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191{ val := x, property := hx }) hr1) \u2208 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r}\n[PROOFSTEP]\nrefine' \u27e8hr1, \u27e8\u27e8x, I_le_J hx\u27e9, rfl\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\n\u22a2 toIdeal (homogeneousHull \ud835\udc9c I) = I\n[PROOFSTEP]\napply le_antisymm _ (Ideal.le_toIdeal_homogeneousHull _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\n\u22a2 toIdeal (homogeneousHull \ud835\udc9c I) \u2264 I\n[PROOFSTEP]\napply Ideal.span_le.2\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\n\u22a2 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r} \u2286 \u2191I\n[PROOFSTEP]\nrintro _ \u27e8i, x, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsHomogeneous \ud835\udc9c I\ni : \u03b9\nx : { x // x \u2208 I }\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) \u2208 \u2191I\n[PROOFSTEP]\nexact h _ x.prop\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 toIdeal (homogeneousHull \ud835\udc9c I) = \u2a06 (i : \u03b9), span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I)\n[PROOFSTEP]\nrw [\u2190 Ideal.span_iUnion]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 toIdeal (homogeneousHull \ud835\udc9c I) = span (\u22c3 (i : \u03b9), \u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I)\n[PROOFSTEP]\napply congr_arg Ideal.span _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r} = \u22c3 (i : \u03b9), \u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nx\u271d : A\n\u22a2 x\u271d \u2208 {r | \u2203 i x, \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191x) i) = r} \u2194 x\u271d \u2208 \u22c3 (i : \u03b9), \u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_image, mem_setOf_eq, GradedRing.proj_apply, SetLike.exists, exists_prop,\n Subtype.coe_mk, SetLike.mem_coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ni : \u03b9\n\u22a2 \u2200 (x : A), x \u2208 \u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I \u2192 Homogeneous \ud835\udc9c x\n[PROOFSTEP]\nrintro _ \u27e8x, -, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ni : \u03b9\nx : A\n\u22a2 Homogeneous \ud835\udc9c (\u2191(GradedRing.proj \ud835\udc9c i) x)\n[PROOFSTEP]\napply SetLike.homogeneous_coe\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 homogeneousHull \ud835\udc9c I =\n \u2a06 (i : \u03b9),\n { toSubmodule := span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I),\n is_homogeneous' := (_ : IsHomogeneous \ud835\udc9c (span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I))) }\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 toIdeal (homogeneousHull \ud835\udc9c I) =\n toIdeal\n (\u2a06 (i : \u03b9),\n { toSubmodule := span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I),\n is_homogeneous' := (_ : IsHomogeneous \ud835\udc9c (span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I))) })\n[PROOFSTEP]\nrw [Ideal.toIdeal_homogeneousHull_eq_iSup, toIdeal_iSup]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\n\u22a2 \u2a06 (i : \u03b9), span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I) =\n \u2a06 (i : \u03b9),\n toIdeal\n { toSubmodule := span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I),\n is_homogeneous' := (_ : IsHomogeneous \ud835\udc9c (span (\u2191(GradedRing.proj \ud835\udc9c i) '' \u2191I))) }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) 0) = 0\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i) \u2208 RingHom.ker (projZeroRingHom \ud835\udc9c)\n[PROOFSTEP]\nchange (decompose \ud835\udc9c (decompose \ud835\udc9c r _ : A) 0 : A) = 0\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) 0) = 0\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i)) 0) = 0\n[PROOFSTEP]\nby_cases h : i = 0\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) 0) = 0\nh : i = 0\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i)) 0) = 0\n[PROOFSTEP]\nrw [h, hr, decompose_zero, zero_apply, ZeroMemClass.coe_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nR : Type u_3\nA : Type u_4\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\ni : \u03b9\nr : A\nhr : \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) 0) = 0\nh : \u00aci = 0\n\u22a2 \u2191(\u2191(\u2191(decompose \ud835\udc9c) \u2191(\u2191(\u2191(decompose \ud835\udc9c) r) i)) 0) = 0\n[PROOFSTEP]\nrw [decompose_of_mem_ne \ud835\udc9c (SetLike.coe_mem _) h]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal", "llama_tokens": 23162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.28861323278484946}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na\u271d b c d : WithTop \u03b1\nx y : \u03b1\na : WithTop \u03b1\n\u22a2 a + \u22a4 = \u22a4\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\n\u22a2 none + \u22a4 = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y val\u271d : \u03b1\n\u22a2 Option.some val\u271d + \u22a4 = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\n\u22a2 a + b = \u22a4 \u2194 a = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\n\u22a2 none + b = \u22a4 \u2194 none = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nb c d : WithTop \u03b1\nx y val\u271d : \u03b1\n\u22a2 Option.some val\u271d + b = \u22a4 \u2194 Option.some val\u271d = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\n\u22a2 none + none = \u22a4 \u2194 none = \u22a4 \u2228 none = \u22a4\n[PROOFSTEP]\nsimp [none_eq_top, some_eq_coe, \u2190 WithTop.coe_add]\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nc d : WithTop \u03b1\nx y val\u271d : \u03b1\n\u22a2 none + Option.some val\u271d = \u22a4 \u2194 none = \u22a4 \u2228 Option.some val\u271d = \u22a4\n[PROOFSTEP]\nsimp [none_eq_top, some_eq_coe, \u2190 WithTop.coe_add]\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nc d : WithTop \u03b1\nx y val\u271d : \u03b1\n\u22a2 Option.some val\u271d + none = \u22a4 \u2194 Option.some val\u271d = \u22a4 \u2228 none = \u22a4\n[PROOFSTEP]\nsimp [none_eq_top, some_eq_coe, \u2190 WithTop.coe_add]\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\nc d : WithTop \u03b1\nx y val\u271d\u00b9 val\u271d : \u03b1\n\u22a2 Option.some val\u271d\u00b9 + Option.some val\u271d = \u22a4 \u2194 Option.some val\u271d\u00b9 = \u22a4 \u2228 Option.some val\u271d = \u22a4\n[PROOFSTEP]\nsimp [none_eq_top, some_eq_coe, \u2190 WithTop.coe_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\na\u271d b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : LT \u03b1\na b : WithTop \u03b1\n\u22a2 a + b < \u22a4 \u2194 a < \u22a4 \u2227 b < \u22a4\n[PROOFSTEP]\nsimp_rw [WithTop.lt_top_iff_ne_top, add_ne_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\nb : WithTop \u03b1\nc : \u03b1\n\u22a2 none + b = \u2191c \u2194 \u2203 a' b', \u2191a' = none \u2227 \u2191b' = b \u2227 a' + b' = c\n[PROOFSTEP]\nsimp [none_eq_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na\u271d b c\u271d d : WithTop \u03b1\nx y a c : \u03b1\n\u22a2 Option.some a + none = \u2191c \u2194 \u2203 a' b', \u2191a' = Option.some a \u2227 \u2191b' = none \u2227 a' + b' = c\n[PROOFSTEP]\nsimp [none_eq_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na\u271d b\u271d c\u271d d : WithTop \u03b1\nx y a b c : \u03b1\n\u22a2 Option.some a + Option.some b = \u2191c \u2194 \u2203 a' b', \u2191a' = Option.some a \u2227 \u2191b' = Option.some b \u2227 a' + b' = c\n[PROOFSTEP]\nsimp only [some_eq_coe, \u2190 coe_add, coe_eq_coe, exists_and_left, exists_eq_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx\u271d y\u271d : \u03b1\nx : WithTop \u03b1\ny : \u03b1\n\u22a2 x + \u2191y = \u22a4 \u2194 x = \u22a4\n[PROOFSTEP]\ninduction x using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y\u271d y : \u03b1\n\u22a2 \u22a4 + \u2191y = \u22a4 \u2194 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp [\u2190 coe_add]\n[GOAL]\ncase coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y\u271d y a\u271d : \u03b1\n\u22a2 \u2191a\u271d + \u2191y = \u22a4 \u2194 \u2191a\u271d = \u22a4\n[PROOFSTEP]\nsimp [\u2190 coe_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y\u271d : \u03b1\ny : WithTop \u03b1\n\u22a2 \u2191x + y = \u22a4 \u2194 y = \u22a4\n[PROOFSTEP]\ninduction y using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\n\u22a2 \u2191x + \u22a4 = \u22a4 \u2194 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp [\u2190 coe_add]\n[GOAL]\ncase coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y a\u271d : \u03b1\n\u22a2 \u2191x + \u2191a\u271d = \u22a4 \u2194 \u2191a\u271d = \u22a4\n[PROOFSTEP]\nsimp [\u2190 coe_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsRightCancelAdd \u03b1\nha : a \u2260 \u22a4\n\u22a2 b + a = c + a \u2194 b = c\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsRightCancelAdd \u03b1\na : \u03b1\n\u22a2 b + \u2191a = c + \u2191a \u2194 b = c\n[PROOFSTEP]\nobtain rfl | hb := (eq_or_ne b \u22a4)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsRightCancelAdd \u03b1\na : \u03b1\n\u22a2 \u22a4 + \u2191a = c + \u2191a \u2194 \u22a4 = c\n[PROOFSTEP]\nrw [top_add, eq_comm, WithTop.add_coe_eq_top_iff, eq_comm]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsRightCancelAdd \u03b1\na : \u03b1\nhb : b \u2260 \u22a4\n\u22a2 b + \u2191a = c + \u2191a \u2194 b = c\n[PROOFSTEP]\nlift b to \u03b1 using hb\n[GOAL]\ncase intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsRightCancelAdd \u03b1\na b : \u03b1\n\u22a2 \u2191b + \u2191a = c + \u2191a \u2194 \u2191b = c\n[PROOFSTEP]\nsimp_rw [\u2190 WithTop.coe_add, eq_comm, WithTop.add_eq_coe, coe_eq_coe, exists_and_left, exists_eq_left, add_left_inj,\n exists_eq_right, eq_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsLeftCancelAdd \u03b1\nha : a \u2260 \u22a4\n\u22a2 a + b = a + c \u2194 b = c\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsLeftCancelAdd \u03b1\na : \u03b1\n\u22a2 \u2191a + b = \u2191a + c \u2194 b = c\n[PROOFSTEP]\nobtain rfl | hb := (eq_or_ne b \u22a4)\n[GOAL]\ncase intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsLeftCancelAdd \u03b1\na : \u03b1\n\u22a2 \u2191a + \u22a4 = \u2191a + c \u2194 \u22a4 = c\n[PROOFSTEP]\nrw [add_top, eq_comm, WithTop.coe_add_eq_top_iff, eq_comm]\n[GOAL]\ncase intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsLeftCancelAdd \u03b1\na : \u03b1\nhb : b \u2260 \u22a4\n\u22a2 \u2191a + b = \u2191a + c \u2194 b = c\n[PROOFSTEP]\nlift b to \u03b1 using hb\n[GOAL]\ncase intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d : IsLeftCancelAdd \u03b1\na b : \u03b1\n\u22a2 \u2191a + \u2191b = \u2191a + c \u2194 \u2191b = c\n[PROOFSTEP]\nsimp_rw [\u2190 WithTop.coe_add, eq_comm, WithTop.add_eq_coe, eq_comm, coe_eq_coe, exists_and_left, exists_eq_left',\n add_right_inj, exists_eq_right']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na\u271d b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : WithTop \u03b1\nh : b \u2264 c\n\u22a2 a + b \u2264 a + c\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb c : WithTop \u03b1\nh : b \u2264 c\n\u22a2 none + b \u2264 none + c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb c : WithTop \u03b1\nh : b \u2264 c\nval\u271d : \u03b1\n\u22a2 Option.some val\u271d + b \u2264 Option.some val\u271d + c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nh : b \u2264 none\n\u22a2 none + b \u2264 none + none\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nh : b \u2264 none\n\u22a2 none + b \u2264 none + none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 none + b \u2264 none + Option.some val\u271d\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 none + b \u2264 none + Option.some val\u271d\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 none\n\u22a2 Option.some val\u271d + b \u2264 Option.some val\u271d + none\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 none\n\u22a2 Option.some val\u271d + b \u2264 Option.some val\u271d + none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 + b \u2264 Option.some val\u271d\u00b9 + Option.some val\u271d\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 + b \u2264 Option.some val\u271d\u00b9 + Option.some val\u271d\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 + b \u2264 Option.some val\u271d\u00b9 + Option.some val\u271d\n[PROOFSTEP]\nrcases le_coe_iff.1 h with \u27e8b, rfl, _\u27e9\n[GOAL]\ncase some.some.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nval\u271d\u00b9 val\u271d b : \u03b1\nright\u271d : b \u2264 val\u271d\nh : \u2191b \u2264 Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 + \u2191b \u2264 Option.some val\u271d\u00b9 + Option.some val\u271d\n[PROOFSTEP]\nexact coe_le_coe.2 (add_le_add_left (coe_le_coe.1 h) _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na\u271d b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : WithTop \u03b1\nh : b \u2264 c\n\u22a2 swap (fun x x_1 => x + x_1) a b \u2264 swap (fun x x_1 => x + x_1) a c\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb c : WithTop \u03b1\nh : b \u2264 c\n\u22a2 swap (fun x x_1 => x + x_1) none b \u2264 swap (fun x x_1 => x + x_1) none c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb c : WithTop \u03b1\nh : b \u2264 c\nval\u271d : \u03b1\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d) c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nh : b \u2264 none\n\u22a2 swap (fun x x_1 => x + x_1) none b \u2264 swap (fun x x_1 => x + x_1) none none\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nh : b \u2264 none\n\u22a2 swap (fun x x_1 => x + x_1) none b \u2264 swap (fun x x_1 => x + x_1) none none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) none b \u2264 swap (fun x x_1 => x + x_1) none (Option.some val\u271d)\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) none b \u2264 swap (fun x x_1 => x + x_1) none (Option.some val\u271d)\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 none\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d) none\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : b \u2264 none\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d) none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d)\n[PROOFSTEP]\ntry exact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d)\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) b \u2264 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d)\n[PROOFSTEP]\nrcases le_coe_iff.1 h with \u27e8b, rfl, _\u27e9\n[GOAL]\ncase some.some.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nval\u271d\u00b9 val\u271d b : \u03b1\nright\u271d : b \u2264 val\u271d\nh : \u2191b \u2264 Option.some val\u271d\n\u22a2 swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) \u2191b \u2264\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d)\n[PROOFSTEP]\nexact coe_le_coe.2 (add_le_add_right (coe_le_coe.1 h) _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na\u271d b\u271d c\u271d d : WithTop \u03b1\nx y : \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 c : WithTop \u03b1\nh : a + b < a + c\n\u22a2 b < c\n[PROOFSTEP]\ninduction a using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \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\nb c : WithTop \u03b1\nh : \u22a4 + b < \u22a4 + c\n\u22a2 b < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \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\nb c : WithTop \u03b1\na\u271d : \u03b1\nh : \u2191a\u271d + b < \u2191a\u271d + c\n\u22a2 b < c\n[PROOFSTEP]\ninduction b using WithTop.recTopCoe\n[GOAL]\ncase coe.top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\na\u271d : \u03b1\nh : \u2191a\u271d + \u22a4 < \u2191a\u271d + c\n\u22a2 \u22a4 < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\na\u271d\u00b9 a\u271d : \u03b1\nh : \u2191a\u271d\u00b9 + \u2191a\u271d < \u2191a\u271d\u00b9 + c\n\u22a2 \u2191a\u271d < c\n[PROOFSTEP]\ninduction c using WithTop.recTopCoe\n[GOAL]\ncase coe.coe.top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\u271d\u00b9 a\u271d : \u03b1\nh : \u2191a\u271d\u00b9 + \u2191a\u271d < \u2191a\u271d\u00b9 + \u22a4\n\u22a2 \u2191a\u271d < \u22a4\n[PROOFSTEP]\nexact coe_lt_top _\n[GOAL]\ncase coe.coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\u271d\u00b2 a\u271d\u00b9 a\u271d : \u03b1\nh : \u2191a\u271d\u00b2 + \u2191a\u271d\u00b9 < \u2191a\u271d\u00b2 + \u2191a\u271d\n\u22a2 \u2191a\u271d\u00b9 < \u2191a\u271d\n[PROOFSTEP]\nexact coe_lt_coe.2 (lt_of_add_lt_add_left <| coe_lt_coe.1 h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na\u271d b\u271d c\u271d d : WithTop \u03b1\nx y : \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 c : WithTop \u03b1\nh : swap (fun x x_1 => x + x_1) a b < swap (fun x x_1 => x + x_1) a c\n\u22a2 b < c\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \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\nb c : WithTop \u03b1\nh : swap (fun x x_1 => x + x_1) none b < swap (fun x x_1 => x + x_1) none c\n\u22a2 b < c\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c\u271d d : WithTop \u03b1\nx y : \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\nb c : WithTop \u03b1\nval\u271d : \u03b1\nh : swap (fun x x_1 => x + x_1) (Option.some val\u271d) b < swap (fun x x_1 => x + x_1) (Option.some val\u271d) c\n\u22a2 b < c\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nh : swap (fun x x_1 => x + x_1) none none < swap (fun x x_1 => x + x_1) none c\n\u22a2 none < c\n[PROOFSTEP]\ntry exact (not_none_lt _ h).elim\n[GOAL]\ncase none.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nh : swap (fun x x_1 => x + x_1) none none < swap (fun x x_1 => x + x_1) none c\n\u22a2 none < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d : \u03b1\nh : swap (fun x x_1 => x + x_1) none (Option.some val\u271d) < swap (fun x x_1 => x + x_1) none c\n\u22a2 Option.some val\u271d < c\n[PROOFSTEP]\ntry exact (not_none_lt _ h).elim\n[GOAL]\ncase none.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d : \u03b1\nh : swap (fun x x_1 => x + x_1) none (Option.some val\u271d) < swap (fun x x_1 => x + x_1) none c\n\u22a2 Option.some val\u271d < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d : \u03b1\nh : swap (fun x x_1 => x + x_1) (Option.some val\u271d) none < swap (fun x x_1 => x + x_1) (Option.some val\u271d) c\n\u22a2 none < c\n[PROOFSTEP]\ntry exact (not_none_lt _ h).elim\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d : \u03b1\nh : swap (fun x x_1 => x + x_1) (Option.some val\u271d) none < swap (fun x x_1 => x + x_1) (Option.some val\u271d) c\n\u22a2 none < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d) < swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) c\n\u22a2 Option.some val\u271d < c\n[PROOFSTEP]\ntry exact (not_none_lt _ h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d) < swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) c\n\u22a2 Option.some val\u271d < c\n[PROOFSTEP]\nexact (not_none_lt _ h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c\u271d d : WithTop \u03b1\nx y : \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\nc : WithTop \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d) < swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) c\n\u22a2 Option.some val\u271d < c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) (Option.some val\u271d) <\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b9) none\n\u22a2 Option.some val\u271d < none\n[PROOFSTEP]\nexact coe_lt_top _\n[GOAL]\ncase some.some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\nval\u271d\u00b2 val\u271d\u00b9 val\u271d : \u03b1\nh :\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b2) (Option.some val\u271d\u00b9) <\n swap (fun x x_1 => x + x_1) (Option.some val\u271d\u00b2) (Option.some val\u271d)\n\u22a2 Option.some val\u271d\u00b9 < Option.some val\u271d\n[PROOFSTEP]\nexact coe_lt_coe.2 (lt_of_add_lt_add_right <| coe_lt_coe.1 h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\nha : a \u2260 \u22a4\nh : a + b \u2264 a + c\n\u22a2 b \u2264 c\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \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 : \u03b1\nh : \u2191a + b \u2264 \u2191a + c\n\u22a2 b \u2264 c\n[PROOFSTEP]\ninduction c using WithTop.recTopCoe\n[GOAL]\ncase intro.top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb d : WithTop \u03b1\nx y : \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 : \u03b1\nh : \u2191a + b \u2264 \u2191a + \u22a4\n\u22a2 b \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase intro.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb d : WithTop \u03b1\nx y : \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 a\u271d : \u03b1\nh : \u2191a + b \u2264 \u2191a + \u2191a\u271d\n\u22a2 b \u2264 \u2191a\u271d\n[PROOFSTEP]\ninduction b using WithTop.recTopCoe\n[GOAL]\ncase intro.coe.top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \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 a\u271d : \u03b1\nh : \u2191a + \u22a4 \u2264 \u2191a + \u2191a\u271d\n\u22a2 \u22a4 \u2264 \u2191a\u271d\n[PROOFSTEP]\nexact (not_top_le_coe _ h).elim\n[GOAL]\ncase intro.coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \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 a\u271d\u00b9 a\u271d : \u03b1\nh : \u2191a + \u2191a\u271d \u2264 \u2191a + \u2191a\u271d\u00b9\n\u22a2 \u2191a\u271d \u2264 \u2191a\u271d\u00b9\n[PROOFSTEP]\nsimp only [\u2190 coe_add, coe_le_coe] at h \u22a2\n[GOAL]\ncase intro.coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \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 a\u271d\u00b9 a\u271d : \u03b1\nh : a + a\u271d \u2264 a + a\u271d\u00b9\n\u22a2 a\u271d \u2264 a\u271d\u00b9\n[PROOFSTEP]\nexact le_of_add_le_add_left h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \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\nha : a \u2260 \u22a4\nh : b + a \u2264 c + a\n\u22a2 b \u2264 c\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \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 : \u03b1\nh : b + \u2191a \u2264 c + \u2191a\n\u22a2 b \u2264 c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase intro.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb d : WithTop \u03b1\nx y : \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 : \u03b1\nh : b + \u2191a \u2264 none + \u2191a\n\u22a2 b \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase intro.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb d : WithTop \u03b1\nx y : \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 val\u271d : \u03b1\nh : b + \u2191a \u2264 Option.some val\u271d + \u2191a\n\u22a2 b \u2264 Option.some val\u271d\n[PROOFSTEP]\ncases b\n[GOAL]\ncase intro.some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \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 val\u271d : \u03b1\nh : none + \u2191a \u2264 Option.some val\u271d + \u2191a\n\u22a2 none \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact (not_top_le_coe _ h).elim\n[GOAL]\ncase intro.some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \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 val\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d + \u2191a \u2264 Option.some val\u271d\u00b9 + \u2191a\n\u22a2 Option.some val\u271d \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact coe_le_coe.2 (le_of_add_le_add_right <| coe_le_coe.1 h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nha : a \u2260 \u22a4\nh : b < c\n\u22a2 a + b < a + c\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : b < c\na : \u03b1\n\u22a2 \u2191a + b < \u2191a + c\n[PROOFSTEP]\nrcases lt_iff_exists_coe.1 h with \u27e8b, rfl, h'\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : \u03b1\nh' h : \u2191b < c\n\u22a2 \u2191a + \u2191b < \u2191a + c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase intro.intro.intro.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : \u03b1\nh' h : \u2191b < none\n\u22a2 \u2191a + \u2191b < \u2191a + none\n[PROOFSTEP]\nexact coe_lt_top _\n[GOAL]\ncase intro.intro.intro.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b val\u271d : \u03b1\nh' h : \u2191b < Option.some val\u271d\n\u22a2 \u2191a + \u2191b < \u2191a + Option.some val\u271d\n[PROOFSTEP]\nexact coe_lt_coe.2 (add_lt_add_left (coe_lt_coe.1 h) _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\nha : a \u2260 \u22a4\nh : b < c\n\u22a2 b + a < c + a\n[PROOFSTEP]\nlift a to \u03b1 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nb c d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : b < c\na : \u03b1\n\u22a2 b + \u2191a < c + \u2191a\n[PROOFSTEP]\nrcases lt_iff_exists_coe.1 h with \u27e8b, rfl, h'\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nc d : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : \u03b1\nh' h : \u2191b < c\n\u22a2 \u2191b + \u2191a < c + \u2191a\n[PROOFSTEP]\ncases c\n[GOAL]\ncase intro.intro.intro.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : \u03b1\nh' h : \u2191b < none\n\u22a2 \u2191b + \u2191a < none + \u2191a\n[PROOFSTEP]\nexact coe_lt_top _\n[GOAL]\ncase intro.intro.intro.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\nd : WithTop \u03b1\nx y : \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b val\u271d : \u03b1\nh' h : \u2191b < Option.some val\u271d\n\u22a2 \u2191b + \u2191a < Option.some val\u271d + \u2191a\n[PROOFSTEP]\nexact coe_lt_coe.2 (add_lt_add_right (coe_lt_coe.1 h) _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na\u271d b\u271d c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\na b : WithTop \u03b1\n\u22a2 map (\u2191f) (a + b) = map (\u2191f) a + map (\u2191f) b\n[PROOFSTEP]\ninduction a using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\nb : WithTop \u03b1\n\u22a2 map (\u2191f) (\u22a4 + b) = map \u2191f \u22a4 + map (\u2191f) b\n[PROOFSTEP]\nexact (top_add _).symm\n[GOAL]\ncase coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b\u271d c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\nb : WithTop \u03b1\na\u271d : \u03b1\n\u22a2 map (\u2191f) (\u2191a\u271d + b) = map \u2191f \u2191a\u271d + map (\u2191f) b\n[PROOFSTEP]\ninduction b using WithTop.recTopCoe\n[GOAL]\ncase coe.top\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\na\u271d : \u03b1\n\u22a2 map (\u2191f) (\u2191a\u271d + \u22a4) = map \u2191f \u2191a\u271d + map \u2191f \u22a4\n[PROOFSTEP]\nexact (add_top _).symm\n[GOAL]\ncase coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\na\u271d\u00b9 a\u271d : \u03b1\n\u22a2 map (\u2191f) (\u2191a\u271d\u00b9 + \u2191a\u271d) = map \u2191f \u2191a\u271d\u00b9 + map \u2191f \u2191a\u271d\n[PROOFSTEP]\nrw [map_coe, map_coe, \u2190 coe_add, \u2190 coe_add, \u2190 map_add]\n[GOAL]\ncase coe.coe\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\nF : Type u_1\ninst\u271d\u00b9 : Add \u03b2\ninst\u271d : AddHomClass F \u03b1 \u03b2\nf : F\na\u271d\u00b9 a\u271d : \u03b1\n\u22a2 map \u2191f \u2191(a\u271d\u00b9 + a\u271d) = \u2191(\u2191f (a\u271d\u00b9 + a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b9 : One (WithTop \u03b1) := one\nsrc\u271d : AddMonoid (WithTop \u03b1) := addMonoid\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nsimp only\n -- Porting note: Had to add this...?\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b9 : One (WithTop \u03b1) := one\nsrc\u271d : AddMonoid (WithTop \u03b1) := addMonoid\n\u22a2 \u2191\u21910 = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, WithTop.coe_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b9 : One (WithTop \u03b1) := one\nsrc\u271d : AddMonoid (WithTop \u03b1) := addMonoid\nn : \u2115\n\u22a2 NatCast.natCast (n + 1) = NatCast.natCast n + 1\n[PROOFSTEP]\nsimp only\n -- Porting note: Had to add this...?\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : AddMonoidWithOne \u03b1\nsrc\u271d\u00b9 : One (WithTop \u03b1) := one\nsrc\u271d : AddMonoid (WithTop \u03b1) := addMonoid\nn : \u2115\n\u22a2 \u2191\u2191(n + 1) = \u2191\u2191n + 1\n[PROOFSTEP]\nrw [Nat.cast_add_one, WithTop.coe_add, WithTop.coe_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\n\u22a2 \u2200 (a b : WithTop \u03b1), a \u2264 b \u2192 \u2200 (c : WithTop \u03b1), c + a \u2264 c + b\n[PROOFSTEP]\nrintro a b h (_ | c)\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\na b : WithTop \u03b1\nh : a \u2264 b\n\u22a2 none + a \u2264 none + b\n[PROOFSTEP]\nsimp [none_eq_top]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\na b : WithTop \u03b1\nh : a \u2264 b\nc : \u03b1\n\u22a2 Option.some c + a \u2264 Option.some c + b\n[PROOFSTEP]\nrcases b with (_ | b)\n[GOAL]\ncase some.none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\na : WithTop \u03b1\nc : \u03b1\nh : a \u2264 none\n\u22a2 Option.some c + a \u2264 Option.some c + none\n[PROOFSTEP]\nsimp [none_eq_top]\n[GOAL]\ncase some.some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\na : WithTop \u03b1\nc b : \u03b1\nh : a \u2264 Option.some b\n\u22a2 Option.some c + a \u2264 Option.some c + Option.some b\n[PROOFSTEP]\nrcases le_coe_iff.1 h with \u27e8a, rfl, _\u27e9\n[GOAL]\ncase some.some.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\nc b a : \u03b1\nright\u271d : a \u2264 b\nh : \u2191a \u2264 Option.some b\n\u22a2 Option.some c + \u2191a \u2264 Option.some c + Option.some b\n[PROOFSTEP]\nsimp only [some_eq_coe, \u2190 coe_add, coe_le_coe] at h \u22a2\n[GOAL]\ncase some.some.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : OrderedAddCommMonoid \u03b1\nsrc\u271d\u00b9 : PartialOrder (WithTop \u03b1) := partialOrder\nsrc\u271d : AddCommMonoid (WithTop \u03b1) := addCommMonoid\nc b a : \u03b1\nright\u271d h : a \u2264 b\n\u22a2 c + a \u2264 c + b\n[PROOFSTEP]\nexact add_le_add_left h c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : Add \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na b : WithTop \u03b1\n\u22a2 \u22a4 \u2264 \u22a4 \u2192 \u2203 c, \u22a4 = \u22a4 + c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : Add \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na\u271d b\u271d : WithTop \u03b1\na b : \u03b1\nh : \u2191a \u2264 \u2191b\n\u22a2 \u2203 c, \u2191b = \u2191a + c\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := exists_add_of_le (WithTop.coe_le_coe.1 h)\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : Add \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na\u271d b : WithTop \u03b1\na c : \u03b1\nh : \u2191a \u2264 \u2191(a + c)\n\u22a2 \u2203 c_1, \u2191(a + c) = \u2191a + c_1\n[PROOFSTEP]\nexact \u27e8c, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : One M\ninst\u271d : One N\nf : OneHom M N\n\u22a2 map (\u2191f) 1 = 1\n[PROOFSTEP]\nrw [WithTop.map_one, map_one, coe_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na\u271d b c d : WithBot \u03b1\nx y : \u03b1\na : WithBot \u03b1\n\u22a2 a + \u22a5 = \u22a5\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithBot \u03b1\nx y : \u03b1\n\u22a2 none + \u22a5 = \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithBot \u03b1\nx y val\u271d : \u03b1\n\u22a2 Option.some val\u271d + \u22a5 = \u22a5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Add \u03b1\na b c d : WithBot \u03b1\nx y : \u03b1\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : One M\ninst\u271d : One N\nf : OneHom M N\n\u22a2 map (\u2191f) 1 = 1\n[PROOFSTEP]\nrw [WithBot.map_one, map_one, coe_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.WithTop", "llama_tokens": 17213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.28837770943708557}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One M\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nl : Filter \u03b1\nhf : f =\u1da0[l \u2293 \ud835\udcdf s] g\nhs : s =\u1da0[l] t\nx : \u03b1\nhst : x \u2208 s \u2194 x \u2208 t\nhfg : x \u2208 s \u2192 f x = g x\nhxs : x \u2208 s\n\u22a2 mulIndicator s f x = mulIndicator t g x\n[PROOFSTEP]\nsimp only [*, hst.1 hxs, mulIndicator_of_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One M\ns t : Set \u03b1\nf g : \u03b1 \u2192 M\na : \u03b1\nl : Filter \u03b1\nhf : f =\u1da0[l \u2293 \ud835\udcdf s] g\nhs : s =\u1da0[l] t\nx : \u03b1\nhst : x \u2208 s \u2194 x \u2208 t\nhfg : x \u2208 s \u2192 f x = g x\nhxs : \u00acx \u2208 s\n\u22a2 mulIndicator s f x = mulIndicator t g x\n[PROOFSTEP]\nsimp only [mulIndicator_of_not_mem, hxs, mt hst.2 hxs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Monotone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c3 (i : \u03b9), s i) f a\n[PROOFSTEP]\nclassical exact hs.piecewise_eventually_eq_iUnion f 1 a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Monotone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c3 (i : \u03b9), s i) f a\n[PROOFSTEP]\nexact hs.piecewise_eventually_eq_iUnion f 1 a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 (i : \u03b9), s i) f a\n[PROOFSTEP]\nclassical exact hs.piecewise_eventually_eq_iInter f 1 a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 (i : \u03b9), s i) f a\n[PROOFSTEP]\nexact hs.piecewise_eventually_eq_iInter f 1 a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun n => mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 n), s i) f a) =\u1da0[atTop] fun x => mulIndicator (iUnion s) f a\n[PROOFSTEP]\nrw [iUnion_eq_iUnion_finset s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun n => mulIndicator (\u22c3 (i : \u03b9) (_ : i \u2208 n), s i) f a) =\u1da0[atTop] fun x =>\n mulIndicator (\u22c3 (t : Finset \u03b9) (i : \u03b9) (_ : i \u2208 t), s i) f a\n[PROOFSTEP]\napply Monotone.mulIndicator_eventuallyEq_iUnion\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 Monotone fun i => \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 i), s i_1\n[PROOFSTEP]\nexact fun _ _ \u21a6 biUnion_subset_biUnion_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhf : f =\u1da0[l] 1\n\u22a2 mulIndicator s 1 =\u1da0[l] 1\n[PROOFSTEP]\nrw [mulIndicator_one']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, f x \u2260 1\ns t : Set \u03b1\nh : mulIndicator s f =\u1da0[l] mulIndicator t f\n\u22a2 s =\u1da0[l] t\n[PROOFSTEP]\nhave : \u2200 {s : Set \u03b1}, Function.mulSupport (s.mulIndicator f) =\u1da0[l] s := fun {s} \u21a6\n by\n rw [mulSupport_mulIndicator]\n exact (hf.mono fun x hx \u21a6 and_iff_left hx).set_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, f x \u2260 1\ns\u271d t : Set \u03b1\nh : mulIndicator s\u271d f =\u1da0[l] mulIndicator t f\ns : Set \u03b1\n\u22a2 Function.mulSupport (mulIndicator s f) =\u1da0[l] s\n[PROOFSTEP]\nrw [mulSupport_mulIndicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, f x \u2260 1\ns\u271d t : Set \u03b1\nh : mulIndicator s\u271d f =\u1da0[l] mulIndicator t f\ns : Set \u03b1\n\u22a2 s \u2229 Function.mulSupport f =\u1da0[l] s\n[PROOFSTEP]\nexact (hf.mono fun x hx \u21a6 and_iff_left hx).set_eq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\ninst\u271d : One \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200\u1da0 (x : \u03b1) in l, f x \u2260 1\ns t : Set \u03b1\nh : mulIndicator s f =\u1da0[l] mulIndicator t f\nthis : \u2200 {s : Set \u03b1}, Function.mulSupport (mulIndicator s f) =\u1da0[l] s\n\u22a2 s =\u1da0[l] t\n[PROOFSTEP]\nexact this.symm.trans <| h.mulSupport.trans this\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.IndicatorFunction", "llama_tokens": 2213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.2878216225833632}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : Arrow C\ninst\u271d : \u2200 (n : \u2115), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX\u271d Y\u271d : SimplexCategory\u1d52\u1d56\ng : X\u271d \u27f6 Y\u271d\n\u22a2 \u2200 (j : Fin (SimplexCategory.len Y\u271d.unop + 1)),\n (fun i => WidePullback.\u03c0 (fun x => f.hom) (\u2191(SimplexCategory.Hom.toOrderHom g.unop) i)) j \u226b f.hom =\n WidePullback.base fun x => f.hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf\u271d : Arrow C\ninst\u271d\u00b2 : \u2200 (n : \u2115), HasWidePullback f\u271d.right (fun x => f\u271d.left) fun x => f\u271d.hom\nf g : Arrow C\ninst\u271d\u00b9 : \u2200 (n : \u2115), HasWidePullback f.right (fun x => f.left) fun x => f.hom\ninst\u271d : \u2200 (n : \u2115), HasWidePullback g.right (fun x => g.left) fun x => g.hom\nF : f \u27f6 g\nn : SimplexCategory\u1d52\u1d56\nj : Fin (SimplexCategory.len n.unop + 1)\n\u22a2 (fun i => WidePullback.\u03c0 (fun x => f.hom) i \u226b F.left) j \u226b g.hom = (WidePullback.base fun x => f.hom) \u226b F.right\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : X \u27f6 Arrow.augmentedCechNerve F\n\u22a2 (\ud835\udfed C).map (NatTrans.app G.left (Opposite.op (SimplexCategory.mk 0)) \u226b WidePullback.\u03c0 (fun x => F.hom) 0) \u226b F.hom =\n (Augmented.toArrow.obj X).hom \u226b (\ud835\udfed C).map G.right\n[PROOFSTEP]\nhave := G.w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : X \u27f6 Arrow.augmentedCechNerve F\nthis : (\ud835\udfed (SimplicialObject C)).map G.left \u226b (Arrow.augmentedCechNerve F).hom = X.hom \u226b (const C).map G.right\n\u22a2 (\ud835\udfed C).map (NatTrans.app G.left (Opposite.op (SimplexCategory.mk 0)) \u226b WidePullback.\u03c0 (fun x => F.hom) 0) \u226b F.hom =\n (Augmented.toArrow.obj X).hom \u226b (\ud835\udfed C).map G.right\n[PROOFSTEP]\napply_fun fun e => e.app (Opposite.op <| SimplexCategory.mk 0) at this \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : X \u27f6 Arrow.augmentedCechNerve F\nthis :\n NatTrans.app ((\ud835\udfed (SimplicialObject C)).map G.left \u226b (Arrow.augmentedCechNerve F).hom)\n (Opposite.op (SimplexCategory.mk 0)) =\n NatTrans.app (X.hom \u226b (const C).map G.right) (Opposite.op (SimplexCategory.mk 0))\n\u22a2 (\ud835\udfed C).map (NatTrans.app G.left (Opposite.op (SimplexCategory.mk 0)) \u226b WidePullback.\u03c0 (fun x => F.hom) 0) \u226b F.hom =\n (Augmented.toArrow.obj X).hom \u226b (\ud835\udfed C).map G.right\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx : SimplexCategory\u1d52\u1d56\ni : Fin (SimplexCategory.len x.unop + 1)\n\u22a2 (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) i \u226b F.hom = NatTrans.app X.hom x \u226b G.right\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx : SimplexCategory\u1d52\u1d56\ni : Fin (SimplexCategory.len x.unop + 1)\n\u22a2 (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right\n[PROOFSTEP]\nerw [Category.assoc, Arrow.w, Augmented.toArrow_obj_hom, NatTrans.naturality_assoc, Functor.const_obj_map,\n Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\n\u22a2 \u2200 \u2983X_1 Y : SimplexCategory\u1d52\u1d56\u2984 (f : X_1 \u27f6 Y),\n X.left.map f \u226b\n (fun x =>\n WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) i \u226b F.hom =\n NatTrans.app X.hom x \u226b G.right))\n Y =\n (fun x =>\n WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) i \u226b F.hom =\n NatTrans.app X.hom x \u226b G.right))\n X_1 \u226b\n (Arrow.augmentedCechNerve F).left.map f\n[PROOFSTEP]\nintro x y f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\n\u22a2 X.left.map f \u226b\n (fun x =>\n WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) i \u226b F.hom =\n NatTrans.app X.hom x \u226b G.right))\n y =\n (fun x =>\n WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) i \u226b F.hom =\n NatTrans.app X.hom x \u226b G.right))\n x \u226b\n (Arrow.augmentedCechNerve F).left.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\n\u22a2 X.left.map f \u226b\n WidePullback.lift (NatTrans.app X.hom y \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const y.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len y.unop + 1)),\n (X.left.map (SimplexCategory.const y.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom y \u226b G.right) =\n WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right) \u226b\n WidePullback.lift (WidePullback.base fun x => F.hom)\n (fun i => WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y.unop + 1)),\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b F.hom =\n WidePullback.base fun x => F.hom)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len y.unop + 1)\n\u22a2 (X.left.map f \u226b\n WidePullback.lift (NatTrans.app X.hom y \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const y.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len y.unop + 1)),\n (X.left.map (SimplexCategory.const y.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom y \u226b G.right)) \u226b\n WidePullback.\u03c0 (fun x => F.hom) j\u271d =\n (WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right) \u226b\n WidePullback.lift (WidePullback.base fun x => F.hom)\n (fun i => WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y.unop + 1)),\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b F.hom =\n WidePullback.base fun x => F.hom)) \u226b\n WidePullback.\u03c0 (fun x => F.hom) j\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len y.unop + 1)\n\u22a2 (X.left.map f \u226b\n WidePullback.lift (NatTrans.app X.hom y \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const y.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len y.unop + 1)),\n (X.left.map (SimplexCategory.const y.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom y \u226b G.right)) \u226b\n WidePullback.\u03c0 (fun x => F.hom) j\u271d =\n (WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right) \u226b\n WidePullback.lift (WidePullback.base fun x => F.hom)\n (fun i => WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y.unop + 1)),\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b F.hom =\n WidePullback.base fun x => F.hom)) \u226b\n WidePullback.\u03c0 (fun x => F.hom) j\u271d\n[PROOFSTEP]\nsimp only [WidePullback.lift_\u03c0, Category.assoc, \u2190 X.left.map_comp_assoc]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len y.unop + 1)\n\u22a2 X.left.map (f \u226b (SimplexCategory.const y.unop j\u271d).op) \u226b G.left =\n X.left.map (SimplexCategory.const x.unop (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j\u271d)).op \u226b G.left\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\n\u22a2 ((X.left.map f \u226b\n WidePullback.lift (NatTrans.app X.hom y \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const y.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len y.unop + 1)),\n (X.left.map (SimplexCategory.const y.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom y \u226b G.right)) \u226b\n WidePullback.base fun x => F.hom) =\n (WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right) \u226b\n WidePullback.lift (WidePullback.base fun x => F.hom)\n (fun i => WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y.unop + 1)),\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b F.hom =\n WidePullback.base fun x => F.hom)) \u226b\n WidePullback.base fun x => F.hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nG : Augmented.toArrow.obj X \u27f6 F\nx y : SimplexCategory\u1d52\u1d56\nf : x \u27f6 y\n\u22a2 ((X.left.map f \u226b\n WidePullback.lift (NatTrans.app X.hom y \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const y.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len y.unop + 1)),\n (X.left.map (SimplexCategory.const y.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom y \u226b G.right)) \u226b\n WidePullback.base fun x => F.hom) =\n (WidePullback.lift (NatTrans.app X.hom x \u226b G.right)\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b G.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (X.left.map (SimplexCategory.const x.unop i).op \u226b G.left) \u226b F.hom = NatTrans.app X.hom x \u226b G.right) \u226b\n WidePullback.lift (WidePullback.base fun x => F.hom)\n (fun i => WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y.unop + 1)),\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b F.hom =\n WidePullback.base fun x => F.hom)) \u226b\n WidePullback.base fun x => F.hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\n\u22a2 Function.LeftInverse (equivalenceRightToLeft X F) (equivalenceLeftToRight X F)\n[PROOFSTEP]\nintro A\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 equivalenceRightToLeft X F (equivalenceLeftToRight X F A) = A\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 (equivalenceRightToLeft X F (equivalenceLeftToRight X F A)).left = A.left\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 WidePullback.lift (NatTrans.app X.hom (Opposite.op (SimplexCategory.mk 0)) \u226b A.right)\n (fun i => X.left.map (SimplexCategory.const (SimplexCategory.mk 0) i).op \u226b A.left)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len (Opposite.op (SimplexCategory.mk 0)).unop + 1)),\n (fun i => X.left.map (SimplexCategory.const (Opposite.op (SimplexCategory.mk 0)).unop i).op \u226b A.left) i \u226b\n F.hom =\n NatTrans.app X.hom (Opposite.op (SimplexCategory.mk 0)) \u226b A.right) \u226b\n WidePullback.\u03c0 (fun x => F.hom) 0 =\n A.left\n[PROOFSTEP]\nerw [WidePullback.lift_\u03c0]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 X.left.map (SimplexCategory.const (SimplexCategory.mk 0) 0).op \u226b A.left = A.left\n[PROOFSTEP]\nnth_rw 2 [\u2190 Category.id_comp A.left]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 X.left.map (SimplexCategory.const (SimplexCategory.mk 0) 0).op \u226b A.left = \ud835\udfd9 (Augmented.toArrow.obj X).left \u226b A.left\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h\u2081.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 X.left.map (SimplexCategory.const (SimplexCategory.mk 0) 0).op = \ud835\udfd9 (Augmented.toArrow.obj X).left\n[PROOFSTEP]\nconvert X.left.map_id _\n[GOAL]\ncase h.e'_2.h.h.e'_8\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 (SimplexCategory.const (SimplexCategory.mk 0) 0).op = \ud835\udfd9 (Opposite.op (SimplexCategory.mk 0))\n[PROOFSTEP]\nrw [\u2190 op_id]\n[GOAL]\ncase h.e'_2.h.h.e'_8\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 (SimplexCategory.const (SimplexCategory.mk 0) 0).op = (\ud835\udfd9 (SimplexCategory.mk 0)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_2.h.h.e'_8.e_f\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 SimplexCategory.const (SimplexCategory.mk 0) 0 = \ud835\udfd9 (SimplexCategory.mk 0)\n[PROOFSTEP]\next \u27e8a, ha\u27e9\n[GOAL]\ncase h.e'_2.h.h.e'_8.e_f.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\na : \u2115\nha : a < SimplexCategory.len (SimplexCategory.mk 0) + 1\n\u22a2 \u2191(\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const (SimplexCategory.mk 0) 0)) { val := a, isLt := ha }) =\n \u2191(\u2191(SimplexCategory.Hom.toOrderHom (\ud835\udfd9 (SimplexCategory.mk 0))) { val := a, isLt := ha })\n[PROOFSTEP]\nchange a < 1 at ha \n[GOAL]\ncase h.e'_2.h.h.e'_8.e_f.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\na : \u2115\nha : a < 1\n\u22a2 \u2191(\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const (SimplexCategory.mk 0) 0)) { val := a, isLt := ha }) =\n \u2191(\u2191(SimplexCategory.Hom.toOrderHom (\ud835\udfd9 (SimplexCategory.mk 0))) { val := a, isLt := ha })\n[PROOFSTEP]\nchange 0 = a\n[GOAL]\ncase h.e'_2.h.h.e'_8.e_f.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\na : \u2115\nha : a < 1\n\u22a2 0 = a\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : Augmented.toArrow.obj X \u27f6 F\n\u22a2 (equivalenceRightToLeft X F (equivalenceLeftToRight X F A)).right = A.right\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\n\u22a2 Function.RightInverse (equivalenceRightToLeft X F) (equivalenceLeftToRight X F)\n[PROOFSTEP]\nintro A\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\n\u22a2 equivalenceLeftToRight X F (equivalenceRightToLeft X F A) = A\n[PROOFSTEP]\next x : 2\n[GOAL]\ncase h\u2081.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\nx : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app (equivalenceLeftToRight X F (equivalenceRightToLeft X F A)).left x = NatTrans.app A.left x\n[PROOFSTEP]\nrefine' WidePullback.hom_ext _ _ _ (fun j => _) _\n[GOAL]\ncase h\u2081.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\nx : SimplexCategory\u1d52\u1d56\nj : Fin (SimplexCategory.len x.unop + 1)\n\u22a2 NatTrans.app (equivalenceLeftToRight X F (equivalenceRightToLeft X F A)).left x \u226b WidePullback.\u03c0 (fun x => F.hom) j =\n NatTrans.app A.left x \u226b WidePullback.\u03c0 (fun x => F.hom) j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\nx : SimplexCategory\u1d52\u1d56\nj : Fin (SimplexCategory.len x.unop + 1)\n\u22a2 WidePullback.lift (NatTrans.app X.hom x \u226b A.right)\n (fun i =>\n X.left.map (SimplexCategory.const x.unop i).op \u226b\n NatTrans.app A.left (Opposite.op (SimplexCategory.mk 0)) \u226b WidePullback.\u03c0 (fun x => F.hom) 0)\n (_ :\n \u2200 (i : Fin (SimplexCategory.len x.unop + 1)),\n (fun i => X.left.map (SimplexCategory.const x.unop i).op \u226b (equivalenceRightToLeft X F A).left) i \u226b F.hom =\n NatTrans.app X.hom x \u226b (equivalenceRightToLeft X F A).right) \u226b\n WidePullback.\u03c0 (fun x => F.hom) j =\n NatTrans.app A.left x \u226b WidePullback.\u03c0 (fun x => F.hom) j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\nx : SimplexCategory\u1d52\u1d56\nj : Fin (SimplexCategory.len x.unop + 1)\n\u22a2 NatTrans.app A.left x \u226b\n WidePullback.\u03c0 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const x.unop j)) 0) =\n NatTrans.app A.left x \u226b WidePullback.\u03c0 (fun x => F.hom) j\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2081.h.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\nx : SimplexCategory\u1d52\u1d56\n\u22a2 (NatTrans.app (equivalenceLeftToRight X F (equivalenceRightToLeft X F A)).left x \u226b WidePullback.base fun x => F.hom) =\n NatTrans.app A.left x \u226b WidePullback.base fun x => F.hom\n[PROOFSTEP]\nsimpa using congr_app A.w.symm x\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\nX : Augmented C\nF : Arrow C\nA : X \u27f6 Arrow.augmentedCechNerve F\n\u22a2 (equivalenceLeftToRight X F (equivalenceRightToLeft X F A)).right = A.right\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\n\u22a2 \u2200 {X' X : Augmented C} {Y : Arrow C} (f : X' \u27f6 X) (g : X \u27f6 augmentedCechNerve.obj Y),\n \u2191(cechNerveEquiv X' Y).symm (f \u226b g) = Augmented.toArrow.map f \u226b \u2191(cechNerveEquiv X Y).symm g\n[PROOFSTEP]\ndsimp [cechNerveEquiv]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\n\u22a2 \u2200 {X' X : Augmented C} {Y : Arrow C} (f : X' \u27f6 X) (g : X \u27f6 augmentedCechNerve.obj Y),\n equivalenceRightToLeft X' Y (f \u226b g) = Augmented.toArrow.map f \u226b equivalenceRightToLeft X Y g\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\n\u22a2 \u2200 {X : Augmented C} {Y Y' : Arrow C} (f : Augmented.toArrow.obj X \u27f6 Y) (g : Y \u27f6 Y'),\n \u2191(cechNerveEquiv X Y') (f \u226b g) = \u2191(cechNerveEquiv X Y) f \u226b augmentedCechNerve.map g\n[PROOFSTEP]\ndsimp [cechNerveEquiv]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom\n\u22a2 \u2200 {X : Augmented C} {Y Y' : Arrow C} (f : Augmented.toArrow.obj X \u27f6 Y) (g : Y \u27f6 Y'),\n equivalenceLeftToRight X Y' (f \u226b g) = equivalenceLeftToRight X Y f \u226b augmentedCechNerve.map g\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : Arrow C\ninst\u271d : \u2200 (n : \u2115), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nx y : SimplexCategory\ng : x \u27f6 y\n\u22a2 (fun n => widePushout f.left (fun x => f.right) fun x => f.hom) x \u27f6\n (fun n => widePushout f.left (fun x => f.right) fun x => f.hom) y\n[PROOFSTEP]\nrefine'\n WidePushout.desc (WidePushout.head _) (fun i => (@WidePushout.\u03b9 _ _ _ _ _ (fun _ => f.hom) ?_ (g.toOrderHom i)))\n (fun j => _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nf : Arrow C\ninst\u271d : \u2200 (n : \u2115), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nx y : SimplexCategory\ng : x \u27f6 y\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 f.hom \u226b (fun i => WidePushout.\u03b9 (fun x => f.hom) (\u2191(SimplexCategory.Hom.toOrderHom g) i)) j =\n WidePushout.head fun x => f.hom\n[PROOFSTEP]\nerw [\u2190 WidePushout.arrow_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf\u271d : Arrow C\ninst\u271d\u00b2 : \u2200 (n : \u2115), HasWidePushout f\u271d.left (fun x => f\u271d.right) fun x => f\u271d.hom\nf g : Arrow C\ninst\u271d\u00b9 : \u2200 (n : \u2115), HasWidePushout f.left (fun x => f.right) fun x => f.hom\ninst\u271d : \u2200 (n : \u2115), HasWidePushout g.left (fun x => g.right) fun x => g.hom\nF : f \u27f6 g\nn : SimplexCategory\ni : Fin (SimplexCategory.len n + 1)\n\u22a2 g.right \u27f6 (cechConerve g).obj n\n[PROOFSTEP]\napply WidePushout.\u03b9 _ i\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf\u271d : Arrow C\ninst\u271d\u00b2 : \u2200 (n : \u2115), HasWidePushout f\u271d.left (fun x => f\u271d.right) fun x => f\u271d.hom\nf g : Arrow C\ninst\u271d\u00b9 : \u2200 (n : \u2115), HasWidePushout f.left (fun x => f.right) fun x => f.hom\ninst\u271d : \u2200 (n : \u2115), HasWidePushout g.left (fun x => g.right) fun x => g.hom\nF : f \u27f6 g\nn : SimplexCategory\ni : Fin (SimplexCategory.len n + 1)\n\u22a2 f.hom \u226b (fun i => F.right \u226b WidePushout.\u03b9 (fun x => g.hom) i) i = F.left \u226b WidePushout.head fun x => g.hom\n[PROOFSTEP]\nrw [\u2190 Arrow.w_assoc F, \u2190 WidePushout.arrow_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 (\ud835\udfed C).map G.left \u226b (Augmented.toArrow.obj X).hom =\n F.hom \u226b (\ud835\udfed C).map (WidePushout.\u03b9 (fun x => F.hom) 0 \u226b NatTrans.app G.right (SimplexCategory.mk 0))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 G.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0) =\n F.hom \u226b WidePushout.\u03b9 (fun x => F.hom) 0 \u226b NatTrans.app G.right (SimplexCategory.mk 0)\n[PROOFSTEP]\nrw [@WidePushout.arrow_\u03b9_assoc _ _ _ _ _ (fun (_ : Fin 1) => F.hom) (by dsimp; infer_instance)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 HasWidePushout ((\ud835\udfed C).obj F.left) (fun x => (\ud835\udfed C).obj F.right) fun x => F.hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 HasWidePushout F.left (fun x => F.right) fun x => F.hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 G.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0) =\n (WidePushout.head fun x => F.hom) \u226b NatTrans.app G.right (SimplexCategory.mk 0)\n[PROOFSTEP]\nexact congr_app G.w (SimplexCategory.mk 0)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\n\u22a2 \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j = G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\nrintro j\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j = G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\nrw [\u2190 Arrow.w_assoc G]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 G.left \u226b (Augmented.toArrow.obj X).hom \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\nhave t := X.hom.naturality (x.const j)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\nt :\n ((const C).obj X.left).map (SimplexCategory.const x j) \u226b NatTrans.app X.hom x =\n NatTrans.app X.hom (SimplexCategory.mk 0) \u226b ((\ud835\udfed (CosimplicialObject C)).obj X.right).map (SimplexCategory.const x j)\n\u22a2 G.left \u226b (Augmented.toArrow.obj X).hom \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\ndsimp at t \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\nt :\n \ud835\udfd9 X.left \u226b NatTrans.app X.hom x = NatTrans.app X.hom (SimplexCategory.mk 0) \u226b X.right.map (SimplexCategory.const x j)\n\u22a2 G.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0) \u226b X.right.map (SimplexCategory.const x j) =\n G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\nsimp only [Category.id_comp] at t \n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\nt : NatTrans.app X.hom x = NatTrans.app X.hom (SimplexCategory.mk 0) \u226b X.right.map (SimplexCategory.const x j)\n\u22a2 G.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0) \u226b X.right.map (SimplexCategory.const x j) =\n G.left \u226b NatTrans.app X.hom x\n[PROOFSTEP]\nrw [\u2190 t]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\n\u22a2 \u2200 \u2983X_1 Y : SimplexCategory\u2984 (f : X_1 \u27f6 Y),\n (Arrow.augmentedCechConerve F).right.map f \u226b\n (fun x =>\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x)\n (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j =\n G.left \u226b NatTrans.app X.hom x))\n Y =\n (fun x =>\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x)\n (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j =\n G.left \u226b NatTrans.app X.hom x))\n X_1 \u226b\n X.right.map f\n[PROOFSTEP]\nintro x y f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\n\u22a2 (Arrow.augmentedCechConerve F).right.map f \u226b\n (fun x =>\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j = G.left \u226b NatTrans.app X.hom x))\n y =\n (fun x =>\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => G.right \u226b X.right.map (SimplexCategory.const x i)) j = G.left \u226b NatTrans.app X.hom x))\n x \u226b\n X.right.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\n\u22a2 WidePushout.desc (WidePushout.head fun x => F.hom)\n (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i)) j =\n WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom y) (fun i => G.right \u226b X.right.map (SimplexCategory.const y i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const y j) = G.left \u226b NatTrans.app X.hom y) =\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x) \u226b\n X.right.map f\n[PROOFSTEP]\next\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len x + 1)\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) j\u271d \u226b\n WidePushout.desc (WidePushout.head fun x => F.hom)\n (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i)) j =\n WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom y) (fun i => G.right \u226b X.right.map (SimplexCategory.const y i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const y j) = G.left \u226b NatTrans.app X.hom y) =\n WidePushout.\u03b9 (fun x => F.hom) j\u271d \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x) \u226b\n X.right.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len x + 1)\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) j\u271d \u226b\n WidePushout.desc (WidePushout.head fun x => F.hom)\n (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i)) j =\n WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom y) (fun i => G.right \u226b X.right.map (SimplexCategory.const y i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const y j) = G.left \u226b NatTrans.app X.hom y) =\n WidePushout.\u03b9 (fun x => F.hom) j\u271d \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x) \u226b\n X.right.map f\n[PROOFSTEP]\nsimp only [WidePushout.\u03b9_desc_assoc, WidePushout.\u03b9_desc]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len x + 1)\n\u22a2 G.right \u226b X.right.map (SimplexCategory.const y (\u2191(SimplexCategory.Hom.toOrderHom f) j\u271d)) =\n (G.right \u226b X.right.map (SimplexCategory.const x j\u271d)) \u226b X.right.map f\n[PROOFSTEP]\nrw [Category.assoc, \u2190 X.right.map_comp]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\nj\u271d : Fin (SimplexCategory.len x + 1)\n\u22a2 G.right \u226b X.right.map (SimplexCategory.const y (\u2191(SimplexCategory.Hom.toOrderHom f) j\u271d)) =\n G.right \u226b X.right.map (SimplexCategory.const x j\u271d \u226b f)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\n\u22a2 (WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (WidePushout.head fun x => F.hom)\n (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i)) j =\n WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom y) (fun i => G.right \u226b X.right.map (SimplexCategory.const y i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const y j) = G.left \u226b NatTrans.app X.hom y) =\n (WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x) \u226b\n X.right.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\n\u22a2 (WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (WidePushout.head fun x => F.hom)\n (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom f) i)) j =\n WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom y) (fun i => G.right \u226b X.right.map (SimplexCategory.const y i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len y + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const y j) = G.left \u226b NatTrans.app X.hom y) =\n (WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (G.left \u226b NatTrans.app X.hom x) (fun i => G.right \u226b X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b G.right \u226b X.right.map (SimplexCategory.const x j) = G.left \u226b NatTrans.app X.hom x) \u226b\n X.right.map f\n[PROOFSTEP]\nsimp only [Functor.const_obj_map, \u2190 NatTrans.naturality, WidePushout.head_desc_assoc, WidePushout.head_desc,\n Category.assoc]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nG : F \u27f6 Augmented.toArrow.obj X\nx y : SimplexCategory\nf : x \u27f6 y\n\u22a2 G.left \u226b NatTrans.app X.hom y = G.left \u226b \ud835\udfd9 X.left \u226b NatTrans.app X.hom y\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\n\u22a2 Function.LeftInverse (equivalenceRightToLeft F X) (equivalenceLeftToRight F X)\n[PROOFSTEP]\nintro A\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 equivalenceRightToLeft F X (equivalenceLeftToRight F X A) = A\n[PROOFSTEP]\next x : 2\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\n\u22a2 (equivalenceRightToLeft F X (equivalenceLeftToRight F X A)).left = A.left\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\n\u22a2 NatTrans.app (equivalenceRightToLeft F X (equivalenceLeftToRight F X A)).right x = NatTrans.app A.right x\n[PROOFSTEP]\nrefine' WidePushout.hom_ext _ _ _ (fun j => _) _\n[GOAL]\ncase h\u2082.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) j \u226b NatTrans.app (equivalenceRightToLeft F X (equivalenceLeftToRight F X A)).right x =\n WidePushout.\u03b9 (fun x => F.hom) j \u226b NatTrans.app A.right x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) j \u226b\n WidePushout.desc (A.left \u226b NatTrans.app X.hom x)\n (fun i =>\n (WidePushout.\u03b9 (fun x => F.hom) 0 \u226b NatTrans.app A.right (SimplexCategory.mk 0)) \u226b\n X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => (equivalenceLeftToRight F X A).right \u226b X.right.map (SimplexCategory.const x i)) j =\n (equivalenceLeftToRight F X A).left \u226b NatTrans.app X.hom x) =\n WidePushout.\u03b9 (fun x => F.hom) j \u226b NatTrans.app A.right x\n[PROOFSTEP]\nsimp only [Category.assoc, \u2190 NatTrans.naturality A.right, Arrow.augmentedCechConerve_right, SimplexCategory.len_mk,\n Arrow.cechConerve_map, colimit.\u03b9_desc, WidePushoutShape.mkCocone_\u03b9_app, colimit.\u03b9_desc_assoc]\n[GOAL]\ncase h\u2082.h.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\nj : Fin (SimplexCategory.len x + 1)\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) (\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const x j)) 0) \u226b\n NatTrans.app A.right x =\n WidePushout.\u03b9 (fun x => F.hom) j \u226b NatTrans.app A.right x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082.h.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\n\u22a2 (WidePushout.head fun x => F.hom) \u226b NatTrans.app (equivalenceRightToLeft F X (equivalenceLeftToRight F X A)).right x =\n (WidePushout.head fun x => F.hom) \u226b NatTrans.app A.right x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082.h.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\n\u22a2 (WidePushout.head fun x => F.hom) \u226b\n WidePushout.desc (A.left \u226b NatTrans.app X.hom x)\n (fun i =>\n (WidePushout.\u03b9 (fun x => F.hom) 0 \u226b NatTrans.app A.right (SimplexCategory.mk 0)) \u226b\n X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => (equivalenceLeftToRight F X A).right \u226b X.right.map (SimplexCategory.const x i)) j =\n (equivalenceLeftToRight F X A).left \u226b NatTrans.app X.hom x) =\n (WidePushout.head fun x => F.hom) \u226b NatTrans.app A.right x\n[PROOFSTEP]\nrw [colimit.\u03b9_desc]\n[GOAL]\ncase h\u2082.h.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : Arrow.augmentedCechConerve F \u27f6 X\nx : SimplexCategory\n\u22a2 NatTrans.app\n (WidePushoutShape.mkCocone (A.left \u226b NatTrans.app X.hom x)\n (fun i =>\n (WidePushout.\u03b9 (fun x => F.hom) 0 \u226b NatTrans.app A.right (SimplexCategory.mk 0)) \u226b\n X.right.map (SimplexCategory.const x i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len x + 1)),\n F.hom \u226b (fun i => (equivalenceLeftToRight F X A).right \u226b X.right.map (SimplexCategory.const x i)) j =\n (equivalenceLeftToRight F X A).left \u226b NatTrans.app X.hom x)).\u03b9\n none =\n (WidePushout.head fun x => F.hom) \u226b NatTrans.app A.right x\n[PROOFSTEP]\nexact congr_app A.w x\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\n\u22a2 Function.RightInverse (equivalenceRightToLeft F X) (equivalenceLeftToRight F X)\n[PROOFSTEP]\nintro A\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 equivalenceLeftToRight F X (equivalenceRightToLeft F X A) = A\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 (equivalenceLeftToRight F X (equivalenceRightToLeft F X A)).left = A.left\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 (equivalenceLeftToRight F X (equivalenceRightToLeft F X A)).right = A.right\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 WidePushout.\u03b9 (fun x => F.hom) 0 \u226b\n WidePushout.desc (A.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0))\n (fun i => A.right \u226b X.right.map (SimplexCategory.const (SimplexCategory.mk 0) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len (SimplexCategory.mk 0) + 1)),\n F.hom \u226b (fun i => A.right \u226b X.right.map (SimplexCategory.const (SimplexCategory.mk 0) i)) j =\n A.left \u226b NatTrans.app X.hom (SimplexCategory.mk 0)) =\n A.right\n[PROOFSTEP]\nerw [WidePushout.\u03b9_desc]\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 A.right \u226b X.right.map (SimplexCategory.const (SimplexCategory.mk 0) 0) = A.right\n[PROOFSTEP]\nnth_rw 2 [\u2190 Category.comp_id A.right]\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 A.right \u226b X.right.map (SimplexCategory.const (SimplexCategory.mk 0) 0) = A.right \u226b \ud835\udfd9 (Augmented.toArrow.obj X).right\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h\u2082.e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 X.right.map (SimplexCategory.const (SimplexCategory.mk 0) 0) = \ud835\udfd9 (Augmented.toArrow.obj X).right\n[PROOFSTEP]\nconvert X.right.map_id _\n[GOAL]\ncase h.e'_2.h.h.e'_8\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\n\u22a2 SimplexCategory.const (SimplexCategory.mk 0) 0 = \ud835\udfd9 (SimplexCategory.mk 0)\n[PROOFSTEP]\next \u27e8a, ha\u27e9\n[GOAL]\ncase h.e'_2.h.h.e'_8.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\na : \u2115\nha : a < SimplexCategory.len (SimplexCategory.mk 0) + 1\n\u22a2 \u2191(\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const (SimplexCategory.mk 0) 0)) { val := a, isLt := ha }) =\n \u2191(\u2191(SimplexCategory.Hom.toOrderHom (\ud835\udfd9 (SimplexCategory.mk 0))) { val := a, isLt := ha })\n[PROOFSTEP]\nchange a < 1 at ha \n[GOAL]\ncase h.e'_2.h.h.e'_8.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\na : \u2115\nha : a < 1\n\u22a2 \u2191(\u2191(SimplexCategory.Hom.toOrderHom (SimplexCategory.const (SimplexCategory.mk 0) 0)) { val := a, isLt := ha }) =\n \u2191(\u2191(SimplexCategory.Hom.toOrderHom (\ud835\udfd9 (SimplexCategory.mk 0))) { val := a, isLt := ha })\n[PROOFSTEP]\nchange 0 = a\n[GOAL]\ncase h.e'_2.h.h.e'_8.a.h.h.mk.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : \u2200 (n : \u2115) (f : Arrow C), HasWidePushout f.left (fun x => f.right) fun x => f.hom\nF : Arrow C\nX : Augmented C\nA : F \u27f6 Augmented.toArrow.obj X\na : \u2115\nha : a < 1\n\u22a2 0 = a\n[PROOFSTEP]\nlinarith\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasTerminal C\n\u03b9 : Type w\nX Y : C\n\u22a2 Unique (Y \u27f6 (wideCospan \u03b9 X).obj none)\n[PROOFSTEP]\ndsimp [wideCospan]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasTerminal C\n\u03b9 : Type w\nX Y : C\n\u22a2 Unique (Y \u27f6 \u22a4_ C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ni j : WidePullbackShape \u03b9\nf : i \u27f6 j\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map f \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n j =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n i \u226b\n (wideCospan \u03b9 X).map f\n[PROOFSTEP]\ncases f\n[GOAL]\ncase id\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ni : WidePullbackShape \u03b9\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.id i) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n i =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n i \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.id i)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase id.none\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.id none) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n none =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n none \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.id none)\ncase id.some\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nval\u271d : \u03b9\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.id (some val\u271d)) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n (some val\u271d) =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n (some val\u271d) \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.id (some val\u271d))\n[PROOFSTEP]\nall_goals dsimp; simp\n[GOAL]\ncase id.none\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.id none) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n none =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n none \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.id none)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase id.none\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\n\u22a2 \ud835\udfd9 (\u220f fun x => X) \u226b terminal.from (\u220f fun x => X) = terminal.from (\u220f fun x => X) \u226b (wideCospan \u03b9 X).map (\ud835\udfd9 none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id.some\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nval\u271d : \u03b9\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.id (some val\u271d)) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n (some val\u271d) =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n (some val\u271d) \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.id (some val\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase id.some\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nval\u271d : \u03b9\n\u22a2 \ud835\udfd9 (\u220f fun x => X) \u226b limit.\u03c0 (Discrete.functor fun x => X) { as := val\u271d } =\n limit.\u03c0 (Discrete.functor fun x => X) { as := val\u271d } \u226b (wideCospan \u03b9 X).map (\ud835\udfd9 (some val\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase term\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nj\u271d : \u03b9\n\u22a2 ((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).map (WidePullbackShape.Hom.term j\u271d) \u226b\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n none =\n (fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i })\n (some j\u271d) \u226b\n (wideCospan \u03b9 X).map (WidePullbackShape.Hom.term j\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase term\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nj\u271d : \u03b9\n\u22a2 \ud835\udfd9 (\u220f fun x => X) \u226b terminal.from (\u220f fun x => X) =\n limit.\u03c0 (Discrete.functor fun x => X) { as := j\u271d } \u226b (wideCospan \u03b9 X).map (WidePullbackShape.Hom.term j\u271d)\n[PROOFSTEP]\nsimp only [terminal.comp_from]\n[GOAL]\ncase term\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nj\u271d : \u03b9\n\u22a2 terminal.from (\u220f fun x => X) =\n limit.\u03c0 (Discrete.functor fun x => X) { as := j\u271d } \u226b (wideCospan \u03b9 X).map (WidePullbackShape.Hom.term j\u271d)\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 f = (fun s => Pi.lift fun j => NatTrans.app s.\u03c0 (some j)) s\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 f = Pi.lift fun j => NatTrans.app s.\u03c0 (some j)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nj : \u03b9\n\u22a2 f \u226b Pi.\u03c0 (fun x => X) j = (Pi.lift fun j => NatTrans.app s.\u03c0 (some j)) \u226b Pi.\u03c0 (fun x => X) j\n[PROOFSTEP]\ndsimp only [Limits.Pi.lift]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nj : \u03b9\n\u22a2 f \u226b Pi.\u03c0 (fun x => X) j =\n limit.lift (Discrete.functor fun b => X) (Fan.mk s.pt fun j => NatTrans.app s.\u03c0 (some j)) \u226b Pi.\u03c0 (fun x => X) j\n[PROOFSTEP]\nrw [limit.lift_\u03c0]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nj : \u03b9\n\u22a2 f \u226b Pi.\u03c0 (fun x => X) j = NatTrans.app (Fan.mk s.pt fun j => NatTrans.app s.\u03c0 (some j)).\u03c0 { as := j }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\ns : Cone (wideCospan \u03b9 X)\nf :\n s.pt \u27f6\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1 (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none))\n fun i => limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.pt\nh :\n \u2200 (j : WidePullbackShape \u03b9),\n f \u226b\n NatTrans.app\n { pt := \u220f fun x => X,\n \u03c0 :=\n NatTrans.mk fun X_1 =>\n Option.casesOn X_1\n (terminal.from (((Functor.const (WidePullbackShape \u03b9)).obj (\u220f fun x => X)).obj none)) fun i =>\n limit.\u03c0 (Discrete.functor fun x => X) { as := i } }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nj : \u03b9\n\u22a2 f \u226b Pi.\u03c0 (fun x => X) j = NatTrans.app s.\u03c0 (some j)\n[PROOFSTEP]\nrw [\u2190 h (some j)]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\n\u22a2 HasWidePullback (Arrow.mk (terminal.from X)).right (fun x => (Arrow.mk (terminal.from X)).left) fun x =>\n (Arrow.mk (terminal.from X)).hom\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nval\u271d : Fintype \u03b9\n\u22a2 HasWidePullback (Arrow.mk (terminal.from X)).right (fun x => (Arrow.mk (terminal.from X)).left) fun x =>\n (Arrow.mk (terminal.from X)).hom\n[PROOFSTEP]\nexact \u27e8\u27e8wideCospan.limitCone \u03b9 X\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasTerminal C\n\u03b9 : Type w\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : Finite \u03b9\nX : C\nj : \u03b9\n\u22a2 (limitIsoPi \u03b9 X).hom \u226b Pi.\u03c0 (fun x => X) j = WidePullback.\u03c0 (fun x => terminal.from X) j\n[PROOFSTEP]\nrw [\u2190 wideCospan.limitIsoPi_inv_comp_pi, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasTerminal C\n\u03b9 : Type w\ninst\u271d : HasFiniteProducts C\nX : C\nm n : SimplexCategory\u1d52\u1d56\nf : m \u27f6 n\n\u22a2 (Arrow.cechNerve (Arrow.mk (terminal.from X))).map f \u226b\n ((fun m => wideCospan.limitIsoPi (Fin (SimplexCategory.len m.unop + 1)) (Arrow.mk (terminal.from X)).left)\n n).hom =\n ((fun m => wideCospan.limitIsoPi (Fin (SimplexCategory.len m.unop + 1)) (Arrow.mk (terminal.from X)).left) m).hom \u226b\n (cechNerveTerminalFrom X).map f\n[PROOFSTEP]\ndsimp only [cechNerveTerminalFrom, Arrow.cechNerve]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasTerminal C\n\u03b9 : Type w\ninst\u271d : HasFiniteProducts C\nX : C\nm n : SimplexCategory\u1d52\u1d56\nf : m \u27f6 n\n\u22a2 WidePullback.lift (WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom)\n (fun i =>\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len n.unop + 1)),\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b\n (Arrow.mk (terminal.from X)).hom =\n WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom) \u226b\n (wideCospan.limitIsoPi (Fin (SimplexCategory.len n.unop + 1)) (Arrow.mk (terminal.from X)).left).hom =\n (wideCospan.limitIsoPi (Fin (SimplexCategory.len m.unop + 1)) (Arrow.mk (terminal.from X)).left).hom \u226b\n Pi.lift fun i => Pi.\u03c0 (fun x => X) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i)\n[PROOFSTEP]\next \u27e8j\u27e9\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasTerminal C\n\u03b9 : Type w\ninst\u271d : HasFiniteProducts C\nX : C\nm n : SimplexCategory\u1d52\u1d56\nf : m \u27f6 n\nj : \u2115\nisLt\u271d : j < SimplexCategory.len n.unop + 1\n\u22a2 (WidePullback.lift (WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom)\n (fun i =>\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len n.unop + 1)),\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b\n (Arrow.mk (terminal.from X)).hom =\n WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom) \u226b\n (wideCospan.limitIsoPi (Fin (SimplexCategory.len n.unop + 1)) (Arrow.mk (terminal.from X)).left).hom) \u226b\n Pi.\u03c0 (fun x => X) { val := j, isLt := isLt\u271d } =\n ((wideCospan.limitIsoPi (Fin (SimplexCategory.len m.unop + 1)) (Arrow.mk (terminal.from X)).left).hom \u226b\n Pi.lift fun i => Pi.\u03c0 (fun x => X) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i)) \u226b\n Pi.\u03c0 (fun x => X) { val := j, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp only [Category.assoc, limit.lift_\u03c0, Fan.mk_\u03c0_app]\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasTerminal C\n\u03b9 : Type w\ninst\u271d : HasFiniteProducts C\nX : C\nm n : SimplexCategory\u1d52\u1d56\nf : m \u27f6 n\nj : \u2115\nisLt\u271d : j < SimplexCategory.len n.unop + 1\n\u22a2 WidePullback.lift (WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom)\n (fun i =>\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len n.unop + 1)),\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b\n (Arrow.mk (terminal.from X)).hom =\n WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom) \u226b\n (wideCospan.limitIsoPi (Fin (SimplexCategory.len n.unop + 1)) (Arrow.mk (terminal.from X)).left).hom \u226b\n Pi.\u03c0 (fun x => X) { val := j, isLt := isLt\u271d } =\n (wideCospan.limitIsoPi (Fin (SimplexCategory.len m.unop + 1)) (Arrow.mk (terminal.from X)).left).hom \u226b\n Pi.\u03c0 (fun x => X) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) { val := j, isLt := isLt\u271d })\n[PROOFSTEP]\nerw [wideCospan.limitIsoPi_hom_comp_pi, wideCospan.limitIsoPi_hom_comp_pi, limit.lift_\u03c0]\n[GOAL]\ncase h.mk\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasTerminal C\n\u03b9 : Type w\ninst\u271d : HasFiniteProducts C\nX : C\nm n : SimplexCategory\u1d52\u1d56\nf : m \u27f6 n\nj : \u2115\nisLt\u271d : j < SimplexCategory.len n.unop + 1\n\u22a2 NatTrans.app\n (WidePullbackShape.mkCone (WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom)\n (fun i =>\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) i))\n (_ :\n \u2200 (j : Fin (SimplexCategory.len n.unop + 1)),\n WidePullback.\u03c0 (fun x => (Arrow.mk (terminal.from X)).hom) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) j) \u226b\n (Arrow.mk (terminal.from X)).hom =\n WidePullback.base fun x => (Arrow.mk (terminal.from X)).hom)).\u03c0\n (some { val := j, isLt := isLt\u271d }) =\n WidePullback.\u03c0 (fun x => terminal.from X) (\u2191(SimplexCategory.Hom.toOrderHom f.unop) { val := j, isLt := isLt\u271d })\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.CechNerve", "llama_tokens": 30265, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.2878216225833632}} {"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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\n\u22a2 Set.Pairwise (\u22c3 (n : \u03ba), f n) r \u2194 \u2200 (n : \u03ba), Set.Pairwise (f n) r\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\n\u22a2 Set.Pairwise (\u22c3 (n : \u03ba), f n) r \u2192 \u2200 (n : \u03ba), Set.Pairwise (f n) r\n[PROOFSTEP]\nintro H n\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nH : Set.Pairwise (\u22c3 (n : \u03ba), f n) r\nn : \u03ba\n\u22a2 Set.Pairwise (f n) r\n[PROOFSTEP]\nexact Pairwise.mono (subset_iUnion _ _) H\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\n\u22a2 (\u2200 (n : \u03ba), Set.Pairwise (f n) r) \u2192 Set.Pairwise (\u22c3 (n : \u03ba), f n) r\n[PROOFSTEP]\nintro H i hi j hj hij\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nH : \u2200 (n : \u03ba), Set.Pairwise (f n) r\ni : \u03b1\nhi : i \u2208 \u22c3 (n : \u03ba), f n\nj : \u03b1\nhj : j \u2208 \u22c3 (n : \u03ba), f n\nhij : i \u2260 j\n\u22a2 r i j\n[PROOFSTEP]\nrcases mem_iUnion.1 hi with \u27e8m, hm\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\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nH : \u2200 (n : \u03ba), Set.Pairwise (f n) r\ni : \u03b1\nhi : i \u2208 \u22c3 (n : \u03ba), f n\nj : \u03b1\nhj : j \u2208 \u22c3 (n : \u03ba), f n\nhij : i \u2260 j\nm : \u03ba\nhm : i \u2208 f m\n\u22a2 r i j\n[PROOFSTEP]\nrcases mem_iUnion.1 hj with \u27e8n, hn\u27e9\n[GOAL]\ncase mpr.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nH : \u2200 (n : \u03ba), Set.Pairwise (f n) r\ni : \u03b1\nhi : i \u2208 \u22c3 (n : \u03ba), f n\nj : \u03b1\nhj : j \u2208 \u22c3 (n : \u03ba), f n\nhij : i \u2260 j\nm : \u03ba\nhm : i \u2208 f m\nn : \u03ba\nhn : j \u2208 f n\n\u22a2 r i j\n[PROOFSTEP]\nrcases h m n with \u27e8p, mp, np\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\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p\u271d q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf\u271d g : \u03b9 \u2192 \u03b1\ns t u : Set \u03b1\na b : \u03b1\nf : \u03ba \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\nH : \u2200 (n : \u03ba), Set.Pairwise (f n) r\ni : \u03b1\nhi : i \u2208 \u22c3 (n : \u03ba), f n\nj : \u03b1\nhj : j \u2208 \u22c3 (n : \u03ba), f n\nhij : i \u2260 j\nm : \u03ba\nhm : i \u2208 f m\nn : \u03ba\nhn : j \u2208 f n\np : \u03ba\nmp : f m \u2286 f p\nnp : f n \u2286 f p\n\u22a2 r i j\n[PROOFSTEP]\nexact H p (mp hm) (np hn) hij\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\n\u03ba : Sort u_6\nr\u271d p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf g : \u03b9 \u2192 \u03b1\ns\u271d t u : Set \u03b1\na b : \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nh : DirectedOn (fun x x_1 => x \u2286 x_1) s\n\u22a2 Set.Pairwise (\u22c3\u2080 s) r \u2194 \u2200 (a : Set \u03b1), a \u2208 s \u2192 Set.Pairwise a r\n[PROOFSTEP]\nrw [sUnion_eq_iUnion, pairwise_iUnion h.directed_val, SetCoe.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\n\u22a2 PairwiseDisjoint (\u22c3 (i : \u03b9') (_ : i \u2208 s), g i) f\n[PROOFSTEP]\nrintro a ha b 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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na : \u03b9\nha : a \u2208 \u22c3 (i : \u03b9') (_ : i \u2208 s), g i\nb : \u03b9\nhb : b \u2208 \u22c3 (i : \u03b9') (_ : i \u2208 s), g i\nhab : a \u2260 b\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nsimp_rw [Set.mem_iUnion] at ha hb \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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na b : \u03b9\nhab : a \u2260 b\nha : \u2203 i i_1, a \u2208 g i\nhb : \u2203 i i_1, b \u2208 g i\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nobtain \u27e8c, hc, ha\u27e9 := ha\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na b : \u03b9\nhab : a \u2260 b\nhb : \u2203 i i_1, b \u2208 g i\nc : \u03b9'\nhc : c \u2208 s\nha : a \u2208 g c\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nobtain \u27e8d, hd, hb\u27e9 := hb\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na b : \u03b9\nhab : a \u2260 b\nc : \u03b9'\nhc : c \u2208 s\nha : a \u2208 g c\nd : \u03b9'\nhd : d \u2208 s\nhb : b \u2208 g d\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nobtain hcd | hcd := eq_or_ne (g c) (g d)\n[GOAL]\ncase 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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na b : \u03b9\nhab : a \u2260 b\nc : \u03b9'\nhc : c \u2208 s\nha : a \u2208 g c\nd : \u03b9'\nhd : d \u2208 s\nhb : b \u2208 g d\nhcd : g c = g d\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nexact hg d hd (hcd.subst ha) hb hab\n[GOAL]\ncase 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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns\u271d : Set \u03b9\nt s : Set \u03b9'\ng : \u03b9' \u2192 Set \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : PairwiseDisjoint s fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 g i'), f i\nhg : \u2200 (i : \u03b9'), i \u2208 s \u2192 PairwiseDisjoint (g i) f\na b : \u03b9\nhab : a \u2260 b\nc : \u03b9'\nhc : c \u2208 s\nha : a \u2208 g c\nd : \u03b9'\nhd : d \u2208 s\nhb : b \u2208 g d\nhcd : g c \u2260 g d\n\u22a2 (Disjoint on f) a b\n[PROOFSTEP]\nexact\n (hs hc hd <| ne_of_apply_ne _ hcd).mono (le_iSup\u2082 (f := fun i (_ : i \u2208 g c) => f i) a ha)\n (le_iSup\u2082 (f := fun i (_ : i \u2208 g d) => f i) b hb)\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\n\u22a2 PairwiseDisjoint (s \u00d7\u02e2 t) f\n[PROOFSTEP]\nrintro \u27e8i, i'\u27e9 hi \u27e8j, j'\u27e9 hj h\n[GOAL]\ncase mk.mk\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i') \u2208 s \u00d7\u02e2 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j') \u2208 s \u00d7\u02e2 t\nh : (i, i') \u2260 (j, j')\n\u22a2 (Disjoint on f) (i, i') (j, j')\n[PROOFSTEP]\nrw [mem_prod] at hi hj \n[GOAL]\ncase mk.mk\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\n\u22a2 (Disjoint on f) (i, i') (j, j')\n[PROOFSTEP]\nobtain rfl | hij := eq_or_ne i j\n[GOAL]\ncase mk.mk.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj' : \u03b9'\nhj : (i, j').fst \u2208 s \u2227 (i, j').snd \u2208 t\nh : (i, i') \u2260 (i, j')\n\u22a2 (Disjoint on f) (i, i') (i, j')\n[PROOFSTEP]\nrefine' (ht hi.2 hj.2 <| (Prod.mk.inj_left _).ne_iff.1 h).mono _ _\n[GOAL]\ncase mk.mk.inl.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj' : \u03b9'\nhj : (i, j').fst \u2208 s \u2227 (i, j').snd \u2208 t\nh : (i, i') \u2260 (i, j')\n\u22a2 f (i, i') \u2264 (fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')) (i, i').snd\n[PROOFSTEP]\nconvert le_iSup\u2082 (\u03b1 := \u03b1) i hi.1\n[GOAL]\ncase h.e'_3\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj' : \u03b9'\nhj : (i, j').fst \u2208 s \u2227 (i, j').snd \u2208 t\nh : (i, i') \u2260 (i, j')\n\u22a2 f (i, i') = f (i, (i, i').snd)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.inl.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj' : \u03b9'\nhj : (i, j').fst \u2208 s \u2227 (i, j').snd \u2208 t\nh : (i, i') \u2260 (i, j')\n\u22a2 f (i, j') \u2264 (fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')) (i, j').snd\n[PROOFSTEP]\nconvert le_iSup\u2082 (\u03b1 := \u03b1) i hj.1\n[GOAL]\ncase h.e'_3\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj' : \u03b9'\nhj : (i, j').fst \u2208 s \u2227 (i, j').snd \u2208 t\nh : (i, i') \u2260 (i, j')\n\u22a2 f (i, j') = f (i, (i, j').snd)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\nhij : i \u2260 j\n\u22a2 (Disjoint on f) (i, i') (j, j')\n[PROOFSTEP]\nrefine' (hs hi.1 hj.1 hij).mono _ _\n[GOAL]\ncase mk.mk.inr.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\nhij : i \u2260 j\n\u22a2 f (i, i') \u2264 (fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')) (i, i').fst\n[PROOFSTEP]\nconvert le_iSup\u2082 (\u03b1 := \u03b1) i' hi.2\n[GOAL]\ncase h.e'_3\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\nhij : i \u2260 j\n\u22a2 f (i, i') = f ((i, i').fst, i')\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.inr.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\nhij : i \u2260 j\n\u22a2 f (j, j') \u2264 (fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')) (j, j').fst\n[PROOFSTEP]\nconvert le_iSup\u2082 (\u03b1 := \u03b1) j' hj.2\n[GOAL]\ncase h.e'_3\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : CompleteLattice \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nhs : PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')\nht : PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\ni : \u03b9\ni' : \u03b9'\nhi : (i, i').fst \u2208 s \u2227 (i, i').snd \u2208 t\nj : \u03b9\nj' : \u03b9'\nhj : (j, j').fst \u2208 s \u2227 (j, j').snd \u2208 t\nh : (i, i') \u2260 (j, j')\nhij : i \u2260 j\n\u22a2 f (j, j') = f ((j, j').fst, j')\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\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\n\u22a2 PairwiseDisjoint (s \u00d7\u02e2 t) f \u2194\n (PairwiseDisjoint s fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')) \u2227\n PairwiseDisjoint t fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8fun i hi j hj hij => _, fun i hi j hj hij => _\u27e9, fun h => h.1.prod_left h.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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => \u2a06 (i' : \u03b9') (_ : i' \u2208 t), f (i, i')) i j\n[PROOFSTEP]\nsimp_rw [Function.onFun, iSup_disjoint_iff, disjoint_iSup_iff]\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9'\nhi : i \u2208 t\nj : \u03b9'\nhj : j \u2208 t\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i' => \u2a06 (i : \u03b9) (_ : i \u2208 s), f (i, i')) i j\n[PROOFSTEP]\nsimp_rw [Function.onFun, iSup_disjoint_iff, disjoint_iSup_iff]\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhij : i \u2260 j\n\u22a2 \u2200 (i_1 : \u03b9'), i_1 \u2208 t \u2192 \u2200 (i_3 : \u03b9'), i_3 \u2208 t \u2192 Disjoint (f (i, i_1)) (f (j, i_3))\n[PROOFSTEP]\nintro i' hi' j' hj'\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9'\nhi : i \u2208 t\nj : \u03b9'\nhj : j \u2208 t\nhij : i \u2260 j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \u2192 \u2200 (i_3 : \u03b9), i_3 \u2208 s \u2192 Disjoint (f (i_1, i)) (f (i_3, j))\n[PROOFSTEP]\nintro i' hi' j' hj'\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhij : i \u2260 j\ni' : \u03b9'\nhi' : i' \u2208 t\nj' : \u03b9'\nhj' : j' \u2208 t\n\u22a2 Disjoint (f (i, i')) (f (j, j'))\n[PROOFSTEP]\nexact h (mk_mem_prod hi hi') (mk_mem_prod hj hj') (ne_of_apply_ne Prod.fst hij)\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Frame \u03b1\ns : Set \u03b9\nt : Set \u03b9'\nf : \u03b9 \u00d7 \u03b9' \u2192 \u03b1\nh : PairwiseDisjoint (s \u00d7\u02e2 t) f\ni : \u03b9'\nhi : i \u2208 t\nj : \u03b9'\nhj : j \u2208 t\nhij : i \u2260 j\ni' : \u03b9\nhi' : i' \u2208 s\nj' : \u03b9\nhj' : j' \u2208 s\n\u22a2 Disjoint (f (i', i)) (f (j', j))\n[PROOFSTEP]\nexact h (mk_mem_prod hi' hi) (mk_mem_prod hj' hj) (ne_of_apply_ne Prod.snd hij)\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\n\u22a2 (\u22c3 (i : \u03b9) (_ : i \u2208 s), f i) \\ \u22c3 (i : \u03b9) (_ : i \u2208 t), f i = \u22c3 (i : \u03b9) (_ : i \u2208 s \\ t), f i\n[PROOFSTEP]\nrefine'\n (biUnion_diff_biUnion_subset f s t).antisymm (iUnion\u2082_subset fun i hi a ha => (mem_diff _).2 \u27e8mem_biUnion hi.1 ha, _\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00aca \u2208 \u22c3 (x : \u03b9) (_ : x \u2208 t), f x\n[PROOFSTEP]\nrw [mem_iUnion\u2082]\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\n\u22a2 \u00ac\u2203 i j, a \u2208 f i\n[PROOFSTEP]\nrintro \u27e8j, hj, haj\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\n\u03b9' : Type u_5\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : PairwiseDisjoint (s \u222a t) f\ni : \u03b9\nhi : i \u2208 s \\ t\na : \u03b1\nha : a \u2208 f i\nj : \u03b9\nhj : j \u2208 t\nhaj : a \u2208 f j\n\u22a2 False\n[PROOFSTEP]\nexact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot \u27e8ha, haj\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\ns t : Set \u03b9\nh\u2080 : PairwiseDisjoint (s \u222a t) f\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Nonempty (f i)\nh : \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\n\u22a2 s \u2286 t\n[PROOFSTEP]\nrintro i hi\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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\ns t : Set \u03b9\nh\u2080 : PairwiseDisjoint (s \u222a t) f\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Nonempty (f i)\nh : \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 i \u2208 t\n[PROOFSTEP]\nobtain \u27e8a, hai\u27e9 := h\u2081 i hi\n[GOAL]\ncase 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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\ns t : Set \u03b9\nh\u2080 : PairwiseDisjoint (s \u222a t) f\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Nonempty (f i)\nh : \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\ni : \u03b9\nhi : i \u2208 s\na : \u03b1\nhai : a \u2208 f i\n\u22a2 i \u2208 t\n[PROOFSTEP]\nobtain \u27e8j, hj, haj\u27e9 := mem_iUnion\u2082.1 (h <| mem_iUnion\u2082_of_mem hi hai)\n[GOAL]\ncase 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\n\u03ba : Sort u_6\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b9 \u2192 Set \u03b1\ns t : Set \u03b9\nh\u2080 : PairwiseDisjoint (s \u222a t) f\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 Set.Nonempty (f i)\nh : \u22c3 (i : \u03b9) (_ : i \u2208 s), f i \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\ni : \u03b9\nhi : i \u2208 s\na : \u03b1\nhai : a \u2208 f i\nj : \u03b9\nhj : j \u2208 t\nhaj : a \u2208 f j\n\u22a2 i \u2208 t\n[PROOFSTEP]\nrwa [h\u2080.eq (subset_union_left _ _ hi) (subset_union_right _ _ hj) (not_disjoint_iff.2 \u27e8a, hai, haj\u27e9)]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Pairwise.Lattice", "llama_tokens": 10807, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.28782162258336313}} {"text": "[GOAL]\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nw :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x i,\n PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n f\n[PROOFSTEP]\napply continuous_iff_continuousAt.2\n[GOAL]\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nw :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x i,\n PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 \u2200 (x : { x // x \u2208 U }), ContinuousAt f x\n[PROOFSTEP]\nintro x\n[GOAL]\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nw :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x i,\n PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\nx : { x // x \u2208 U }\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nspecialize w x\n[GOAL]\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nx : { x // x \u2208 U }\nw :\n \u2203 V x i,\n PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nrcases w with \u27e8V, m, i, w\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\nw :\n PrelocalPredicate.pred\n { pred := fun {U} => PrelocalPredicate.pred src\u271d,\n res :=\n (_ :\n \u2200 {U V : Opens \u2191X} (i : U \u27f6 V) (f : { x // x \u2208 V } \u2192 \u2191T),\n PrelocalPredicate.pred src\u271d f \u2192\n PrelocalPredicate.pred src\u271d fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\ndsimp at w \n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\nw : Continuous fun x => f { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nrw [continuous_iff_continuousAt] at w \n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\nw : \u2200 (x : { x // x \u2208 V }), ContinuousAt (fun x => f { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nspecialize w \u27e8x, m\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d : \u2191X \u2192 Type v\nT : TopCat\nsrc\u271d : PrelocalPredicate fun x => \u2191T := continuousPrelocal X T\nU : Opens \u2191X\nf : { x // x \u2208 U } \u2192 \u2191T\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\nw : ContinuousAt (fun x => f { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) { val := \u2191x, property := m }\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nsimpa using (Opens.openEmbedding_of_le i.le).continuousAt_iff.1 w\n[GOAL]\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nV U : Opens \u2191X\ni : V \u27f6 U\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nw :\n (fun {U} f =>\n \u2200 (x : { x // x \u2208 U }), \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n f\nx : { x // x \u2208 V }\n\u22a2 \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n[PROOFSTEP]\nspecialize w (i x)\n[GOAL]\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nV U : Opens \u2191X\ni : V \u27f6 U\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 V }\nw : \u2203 V_1 x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n[PROOFSTEP]\nrcases w with \u27e8V', m', i', p\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nV U : Opens \u2191X\ni : V \u27f6 U\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 V }\nV' : Opens \u2191X\nm' : \u2191((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x) \u2208 V'\ni' : V' \u27f6 U\np : pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n[PROOFSTEP]\nrefine' \u27e8V \u2293 V', \u27e8x.2, m'\u27e9, Opens.infLELeft _ _, _\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nV U : Opens \u2191X\ni : V \u27f6 U\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 V }\nV' : Opens \u2191X\nm' : \u2191((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x) \u2208 V'\ni' : V' \u27f6 U\np : pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n[PROOFSTEP]\nconvert P.res (Opens.infLERight V V') _ p\n[GOAL]\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nw :\n \u2200 (x : { x // x \u2208 U }),\n \u2203 V x i,\n pred\n {\n pred := fun {U} f =>\n \u2200 (x : { x // x \u2208 U }), \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x),\n res :=\n (_ :\n \u2200 {V U : Opens \u2191X} (i : V \u27f6 U) (f : (x : { x // x \u2208 U }) \u2192 T \u2191x),\n (\u2200 (x : { x // x \u2208 U }),\n \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) \u2192\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\nx : { x // x \u2208 U }\n\u22a2 \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nspecialize w x\n[GOAL]\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 U }\nw :\n \u2203 V x i,\n pred\n {\n pred := fun {U} f =>\n \u2200 (x : { x // x \u2208 U }), \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x),\n res :=\n (_ :\n \u2200 {V U : Opens \u2191X} (i : V \u27f6 U) (f : (x : { x // x \u2208 U }) \u2192 T \u2191x),\n (\u2200 (x : { x // x \u2208 U }),\n \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) \u2192\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nrcases w with \u27e8V, m, i, p\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\np :\n pred\n {\n pred := fun {U} f =>\n \u2200 (x : { x // x \u2208 U }), \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x),\n res :=\n (_ :\n \u2200 {V U : Opens \u2191X} (i : V \u27f6 U) (f : (x : { x // x \u2208 U }) \u2192 T \u2191x),\n (\u2200 (x : { x // x \u2208 U }),\n \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) \u2192\n \u2200 (x : { x // x \u2208 V }),\n \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) }\n fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n\u22a2 \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nspecialize p \u27e8x.1, m\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\np :\n \u2203 V_1 x i_1,\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n\u22a2 \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nrcases p with \u27e8V', m', i', p'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nx : { x // x \u2208 U }\nV : Opens \u2191X\nm : \u2191x \u2208 V\ni : V \u27f6 U\nV' : Opens \u2191X\nm' : \u2191{ val := \u2191x, property := m } \u2208 V'\ni' : V' \u27f6 V\np' :\n pred P fun x =>\n (fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x))\n ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)\n\u22a2 \u2203 V x i, pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nexact \u27e8V', m', i' \u226b i, p'\u27e9\n[GOAL]\nX : TopCat\nT\u271d T : \u2191X \u2192 Type v\nP : PrelocalPredicate T\nU : Opens \u2191X\nf : (x : { x // x \u2208 U }) \u2192 T \u2191x\nh : pred P f\nx : { x // x \u2208 U }\n\u22a2 pred P fun x => f ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)\n[PROOFSTEP]\nconvert h\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\n\u22a2 \u2203! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s\n[PROOFSTEP]\nlet sf' : \u2200 i : \u03b9, (presheafToTypes X T).obj (op (U i)) := fun i => (sf i).val\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\n\u22a2 \u2203! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s\n[PROOFSTEP]\nhave sf'_comp : (presheafToTypes X T).IsCompatible U sf' := fun i j =>\n congr_arg Subtype.val\n (sf_comp i j)\n -- So, we can obtain a unique gluing\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\n\u22a2 \u2203! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s\n[PROOFSTEP]\nobtain \u27e8gl, gl_spec, gl_uniq\u27e9 := (sheafToTypes X T).existsUnique_gluing U sf' sf'_comp\n[GOAL]\ncase intro.intro\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\n\u22a2 \u2203! s, IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s\n[PROOFSTEP]\nrefine' \u27e8\u27e8gl, _\u27e9, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\n\u22a2 PrelocalPredicate.pred P.toPrelocalPredicate gl\n[PROOFSTEP]\napply P.locality\n[GOAL]\ncase intro.intro.refine'_1.x\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\n\u22a2 \u2200 (x : { x // x \u2208 (op (iSup U)).unop }),\n \u2203 V x i,\n PrelocalPredicate.pred P.toPrelocalPredicate fun x =>\n gl ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191(op (iSup U)).unop) }) x)\n[PROOFSTEP]\nrintro\n \u27e8x, mem\u27e9\n -- Once we're at a particular point `x`, we can select some open set `x \u2208 U i`.\n[GOAL]\ncase intro.intro.refine'_1.x.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\nx : \u2191X\nmem : x \u2208 (op (iSup U)).unop\n\u22a2 \u2203 V x i,\n PrelocalPredicate.pred P.toPrelocalPredicate fun x =>\n gl ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191(op (iSup U)).unop) }) x)\n[PROOFSTEP]\nchoose i hi using Opens.mem_iSup.mp mem\n[GOAL]\ncase intro.intro.refine'_1.x.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\nx : \u2191X\nmem : x \u2208 (op (iSup U)).unop\ni : \u03b9\nhi : x \u2208 U i\n\u22a2 \u2203 V x i,\n PrelocalPredicate.pred P.toPrelocalPredicate fun x =>\n gl ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191(op (iSup U)).unop) }) x)\n[PROOFSTEP]\nuse U i, hi, Opens.leSupr U i\n[GOAL]\ncase h\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\nx : \u2191X\nmem : x \u2208 (op (iSup U)).unop\ni : \u03b9\nhi : x \u2208 U i\n\u22a2 PrelocalPredicate.pred P.toPrelocalPredicate fun x =>\n gl ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191(op (iSup U)).unop) }) x)\n[PROOFSTEP]\nconvert (sf i).property using 1\n[GOAL]\ncase h.e'_5.h\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\nx : \u2191X\nmem : x \u2208 (op (iSup U)).unop\ni : \u03b9\nhi : x \u2208 U i\ne_4\u271d : U i = (op (U i)).unop\n\u22a2 (fun x => gl ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191(op (iSup U)).unop) }) x)) = \u2191(sf i)\n[PROOFSTEP]\nexact gl_spec i\n[GOAL]\ncase intro.intro.refine'_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\n\u22a2 (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s)\n { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) }\n[PROOFSTEP]\nexact fun i => Subtype.ext (gl_spec i)\n[GOAL]\ncase intro.intro.refine'_3\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\n\u22a2 \u2200 (y : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U)))),\n (fun s => IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf s) y \u2192\n y = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) }\n[PROOFSTEP]\nintro gl' hgl'\n[GOAL]\ncase intro.intro.refine'_3\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\ngl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U)))\nhgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl'\n\u22a2 gl' = { val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) }\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase intro.intro.refine'_3\nX : TopCat\nT : \u2191X \u2192 Type v\nP\u271d : PrelocalPredicate T\nP : LocalPredicate T\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nsf : (i : \u03b9) \u2192 (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : IsCompatible (subpresheafToTypes P.toPrelocalPredicate) U sf\nsf' : (i : \u03b9) \u2192 (presheafToTypes X T).obj (op (U i)) := fun i => \u2191(sf i)\nsf'_comp : IsCompatible (presheafToTypes X T) U sf'\ngl : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))\ngl_spec : IsGluing (sheafToTypes X T).val U sf' gl\ngl_uniq :\n \u2200 (y : (forget (Type v)).obj ((sheafToTypes X T).val.obj (op (iSup U)))),\n (fun s => IsGluing (sheafToTypes X T).val U sf' s) y \u2192 y = gl\ngl' : (forget (Type v)).obj ((subpresheafToTypes P.toPrelocalPredicate).obj (op (iSup U)))\nhgl' : IsGluing (subpresheafToTypes P.toPrelocalPredicate) U sf gl'\n\u22a2 \u2191gl' = \u2191{ val := gl, property := (_ : PrelocalPredicate.pred P.toPrelocalPredicate gl) }\n[PROOFSTEP]\nexact gl_uniq gl'.1 fun i => congr_arg Subtype.val (hgl' i)\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\n\u22a2 Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x \u27f6 T x\n[PROOFSTEP]\nrefine'\n colimit.desc _\n { pt := T x\n \u03b9 :=\n { app := fun U f => _\n naturality := _ } }\n[GOAL]\ncase refine'_1\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nf :\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P))).obj\n U\n\u22a2 ((Functor.const (OpenNhds x)\u1d52\u1d56).obj (T x)).obj U\n[PROOFSTEP]\nexact f.1 \u27e8x, (unop U).2\u27e9\n[GOAL]\ncase refine'_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\n\u22a2 \u2200 \u2983X_1 Y : (OpenNhds x)\u1d52\u1d56\u2984 (f : X_1 \u27f6 Y),\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P))).map\n f \u226b\n (fun U f => \u2191f { val := x, property := (_ : x \u2208 U.unop.obj) }) Y =\n (fun U f => \u2191f { val := x, property := (_ : x \u2208 U.unop.obj) }) X_1 \u226b\n ((Functor.const (OpenNhds x)\u1d52\u1d56).obj (T x)).map f\n[PROOFSTEP]\naesop\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nU : Opens \u2191X\nx : { x // x \u2208 U }\nf : (Sheaf.presheaf (subsheafToTypes P)).obj (op U)\n\u22a2 stalkToFiber P (\u2191x) (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) x f) = \u2191f x\n[PROOFSTEP]\ndsimp [Presheaf.germ, stalkToFiber]\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nU : Opens \u2191X\nx : { x // x \u2208 U }\nf : (Sheaf.presheaf (subsheafToTypes P)).obj (op U)\n\u22a2 colimit.desc ((OpenNhds.inclusion \u2191x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate)\n { pt := T \u2191x, \u03b9 := NatTrans.mk fun U_1 f => \u2191f { val := \u2191x, property := (_ : \u2191x \u2208 U_1.unop.obj) } }\n (colimit.\u03b9 ((OpenNhds.inclusion \u2191x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate)\n (op { obj := U, property := (_ : \u2191x \u2208 U) }) f) =\n \u2191f x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nU : Opens \u2191X\nf : (Sheaf.presheaf (subsheafToTypes P)).obj (op U)\nval\u271d : \u2191X\nproperty\u271d : val\u271d \u2208 U\n\u22a2 colimit.desc\n ((OpenNhds.inclusion \u2191{ val := val\u271d, property := property\u271d }).op \u22d9 subpresheafToTypes P.toPrelocalPredicate)\n { pt := T \u2191{ val := val\u271d, property := property\u271d },\n \u03b9 :=\n NatTrans.mk fun U_1 f =>\n \u2191f\n { val := \u2191{ val := val\u271d, property := property\u271d },\n property := (_ : \u2191{ val := val\u271d, property := property\u271d } \u2208 U_1.unop.obj) } }\n (colimit.\u03b9\n ((OpenNhds.inclusion \u2191{ val := val\u271d, property := property\u271d }).op \u22d9 subpresheafToTypes P.toPrelocalPredicate)\n (op { obj := U, property := (_ : \u2191{ val := val\u271d, property := property\u271d } \u2208 U) }) f) =\n \u2191f { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw : \u2200 (t : T x), \u2203 U f x_1, f { val := x, property := (_ : x \u2208 U.obj) } = t\nt : T x\n\u22a2 \u2203 a, stalkToFiber P x a = t\n[PROOFSTEP]\nrcases w t with \u27e8U, f, h, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw : \u2200 (t : T x), \u2203 U f x_1, f { val := x, property := (_ : x \u2208 U.obj) } = t\nU : OpenNhds x\nf : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y\nh : PrelocalPredicate.pred P.toPrelocalPredicate f\n\u22a2 \u2203 a, stalkToFiber P x a = f { val := x, property := (_ : x \u2208 U.obj) }\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase intro.intro.intro.w\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw : \u2200 (t : T x), \u2203 U f x_1, f { val := x, property := (_ : x \u2208 U.obj) } = t\nU : OpenNhds x\nf : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y\nh : PrelocalPredicate.pred P.toPrelocalPredicate f\n\u22a2 Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\n[PROOFSTEP]\nexact (subsheafToTypes P).presheaf.germ \u27e8x, U.2\u27e9 \u27e8f, h\u27e9\n[GOAL]\ncase intro.intro.intro.h\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw : \u2200 (t : T x), \u2203 U f x_1, f { val := x, property := (_ : x \u2208 U.obj) } = t\nU : OpenNhds x\nf : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y\nh : PrelocalPredicate.pred P.toPrelocalPredicate f\n\u22a2 stalkToFiber P x\n (Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 U.obj) }\n { val := f, property := h }) =\n f { val := x, property := (_ : x \u2208 U.obj) }\n[PROOFSTEP]\nexact stalkToFiber_germ _ U.1 \u27e8x, U.2\u27e9 \u27e8f, h\u27e9\n[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\ntU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\nh : stalkToFiber P x tU = stalkToFiber P x tV\n\u22a2 tU = tV\n[PROOFSTEP]\nlet Q :\n \u2203 (W : (OpenNhds x)\u1d52\u1d56) (s : \u2200 w : (unop W).1, T w) (hW : P.pred s),\n tU = (subsheafToTypes P).presheaf.germ \u27e8x, (unop W).2\u27e9 \u27e8s, hW\u27e9 \u2227\n tV = (subsheafToTypes P).presheaf.germ \u27e8x, (unop W).2\u27e9 \u27e8s, hW\u27e9 :=\n ?_\n[GOAL]\ncase refine_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\ntU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\nh : stalkToFiber P x tU = stalkToFiber P x tV\nQ : \u2203 W s hW,\n tU =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n tV =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } :=\n ?refine_1\n\u22a2 tU = tV\n[PROOFSTEP]\nchoose W s hW e using Q\n[GOAL]\ncase refine_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\ntU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\nh : stalkToFiber P x tU = stalkToFiber P x tV\nW : (OpenNhds x)\u1d52\u1d56\ns : (w : { x_1 // x_1 \u2208 W.unop.obj }) \u2192 T \u2191w\nhW : PrelocalPredicate.pred P.toPrelocalPredicate s\ne :\n tU =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n tV =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n\u22a2 tU = tV\n[PROOFSTEP]\nexact e.1.trans e.2.symm\n[GOAL]\ncase refine_1\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\ntU tV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\nh : stalkToFiber P x tU = stalkToFiber P x tV\n\u22a2 \u2203 W s hW,\n tU =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n tV =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nobtain \u27e8U, \u27e8fU, hU\u27e9, rfl\u27e9 := jointly_surjective'.{v, v} tU\n[GOAL]\ncase refine_1.intro.intro.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\ntV : Presheaf.stalk (Sheaf.presheaf (subsheafToTypes P)) x\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nh :\n stalkToFiber P x\n (colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n U { val := fU, property := hU }) =\n stalkToFiber P x tV\n\u22a2 \u2203 W s hW,\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n U { val := fU, property := hU } =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n tV =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nobtain \u27e8V, \u27e8fV, hV\u27e9, rfl\u27e9 := jointly_surjective'.{v, v} tV\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh :\n stalkToFiber P x\n (colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n U { val := fU, property := hU }) =\n stalkToFiber P x\n (colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n V { val := fV, property := hV })\n\u22a2 \u2203 W s hW,\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n U { val := fU, property := hU } =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n V { val := fV, property := hV } =\n Presheaf.germ (Sheaf.presheaf (subsheafToTypes P)) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh :\n stalkToFiber P x\n (colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n U { val := fU, property := hU }) =\n stalkToFiber P x\n (colimit.\u03b9\n (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 (Type v)).obj (OpenNhds.inclusion x).op).obj\n (Sheaf.presheaf (subsheafToTypes P)))\n V { val := fV, property := hV })\n\u22a2 \u2203 W s hW,\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nsimp only [stalkToFiber, Types.Colimit.\u03b9_desc_apply'] at h \n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nw :\n \u2200 (U V : OpenNhds x) (fU : (y : { x_1 // x_1 \u2208 U.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fU \u2192\n \u2200 (fV : (y : { x_2 // x_2 \u2208 V.obj }) \u2192 T \u2191y),\n PrelocalPredicate.pred P.toPrelocalPredicate fV \u2192\n fU { val := x, property := (_ : x \u2208 U.obj) } = fV { val := x, property := (_ : x \u2208 V.obj) } \u2192\n \u2203 W iU iV,\n \u2200 (w : { x_4 // x_4 \u2208 W.obj }),\n fU ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191U.obj) }) w) =\n fV ((fun x_4 => { val := \u2191x_4, property := (_ : \u2191x_4 \u2208 \u2191V.obj) }) w)\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\n\u22a2 \u2203 W s hW,\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nspecialize w (unop U) (unop V) fU hU fV hV h\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nw :\n \u2203 W iU iV,\n \u2200 (w : { x_1 // x_1 \u2208 W.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 \u2203 W s hW,\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nrcases w with\n \u27e8W, iU, iV, w\u27e9\n -- and put it back together again in the correct order.\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nW : OpenNhds x\niU : W \u27f6 U.unop\niV : W \u27f6 V.unop\nw :\n \u2200 (w : { x_1 // x_1 \u2208 W.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 \u2203 W s hW,\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 W.unop.obj) }\n { val := s, property := hW }\n[PROOFSTEP]\nrefine' \u27e8op W, fun w => fU (iU w : (unop U).1), P.res _ _ hU, _\u27e9\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nW : OpenNhds x\niU : W \u27f6 U.unop\niV : W \u27f6 V.unop\nw :\n \u2200 (w : { x_1 // x_1 \u2208 W.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 (op W).unop.obj \u27f6 U.unop.obj\ncase refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nW : OpenNhds x\niU : W \u27f6 U.unop\niV : W \u27f6 V.unop\nw :\n \u2200 (w : { x_1 // x_1 \u2208 W.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 (op W).unop.obj) }\n { val := fun w => fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w),\n property :=\n (_ :\n PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>\n fU ((fun x_2 => { val := \u2191x_2, property := (_ : \u2191x_2 \u2208 \u2191U.unop.obj) }) x_1)) } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 (op W).unop.obj) }\n { val := fun w => fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w),\n property :=\n (_ :\n PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>\n fU ((fun x_2 => { val := \u2191x_2, property := (_ : \u2191x_2 \u2208 \u2191U.unop.obj) }) x_1)) }\n[PROOFSTEP]\nrcases W with \u27e8W, m\u27e9\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1.mk\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nW : Opens \u2191X\nm : x \u2208 W\niU : { obj := W, property := m } \u27f6 U.unop\niV : { obj := W, property := m } \u27f6 V.unop\nw :\n \u2200 (w : { x_1 // x_1 \u2208 { obj := W, property := m }.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 (op { obj := W, property := m }).unop.obj \u27f6 U.unop.obj\n[PROOFSTEP]\nexact iU\n[GOAL]\ncase refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2\nX : TopCat\nT : \u2191X \u2192 Type v\nP : LocalPredicate T\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\nfU : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj U).unop }) \u2192 T \u2191x_1\nhU : PrelocalPredicate.pred P.toPrelocalPredicate fU\nV : (OpenNhds x)\u1d52\u1d56\nfV : (x_1 : { x_1 // x_1 \u2208 ((OpenNhds.inclusion x).op.obj V).unop }) \u2192 T \u2191x_1\nhV : PrelocalPredicate.pred P.toPrelocalPredicate fV\nh : fU { val := x, property := (_ : x \u2208 U.unop.obj) } = fV { val := x, property := (_ : x \u2208 V.unop.obj) }\nW : OpenNhds x\niU : W \u27f6 U.unop\niV : W \u27f6 V.unop\nw :\n \u2200 (w : { x_1 // x_1 \u2208 W.obj }),\n fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w) =\n fV ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191V.unop.obj) }) w)\n\u22a2 colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 (op W).unop.obj) }\n { val := fun w => fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w),\n property :=\n (_ :\n PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>\n fU ((fun x_2 => { val := \u2191x_2, property := (_ : \u2191x_2 \u2208 \u2191U.unop.obj) }) x_1)) } \u2227\n colimit.\u03b9 ((OpenNhds.inclusion x).op \u22d9 subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =\n Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x \u2208 (op W).unop.obj) }\n { val := fun w => fU ((fun x_1 => { val := \u2191x_1, property := (_ : \u2191x_1 \u2208 \u2191U.unop.obj) }) w),\n property :=\n (_ :\n PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>\n fU ((fun x_2 => { val := \u2191x_2, property := (_ : \u2191x_2 \u2208 \u2191U.unop.obj) }) x_1)) }\n[PROOFSTEP]\nexact \u27e8colimit_sound iU.op (Subtype.eq rfl), colimit_sound iV.op (Subtype.eq (funext w).symm)\u27e9\n[GOAL]\nX\u271d : TopCat\nT\u271d : \u2191X\u271d \u2192 Type v\nT : TopCat\nX : (Opens \u2191X\u271d)\u1d52\u1d56\n\u22a2 (subpresheafToTypes (continuousPrelocal X\u271d T)).obj X \u27f6 (presheafToTop X\u271d T).obj X\n[PROOFSTEP]\nrintro \u27e8f, c\u27e9\n[GOAL]\ncase mk\nX\u271d : TopCat\nT\u271d : \u2191X\u271d \u2192 Type v\nT : TopCat\nX : (Opens \u2191X\u271d)\u1d52\u1d56\nf : (x : { x // x \u2208 X.unop }) \u2192 (fun x => \u2191T) \u2191x\nc : PrelocalPredicate.pred (continuousPrelocal X\u271d T) f\n\u22a2 (presheafToTop X\u271d T).obj X\n[PROOFSTEP]\nexact \u27e8f, c\u27e9\n[GOAL]\nX\u271d : TopCat\nT\u271d : \u2191X\u271d \u2192 Type v\nT : TopCat\nX : (Opens \u2191X\u271d)\u1d52\u1d56\n\u22a2 (presheafToTop X\u271d T).obj X \u27f6 (subpresheafToTypes (continuousPrelocal X\u271d T)).obj X\n[PROOFSTEP]\nrintro \u27e8f, c\u27e9\n[GOAL]\ncase mk\nX\u271d : TopCat\nT\u271d : \u2191X\u271d \u2192 Type v\nT : TopCat\nX : (Opens \u2191X\u271d)\u1d52\u1d56\nf : \u2191((Opens.toTopCat X\u271d).op.obj X).unop \u2192 \u2191T\nc : Continuous f\n\u22a2 (subpresheafToTypes (continuousPrelocal X\u271d T)).obj X\n[PROOFSTEP]\nexact \u27e8f, c\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.LocalPredicate", "llama_tokens": 22318, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.2870758976339061}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (G.obj (op U))\n\u22a2 \u2200 {Y Z : C} {f_1 : Y \u27f6 U},\n (fun V i => \u2203 t, \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) s) Y f_1 \u2192\n \u2200 (g : Z \u27f6 Y), (fun V i => \u2203 t, \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) s) Z (g \u226b f_1)\n[PROOFSTEP]\nrintro V W i \u27e8t, ht\u27e9 j\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (G.obj (op U))\nV W : C\ni : V \u27f6 U\nt : (forget A).obj (F.obj (op V))\nht : \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) s\nj : W \u27f6 V\n\u22a2 \u2203 t, \u2191(NatTrans.app f (op W)) t = \u2191(G.map (j \u226b i).op) s\n[PROOFSTEP]\nrefine' \u27e8F.map j.op t, _\u27e9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (G.obj (op U))\nV W : C\ni : V \u27f6 U\nt : (forget A).obj (F.obj (op V))\nht : \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) s\nj : W \u27f6 V\n\u22a2 \u2191(NatTrans.app f (op W)) (\u2191(F.map j.op) t) = \u2191(G.map (j \u226b i).op) s\n[PROOFSTEP]\nrw [op_comp, G.map_comp, comp_apply, \u2190 ht, elementwise_of% f.naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj (op U))\n\u22a2 imageSieve f (\u2191(NatTrans.app f (op U)) s) = \u22a4\n[PROOFSTEP]\next V i\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V \u27f6 U\n\u22a2 (imageSieve f (\u2191(NatTrans.app f (op U)) s)).arrows i \u2194 \u22a4.arrows i\n[PROOFSTEP]\nsimp only [Sieve.top_apply, iff_true_iff, imageSieve_apply]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V \u27f6 U\n\u22a2 \u2203 t, \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) (\u2191(NatTrans.app f (op U)) s)\n[PROOFSTEP]\nhave := elementwise_of% (f.naturality i.op)\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj (op U))\nV : C\ni : V \u27f6 U\nthis :\n \u2200 (x : (forget A).obj (F.obj (op U))),\n \u2191(NatTrans.app f (op V)) (\u2191(F.map i.op) x) = \u2191(G.map i.op) (\u2191(NatTrans.app f (op U)) x)\n\u22a2 \u2203 t, \u2191(NatTrans.app f (op V)) t = \u2191(G.map i.op) (\u2191(NatTrans.app f (op U)) s)\n[PROOFSTEP]\nexact \u27e8F.map i.op s, this s\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A))) = \u22a4\n[PROOFSTEP]\nsimp only [Subpresheaf.ext_iff, Function.funext_iff, Set.ext_iff, top_subpresheaf_obj, Set.top_eq_univ, Set.mem_univ,\n iff_true_iff]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194\n \u2200 (a : C\u1d52\u1d56) (x : (G \u22d9 forget A).obj a),\n x \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A)))) a\n[PROOFSTEP]\nexact \u27e8fun H U => H (unop U), fun H U => H (op U)\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 Type w\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 Subpresheaf.sheafify J (imagePresheaf f) = \u22a4\n[PROOFSTEP]\nsimp only [Subpresheaf.ext_iff, Function.funext_iff, Set.ext_iff, top_subpresheaf_obj, Set.top_eq_univ, Set.mem_univ,\n iff_true_iff]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 Type w\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 \u2200 (a : C\u1d52\u1d56) (x : G.obj a), x \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f)) a\n[PROOFSTEP]\nexact \u27e8fun H U => H (unop U), fun H U => H (op U)\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : Sheaf J (Type w)\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f.val \u2194 IsIso (imageSheaf\u03b9 f)\n[PROOFSTEP]\nrw [imageSheaf\u03b9, isLocallySurjective_iff_imagePresheaf_sheafify_eq_top', Subpresheaf.eq_top_iff_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : Sheaf J (Type w)\nf : F \u27f6 G\n\u22a2 IsIso (Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf f.val))) \u2194\n IsIso { val := Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf f.val)) }\n[PROOFSTEP]\nexact\n \u27e8fun h => @isIso_of_reflects_iso _ _ _ _ _ _ (imageSheaf\u03b9 f) (sheafToPresheaf J _) h _, fun h =>\n @Functor.map_isIso _ _ _ _ _ _ (sheafToPresheaf J _) _ h\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 IsLocallySurjective J (whiskerRight f (forget A))\n[PROOFSTEP]\nsimp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A))) = \u22a4 \u2194\n Subpresheaf.sheafify J (imagePresheaf (whiskerRight (whiskerRight f (forget A)) (forget (Type w')))) = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nH : \u2200 (U : C\u1d52\u1d56), Function.Surjective \u2191(NatTrans.app f U)\n\u22a2 IsLocallySurjective J f\n[PROOFSTEP]\nintro U s\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nH : \u2200 (U : C\u1d52\u1d56), Function.Surjective \u2191(NatTrans.app f U)\nU : C\ns : (forget A).obj (G.obj (op U))\n\u22a2 imageSieve f s \u2208 sieves J U\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 := H _ s\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nH : \u2200 (U : C\u1d52\u1d56), Function.Surjective \u2191(NatTrans.app f U)\nU : C\nt : (forget A).obj (F.obj (op U))\n\u22a2 imageSieve f (\u2191(NatTrans.app f (op U)) t) \u2208 sieves J U\n[PROOFSTEP]\nrw [imageSieve_app]\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nH : \u2200 (U : C\u1d52\u1d56), Function.Surjective \u2191(NatTrans.app f U)\nU : C\nt : (forget A).obj (F.obj (op U))\n\u22a2 \u22a4 \u2208 sieves J U\n[PROOFSTEP]\nexact J.top_mem _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b2 : Category.{v', u'} A\ninst\u271d\u00b9 : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\ninst\u271d : IsIso f\n\u22a2 IsLocallySurjective J f\n[PROOFSTEP]\napply isLocallySurjective_of_surjective\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b2 : Category.{v', u'} A\ninst\u271d\u00b9 : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\ninst\u271d : IsIso f\n\u22a2 \u2200 (U : C\u1d52\u1d56), Function.Surjective \u2191(NatTrans.app f U)\n[PROOFSTEP]\nintro U\n[GOAL]\ncase H\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b2 : Category.{v', u'} A\ninst\u271d\u00b9 : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\ninst\u271d : IsIso f\nU : C\u1d52\u1d56\n\u22a2 Function.Surjective \u2191(NatTrans.app f U)\n[PROOFSTEP]\napply Function.Bijective.surjective\n[GOAL]\ncase H.hf\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b2 : Category.{v', u'} A\ninst\u271d\u00b9 : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\ninst\u271d : IsIso f\nU : C\u1d52\u1d56\n\u22a2 Function.Bijective \u2191(NatTrans.app f U)\n[PROOFSTEP]\nrw [\u2190 isIso_iff_bijective, \u2190 forget_map_eq_coe]\n[GOAL]\ncase H.hf\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b2 : Category.{v', u'} A\ninst\u271d\u00b9 : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\ninst\u271d : IsIso f\nU : C\u1d52\u1d56\n\u22a2 IsIso ((forget A).map (NatTrans.app f U))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\n\u22a2 IsLocallySurjective J (f\u2081 \u226b f\u2082)\n[PROOFSTEP]\nintro U s\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\n\u22a2 imageSieve (f\u2081 \u226b f\u2082) s \u2208 sieves J U\n[PROOFSTEP]\nhave : (Sieve.bind (imageSieve f\u2082 s) fun _ _ h => imageSieve f\u2081 h.choose) \u2264 imageSieve (f\u2081 \u226b f\u2082) s :=\n by\n rintro V i \u27e8W, i, j, H, \u27e8t', ht'\u27e9, rfl\u27e9\n refine' \u27e8t', _\u27e9\n rw [op_comp, F\u2083.map_comp, NatTrans.comp_app, comp_apply, comp_apply, ht', elementwise_of% f\u2082.naturality,\n H.choose_spec]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\n\u22a2 (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\n[PROOFSTEP]\nrintro V i \u27e8W, i, j, H, \u27e8t', ht'\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nV W : C\ni : V \u27f6 W\nj : W \u27f6 U\nH : (imageSieve f\u2082 s).arrows j\nt' : (forget A).obj (F\u2081.obj (op V))\nht' : \u2191(NatTrans.app f\u2081 (op V)) t' = \u2191(F\u2082.map i.op) (Exists.choose H)\n\u22a2 (imageSieve (f\u2081 \u226b f\u2082) s).arrows (i \u226b j)\n[PROOFSTEP]\nrefine' \u27e8t', _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nV W : C\ni : V \u27f6 W\nj : W \u27f6 U\nH : (imageSieve f\u2082 s).arrows j\nt' : (forget A).obj (F\u2081.obj (op V))\nht' : \u2191(NatTrans.app f\u2081 (op V)) t' = \u2191(F\u2082.map i.op) (Exists.choose H)\n\u22a2 \u2191(NatTrans.app (f\u2081 \u226b f\u2082) (op V)) t' = \u2191(F\u2083.map (i \u226b j).op) s\n[PROOFSTEP]\nrw [op_comp, F\u2083.map_comp, NatTrans.comp_app, comp_apply, comp_apply, ht', elementwise_of% f\u2082.naturality, H.choose_spec]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nthis : (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\n\u22a2 imageSieve (f\u2081 \u226b f\u2082) s \u2208 sieves J U\n[PROOFSTEP]\napply J.superset_covering this\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nthis : (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\n\u22a2 (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2208 sieves J U\n[PROOFSTEP]\napply J.bind_covering\n[GOAL]\ncase hS\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nthis : (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\n\u22a2 imageSieve f\u2082 s \u2208 sieves J U\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase hR\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nthis : (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U\u2984 (H : (imageSieve f\u2082 s).arrows f), imageSieve f\u2081 (Exists.choose H) \u2208 sieves J Y\n[PROOFSTEP]\nintros\n[GOAL]\ncase hR\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A\nf\u2081 : F\u2081 \u27f6 F\u2082\nf\u2082 : F\u2082 \u27f6 F\u2083\nh\u2081 : IsLocallySurjective J f\u2081\nh\u2082 : IsLocallySurjective J f\u2082\nU : C\ns : (forget A).obj (F\u2083.obj (op U))\nthis : (Sieve.bind (imageSieve f\u2082 s).arrows fun x x_1 h => imageSieve f\u2081 (Exists.choose h)) \u2264 imageSieve (f\u2081 \u226b f\u2082) s\nY\u271d : C\nf\u271d : Y\u271d \u27f6 U\nH\u271d : (imageSieve f\u2082 s).arrows f\u271d\n\u22a2 imageSieve f\u2081 (Exists.choose H\u271d) \u2208 sieves J Y\u271d\n[PROOFSTEP]\napply h\u2081\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 toSheafify J F \u226b\n sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) \u226b\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) =\n toSheafify J F \u226b \ud835\udfd9 (sheafify J F)\n[PROOFSTEP]\nsimp [toImagePresheafSheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) \u226b\n sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) =\n \ud835\udfd9 (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))\n[PROOFSTEP]\nrw [\u2190 cancel_mono (Subpresheaf.\u03b9 _), Category.id_comp, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) \u226b\n sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) \u226b\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) =\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F)))\n[PROOFSTEP]\nrefine' Eq.trans _ (Category.comp_id _)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) \u226b\n sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) \u226b\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) =\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) \u226b \ud835\udfd9 (sheafify J F)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) \u226b\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) =\n \ud835\udfd9 (sheafify J F)\n[PROOFSTEP]\nexact J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [toImagePresheafSheafify])\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF : C\u1d52\u1d56 \u2964 Type (max u v)\n\u22a2 toSheafify J F \u226b\n sheafifyLift J (toImagePresheafSheafify J (toSheafify J F))\n (_ : Presheaf.IsSheaf J (Subpresheaf.toPresheaf (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))))) \u226b\n Subpresheaf.\u03b9 (Subpresheaf.sheafify J (imagePresheaf (toSheafify J F))) =\n toSheafify J F \u226b \ud835\udfd9 (sheafify J F)\n[PROOFSTEP]\nsimp [toImagePresheafSheafify]\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2078 : Category.{v', u'} A\ninst\u271d\u2077 : ConcreteCategory A\nF\u271d : C\u1d52\u1d56 \u2964 Type (max u v)\nB : Type w\ninst\u271d\u2076 : Category.{max u v, w} B\ninst\u271d\u2075 : ConcreteCategory B\ninst\u271d\u2074 : \u2200 (X : C), Limits.HasColimitsOfShape (Cover J X)\u1d52\u1d56 B\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : Cover J X), Limits.HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 :\n (X : C) \u2192\n (W : Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 B) \u2192 Limits.PreservesLimit (Limits.MulticospanIndex.multicospan (Cover.index W P)) (forget B)\ninst\u271d\u00b9 : (X : C) \u2192 Limits.PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget B)\ninst\u271d : \u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B\nF : C\u1d52\u1d56 \u2964 B\n\u22a2 IsLocallySurjective J (toSheafify J F)\n[PROOFSTEP]\nrw [isLocallySurjective_iff_whisker_forget, \u2190 toSheafify_comp_sheafifyCompIso_inv]\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2078 : Category.{v', u'} A\ninst\u271d\u2077 : ConcreteCategory A\nF\u271d : C\u1d52\u1d56 \u2964 Type (max u v)\nB : Type w\ninst\u271d\u2076 : Category.{max u v, w} B\ninst\u271d\u2075 : ConcreteCategory B\ninst\u271d\u2074 : \u2200 (X : C), Limits.HasColimitsOfShape (Cover J X)\u1d52\u1d56 B\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : Cover J X), Limits.HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 :\n (X : C) \u2192\n (W : Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 B) \u2192 Limits.PreservesLimit (Limits.MulticospanIndex.multicospan (Cover.index W P)) (forget B)\ninst\u271d\u00b9 : (X : C) \u2192 Limits.PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget B)\ninst\u271d : \u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B\nF : C\u1d52\u1d56 \u2964 B\n\u22a2 IsLocallySurjective J (toSheafify J (F \u22d9 forget B) \u226b (sheafifyCompIso J (forget B) F).inv)\n[PROOFSTEP]\napply IsLocallySurjective.comp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2078 : Category.{v', u'} A\ninst\u271d\u2077 : ConcreteCategory A\nF\u271d : C\u1d52\u1d56 \u2964 Type (max u v)\nB : Type w\ninst\u271d\u2076 : Category.{max u v, w} B\ninst\u271d\u2075 : ConcreteCategory B\ninst\u271d\u2074 : \u2200 (X : C), Limits.HasColimitsOfShape (Cover J X)\u1d52\u1d56 B\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : Cover J X), Limits.HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 :\n (X : C) \u2192\n (W : Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 B) \u2192 Limits.PreservesLimit (Limits.MulticospanIndex.multicospan (Cover.index W P)) (forget B)\ninst\u271d\u00b9 : (X : C) \u2192 Limits.PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget B)\ninst\u271d : \u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B\nF : C\u1d52\u1d56 \u2964 B\n\u22a2 IsLocallySurjective J (toSheafify J (F \u22d9 forget B))\n[PROOFSTEP]\nrw [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top, Subpresheaf.eq_top_iff_isIso]\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2078 : Category.{v', u'} A\ninst\u271d\u2077 : ConcreteCategory A\nF\u271d : C\u1d52\u1d56 \u2964 Type (max u v)\nB : Type w\ninst\u271d\u2076 : Category.{max u v, w} B\ninst\u271d\u2075 : ConcreteCategory B\ninst\u271d\u2074 : \u2200 (X : C), Limits.HasColimitsOfShape (Cover J X)\u1d52\u1d56 B\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : Cover J X), Limits.HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 :\n (X : C) \u2192\n (W : Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 B) \u2192 Limits.PreservesLimit (Limits.MulticospanIndex.multicospan (Cover.index W P)) (forget B)\ninst\u271d\u00b9 : (X : C) \u2192 Limits.PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget B)\ninst\u271d : \u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B\nF : C\u1d52\u1d56 \u2964 B\n\u22a2 IsIso\n (Subpresheaf.\u03b9\n (Subpresheaf.sheafify J (imagePresheaf (whiskerRight (toSheafify J (F \u22d9 forget B)) (forget (Type (max u v)))))))\n[PROOFSTEP]\nexact IsIso.of_iso_inv (sheafificationIsoImagePresheaf J (F \u22d9 forget B))\n[GOAL]\ncase h\u2082\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u2078 : Category.{v', u'} A\ninst\u271d\u2077 : ConcreteCategory A\nF\u271d : C\u1d52\u1d56 \u2964 Type (max u v)\nB : Type w\ninst\u271d\u2076 : Category.{max u v, w} B\ninst\u271d\u2075 : ConcreteCategory B\ninst\u271d\u2074 : \u2200 (X : C), Limits.HasColimitsOfShape (Cover J X)\u1d52\u1d56 B\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : Cover J X), Limits.HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 :\n (X : C) \u2192\n (W : Cover J X) \u2192\n (P : C\u1d52\u1d56 \u2964 B) \u2192 Limits.PreservesLimit (Limits.MulticospanIndex.multicospan (Cover.index W P)) (forget B)\ninst\u271d\u00b9 : (X : C) \u2192 Limits.PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget B)\ninst\u271d : \u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1), Limits.HasLimitsOfShape (Limits.WalkingMulticospan fst snd) B\nF : C\u1d52\u1d56 \u2964 B\n\u22a2 IsLocallySurjective J (sheafifyCompIso J (forget B) F).inv\n[PROOFSTEP]\nexact isLocallySurjective_of_iso _ _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Surjective", "llama_tokens": 10873, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.2867283018206714}} {"text": "[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\n\u22a2 Function.Injective fun S => S.carrier\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\ntoSubsemiring\u271d\u00b9 : Subsemiring L\nalgebraMap_mem'\u271d\u00b9 : \u2200 (r : K), \u2191(algebraMap K L) r \u2208 toSubsemiring\u271d\u00b9.carrier\ninv_mem'\u271d\u00b9 :\n \u2200 (x : L),\n x \u2208\n { toSubsemiring := toSubsemiring\u271d\u00b9,\n algebraMap_mem' := algebraMap_mem'\u271d\u00b9 }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := toSubsemiring\u271d\u00b9,\n algebraMap_mem' := algebraMap_mem'\u271d\u00b9 }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\ntoSubsemiring\u271d : Subsemiring L\nalgebraMap_mem'\u271d : \u2200 (r : K), \u2191(algebraMap K L) r \u2208 toSubsemiring\u271d.carrier\ninv_mem'\u271d :\n \u2200 (x : L),\n x \u2208\n { toSubsemiring := toSubsemiring\u271d,\n algebraMap_mem' := algebraMap_mem'\u271d }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := toSubsemiring\u271d,\n algebraMap_mem' := algebraMap_mem'\u271d }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 (fun S => S.carrier)\n { toSubalgebra := { toSubsemiring := toSubsemiring\u271d\u00b9, algebraMap_mem' := algebraMap_mem'\u271d\u00b9 },\n inv_mem' := inv_mem'\u271d\u00b9 } =\n (fun S => S.carrier)\n { toSubalgebra := { toSubsemiring := toSubsemiring\u271d, algebraMap_mem' := algebraMap_mem'\u271d },\n inv_mem' := inv_mem'\u271d } \u2192\n { toSubalgebra := { toSubsemiring := toSubsemiring\u271d\u00b9, algebraMap_mem' := algebraMap_mem'\u271d\u00b9 },\n inv_mem' := inv_mem'\u271d\u00b9 } =\n { toSubalgebra := { toSubsemiring := toSubsemiring\u271d, algebraMap_mem' := algebraMap_mem'\u271d },\n inv_mem' := inv_mem'\u271d }\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : L\nhx : x \u2208 S\n\u22a2 -x \u2208 S\n[PROOFSTEP]\nshow -x \u2208 S.toSubalgebra\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : L\nhx : x \u2208 S\n\u22a2 -x \u2208 S.toSubalgebra\n[PROOFSTEP]\nsimpa\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nn : \u2115\n\u22a2 \u2191n \u2208 S\n[PROOFSTEP]\nsimpa using coe_int_mem S n\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\ninv_mem : \u2200 (x : L), x \u2208 S \u2192 x\u207b\u00b9 \u2208 S\n\u22a2 (Subalgebra.toIntermediateField S inv_mem).toSubalgebra = S\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\ninv_mem : \u2200 (x : L), x \u2208 S \u2192 x\u207b\u00b9 \u2208 S\nx\u271d : L\n\u22a2 x\u271d \u2208 (Subalgebra.toIntermediateField S inv_mem).toSubalgebra \u2194 x\u271d \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S : IntermediateField K L\n\u22a2 Subalgebra.toIntermediateField S.toSubalgebra (_ : \u2200 (x : L), x \u2208 S \u2192 x\u207b\u00b9 \u2208 S) = S\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S : IntermediateField K L\nx\u271d : L\n\u22a2 x\u271d \u2208 Subalgebra.toIntermediateField S.toSubalgebra (_ : \u2200 (x : L), x \u2208 S \u2192 x\u207b\u00b9 \u2208 S) \u2194 x\u271d \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nby_cases hx0 : x = 0\n[GOAL]\ncase pos\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : x = 0\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nrw [hx0, inv_zero]\n[GOAL]\ncase pos\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : x = 0\n\u22a2 0 \u2208 S\n[PROOFSTEP]\nexact S.zero_mem\n[GOAL]\ncase neg\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : \u00acx = 0\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nletI hS' := hS.toField\n[GOAL]\ncase neg\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : \u00acx = 0\nhS' : Field { x // x \u2208 S } := IsField.toField hS\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hS.mul_inv_cancel (show (\u27e8x, hx\u27e9 : S) \u2260 0 from Subtype.ne_of_val_ne hx0)\n[GOAL]\ncase neg.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : \u00acx = 0\nhS' : Field { x // x \u2208 S } := IsField.toField hS\ny : { x // x \u2208 S }\nhy : { val := x, property := hx } * y = 1\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nrw [Subtype.ext_iff, S.coe_mul, S.coe_one, Subtype.coe_mk, mul_eq_one_iff_inv_eq\u2080 hx0] at hy \n[GOAL]\ncase neg.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx : L\nhx : x \u2208 S\nhx0 : \u00acx = 0\nhS' : Field { x // x \u2208 S } := IsField.toField hS\ny : { x // x \u2208 S }\nhy : x\u207b\u00b9 = \u2191y\n\u22a2 x\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nexact hy.symm \u25b8 y.2\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\n\u22a2 (Subalgebra.toIntermediateField' S hS).toSubalgebra = S\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nS : Subalgebra K L\nhS : IsField { x // x \u2208 S }\nx\u271d : L\n\u22a2 x\u271d \u2208 (Subalgebra.toIntermediateField' S hS).toSubalgebra \u2194 x\u271d \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S : IntermediateField K L\n\u22a2 Subalgebra.toIntermediateField' S.toSubalgebra (_ : IsField { x // x \u2208 S }) = S\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S : IntermediateField K L\nx\u271d : L\n\u22a2 x\u271d \u2208 Subalgebra.toIntermediateField' S.toSubalgebra (_ : IsField { x // x \u2208 S }) \u2194 x\u271d \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u2211 i : \u03b9, f i) = \u2211 i : \u03b9, \u2191(f i)\n[PROOFSTEP]\nclassical\ninduction' (Finset.univ : Finset \u03b9) using Finset.induction_on with i s hi H\n\u00b7 simp\n\u00b7 rw [Finset.sum_insert hi, AddMemClass.coe_add, H, Finset.sum_insert hi]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u2211 i : \u03b9, f i) = \u2211 i : \u03b9, \u2191(f i)\n[PROOFSTEP]\ninduction' (Finset.univ : Finset \u03b9) using Finset.induction_on with i s hi H\n[GOAL]\ncase empty\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u2211 i in \u2205, f i) = \u2211 i in \u2205, \u2191(f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nH : \u2191(\u2211 i in s, f i) = \u2211 i in s, \u2191(f i)\n\u22a2 \u2191(\u2211 i in insert i s, f i) = \u2211 i in insert i s, \u2191(f i)\n[PROOFSTEP]\nrw [Finset.sum_insert hi, AddMemClass.coe_add, H, Finset.sum_insert hi]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u220f i : \u03b9, f i) = \u220f i : \u03b9, \u2191(f i)\n[PROOFSTEP]\nclassical\ninduction' (Finset.univ : Finset \u03b9) using Finset.induction_on with i s hi H\n\u00b7 simp\n\u00b7 rw [Finset.prod_insert hi, MulMemClass.coe_mul, H, Finset.prod_insert hi]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u220f i : \u03b9, f i) = \u220f i : \u03b9, \u2191(f i)\n[PROOFSTEP]\ninduction' (Finset.univ : Finset \u03b9) using Finset.induction_on with i s hi H\n[GOAL]\ncase empty\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\n\u22a2 \u2191(\u220f i in \u2205, f i) = \u220f i in \u2205, \u2191(f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS : IntermediateField K L\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 { x // x \u2208 S }\ni : \u03b9\ns : Finset \u03b9\nhi : \u00aci \u2208 s\nH : \u2191(\u220f i in s, f i) = \u220f i in s, \u2191(f i)\n\u22a2 \u2191(\u220f i in insert i s, f i) = \u220f i in insert i s, \u2191(f i)\n[PROOFSTEP]\nrw [Finset.prod_insert hi, MulMemClass.coe_mul, H, Finset.prod_insert hi]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nf : L \u2192\u2090[K] L'\nS : IntermediateField K L\nsrc\u271d : Subalgebra K L' := Subalgebra.map f S.toSubalgebra\n\u22a2 \u2200 (x : L'),\n x \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n algebraMap_mem' :=\n (_ :\n \u2200 (r : K),\n \u2191(algebraMap K L') r \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n algebraMap_mem' :=\n (_ :\n \u2200 (r : K), \u2191(algebraMap K L') r \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d : IntermediateField K L\nf : L \u2192\u2090[K] L'\nS : IntermediateField K L\nsrc\u271d : Subalgebra K L' := Subalgebra.map f S.toSubalgebra\nx : L\nhx : x \u2208 \u2191S.toSubsemiring\n\u22a2 (\u2191\u2191f x)\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n algebraMap_mem' :=\n (_ : \u2200 (r : K), \u2191(algebraMap K L') r \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_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\n\u22a2 \u2191(aeval \u2191x) P = \u2191(\u2191(aeval x) P)\n[PROOFSTEP]\nrefine' Polynomial.induction_on' P (fun f g hf hg => _) fun n r => _\n[GOAL]\ncase refine'_1\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP f g : R[X]\nhf : \u2191(aeval \u2191x) f = \u2191(\u2191(aeval x) f)\nhg : \u2191(aeval \u2191x) g = \u2191(\u2191(aeval x) g)\n\u22a2 \u2191(aeval \u2191x) (f + g) = \u2191(\u2191(aeval x) (f + g))\n[PROOFSTEP]\nrw [aeval_add, aeval_add, AddMemClass.coe_add, hf, hg]\n[GOAL]\ncase refine'_2\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(aeval \u2191x) (\u2191(monomial n) r) = \u2191(\u2191(aeval x) (\u2191(monomial n) r))\n[PROOFSTEP]\nsimp only [MulMemClass.coe_mul, aeval_monomial, SubmonoidClass.coe_pow, mul_eq_mul_right_iff]\n[GOAL]\ncase refine'_2\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(algebraMap R L) r = \u2191(\u2191(algebraMap R { x // x \u2208 S }) r) \u2228 \u2191x ^ n = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase refine'_2.h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191(algebraMap R L) r = \u2191(\u2191(algebraMap R { x // x \u2208 S }) r)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\n\u22a2 IsIntegral R \u2191x \u2194 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nh : IsIntegral R \u2191x\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nobtain \u27e8P, hPmo, hProot\u27e9 := h\n[GOAL]\ncase refine'_1.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R L) (\u2191x) P = 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8P, hPmo, (injective_iff_map_eq_zero _).1 (algebraMap (\u21a5S) L).injective _ _\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R L) (\u2191x) P = 0\n\u22a2 \u2191(algebraMap { x // x \u2208 S } L) (eval\u2082 (algebraMap R { x // x \u2208 S }) x P) = 0\n[PROOFSTEP]\nletI : IsScalarTower R S L := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n[GOAL]\ncase refine'_1.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R L) (\u2191x) P = 0\nthis : IsScalarTower R { x // x \u2208 S } L := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n\u22a2 \u2191(algebraMap { x // x \u2208 S } L) (eval\u2082 (algebraMap R { x // x \u2208 S }) x P) = 0\n[PROOFSTEP]\nrw [eval\u2082_eq_eval_map, \u2190 eval\u2082_at_apply, eval\u2082_eq_eval_map, Polynomial.map_map, \u2190\n --Porting note: very strange that I have to `rw` twice with `eval\u2082_eq_eval_map`.\n -- The first `rw` does nothingIsScalarTower.algebraMap_eq, \u2190 eval\u2082_eq_eval_map, \u2190 eval\u2082_eq_eval_map]\n[GOAL]\ncase refine'_1.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R L) (\u2191x) P = 0\nthis : IsScalarTower R { x // x \u2208 S } L := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n\u22a2 eval\u2082 (algebraMap R L) (\u2191(algebraMap { x // x \u2208 S } L) x) P = 0\n[PROOFSTEP]\nexact hProot\n[GOAL]\ncase refine'_2\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nh : IsIntegral R x\n\u22a2 IsIntegral R \u2191x\n[PROOFSTEP]\nobtain \u27e8P, hPmo, hProot\u27e9 := h\n[GOAL]\ncase refine'_2.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R { x // x \u2208 S }) x P = 0\n\u22a2 IsIntegral R \u2191x\n[PROOFSTEP]\nrefine' \u27e8P, hPmo, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K L'\nS : IntermediateField K L\nR : Type u_4\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Algebra R K\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nx : { x // x \u2208 S }\nP : R[X]\nhPmo : Monic P\nhProot : eval\u2082 (algebraMap R { x // x \u2208 S }) x P = 0\n\u22a2 eval\u2082 (algebraMap R L) (\u2191x) P = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def, aeval_coe, aeval_def, hProot, ZeroMemClass.coe_zero]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\n\u22a2 S = S'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS\u271d S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\nx\u271d : L\n\u22a2 x\u271d \u2208 S \u2194 x\u271d \u2208 S'\n[PROOFSTEP]\nrw [\u2190 mem_toSubalgebra, \u2190 mem_toSubalgebra, h]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : L\nhx : x \u2208 RingHom.fieldRange (algebraMap K L)\n\u22a2 x \u2208 Set.range \u2191(algebraMap K L)\n[PROOFSTEP]\nrwa [Set.mem_range, \u2190 RingHom.mem_fieldRange]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field L'\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K L'\nS : IntermediateField K L\ninst\u271d\u00b9 : Algebra L' L\ninst\u271d : IsScalarTower K L' L\nU V : IntermediateField L' L\nH : restrictScalars K U = restrictScalars K V\nx : L\n\u22a2 x \u2208 U \u2194 x \u2208 V\n[PROOFSTEP]\nrw [\u2190 mem_restrictScalars K, H, mem_restrictScalars]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS F : IntermediateField K L\nE : IntermediateField { x // x \u2208 F } L\n\u22a2 Algebra K { x // x \u2208 E }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS F E : IntermediateField K L\n\u22a2 F.toSubalgebra = E.toSubalgebra \u2194 F = E\n[PROOFSTEP]\nrw [SetLike.ext_iff, SetLike.ext'_iff, Set.ext_iff]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS F E : IntermediateField K L\n\u22a2 (\u2200 (x : L), x \u2208 F.toSubalgebra \u2194 x \u2208 E.toSubalgebra) \u2194 \u2200 (x : L), x \u2208 \u2191F \u2194 x \u2208 \u2191E\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS F E : IntermediateField K L\ninst\u271d : FiniteDimensional K L\nh_le : F \u2264 E\nh_finrank : finrank { x // x \u2208 F } L \u2264 finrank { x // x \u2208 E } L\n\u22a2 F = E\n[PROOFSTEP]\napply eq_of_le_of_finrank_le h_le\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS F E : IntermediateField K L\ninst\u271d : FiniteDimensional K L\nh_le : F \u2264 E\nh_finrank : finrank { x // x \u2208 F } L \u2264 finrank { x // x \u2208 E } L\n\u22a2 finrank K { x // x \u2208 E } \u2264 finrank K { x // x \u2208 F }\n[PROOFSTEP]\nhave h1 := finrank_mul_finrank K F L\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS F E : IntermediateField K L\ninst\u271d : FiniteDimensional K L\nh_le : F \u2264 E\nh_finrank : finrank { x // x \u2208 F } L \u2264 finrank { x // x \u2208 E } L\nh1 : finrank K { x // x \u2208 F } * finrank { x // x \u2208 F } L = finrank K L\n\u22a2 finrank K { x // x \u2208 E } \u2264 finrank K { x // x \u2208 F }\n[PROOFSTEP]\nhave h2 := finrank_mul_finrank K E L\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS F E : IntermediateField K L\ninst\u271d : FiniteDimensional K L\nh_le : F \u2264 E\nh_finrank : finrank { x // x \u2208 F } L \u2264 finrank { x // x \u2208 E } L\nh1 : finrank K { x // x \u2208 F } * finrank { x // x \u2208 F } L = finrank K L\nh2 : finrank K { x // x \u2208 E } * finrank { x // x \u2208 E } L = finrank K L\n\u22a2 finrank K { x // x \u2208 E } \u2264 finrank K { x // x \u2208 F }\n[PROOFSTEP]\nhave h3 : 0 < finrank E L := finrank_pos\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field L'\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K L'\nS F E : IntermediateField K L\ninst\u271d : FiniteDimensional K L\nh_le : F \u2264 E\nh_finrank : finrank { x // x \u2208 F } L \u2264 finrank { x // x \u2208 E } L\nh1 : finrank K { x // x \u2208 F } * finrank { x // x \u2208 F } L = finrank K L\nh2 : finrank K { x // x \u2208 E } * finrank { x // x \u2208 E } L = finrank K L\nh3 : 0 < finrank { x // x \u2208 E } L\n\u22a2 finrank K { x // x \u2208 E } \u2264 finrank K { x // x \u2208 F }\n[PROOFSTEP]\nnlinarith\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : { x // x \u2208 S }\n\u22a2 IsIntegral K x \u2194 IsIntegral K \u2191x\n[PROOFSTEP]\nrw [\u2190 isAlgebraic_iff_isIntegral, isAlgebraic_iff, isAlgebraic_iff_isIntegral]\n[GOAL]\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : { x // x \u2208 S }\n\u22a2 minpoly K x = minpoly K \u2191x\n[PROOFSTEP]\nby_cases hx : IsIntegral K x\n[GOAL]\ncase pos\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : { x // x \u2208 S }\nhx : IsIntegral K x\n\u22a2 minpoly K x = minpoly K \u2191x\n[PROOFSTEP]\nexact minpoly.eq_of_algebraMap_eq (algebraMap S L).injective hx rfl\n[GOAL]\ncase neg\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nx : { x // x \u2208 S }\nhx : \u00acIsIntegral K x\n\u22a2 minpoly K x = minpoly K \u2191x\n[PROOFSTEP]\nexact (minpoly.eq_zero hx).trans (minpoly.eq_zero (mt isIntegral_iff.mpr hx)).symm\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.IntermediateField", "llama_tokens": 12096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.28657340276781285}} {"text": "[GOAL]\nX : Scheme\n\u22a2 Continuous fun x => PEmpty.elim x\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX : Scheme\nf g : \u2205 \u27f6 X\n\u22a2 f.val.base = g.val.base\n[PROOFSTEP]\next a\n[GOAL]\ncase w\nX : Scheme\nf g : \u2205 \u27f6 X\na : (forget TopCat).obj \u2191\u2205.toPresheafedSpace\n\u22a2 \u2191f.val.base a = \u2191g.val.base a\n[PROOFSTEP]\nexact PEmpty.elim a\n[GOAL]\nX : Scheme\nf g : \u2205 \u27f6 X\na : (Opens \u2191\u2191X.toPresheafedSpace)\u1d52\u1d56\n\u22a2 NatTrans.app (f.val.c \u226b whiskerRight (eqToHom (_ : (Opens.map f.val.base).op = (Opens.map g.val.base).op)) \u2205.presheaf)\n a =\n NatTrans.app g.val.c a\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 IsEmpty PEmpty\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 IsOpenImmersion f\n[PROOFSTEP]\napply (config := { allowSynthFailures := true }) IsOpenImmersion.of_stalk_iso\n[GOAL]\ncase hf\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 OpenEmbedding \u2191f.val.base\n[PROOFSTEP]\napply openEmbedding_of_continuous_injective_open\n[GOAL]\ncase hf.h\u2081\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 Continuous \u2191f.val.base\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase hf.h\u2082\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Injective \u2191f.val.base\n[PROOFSTEP]\nrintro (i : X.carrier)\n[GOAL]\ncase hf.h\u2082\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\ni : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 \u2983a\u2082 : (forget TopCat).obj \u2191X.toPresheafedSpace\u2984, \u2191f.val.base i = \u2191f.val.base a\u2082 \u2192 i = a\u2082\n[PROOFSTEP]\nexact isEmptyElim i\n[GOAL]\ncase hf.h\u2083\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 IsOpenMap \u2191f.val.base\n[PROOFSTEP]\nintro U _\n[GOAL]\ncase hf.h\u2083\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\nU : Set ((forget TopCat).obj \u2191X.toPresheafedSpace)\na\u271d : IsOpen U\n\u22a2 IsOpen (\u2191f.val.base '' U)\n[PROOFSTEP]\nconvert isOpen_empty (\u03b1 := Y)\n[GOAL]\ncase h.e'_3.h\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\nU : Set ((forget TopCat).obj \u2191X.toPresheafedSpace)\na\u271d : IsOpen U\ne_1\u271d : (forget TopCat).obj \u2191Y.toPresheafedSpace = \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base '' U = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h.h\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\nU : Set ((forget TopCat).obj \u2191X.toPresheafedSpace)\na\u271d : IsOpen U\ne_1\u271d : (forget TopCat).obj \u2191Y.toPresheafedSpace = \u2191\u2191Y.toPresheafedSpace\nx\u271d : (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 x\u271d \u2208 \u2191f.val.base '' U \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nrw [Set.mem_empty_iff_false, iff_false_iff]\n[GOAL]\ncase h.e'_3.h.h\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\nU : Set ((forget TopCat).obj \u2191X.toPresheafedSpace)\na\u271d : IsOpen U\ne_1\u271d : (forget TopCat).obj \u2191Y.toPresheafedSpace = \u2191\u2191Y.toPresheafedSpace\nx\u271d : (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 \u00acx\u271d \u2208 \u2191f.val.base '' U\n[PROOFSTEP]\nexact fun x => isEmptyElim (show X.carrier from x.choose)\n[GOAL]\ncase inst\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n[PROOFSTEP]\nrintro (i : X.carrier)\n[GOAL]\ncase inst\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\ni : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIso (PresheafedSpace.stalkMap f.val i)\n[PROOFSTEP]\nexact isEmptyElim i\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : IsEmpty X.carrier := \u27e8fun x => isEmptyElim (show Y.carrier from f.1.base x)\u27e9\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\nthis : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIso f\n[PROOFSTEP]\nhave : Epi f.1.base\n[GOAL]\ncase this\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\nthis : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 Epi f.val.base\n[PROOFSTEP]\nrw [TopCat.epi_iff_surjective]\n[GOAL]\ncase this\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\nthis : IsEmpty \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.Surjective \u2191f.val.base\n[PROOFSTEP]\nrintro (x : Y.carrier)\n[GOAL]\ncase this\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\nthis : IsEmpty \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2203 a, \u2191f.val.base a = x\n[PROOFSTEP]\nexact isEmptyElim x\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsEmpty \u2191\u2191Y.toPresheafedSpace\nthis\u271d : IsEmpty \u2191\u2191X.toPresheafedSpace\nthis : Epi f.val.base\n\u22a2 IsIso f\n[PROOFSTEP]\napply IsOpenImmersion.to_iso\n[GOAL]\nX : Scheme\n\u22a2 IsAffineOpen \u22a5\n[PROOFSTEP]\nconvert rangeIsAffineOpenOfOpenImmersion (initial.to X)\n[GOAL]\ncase h.e'_2\nX : Scheme\n\u22a2 \u22a5 = Scheme.Hom.opensRange (initial.to X)\n[PROOFSTEP]\next\n -- Porting note : added this `erw` to turn LHS to `False`\n[GOAL]\ncase h.e'_2.h.h\nX : Scheme\nx\u271d : \u2191\u2191X.toPresheafedSpace\n\u22a2 x\u271d \u2208 \u2191\u22a5 \u2194 x\u271d \u2208 \u2191(Scheme.Hom.opensRange (initial.to X))\n[PROOFSTEP]\nerw [Set.mem_empty_iff_false]\n[GOAL]\ncase h.e'_2.h.h\nX : Scheme\nx\u271d : \u2191\u2191X.toPresheafedSpace\n\u22a2 False \u2194 x\u271d \u2208 \u2191(Scheme.Hom.opensRange (initial.to X))\n[PROOFSTEP]\nrw [false_iff_iff]\n[GOAL]\ncase h.e'_2.h.h\nX : Scheme\nx\u271d : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u00acx\u271d \u2208 \u2191(Scheme.Hom.opensRange (initial.to X))\n[PROOFSTEP]\nexact fun x => isEmptyElim (show (\u22a5_ Scheme).carrier from x.choose)\n[GOAL]\nA : Scheme\nf : A \u27f6 \u22a5_ Scheme\n\u22a2 IsIso f\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Limits", "llama_tokens": 2545, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.28649188721673646}} {"text": "[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX\u271d Y\u271d : ModuleCat S\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : (restrictScalars f).map a\u2081\u271d = (restrictScalars f).map a\u2082\u271d\nx : \u2191X\u271d\n\u22a2 \u2191a\u2081\u271d x = \u2191a\u2082\u271d x\n[PROOFSTEP]\nsimpa only using FunLike.congr_fun h x\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nM : Type v\nI : AddCommGroup M\ninst\u271d : Module S M\nthis : SMul R M\nr : R\ns : S\nm : M\n\u22a2 \u2191f r \u2022 s \u2022 m = s \u2022 \u2191f r \u2022 m\n[PROOFSTEP]\nsimp [\u2190 mul_smul, mul_comm]\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M1 M2 : ModuleCat R\nl : M1 \u27f6 M2\n\u22a2 obj' f M1 \u27f6 obj' f M2\n[PROOFSTEP]\napply @LinearMap.baseChange R S M1 M2 _ _ ((algebraMap S _).comp f).toAlgebra _ _ _ _ l\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(map' f (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x\n[PROOFSTEP]\ndsimp only [map']\n -- Porting note: this got put in the dsimp by mathport\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x\n[PROOFSTEP]\nrw [ModuleCat.id_apply]\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = x\n[PROOFSTEP]\ninduction' x using TensorProduct.induction_on with _ _ m s ihx ihy\n[GOAL]\ncase zero\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) 0 = 0\n[PROOFSTEP]\nrw [map_zero]\n -- Porting note: simp only [map_zero] failed\n[GOAL]\ncase tmul\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny\u271d : \u2191M\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (x\u271d \u2297\u209c[R] y\u271d) = x\u271d \u2297\u209c[R] y\u271d\n[PROOFSTEP]\nerw [@LinearMap.baseChange_tmul R S M M _ _ (_), ModuleCat.id_apply]\n[GOAL]\ncase add\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nm s : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191M\nihx : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) m = m\nihy : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) s = s\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (m + s) = m + s\n[PROOFSTEP]\nrw [map_add, ihx, ihy]\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M\u2081 M\u2082 M\u2083 : ModuleCat R\nl\u2081\u2082 : M\u2081 \u27f6 M\u2082\nl\u2082\u2083 : M\u2082 \u27f6 M\u2083\nx : \u2191(obj' f M\u2081)\n\u22a2 \u2191(map' f (l\u2081\u2082 \u226b l\u2082\u2083)) x = \u2191(map' f l\u2081\u2082 \u226b map' f l\u2082\u2083) x\n[PROOFSTEP]\ndsimp only [map']\n[GOAL]\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M\u2081 M\u2082 M\u2083 : ModuleCat R\nl\u2081\u2082 : M\u2081 \u27f6 M\u2082\nl\u2082\u2083 : M\u2082 \u27f6 M\u2083\nx : \u2191(obj' f M\u2081)\n\u22a2 \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) x = \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) x\n[PROOFSTEP]\ninduction' x using TensorProduct.induction_on with _ _ x y ihx ihy\n[GOAL]\ncase zero\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M\u2081 M\u2082 M\u2083 : ModuleCat R\nl\u2081\u2082 : M\u2081 \u27f6 M\u2082\nl\u2082\u2083 : M\u2082 \u27f6 M\u2083\n\u22a2 \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) 0 = \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase tmul\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M\u2081 M\u2082 M\u2083 : ModuleCat R\nl\u2081\u2082 : M\u2081 \u27f6 M\u2082\nl\u2082\u2083 : M\u2082 \u27f6 M\u2083\nx\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny\u271d : \u2191M\u2081\n\u22a2 \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) (x\u271d \u2297\u209c[R] y\u271d) =\n \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) (x\u271d \u2297\u209c[R] y\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase add\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM M\u2081 M\u2082 M\u2083 : ModuleCat R\nl\u2081\u2082 : M\u2081 \u27f6 M\u2082\nl\u2082\u2083 : M\u2082 \u27f6 M\u2083\nx y : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191M\u2081\nihx : \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) x = \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) x\nihy : \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) y = \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) y\n\u22a2 \u2191(LinearMap.baseChange S (l\u2081\u2082 \u226b l\u2082\u2083)) (x + y) = \u2191(LinearMap.baseChange S l\u2081\u2082 \u226b LinearMap.baseChange S l\u2082\u2083) (x + y)\n[PROOFSTEP]\nrw [map_add, map_add, ihx, ihy]\n -- Porting note: simp again failing where rw succeeds\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nx y : S\n\u22a2 (fun s' => \u2191g (s' * s)) (x + y) = (fun s' => \u2191g (s' * s)) x + (fun s' => \u2191g (s' * s)) y\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nx y : S\n\u22a2 \u2191g ((x + y) * s) = \u2191g (x * s) + \u2191g (y * s)\n[PROOFSTEP]\nrw [add_mul, map_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nr : R\nt : S\n\u22a2 AddHom.toFun\n { toFun := fun s' => \u2191g (s' * s),\n map_add' :=\n (_ : \u2200 (x y : S), (fun s' => \u2191g (s' * s)) (x + y) = (fun s' => \u2191g (s' * s)) x + (fun s' => \u2191g (s' * s)) y) }\n (r \u2022 t) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s' => \u2191g (s' * s),\n map_add' :=\n (_ : \u2200 (x y : S), (fun s' => \u2191g (s' * s)) (x + y) = (fun s' => \u2191g (s' * s)) x + (fun s' => \u2191g (s' * s)) y) }\n t\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nr : R\nt : S\n\u22a2 \u2191g (r \u2022 t * s) = r \u2022 \u2191g (t * s)\n[PROOFSTEP]\nrw [\u2190 LinearMap.map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nr : R\nt : S\n\u22a2 \u2191g (r \u2022 t * s) = \u2191g (r \u2022 (t * s))\n[PROOFSTEP]\nerw [smul_eq_mul, smul_eq_mul, mul_assoc]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : SMul S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := hasSMul f M\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\ns : S\n\u22a2 \u2191(1 \u2022 g) s = \u2191g s\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : SMul S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := hasSMul f M\ns t : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nx : S\n\u22a2 \u2191((s * t) \u2022 g) x = \u2191(s \u2022 t \u2022 g) x\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : MulAction S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := mulAction f M\ns t : S\n\u22a2 \u2191(s \u2022 0) t = \u21910 t\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : MulAction S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := mulAction f M\ns : S\ng h : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nt : S\n\u22a2 \u2191(s \u2022 (g + h)) t = \u2191(s \u2022 g + s \u2022 h) t\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : DistribMulAction S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := distribMulAction f M\ns1 s2 : S\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nx : S\n\u22a2 \u2191((s1 + s2) \u2022 g) x = \u2191(s1 \u2022 g + s2 \u2022 g) x\n[PROOFSTEP]\nsimp [mul_add, LinearMap.map_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nf : R \u2192+* S\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nsrc\u271d : DistribMulAction S (\u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M) := distribMulAction f M\ng : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] M\nx : S\n\u22a2 \u2191(0 \u2022 g) x = \u21910 x\n[PROOFSTEP]\nsimp [LinearMap.map_zero]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nM\u271d : ModuleCat R\nM M' : ModuleCat R\ng : M \u27f6 M'\ns : S\nh : \u2191(obj' f M)\nt : S\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun h => LinearMap.comp g h,\n map_add' :=\n (_ : \u2200 (x x_1 : \u2191(obj' f M)), LinearMap.comp g (x + x_1) = LinearMap.comp g x + LinearMap.comp g x_1) }\n (s \u2022 h))\n t =\n \u2191(\u2191(RingHom.id S) s \u2022\n AddHom.toFun\n { toFun := fun h => LinearMap.comp g h,\n map_add' :=\n (_ : \u2200 (x x_1 : \u2191(obj' f M)), LinearMap.comp g (x + x_1) = LinearMap.comp g x + LinearMap.comp g x_1) }\n h)\n t\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nM\u271d : ModuleCat R\nM M' : ModuleCat R\ng : M \u27f6 M'\ns : S\nh : \u2191(obj' f M)\nt : S\n\u22a2 \u2191g (\u2191(s \u2022 h) t) = \u2191(s \u2022 LinearMap.comp g h) t\n[PROOFSTEP]\nrw [smul_apply', smul_apply']\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nM\u271d : ModuleCat R\nM M' : ModuleCat R\ng : M \u27f6 M'\ns : S\nh : \u2191(obj' f M)\nt : S\n\u22a2 \u2191g (\u2191h (t * s)) = \u2191(LinearMap.comp g h) (t * s)\n[PROOFSTEP]\nsimp\n -- Porting note: smul_apply' not working in simp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\ns1 s2 : S\n\u22a2 (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2\n[PROOFSTEP]\nsimp only [add_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\ns1 s2 : S\n\u22a2 \u2191g (s1 \u2022 y + s2 \u2022 y) = \u2191g (s1 \u2022 y) + \u2191g (s2 \u2022 y)\n[PROOFSTEP]\nrw [LinearMap.map_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\nr : R\ns : S\n\u22a2 AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ : \u2200 (s1 s2 : S), (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ : \u2200 (s1 s2 : S), (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\nr : R\ns : S\n\u22a2 \u2191g ((r \u2022 s) \u2022 y) = r \u2022 \u2191g (s \u2022 y)\n[PROOFSTEP]\nrw [\u2190 g.map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\nr : R\ns : S\n\u22a2 \u2191g ((r \u2022 s) \u2022 y) = \u2191g (r \u2022 s \u2022 y)\n[PROOFSTEP]\nerw [smul_eq_mul, mul_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny : \u2191Y\nr : R\ns : S\n\u22a2 \u2191g (\u2191f r \u2022 s \u2022 y) = \u2191g (r \u2022 s \u2022 y)\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191((fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n (y1 + y2))\n s =\n \u2191((fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y2)\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191g (s \u2022 (y1 + y2)) =\n \u2191({\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y1),\n map_add' := (_ : \u2200 (s1 s2 : S), \u2191g ((s1 + s2) \u2022 y1) = \u2191g (s1 \u2022 y1) + \u2191g (s2 \u2022 y1)) },\n map_smul' := (_ : \u2200 (r : R) (s : S), \u2191g ((r \u2022 s) \u2022 y1) = r \u2022 \u2191g (s \u2022 y1)) } +\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y2),\n map_add' := (_ : \u2200 (s1 s2 : S), \u2191g ((s1 + s2) \u2022 y2) = \u2191g (s1 \u2022 y2) + \u2191g (s2 \u2022 y2)) },\n map_smul' := (_ : \u2200 (r : R) (s : S), \u2191g ((r \u2022 s) \u2022 y2) = r \u2022 \u2191g (s \u2022 y2)) })\n s\n[PROOFSTEP]\nrw [LinearMap.add_apply, LinearMap.coe_mk, LinearMap.coe_mk]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191g (s \u2022 (y1 + y2)) =\n \u2191{ toFun := fun s => \u2191g (s \u2022 y1),\n map_add' := (_ : \u2200 (s1 s2 : S), \u2191g ((s1 + s2) \u2022 y1) = \u2191g (s1 \u2022 y1) + \u2191g (s2 \u2022 y1)) }\n s +\n \u2191{ toFun := fun s => \u2191g (s \u2022 y2),\n map_add' := (_ : \u2200 (s1 s2 : S), \u2191g ((s1 + s2) \u2022 y2) = \u2191g (s1 \u2022 y2) + \u2191g (s2 \u2022 y2)) }\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191g (s \u2022 (y1 + y2)) = \u2191g (s \u2022 y1) + \u2191g (s \u2022 y2)\n[PROOFSTEP]\nrw [smul_add, map_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ns : S\ny : \u2191Y\nt : S\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) },\n map_add' :=\n (_ :\n \u2200 (y1 y2 : \u2191Y),\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n (y1 + y2) =\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y2) }\n (s \u2022 y))\n t =\n \u2191(\u2191(RingHom.id S) s \u2022\n AddHom.toFun\n {\n toFun := fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) },\n map_add' :=\n (_ :\n \u2200 (y1 y2 : \u2191Y),\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n (y1 + y2) =\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s => \u2191g (s \u2022 y)) (s1 + s2) =\n (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n s) })\n y2) }\n y)\n t\n[PROOFSTEP]\nrw [RingHom.id_apply, LinearMap.coe_mk, CategoryTheory.ModuleCat.CoextendScalars.smul_apply', LinearMap.coe_mk]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ns : S\ny : \u2191Y\nt : S\n\u22a2 \u2191{ toFun := fun s_1 => \u2191g (s_1 \u2022 s \u2022 y),\n map_add' :=\n (_ :\n \u2200 (s1 s2 : S),\n (fun s_1 => \u2191g (s_1 \u2022 s \u2022 y)) (s1 + s2) =\n (fun s_1 => \u2191g (s_1 \u2022 s \u2022 y)) s1 + (fun s_1 => \u2191g (s_1 \u2022 s \u2022 y)) s2) }\n t =\n \u2191{ toFun := fun s => \u2191g (s \u2022 y),\n map_add' :=\n (_ : \u2200 (s1 s2 : S), (fun s => \u2191g (s \u2022 y)) (s1 + s2) = (fun s => \u2191g (s \u2022 y)) s1 + (fun s => \u2191g (s \u2022 y)) s2) }\n (t * s)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y \u27f6 X\ns : S\ny : \u2191Y\nt : S\n\u22a2 \u2191g (t \u2022 s \u2022 y) = \u2191g ((t * s) \u2022 y)\n[PROOFSTEP]\nrw [mul_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nx y : \u2191((restrictScalars f).obj Y)\n\u22a2 (fun y => AddHom.toFun (\u2191g y).toAddHom 1) (x + y) =\n (fun y => AddHom.toFun (\u2191g y).toAddHom 1) x + (fun y => AddHom.toFun (\u2191g y).toAddHom 1) y\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nx y : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191(\u2191g (x + y)) 1 = \u2191(\u2191g x) 1 + \u2191(\u2191g y) 1\n[PROOFSTEP]\nrw [g.map_add, LinearMap.add_apply]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 AddHom.toFun\n { toFun := fun y => AddHom.toFun (\u2191g y).toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f).obj Y)),\n (fun y => AddHom.toFun (\u2191g y).toAddHom 1) (x + y) =\n (fun y => AddHom.toFun (\u2191g y).toAddHom 1) x + (fun y => AddHom.toFun (\u2191g y).toAddHom 1) y) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => AddHom.toFun (\u2191g y).toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f).obj Y)),\n (fun y => AddHom.toFun (\u2191g y).toAddHom 1) (x + y) =\n (fun y => AddHom.toFun (\u2191g y).toAddHom 1) x + (fun y => AddHom.toFun (\u2191g y).toAddHom 1) y) }\n y\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 \u2191(\u2191g (\u2191f r \u2022 y)) 1 = r \u2022 \u2191(\u2191g y) 1\n[PROOFSTEP]\nrw [\u2190 LinearMap.map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 \u2191(\u2191g (\u2191f r \u2022 y)) 1 = \u2191(\u2191g y) (r \u2022 1)\n[PROOFSTEP]\nerw [smul_eq_mul, mul_one, LinearMap.map_smul]\n -- Porting note: should probably change CoeFun for obj above\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 \u2191(\u2191f r \u2022 \u2191g y) 1 = \u2191(\u2191g y) (\u2191f r)\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 AddHom.toFun (\u2191f r \u2022 \u2191g y).toAddHom 1 = \u2191(\u2191g y) (\u2191f r)\n[PROOFSTEP]\nrw [CoextendScalars.smul_apply (s := f r) (g := g y) (s' := 1), one_mul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : Y \u27f6 (coextendScalars f).obj X\nr : R\ny : \u2191Y\n\u22a2 AddHom.toFun (\u2191g y).toAddHom (\u2191f r) = \u2191(\u2191g y) (\u2191f r)\n[PROOFSTEP]\nrw [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny : \u2191Y\nr : R\ns : S\n\u22a2 AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' := (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' := (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny : \u2191Y\nr : R\ns : S\n\u22a2 (r \u2022 s) \u2022 y = \u2191f r \u2022 s \u2022 y\n[PROOFSTEP]\nerw [smul_eq_mul, mul_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191((fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n (y1 + y2))\n s =\n \u2191((fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y2)\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191{\n toAddHom :=\n { toFun := fun s => s \u2022 (y1 + y2),\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 (y1 + y2) = s \u2022 (y1 + y2) + s' \u2022 (y1 + y2)) },\n map_smul' := (_ : \u2200 (r : R) (s : S), (\u2191f r * s) \u2022 (y1 + y2) = \u2191f r \u2022 s \u2022 (y1 + y2)) }\n s =\n \u2191({\n toAddHom :=\n { toFun := fun s => s \u2022 y1,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y1 = s \u2022 y1 + s' \u2022 y1) },\n map_smul' := (_ : \u2200 (r : R) (s : S), (\u2191f r * s) \u2022 y1 = \u2191f r \u2022 s \u2022 y1) } +\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y2,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y2 = s \u2022 y2 + s' \u2022 y2) },\n map_smul' := (_ : \u2200 (r : R) (s : S), (\u2191f r * s) \u2022 y2 = \u2191f r \u2022 s \u2022 y2) })\n s\n[PROOFSTEP]\nrw [LinearMap.add_apply, LinearMap.coe_mk, LinearMap.coe_mk, LinearMap.coe_mk]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny1 y2 : \u2191Y\ns : S\n\u22a2 \u2191{ toFun := fun s => s \u2022 (y1 + y2),\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 (y1 + y2) = s \u2022 (y1 + y2) + s' \u2022 (y1 + y2)) }\n s =\n \u2191{ toFun := fun s => s \u2022 y1,\n map_add' := (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y1 = s \u2022 y1 + s' \u2022 y1) }\n s +\n \u2191{ toFun := fun s => s \u2022 y2,\n map_add' := (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y2 = s \u2022 y2 + s' \u2022 y2) }\n s\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ny1 y2 : \u2191Y\ns : S\n\u22a2 s \u2022 (y1 + y2) = s \u2022 y1 + s \u2022 y2\n[PROOFSTEP]\nrw [smul_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ns : S\ny : \u2191Y\nt : S\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) },\n map_add' :=\n (_ :\n \u2200 (y1 y2 : \u2191Y),\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n (y1 + y2) =\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y2) }\n (s \u2022 y))\n t =\n \u2191(\u2191(RingHom.id S) s \u2022\n AddHom.toFun\n {\n toFun := fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) },\n map_add' :=\n (_ :\n \u2200 (y1 y2 : \u2191Y),\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n (y1 + y2) =\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y1 +\n (fun y =>\n {\n toAddHom :=\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))),\n (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n s) })\n y2) }\n y)\n t\n[PROOFSTEP]\nrw [RingHom.id_apply, LinearMap.coe_mk, CoextendScalars.smul_apply', LinearMap.coe_mk]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ns : S\ny : \u2191Y\nt : S\n\u22a2 \u2191{ toFun := fun s_1 => s_1 \u2022 s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (s_1 s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s_1 + s') \u2022 s \u2022 y = s_1 \u2022 s \u2022 y + s' \u2022 s \u2022 y) }\n t =\n \u2191{ toFun := fun s => s \u2022 y,\n map_add' := (_ : \u2200 (s s' : \u2191((restrictScalars f).obj (ModuleCat.mk S))), (s + s') \u2022 y = s \u2022 y + s' \u2022 y) }\n (t * s)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY : ModuleCat S\ns : S\ny : \u2191Y\nt : S\n\u22a2 t \u2022 s \u2022 y = (t * s) \u2022 y\n[PROOFSTEP]\nrw [mul_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 \u2191(\u2191((\ud835\udfed (ModuleCat S)).map g \u226b (fun Y => app' f Y) Y') y) s =\n \u2191(\u2191((fun Y => app' f Y) Y \u226b (restrictScalars f \u22d9 coextendScalars f).map g) y) s\n[PROOFSTEP]\nsimp only [ModuleCat.coe_comp, Functor.id_map, Functor.id_obj, Functor.comp_obj, Functor.comp_map]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 \u2191(\u2191(g \u226b app' f Y') y) s = \u2191(\u2191(app' f Y \u226b (coextendScalars f).map ((restrictScalars f).map g)) y) s\n[PROOFSTEP]\nrw [coe_comp, coe_comp, Function.comp, Function.comp]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 \u2191(\u2191(app' f Y') (\u2191g y)) s = \u2191(\u2191((coextendScalars f).map ((restrictScalars f).map g)) (\u2191(app' f Y) y)) s\n[PROOFSTEP]\nconv_rhs => rw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n| \u2191(\u2191((coextendScalars f).map ((restrictScalars f).map g)) (\u2191(app' f Y) y)) s\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n| \u2191(\u2191((coextendScalars f).map ((restrictScalars f).map g)) (\u2191(app' f Y) y)) s\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n| \u2191(\u2191((coextendScalars f).map ((restrictScalars f).map g)) (\u2191(app' f Y) y)) s\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 \u2191(\u2191(app' f Y') (\u2191g y)) s =\n AddHom.toFun (\u2191((coextendScalars f).map ((restrictScalars f).map g)) (\u2191(app' f Y) y)).toAddHom s\n[PROOFSTEP]\nerw [CoextendScalars.map_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, restrictScalars.map_apply f]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 \u2191(\u2191(app' f Y') (\u2191g y)) s = \u2191((restrictScalars f).map g) (\u2191(\u2191(app' f Y) y).toAddHom s)\n[PROOFSTEP]\nchange s \u2022 (g y) = g (s \u2022 y)\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\ny : \u2191Y\ns : S\n\u22a2 s \u2022 \u2191g y = \u2191g (s \u2022 y)\n[PROOFSTEP]\nrw [map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nx1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)\n\u22a2 (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nx1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)\n\u22a2 \u2191(x1 + x2) 1 = \u2191x1 1 + \u2191x2 1\n[PROOFSTEP]\nrw [LinearMap.add_apply]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n (r \u2022 g) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n g\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 \u2191(\u2191f r \u2022 g) 1 = r \u2022 \u2191g 1\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 AddHom.toFun (\u2191f r \u2022 g).toAddHom 1 = r \u2022 \u2191g 1\n[PROOFSTEP]\nrw [CoextendScalars.smul_apply (s := f r) (g := g) (s' := 1), one_mul, \u2190 LinearMap.map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 AddHom.toFun g.toAddHom (\u2191f r) = \u2191g (r \u2022 1)\n[PROOFSTEP]\nrw [\u2190 LinearMap.coe_toAddHom, \u2190 AddHom.toFun_eq_coe]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 AddHom.toFun g.toAddHom (\u2191f r) = AddHom.toFun g.toAddHom (r \u2022 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 \u2191f r = r \u2022 1\n[PROOFSTEP]\nchange f r = (f r) \u2022 (1 : S)\n[GOAL]\ncase e_a\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\ng : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))\n\u22a2 \u2191f r = \u2191f r \u2022 1\n[PROOFSTEP]\nrw [smul_eq_mul (a := f r) (a' := 1), mul_one]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX X' : ModuleCat R\ng : X \u27f6 X'\nh : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)\n\u22a2 \u2191((coextendScalars f \u22d9 restrictScalars f).map g \u226b\n (fun X =>\n {\n toAddHom :=\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (g : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))),\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n (r \u2022 g) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 +\n (fun g => AddHom.toFun g.toAddHom 1) x2) }\n g) })\n X')\n h =\n \u2191((fun X =>\n {\n toAddHom :=\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (g : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))),\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n (r \u2022 g) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 +\n (fun g => AddHom.toFun g.toAddHom 1) x2) }\n g) })\n X \u226b\n (\ud835\udfed (ModuleCat R)).map g)\n h\n[PROOFSTEP]\nrw [ModuleCat.coe_comp]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX X' : ModuleCat R\ng : X \u27f6 X'\nh : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)\n\u22a2 (\u2191((fun X =>\n {\n toAddHom :=\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (g : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))),\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n (r \u2022 g) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 +\n (fun g => AddHom.toFun g.toAddHom 1) x2) }\n g) })\n X') \u2218\n \u2191((coextendScalars f \u22d9 restrictScalars f).map g))\n h =\n \u2191((fun X =>\n {\n toAddHom :=\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (g : \u2191((restrictScalars f).obj ((coextendScalars f).obj X))),\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 + (fun g => AddHom.toFun g.toAddHom 1) x2) }\n (r \u2022 g) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun g => AddHom.toFun g.toAddHom 1,\n map_add' :=\n (_ :\n \u2200 (x1 x2 : \u2191((coextendScalars f \u22d9 restrictScalars f).obj X)),\n (fun g => AddHom.toFun g.toAddHom 1) (x1 + x2) =\n (fun g => AddHom.toFun g.toAddHom 1) x1 +\n (fun g => AddHom.toFun g.toAddHom 1) x2) }\n g) })\n X \u226b\n (\ud835\udfed (ModuleCat R)).map g)\n h\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : (restrictScalars f).obj X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(RestrictionCoextensionAdj.HomEquiv.toRestriction f (RestrictionCoextensionAdj.HomEquiv.fromRestriction f g)) x =\n \u2191g x\n[PROOFSTEP]\nrw [RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,\n RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply, one_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\nx : \u2191X\ns : S\n\u22a2 \u2191(\u2191(RestrictionCoextensionAdj.HomEquiv.fromRestriction f (RestrictionCoextensionAdj.HomEquiv.toRestriction f g)) x)\n s =\n \u2191(\u2191g x) s\n[PROOFSTEP]\nrw [RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply,\n RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,\n LinearMap.map_smul\u209b\u2097, RingHom.id_apply, CoextendScalars.smul_apply', one_mul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\n\u22a2 \u2200 (x : \u2191((restrictScalars f).obj X)),\n \u2191(\u2191((fun X Y =>\n { toFun := RestrictionCoextensionAdj.HomEquiv.fromRestriction f,\n invFun := RestrictionCoextensionAdj.HomEquiv.toRestriction f,\n left_inv :=\n (_ :\n \u2200 (g : (restrictScalars f).obj X \u27f6 Y),\n RestrictionCoextensionAdj.HomEquiv.toRestriction f\n (RestrictionCoextensionAdj.HomEquiv.fromRestriction f g) =\n g),\n right_inv :=\n (_ :\n \u2200 (g : X \u27f6 (coextendScalars f).obj Y),\n RestrictionCoextensionAdj.HomEquiv.fromRestriction f\n (RestrictionCoextensionAdj.HomEquiv.toRestriction f g) =\n g) })\n X Y).symm\n g)\n x =\n \u2191((restrictScalars f).map g \u226b NatTrans.app (RestrictionCoextensionAdj.counit' f) Y) x\n[PROOFSTEP]\nintro x\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\nx : \u2191((restrictScalars f).obj X)\n\u22a2 \u2191(\u2191((fun X Y =>\n { toFun := RestrictionCoextensionAdj.HomEquiv.fromRestriction f,\n invFun := RestrictionCoextensionAdj.HomEquiv.toRestriction f,\n left_inv :=\n (_ :\n \u2200 (g : (restrictScalars f).obj X \u27f6 Y),\n RestrictionCoextensionAdj.HomEquiv.toRestriction f\n (RestrictionCoextensionAdj.HomEquiv.fromRestriction f g) =\n g),\n right_inv :=\n (_ :\n \u2200 (g : X \u27f6 (coextendScalars f).obj Y),\n RestrictionCoextensionAdj.HomEquiv.fromRestriction f\n (RestrictionCoextensionAdj.HomEquiv.toRestriction f g) =\n g) })\n X Y).symm\n g)\n x =\n \u2191((restrictScalars f).map g \u226b NatTrans.app (RestrictionCoextensionAdj.counit' f) Y) x\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\nx : \u2191((restrictScalars f).obj X)\n\u22a2 \u2191(\u2191g x) 1 = \u2191((restrictScalars f).map g \u226b NatTrans.app (RestrictionCoextensionAdj.counit' f) Y) x\n[PROOFSTEP]\nrw [coe_comp, Function.comp]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\nx : \u2191((restrictScalars f).obj X)\n\u22a2 \u2191(\u2191g x) 1 = \u2191(NatTrans.app (RestrictionCoextensionAdj.counit' f) Y) (\u2191((restrictScalars f).map g) x)\n[PROOFSTEP]\nchange _ = (((restrictScalars f).map g) x).toFun (1 : S)\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\nX : ModuleCat S\nY : ModuleCat R\ng : X \u27f6 (coextendScalars f).obj Y\nx : \u2191((restrictScalars f).obj X)\n\u22a2 \u2191(\u2191g x) 1 = AddHom.toFun (\u2191((restrictScalars f).map g) x).toAddHom 1\n[PROOFSTEP]\nrw [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, restrictScalars.map_apply]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nx\u271d\u00b9 x\u271d : \u2191X\n\u22a2 (fun x => \u2191g (1 \u2297\u209c[R] x)) (x\u271d\u00b9 + x\u271d) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x\u271d\u00b9 + (fun x => \u2191g (1 \u2297\u209c[R] x)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nx\u271d\u00b9 x\u271d : \u2191X\n\u22a2 \u2191g (1 \u2297\u209c[R] (x\u271d\u00b9 + x\u271d)) = \u2191g (1 \u2297\u209c[R] x\u271d\u00b9) + \u2191g (1 \u2297\u209c[R] x\u271d)\n[PROOFSTEP]\nrw [tmul_add, map_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\n\u22a2 AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n x\n[PROOFSTEP]\nletI : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\nthis : Module R S := Module.compHom S f\n\u22a2 AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n x\n[PROOFSTEP]\nletI : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\nthis\u271d : Module R S := Module.compHom S f\nthis : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => \u2191g (1 \u2297\u209c[R] x),\n map_add' :=\n (_ :\n \u2200 (x x_1 : \u2191X),\n (fun x => \u2191g (1 \u2297\u209c[R] x)) (x + x_1) = (fun x => \u2191g (1 \u2297\u209c[R] x)) x + (fun x => \u2191g (1 \u2297\u209c[R] x)) x_1) }\n x\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\nthis\u271d : Module R S := Module.compHom S f\nthis : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191g (1 \u2297\u209c[R] (r \u2022 x)) = \u2191f r \u2022 \u2191g (1 \u2297\u209c[R] x)\n[PROOFSTEP]\nrw [RestrictScalars.smul_def, \u2190 LinearMap.map_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\nthis\u271d : Module R S := Module.compHom S f\nthis : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191g\n (1 \u2297\u209c[R]\n \u2191(AddEquiv.symm (RestrictScalars.addEquiv R R \u2191X))\n (\u2191(algebraMap R R) r \u2022 \u2191(RestrictScalars.addEquiv R R \u2191X) x)) =\n \u2191g (\u2191f r \u2022 1 \u2297\u209c[R] x)\n[PROOFSTEP]\nerw [tmul_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nr : R\nx : \u2191X\nthis\u271d : Module R S := Module.compHom S f\nthis : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191g (\u2191(algebraMap R R) r \u2022 1 \u2297\u209c[R] \u2191(RestrictScalars.addEquiv R R \u2191X) x) = \u2191g (\u2191f r \u2022 1 \u2297\u209c[R] x)\n[PROOFSTEP]\ncongr\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X \u27f6 (restrictScalars f).obj Y\n\u22a2 \u2200 (x y : \u2191X), (fun x => s \u2022 \u2191g x) (x + y) = (fun x => s \u2022 \u2191g x) x + (fun x => s \u2022 \u2191g x) y\n[PROOFSTEP]\nintros\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X \u27f6 (restrictScalars f).obj Y\nx\u271d y\u271d : \u2191X\n\u22a2 (fun x => s \u2022 \u2191g x) (x\u271d + y\u271d) = (fun x => s \u2022 \u2191g x) x\u271d + (fun x => s \u2022 \u2191g x) y\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X \u27f6 (restrictScalars f).obj Y\nx\u271d y\u271d : \u2191X\n\u22a2 s \u2022 \u2191g (x\u271d + y\u271d) = s \u2022 \u2191g x\u271d + s \u2022 \u2191g y\u271d\n[PROOFSTEP]\nrw [map_add, smul_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X \u27f6 (restrictScalars f).obj Y\n\u22a2 \u2200 (r : R) (x : \u2191X),\n AddHom.toFun\n { toFun := fun x => s \u2022 \u2191g x,\n map_add' := (_ : \u2200 (x y : \u2191X), (fun x => s \u2022 \u2191g x) (x + y) = (fun x => s \u2022 \u2191g x) x + (fun x => s \u2022 \u2191g x) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => s \u2022 \u2191g x,\n map_add' :=\n (_ : \u2200 (x y : \u2191X), (fun x => s \u2022 \u2191g x) (x + y) = (fun x => s \u2022 \u2191g x) x + (fun x => s \u2022 \u2191g x) y) }\n x\n[PROOFSTEP]\nintros r x\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X \u27f6 (restrictScalars f).obj Y\nr : R\nx : \u2191X\n\u22a2 AddHom.toFun\n { toFun := fun x => s \u2022 \u2191g x,\n map_add' := (_ : \u2200 (x y : \u2191X), (fun x => s \u2022 \u2191g x) (x + y) = (fun x => s \u2022 \u2191g x) x + (fun x => s \u2022 \u2191g x) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => s \u2022 \u2191g x,\n map_add' := (_ : \u2200 (x y : \u2191X), (fun x => s \u2022 \u2191g x) (x + y) = (fun x => s \u2022 \u2191g x) x + (fun x => s \u2022 \u2191g x) y) }\n x\n[PROOFSTEP]\nrw [AddHom.toFun_eq_coe, AddHom.coe_mk, RingHom.id_apply, LinearMap.map_smul, smul_comm r s (g x : Y)]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\n\u22a2 (extendScalars f).obj X \u27f6 Y\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\n\u22a2 (extendScalars f).obj X \u27f6 Y\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 (extendScalars f).obj X \u27f6 Y\n[PROOFSTEP]\nrefine { toFun := fun z => TensorProduct.lift ?_ z, map_add' := ?_, map_smul' := ?_ }\n[GOAL]\ncase refine_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\n\u22a2 \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2192\u2097[R] \u2191X \u2192\u2097[R] \u2191Y\n[PROOFSTEP]\nrefine\n { toFun := fun s => HomEquiv.evalAt f s g, map_add' := fun (s\u2081 s\u2082 : S) => ?_, map_smul' := fun (r : R) (s : S) => ?_ }\n[GOAL]\ncase refine_1.refine_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\ns\u2081 s\u2082 : S\n\u22a2 (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase refine_1.refine_1.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\ns\u2081 s\u2082 : S\nx\u271d : \u2191X\n\u22a2 \u2191((fun s => evalAt f s g) (s\u2081 + s\u2082)) x\u271d = \u2191((fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1.refine_1.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\ns\u2081 s\u2082 : S\nx\u271d : \u2191X\n\u22a2 (s\u2081 + s\u2082) \u2022 \u2191g x\u271d = s\u2081 \u2022 \u2191g x\u271d + s\u2082 \u2022 \u2191g x\u271d\n[PROOFSTEP]\nrw [\u2190 add_smul]\n[GOAL]\ncase refine_1.refine_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\nr : R\ns : S\n\u22a2 AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s\n[PROOFSTEP]\next x\n[GOAL]\ncase refine_1.refine_2.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\nr : R\ns : S\nx : \u2191X\n\u22a2 \u2191(AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s))\n x =\n \u2191(\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s)\n x\n[PROOFSTEP]\napply mul_smul (f r) s (g x)\n[GOAL]\ncase refine_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (x y : \u2191((extendScalars f).obj X)),\n (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n (x + y) =\n (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n x +\n (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n y\n[PROOFSTEP]\nintros z\u2081 z\u2082\n[GOAL]\ncase refine_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz\u2081 z\u2082 : \u2191((extendScalars f).obj X)\n\u22a2 (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n (z\u2081 + z\u2082) =\n (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n z\u2081 +\n (fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z)\n z\u2082\n[PROOFSTEP]\nchange lift _ (z\u2081 + z\u2082) = lift _ z\u2081 + lift _ z\u2082\n[GOAL]\ncase refine_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz\u2081 z\u2082 : \u2191((extendScalars f).obj X)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (z\u2081 + z\u2082) =\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2081 +\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2082\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\ncase refine_3\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (r : S) (x : \u2191((extendScalars f).obj X)),\n AddHom.toFun\n {\n toFun := fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z,\n map_add' :=\n (_ :\n \u2200 (z\u2081 z\u2082 : \u2191((extendScalars f).obj X)),\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (z\u2081 + z\u2082) =\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2081 +\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2082) }\n (r \u2022 x) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n {\n toFun := fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z,\n map_add' :=\n (_ :\n \u2200 (z\u2081 z\u2082 : \u2191((extendScalars f).obj X)),\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (z\u2081 + z\u2082) =\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2081 +\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2082) }\n x\n[PROOFSTEP]\nintro s z\n[GOAL]\ncase refine_3\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nz : \u2191((extendScalars f).obj X)\n\u22a2 AddHom.toFun\n {\n toFun := fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z,\n map_add' :=\n (_ :\n \u2200 (z\u2081 z\u2082 : \u2191((extendScalars f).obj X)),\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (z\u2081 + z\u2082) =\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2081 +\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2082) }\n (s \u2022 z) =\n \u2191(RingHom.id S) s \u2022\n AddHom.toFun\n {\n toFun := fun z =>\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z,\n map_add' :=\n (_ :\n \u2200 (z\u2081 z\u2082 : \u2191((extendScalars f).obj X)),\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (z\u2081 + z\u2082) =\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2081 +\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\u2082) }\n z\n[PROOFSTEP]\nchange lift _ (s \u2022 z) = s \u2022 lift _ z\n[GOAL]\ncase refine_3\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nz : \u2191((extendScalars f).obj X)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 z) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n z\n[PROOFSTEP]\ninduction' z using TensorProduct.induction_on with s' x x y ih1 ih2\n[GOAL]\ncase refine_3.zero\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 0) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n 0\n[PROOFSTEP]\nrw [smul_zero, map_zero, smul_zero]\n[GOAL]\ncase refine_3.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 s' \u2297\u209c[R] x) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s' \u2297\u209c[R] x)\n[PROOFSTEP]\nrw [LinearMap.coe_mk, ExtendScalars.smul_tmul]\n[GOAL]\ncase refine_3.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }).toAddHom\n ((s * s') \u2297\u209c[R] x) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }).toAddHom\n (s' \u2297\u209c[R] x)\n[PROOFSTEP]\nerw [lift.tmul, lift.tmul]\n[GOAL]\ncase refine_3.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 \u2191(\u2191{\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }\n (s * s'))\n x =\n s \u2022\n \u2191(\u2191{\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }\n s')\n x\n[PROOFSTEP]\nset s' : S := s'\n[GOAL]\ncase refine_3.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\ns' : S := s'\u271d\n\u22a2 \u2191(\u2191{\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }\n (s * s'))\n x =\n s \u2022\n \u2191(\u2191{\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) }\n s')\n x\n[PROOFSTEP]\nchange (s * s') \u2022 (g x) = s \u2022 s' \u2022 (g x)\n[GOAL]\ncase refine_3.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\ns' : S := s'\u271d\n\u22a2 (s * s') \u2022 \u2191g x = s \u2022 s' \u2022 \u2191g x\n[PROOFSTEP]\nrw [mul_smul]\n[GOAL]\ncase refine_3.add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nx y : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191X\nih1 :\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 x) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n x\nih2 :\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 y) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n y\n\u22a2 \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (s \u2022 (x + y)) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) = (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => evalAt f s g) s\u2081 + (fun s => evalAt f s g) s\u2082) }\n s) })\n (x + y)\n[PROOFSTEP]\nrw [smul_add, map_add, ih1, ih2, map_add, smul_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\n\u22a2 HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g) = g\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\n\u22a2 HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g) = g\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g) = g\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (x : \u2191((extendScalars f).obj X)), \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) x = \u2191g x\n[PROOFSTEP]\nintro z\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz : \u2191((extendScalars f).obj X)\n\u22a2 \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) z = \u2191g z\n[PROOFSTEP]\ninduction' z using TensorProduct.induction_on with x s z1 z2 ih1 ih2\n[GOAL]\ncase h.zero\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) 0 = \u2191g 0\n[PROOFSTEP]\nrw [map_zero, map_zero]\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ns : \u2191X\n\u22a2 \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) (x \u2297\u209c[R] s) = \u2191g (x \u2297\u209c[R] s)\n[PROOFSTEP]\nerw [TensorProduct.lift.tmul]\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ns : \u2191X\n\u22a2 \u2191(\u2191{\n toAddHom :=\n { toFun := fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g),\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2081 +\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n { toFun := fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g),\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2081 +\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g),\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2081 +\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2082) }\n s) }\n x)\n s =\n \u2191g (x \u2297\u209c[R] s)\n[PROOFSTEP]\nsimp only [LinearMap.coe_mk]\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ns : \u2191X\n\u22a2 \u2191(\u2191{ toFun := fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g),\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2081 +\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2082) }\n x)\n s =\n \u2191g (x \u2297\u209c[R] s)\n[PROOFSTEP]\nchange S at x \n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191X\nx : S\n\u22a2 \u2191(\u2191{ toFun := fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g),\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2081 +\n (fun s => HomEquiv.evalAt f s (HomEquiv.toRestrictScalars f g)) s\u2082) }\n x)\n s =\n \u2191g (x \u2297\u209c[R] s)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191X\nx : S\n\u22a2 x \u2022 \u2191(HomEquiv.toRestrictScalars f g) s = \u2191g (x \u2297\u209c[R] s)\n[PROOFSTEP]\nerw [\u2190 LinearMap.map_smul, ExtendScalars.smul_tmul, mul_one x]\n[GOAL]\ncase h.add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nz1 z2 : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191X\nih1 : \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) z1 = \u2191g z1\nih2 : \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) z2 = \u2191g z2\n\u22a2 \u2191(HomEquiv.fromExtendScalars f (HomEquiv.toRestrictScalars f g)) (z1 + z2) = \u2191g (z1 + z2)\n[PROOFSTEP]\nrw [map_add, map_add, ih1, ih2]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\n\u22a2 HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g) = g\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\n\u22a2 HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g) = g\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g) = g\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (x : \u2191X), \u2191(HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g)) x = \u2191g x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191X\n\u22a2 \u2191(HomEquiv.toRestrictScalars f (HomEquiv.fromExtendScalars f g)) x = \u2191g x\n[PROOFSTEP]\nrw [HomEquiv.toRestrictScalars_apply, HomEquiv.fromExtendScalars_apply, lift.tmul, LinearMap.coe_mk, LinearMap.coe_mk]\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191X\n\u22a2 \u2191(\u2191{ toFun := fun s => HomEquiv.evalAt f s g,\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s => HomEquiv.evalAt f s g) (s\u2081 + s\u2082) =\n (fun s => HomEquiv.evalAt f s g) s\u2081 + (fun s => HomEquiv.evalAt f s g) s\u2082) }\n 1).toAddHom\n x =\n \u2191g x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx : \u2191X\n\u22a2 1 \u2022 \u2191g x = \u2191g x\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nx x' : \u2191X\n\u22a2 (fun x => 1 \u2297\u209c[R] x) (x + x') = (fun x => 1 \u2297\u209c[R] x) x + (fun x => 1 \u2297\u209c[R] x) x'\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nx x' : \u2191X\n\u22a2 1 \u2297\u209c[R] (x + x') = 1 \u2297\u209c[R] x + 1 \u2297\u209c[R] x'\n[PROOFSTEP]\nrw [TensorProduct.tmul_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\nx : \u2191X\n\u22a2 AddHom.toFun\n { toFun := fun x => 1 \u2297\u209c[R] x,\n map_add' :=\n (_ : \u2200 (x x' : \u2191X), (fun x => 1 \u2297\u209c[R] x) (x + x') = (fun x => 1 \u2297\u209c[R] x) x + (fun x => 1 \u2297\u209c[R] x) x') }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => 1 \u2297\u209c[R] x,\n map_add' :=\n (_ : \u2200 (x x' : \u2191X), (fun x => 1 \u2297\u209c[R] x) (x + x') = (fun x => 1 \u2297\u209c[R] x) x + (fun x => 1 \u2297\u209c[R] x) x') }\n x\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\nx : \u2191X\nm1 : Module R S := Module.compHom S f\n\u22a2 AddHom.toFun\n { toFun := fun x => 1 \u2297\u209c[R] x,\n map_add' :=\n (_ : \u2200 (x x' : \u2191X), (fun x => 1 \u2297\u209c[R] x) (x + x') = (fun x => 1 \u2297\u209c[R] x) x + (fun x => 1 \u2297\u209c[R] x) x') }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x => 1 \u2297\u209c[R] x,\n map_add' :=\n (_ : \u2200 (x x' : \u2191X), (fun x => 1 \u2297\u209c[R] x) (x + x') = (fun x => 1 \u2297\u209c[R] x) x + (fun x => 1 \u2297\u209c[R] x) x') }\n x\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nr : R\nx : \u2191X\nm1 : Module R S := Module.compHom S f\n\u22a2 1 \u2297\u209c[R] (r \u2022 x) = r \u2022 1 \u2297\u209c[R] x\n[PROOFSTEP]\nrw [\u2190 TensorProduct.smul_tmul, TensorProduct.smul_tmul']\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\n\u22a2 (restrictScalars f \u22d9 extendScalars f).obj Y \u27f6 Y\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\n\u22a2 (restrictScalars f \u22d9 extendScalars f).obj Y \u27f6 Y\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 (restrictScalars f \u22d9 extendScalars f).obj Y \u27f6 Y\n[PROOFSTEP]\nrefine'\n {\n toFun :=\n TensorProduct.lift\n { toFun := fun s : S => { toFun := fun y : Y => s \u2022 y, map_add' := smul_add _, map_smul' := _ }, map_add' := _,\n map_smul' := _ },\n map_add' := _, map_smul' := _ }\n[GOAL]\ncase refine'_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\n\u22a2 \u2200 (r : R) (x : \u2191Y),\n AddHom.toFun { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) } x\n[PROOFSTEP]\nintros r y\n[GOAL]\ncase refine'_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nr : R\ny : \u2191Y\n\u22a2 AddHom.toFun { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) } (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) } y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nr : R\ny : \u2191Y\n\u22a2 s \u2022 r \u2022 y = r \u2022 s \u2022 y\n[PROOFSTEP]\nchange s \u2022 f r \u2022 y = f r \u2022 s \u2022 y\n[GOAL]\ncase refine'_1\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nr : R\ny : \u2191Y\n\u22a2 s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y\n[PROOFSTEP]\nrw [\u2190 mul_smul, mul_comm, mul_smul]\n[GOAL]\ncase refine'_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (x y : S),\n (fun s =>\n { toAddHom := { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (x + y) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n x +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n y\n[PROOFSTEP]\nintros s\u2081 s\u2082\n[GOAL]\ncase refine'_2\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns\u2081 s\u2082 : S\n\u22a2 (fun s =>\n { toAddHom := { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n { toAddHom := { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n { toAddHom := { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082\n[PROOFSTEP]\next y\n[GOAL]\ncase refine'_2.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns\u2081 s\u2082 : S\ny : \u2191Y\n\u22a2 \u2191((fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082))\n y =\n \u2191((fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082)\n y\n[PROOFSTEP]\nchange (s\u2081 + s\u2082) \u2022 y = s\u2081 \u2022 y + s\u2082 \u2022 y\n[GOAL]\ncase refine'_2.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns\u2081 s\u2082 : S\ny : \u2191Y\n\u22a2 (s\u2081 + s\u2082) \u2022 y = s\u2081 \u2022 y + s\u2082 \u2022 y\n[PROOFSTEP]\nrw [add_smul]\n[GOAL]\ncase refine'_3\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (r : R) (x : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n x\n[PROOFSTEP]\nintros r s\n[GOAL]\ncase refine'_3\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nr : R\ns : S\n\u22a2 AddHom.toFun\n {\n toFun := fun s =>\n { toAddHom := { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s\n[PROOFSTEP]\next y\n[GOAL]\ncase refine'_3.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nr : R\ns : S\ny : \u2191Y\n\u22a2 \u2191(AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s))\n y =\n \u2191(\u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s)\n y\n[PROOFSTEP]\nchange (f r \u2022 s) \u2022 y = (f r) \u2022 s \u2022 y\n[GOAL]\ncase refine'_3.h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nr : R\ns : S\ny : \u2191Y\n\u22a2 (\u2191f r \u2022 s) \u2022 y = \u2191f r \u2022 s \u2022 y\n[PROOFSTEP]\nrw [smul_eq_mul, mul_smul]\n[GOAL]\ncase refine'_4\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (x y : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x + y) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx\u271d y\u271d : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x\u271d + y\u271d) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x\u271d +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y\u271d\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\ncase refine'_5\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2200 (r : S) (x : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n AddHom.toFun\n {\n toFun :=\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x + y) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y) }\n (r \u2022 x) =\n \u2191(RingHom.id S) r \u2022\n AddHom.toFun\n {\n toFun :=\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x + y) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y) }\n x\n[PROOFSTEP]\nintro s z\n[GOAL]\ncase refine'_5\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nz : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)\n\u22a2 AddHom.toFun\n {\n toFun :=\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x + y) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y) }\n (s \u2022 z) =\n \u2191(RingHom.id S) s \u2022\n AddHom.toFun\n {\n toFun :=\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }),\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (x + y) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n x +\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ :\n \u2200 (b\u2081 b\u2082 : \u2191Y),\n s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n y) }\n z\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_5\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nz : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 z) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n z\n[PROOFSTEP]\ninduction' z using TensorProduct.induction_on with s' y z1 z2 ih1 ih2\n[GOAL]\ncase refine'_5.zero\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 0) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n 0\n[PROOFSTEP]\nrw [smul_zero, map_zero, smul_zero]\n[GOAL]\ncase refine'_5.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 s' \u2297\u209c[R] y) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s' \u2297\u209c[R] y)\n[PROOFSTEP]\nrw [ExtendScalars.smul_tmul, LinearMap.coe_mk]\n[GOAL]\ncase refine'_5.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }).toAddHom\n ((s * s') \u2297\u209c[R] y) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }).toAddHom\n (s' \u2297\u209c[R] y)\n[PROOFSTEP]\nerw [TensorProduct.lift.tmul, TensorProduct.lift.tmul]\n[GOAL]\ncase refine'_5.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }\n (s * s'))\n y =\n s \u2022\n \u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }\n s')\n y\n[PROOFSTEP]\nset s' : S := s'\n[GOAL]\ncase refine'_5.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\ns' : S := s'\u271d\n\u22a2 \u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }\n (s * s'))\n y =\n s \u2022\n \u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) }\n s')\n y\n[PROOFSTEP]\nchange (s * s') \u2022 y = s \u2022 s' \u2022 y\n[GOAL]\ncase refine'_5.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\ns' : S := s'\u271d\n\u22a2 (s * s') \u2022 y = s \u2022 s' \u2022 y\n[PROOFSTEP]\nrw [mul_smul]\n[GOAL]\ncase refine'_5.add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY : ModuleCat S\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : S\nz1 z2 : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191((restrictScalars f).obj Y)\nih1 :\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 z1) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n z1\nih2 :\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 z2) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n z2\n\u22a2 \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (s \u2022 (z1 + z2)) =\n s \u2022\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n {\n toAddHom :=\n { toFun := fun y => (s\u2081 + s\u2082) \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (s\u2081 + s\u2082) \u2022 (b\u2081 + b\u2082) = (s\u2081 + s\u2082) \u2022 b\u2081 + (s\u2081 + s\u2082) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), (s\u2081 + s\u2082) \u2022 \u2191f r \u2022 y = \u2191f r \u2022 (s\u2081 + s\u2082) \u2022 y) } =\n {\n toAddHom :=\n { toFun := fun y => s\u2081 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2081 \u2022 (b\u2081 + b\u2082) = s\u2081 \u2022 b\u2081 + s\u2081 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2081 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2081 \u2022 y) } +\n {\n toAddHom :=\n { toFun := fun y => s\u2082 \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s\u2082 \u2022 (b\u2081 + b\u2082) = s\u2082 \u2022 b\u2081 + s\u2082 \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s\u2082 \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s\u2082 \u2022 y) }) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n {\n toAddHom :=\n { toFun := fun y => (r \u2022 s) \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), (r \u2022 s) \u2022 (b\u2081 + b\u2082) = (r \u2022 s) \u2022 b\u2081 + (r \u2022 s) \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r_1 : R) (y : \u2191Y), (r \u2022 s) \u2022 \u2191f r_1 \u2022 y = \u2191f r_1 \u2022 (r \u2022 s) \u2022 y) } =\n r \u2022\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' := (_ : \u2200 (r : R) (y : \u2191Y), s \u2022 \u2191f r \u2022 y = \u2191f r \u2022 s \u2022 y) }) })\n (z1 + z2)\n[PROOFSTEP]\nrw [smul_add, map_add, map_add, ih1, ih2, smul_add]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\n\u22a2 (restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y' =\n (fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\n\u22a2 (restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y' =\n (fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 (restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y' =\n (fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g\n[PROOFSTEP]\nletI m2 : Module R Y' := Module.compHom Y' f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\n\u22a2 (restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y' =\n (fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\n\u22a2 \u2200 (x : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)),\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') x =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) x\n[PROOFSTEP]\nintro z\n[GOAL]\ncase h\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\nz : \u2191((restrictScalars f \u22d9 extendScalars f).obj Y)\n\u22a2 \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\n[PROOFSTEP]\ninduction' z using TensorProduct.induction_on with s' y z\u2081 z\u2082 ih\u2081 ih\u2082\n[GOAL]\ncase h.zero\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\n\u22a2 \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') 0 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) 0\n[PROOFSTEP]\nrw [map_zero, map_zero]\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') (s' \u2297\u209c[R] y) =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) (s' \u2297\u209c[R] y)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191((extendScalars f).map ((restrictScalars f).map g) \u226b Counit.map f) (s' \u2297\u209c[R] y) = \u2191(Counit.map f \u226b g) (s' \u2297\u209c[R] y)\n[PROOFSTEP]\nrw [ModuleCat.coe_comp, ModuleCat.coe_comp, Function.comp, Function.comp, ExtendScalars.map_tmul,\n restrictScalars.map_apply, Counit.map_apply, Counit.map_apply, lift.tmul, lift.tmul, LinearMap.coe_mk,\n LinearMap.coe_mk]\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\ns' : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\n\u22a2 \u2191(\u2191{\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s').toAddHom\n (\u2191g y) =\n \u2191g\n (\u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }\n s')\n y)\n[PROOFSTEP]\nset s' : S := s'\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\ns' : S := s'\u271d\n\u22a2 \u2191(\u2191{\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y'),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y'), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s').toAddHom\n (\u2191g y) =\n \u2191g\n (\u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }\n s')\n y)\n[PROOFSTEP]\nchange s' \u2022 g y = g (s' \u2022 y)\n[GOAL]\ncase h.tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\ns'\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S))\ny : \u2191((restrictScalars f).obj Y)\ns' : S := s'\u271d\n\u22a2 s' \u2022 \u2191g y = \u2191g (s' \u2022 y)\n[PROOFSTEP]\nrw [map_smul]\n[GOAL]\ncase h.add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\nz\u2081 z\u2082 : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191((restrictScalars f).obj Y)\nih\u2081 :\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2081 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2081\nih\u2082 :\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2082 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2082\n\u22a2 \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') (z\u2081 + z\u2082) =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) (z\u2081 + z\u2082)\n[PROOFSTEP]\nrw [map_add, map_add]\n[GOAL]\ncase h.add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nY Y' : ModuleCat S\ng : Y \u27f6 Y'\nm1 : Module R S := Module.compHom S f\nm2\u271d : Module R \u2191Y := Module.compHom (\u2191Y) f\nm2 : Module R \u2191Y' := Module.compHom (\u2191Y') f\nz\u2081 z\u2082 : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191((restrictScalars f).obj Y)\nih\u2081 :\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2081 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2081\nih\u2082 :\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2082 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2082\n\u22a2 \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2081 +\n \u2191((restrictScalars f \u22d9 extendScalars f).map g \u226b (fun x => Counit.map f) Y') z\u2082 =\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2081 +\n \u2191((fun x => Counit.map f) Y \u226b (\ud835\udfed (ModuleCat S)).map g) z\u2082\n[PROOFSTEP]\ncongr 1\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y) g) x =\n \u2191(NatTrans.app (ExtendRestrictScalarsAdj.unit f) X \u226b (restrictScalars f).map g) x\n[PROOFSTEP]\ndsimp\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(\u2191(ExtendRestrictScalarsAdj.homEquiv f) g) x = \u2191(ExtendRestrictScalarsAdj.Unit.map f \u226b (restrictScalars f).map g) x\n[PROOFSTEP]\nrw [ModuleCat.coe_comp, Function.comp, restrictScalars.map_apply]\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X \u27f6 Y\nx : \u2191X\n\u22a2 \u2191(\u2191(ExtendRestrictScalarsAdj.homEquiv f) g) x = \u2191g (\u2191(ExtendRestrictScalarsAdj.Unit.map f) x)\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nx : \u2191((extendScalars f).obj X)\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\n[PROOFSTEP]\nletI m1 : Module R S := Module.compHom S f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nx : \u2191((extendScalars f).obj X)\nm1 : Module R S := Module.compHom S f\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\n[PROOFSTEP]\nletI m2 : Module R Y := Module.compHom Y f\n[GOAL]\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nx : \u2191((extendScalars f).obj X)\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\n[PROOFSTEP]\ninduction' x using TensorProduct.induction_on with s x _ _ _ _\n[GOAL]\ncase zero\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) 0 =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) 0\n[PROOFSTEP]\nrw [map_zero, map_zero]\n[GOAL]\ncase tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) (s \u2297\u209c[R] x) =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) (s \u2297\u209c[R] x)\n[PROOFSTEP]\nrw [ExtendRestrictScalarsAdj.homEquiv_symm_apply, ModuleCat.coe_comp, Function.comp_apply,\n ExtendRestrictScalarsAdj.counit_app, ExtendRestrictScalarsAdj.Counit.map_apply]\n[GOAL]\ncase tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 \u2191(ExtendRestrictScalarsAdj.HomEquiv.fromExtendScalars f g) (s \u2297\u209c[R] x) =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (\u2191((extendScalars f).map g) (s \u2297\u209c[R] x))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 s \u2022 \u2191g x =\n \u2191(lift\n {\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) })\n (s \u2297\u209c[R] \u2191g x)\n[PROOFSTEP]\nrw [TensorProduct.lift.tmul]\n[GOAL]\ncase tmul\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\ns : \u2191((restrictScalars f).obj (ModuleCat.mk S))\nx : \u2191X\n\u22a2 s \u2022 \u2191g x =\n \u2191(\u2191{\n toAddHom :=\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y, map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (s : S),\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n (r \u2022 s) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n {\n toFun := fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) },\n map_add' :=\n (_ :\n \u2200 (s\u2081 s\u2082 : S),\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n (s\u2081 + s\u2082) =\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2081 +\n (fun s =>\n {\n toAddHom :=\n { toFun := fun y => s \u2022 y,\n map_add' := (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) },\n map_smul' :=\n (_ :\n \u2200 (r : R) (y : \u2191Y),\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n (r \u2022 y) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun y => s \u2022 y,\n map_add' :=\n (_ : \u2200 (b\u2081 b\u2082 : \u2191Y), s \u2022 (b\u2081 + b\u2082) = s \u2022 b\u2081 + s \u2022 b\u2082) }\n y) })\n s\u2082) }\n s) }\n s)\n (\u2191g x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx\u271d y\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191X\na\u271d\u00b9 :\n \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x\u271d =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\u271d\na\u271d :\n \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) y\u271d =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) y\u271d\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) (x\u271d + y\u271d) =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) (x\u271d + y\u271d)\n[PROOFSTEP]\nrw [map_add, map_add]\n[GOAL]\ncase add\nR\u271d : Type u\u2081\nS\u271d : Type u\u2082\ninst\u271d\u00b3 : CommRing R\u271d\ninst\u271d\u00b2 : CommRing S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X \u27f6 (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R \u2191Y := Module.compHom (\u2191Y) f\nx\u271d y\u271d : \u2191((restrictScalars f).obj (ModuleCat.mk S)) \u2297[R] \u2191X\na\u271d\u00b9 :\n \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x\u271d =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\u271d\na\u271d :\n \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) y\u271d =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) y\u271d\n\u22a2 \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) x\u271d +\n \u2191(\u2191((fun x x_1 => ExtendRestrictScalarsAdj.homEquiv f) X Y).symm g) y\u271d =\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) x\u271d +\n \u2191((extendScalars f).map g \u226b NatTrans.app (ExtendRestrictScalarsAdj.counit f) Y) y\u271d\n[PROOFSTEP]\ncongr 1\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.ChangeOfRings", "llama_tokens": 209043, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.2859553171226515}} {"text": "[GOAL]\n\u22a2 ConcreteCategory GroupCat\n[PROOFSTEP]\ndsimp only [GroupCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled Group)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR S : GroupCat\ni : R \u27f6 S\nr : \u2191R\nh : r = 1\n\u22a2 \u2191i r = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u22a2 ConcreteCategory CommGroupCat\n[PROOFSTEP]\ndsimp only [CommGroupCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled CommGroup)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR S : CommGroupCat\ni : R \u27f6 S\nr : \u2191R\nh : r = 1\n\u22a2 \u2191i r = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nG : AddCommGroupCat\nh k : \u2191G\nw : asHom h = asHom k\n\u22a2 h = k\n[PROOFSTEP]\nconvert congr_arg (fun k : AddCommGroupCat.of \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w\n[GOAL]\ncase h.e'_2\nG : AddCommGroupCat\nh k : \u2191G\nw : asHom h = asHom k\n\u22a2 h = \u2191(asHom h) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nG : AddCommGroupCat\nh k : \u2191G\nw : asHom h = asHom k\n\u22a2 k = \u2191(asHom k) 1\n[PROOFSTEP]\nsimp\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ninst\u271d : Mono f\ng\u2081 g\u2082 : \u2191G\nh : \u2191f g\u2081 = \u2191f g\u2082\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nhave t0 : asHom g\u2081 \u226b f = asHom g\u2082 \u226b f := by aesop_cat\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ninst\u271d : Mono f\ng\u2081 g\u2082 : \u2191G\nh : \u2191f g\u2081 = \u2191f g\u2082\n\u22a2 asHom g\u2081 \u226b f = asHom g\u2082 \u226b f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ninst\u271d : Mono f\ng\u2081 g\u2082 : \u2191G\nh : \u2191f g\u2081 = \u2191f g\u2082\nt0 : asHom g\u2081 \u226b f = asHom g\u2082 \u226b f\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nhave t1 : asHom g\u2081 = asHom g\u2082 := (cancel_mono _).1 t0\n[GOAL]\nG H : AddCommGroupCat\nf : G \u27f6 H\ninst\u271d : Mono f\ng\u2081 g\u2082 : \u2191G\nh : \u2191f g\u2081 = \u2191f g\u2082\nt0 : asHom g\u2081 \u226b f = asHom g\u2082 \u226b f\nt1 : asHom g\u2081 = asHom g\u2082\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\napply asHom_injective t1\n[GOAL]\n\u03b1 : Type u\n\u22a2 (fun g => g.toEquiv) 1 = 1\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u\n\u22a2 \u2200 (x y : \u2191(GroupCat.of (Aut \u03b1))),\n OneHom.toFun { toFun := fun g => g.toEquiv, map_one' := (_ : (fun g => g.toEquiv) 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun g => g.toEquiv, map_one' := (_ : (fun g => g.toEquiv) 1 = 1) } x *\n OneHom.toFun { toFun := fun g => g.toEquiv, map_one' := (_ : (fun g => g.toEquiv) 1 = 1) } y\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u\n\u22a2 (fun g => Equiv.toIso g) 1 = 1\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u\n\u22a2 \u2200 (x y : \u2191(GroupCat.of (Equiv.Perm \u03b1))),\n OneHom.toFun { toFun := fun g => Equiv.toIso g, map_one' := (_ : (fun g => Equiv.toIso g) 1 = 1) } (x * y) =\n OneHom.toFun { toFun := fun g => Equiv.toIso g, map_one' := (_ : (fun g => Equiv.toIso g) 1 = 1) } x *\n OneHom.toFun { toFun := fun g => Equiv.toIso g, map_one' := (_ : (fun g => Equiv.toIso g) 1 = 1) } y\n[PROOFSTEP]\naesop\n[GOAL]\nX Y : GroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget GroupCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget GroupCat).map f)\n[GOAL]\nX Y : GroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget GroupCat).map f)\ni : (forget GroupCat).obj X \u2245 (forget GroupCat).obj Y := asIso ((forget GroupCat).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 Y : GroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget GroupCat).map f)\ni : (forget GroupCat).obj X \u2245 (forget GroupCat).obj Y := asIso ((forget GroupCat).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.toGroupCatIso\n[GOAL]\nX Y : CommGroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommGroupCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget CommGroupCat).map f)\n[GOAL]\nX Y : CommGroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommGroupCat).map f)\ni : (forget CommGroupCat).obj X \u2245 (forget CommGroupCat).obj Y := asIso ((forget CommGroupCat).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 Y : CommGroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget CommGroupCat).map f)\ni : (forget CommGroupCat).obj X \u2245 (forget CommGroupCat).obj Y := asIso ((forget CommGroupCat).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.toCommGroupCatIso\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Basic", "llama_tokens": 2076, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6406358548398979, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.2854221945364989}} {"text": "[GOAL]\n\u22a2 LawfulFunctor Multiset\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.250}, Functor.mapConst = Functor.map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u22a2 \u2200 {\u03b1 : Type ?u.250} (x : Multiset \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.250} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : Multiset \u03b1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2\u271d : Type ?u.250\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2\u271d : Type ?u.250\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type ?u.250\nx\u271d : Multiset \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type ?u.250\nx\u271d : Multiset \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.250\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : Multiset \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_3\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.250\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : Multiset \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2\u271d : Type ?u.250\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u22a2 Multiset \u03b1' \u2192 F (Multiset \u03b2')\n[PROOFSTEP]\nrefine' Quotient.lift (Functor.map Coe.coe \u2218 Traversable.traverse f) _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u22a2 \u2200 (a b : List \u03b1'),\n a \u2248 b \u2192 (Functor.map Coe.coe \u2218 Traversable.traverse f) a = (Functor.map Coe.coe \u2218 Traversable.traverse f) b\n[PROOFSTEP]\nintrov p\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\np : a \u2248 b\n\u22a2 (Functor.map Coe.coe \u2218 Traversable.traverse f) a = (Functor.map Coe.coe \u2218 Traversable.traverse f) b\n[PROOFSTEP]\nunfold Function.comp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\np : a \u2248 b\n\u22a2 Coe.coe <$> Traversable.traverse f a = Coe.coe <$> Traversable.traverse f b\n[PROOFSTEP]\ninduction p\n[GOAL]\ncase nil\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f [] = Coe.coe <$> Traversable.traverse f []\ncase cons\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx\u271d : \u03b1'\nl\u2081\u271d l\u2082\u271d : List \u03b1'\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f (x\u271d :: l\u2081\u271d) = Coe.coe <$> Traversable.traverse f (x\u271d :: l\u2082\u271d)\ncase swap\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx\u271d y\u271d : \u03b1'\nl\u271d : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f (y\u271d :: x\u271d :: l\u271d) = Coe.coe <$> Traversable.traverse f (x\u271d :: y\u271d :: l\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f [] = Coe.coe <$> Traversable.traverse f []\n[PROOFSTEP]\ncase nil => rfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f [] = Coe.coe <$> Traversable.traverse f []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx\u271d : \u03b1'\nl\u2081\u271d l\u2082\u271d : List \u03b1'\na\u271d : l\u2081\u271d ~ l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f (x\u271d :: l\u2081\u271d) = Coe.coe <$> Traversable.traverse f (x\u271d :: l\u2082\u271d)\ncase swap\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx\u271d y\u271d : \u03b1'\nl\u271d : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f (y\u271d :: x\u271d :: l\u271d) = Coe.coe <$> Traversable.traverse f (x\u271d :: y\u271d :: l\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\ncase cons x l\u2081 l\u2082 _\n h =>\n have :\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2081 =\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2082 :=\n by rw [h]\n simpa [functor_norm] using this\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx : \u03b1'\nl\u2081 l\u2082 : List \u03b1'\na\u271d : l\u2081 ~ l\u2082\nh : Coe.coe <$> Traversable.traverse f l\u2081 = Coe.coe <$> Traversable.traverse f l\u2082\n\u22a2 Coe.coe <$> Traversable.traverse f (x :: l\u2081) = Coe.coe <$> Traversable.traverse f (x :: l\u2082)\n[PROOFSTEP]\ncase cons x l\u2081 l\u2082 _\n h =>\n have :\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2081 =\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2082 :=\n by rw [h]\n simpa [functor_norm] using this\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx : \u03b1'\nl\u2081 l\u2082 : List \u03b1'\na\u271d : l\u2081 ~ l\u2082\nh : Coe.coe <$> Traversable.traverse f l\u2081 = Coe.coe <$> Traversable.traverse f l\u2082\n\u22a2 Coe.coe <$> Traversable.traverse f (x :: l\u2081) = Coe.coe <$> Traversable.traverse f (x :: l\u2082)\n[PROOFSTEP]\nhave :\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2081 =\n Multiset.cons <$> f x <*> Coe.coe <$> Traversable.traverse f l\u2082 :=\n by rw [h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx : \u03b1'\nl\u2081 l\u2082 : List \u03b1'\na\u271d : l\u2081 ~ l\u2082\nh : Coe.coe <$> Traversable.traverse f l\u2081 = Coe.coe <$> Traversable.traverse f l\u2082\n\u22a2 (Seq.seq (cons <$> f x) fun x => Coe.coe <$> Traversable.traverse f l\u2081) =\n Seq.seq (cons <$> f x) fun x => Coe.coe <$> Traversable.traverse f l\u2082\n[PROOFSTEP]\nrw [h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx : \u03b1'\nl\u2081 l\u2082 : List \u03b1'\na\u271d : l\u2081 ~ l\u2082\nh : Coe.coe <$> Traversable.traverse f l\u2081 = Coe.coe <$> Traversable.traverse f l\u2082\nthis :\n (Seq.seq (cons <$> f x) fun x => Coe.coe <$> Traversable.traverse f l\u2081) =\n Seq.seq (cons <$> f x) fun x => Coe.coe <$> Traversable.traverse f l\u2082\n\u22a2 Coe.coe <$> Traversable.traverse f (x :: l\u2081) = Coe.coe <$> Traversable.traverse f (x :: l\u2082)\n[PROOFSTEP]\nsimpa [functor_norm] using this\n[GOAL]\ncase swap\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx\u271d y\u271d : \u03b1'\nl\u271d : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f (y\u271d :: x\u271d :: l\u271d) = Coe.coe <$> Traversable.traverse f (x\u271d :: y\u271d :: l\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\ncase swap x y\n l =>\n have :\n (fun a b (l : List \u03b2') \u21a6 (\u2191(a :: b :: l) : Multiset \u03b2')) <$> f y <*> f x =\n (fun a b l \u21a6 \u2191(a :: b :: l)) <$> f x <*> f y :=\n by\n rw [CommApplicative.commutative_map]\n congr\n funext a b l\n simpa [flip] using Perm.swap a b l\n simp [(\u00b7 \u2218 \u00b7), this, functor_norm, Coe.coe]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f (y :: x :: l) = Coe.coe <$> Traversable.traverse f (x :: y :: l)\n[PROOFSTEP]\ncase swap x y\n l =>\n have :\n (fun a b (l : List \u03b2') \u21a6 (\u2191(a :: b :: l) : Multiset \u03b2')) <$> f y <*> f x =\n (fun a b l \u21a6 \u2191(a :: b :: l)) <$> f x <*> f y :=\n by\n rw [CommApplicative.commutative_map]\n congr\n funext a b l\n simpa [flip] using Perm.swap a b l\n simp [(\u00b7 \u2218 \u00b7), this, functor_norm, Coe.coe]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\n\u22a2 Coe.coe <$> Traversable.traverse f (y :: x :: l) = Coe.coe <$> Traversable.traverse f (x :: y :: l)\n[PROOFSTEP]\nhave :\n (fun a b (l : List \u03b2') \u21a6 (\u2191(a :: b :: l) : Multiset \u03b2')) <$> f y <*> f x =\n (fun a b l \u21a6 \u2191(a :: b :: l)) <$> f x <*> f y :=\n by\n rw [CommApplicative.commutative_map]\n congr\n funext a b l\n simpa [flip] using Perm.swap a b l\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\n\u22a2 (Seq.seq ((fun a b l => \u2191(a :: b :: l)) <$> f y) fun x_1 => f x) =\n Seq.seq ((fun a b l => \u2191(a :: b :: l)) <$> f x) fun x => f y\n[PROOFSTEP]\nrw [CommApplicative.commutative_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\n\u22a2 (Seq.seq ((flip fun a b l => \u2191(a :: b :: l)) <$> f x) fun x => f y) =\n Seq.seq ((fun a b l => \u2191(a :: b :: l)) <$> f x) fun x => f y\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\n\u22a2 (flip fun a b l => \u2191(a :: b :: l)) = fun a b l => \u2191(a :: b :: l)\n[PROOFSTEP]\nfunext a b l\n[GOAL]\ncase e_a.e_a.h.h.h\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na\u271d b\u271d : List \u03b1'\nx y : \u03b1'\nl\u271d : List \u03b1'\na b : \u03b2'\nl : List \u03b2'\n\u22a2 flip (fun a b l => \u2191(a :: b :: l)) a b l = \u2191(a :: b :: l)\n[PROOFSTEP]\nsimpa [flip] using Perm.swap a b l\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b : List \u03b1'\nx y : \u03b1'\nl : List \u03b1'\nthis :\n (Seq.seq ((fun a b l => \u2191(a :: b :: l)) <$> f y) fun x_1 => f x) =\n Seq.seq ((fun a b l => \u2191(a :: b :: l)) <$> f x) fun x => f y\n\u22a2 Coe.coe <$> Traversable.traverse f (y :: x :: l) = Coe.coe <$> Traversable.traverse f (x :: y :: l)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7), this, functor_norm, Coe.coe]\n[GOAL]\ncase trans\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\ncase trans => simp [*]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\ncase trans => simp [*]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\na b l\u2081\u271d l\u2082\u271d l\u2083\u271d : List \u03b1'\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\na_ih\u271d\u00b9 : Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2082\u271d\na_ih\u271d : Coe.coe <$> Traversable.traverse f l\u2082\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n\u22a2 Coe.coe <$> Traversable.traverse f l\u2081\u271d = Coe.coe <$> Traversable.traverse f l\u2083\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1\u271d : Type ?u.36389\nx\u271d : Multiset \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp only [fmap_def, id_eq, map_id']\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1\u271d \u03b2\u271d : Type ?u.36389\nx\u271d\u00b9 : \u03b1\u271d\nx\u271d : \u03b1\u271d \u2192 Multiset \u03b2\u271d\n\u22a2 pure x\u271d\u00b9 >>= x\u271d = x\u271d x\u271d\u00b9\n[PROOFSTEP]\nsimp only [pure_def, bind_def, singleton_bind]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1\u271d \u03b2\u271d : Type ?u.36389\nx\u271d\u00b9 : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Multiset \u03b1\u271d\n\u22a2 (do\n let y \u2190 x\u271d\n pure (x\u271d\u00b9 y)) =\n x\u271d\u00b9 <$> x\u271d\n[PROOFSTEP]\nsimp only [pure_def, bind_def, bind_singleton, fmap_def]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1 \u03b2 : Type u_1\nh : \u03b1 \u2192 \u03b2\n\u22a2 Functor.map h \u2218 Coe.coe = Coe.coe \u2218 Functor.map h\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1 \u03b2 : Type u_1\nh : \u03b1 \u2192 \u03b2\nx\u271d : List \u03b1\n\u22a2 (Functor.map h \u2218 Coe.coe) x\u271d = (Coe.coe \u2218 Functor.map h) x\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, Coe.coe, fmap_def, coe_map, List.map_eq_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1 : Type u_1\nx : Multiset \u03b1\n\u22a2 traverse pure x = x\n[PROOFSTEP]\nrefine' Quotient.inductionOn x _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1 : Type u_1\nx : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), traverse pure (Quotient.mk (isSetoid \u03b1) a) = Quotient.mk (isSetoid \u03b1) a\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\n\u03b1 : Type u_1\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 traverse pure (Quotient.mk (isSetoid \u03b1) a\u271d) = Quotient.mk (isSetoid \u03b1) a\u271d\n[PROOFSTEP]\nsimp [traverse, Coe.coe]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 H \u03b3\nx : Multiset \u03b1\n\u22a2 traverse (Comp.mk \u2218 Functor.map h \u2218 g) x = Comp.mk (traverse h <$> traverse g x)\n[PROOFSTEP]\nrefine' Quotient.inductionOn x _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 H \u03b3\nx : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n traverse (Comp.mk \u2218 Functor.map h \u2218 g) (Quotient.mk (isSetoid \u03b1) a) =\n Comp.mk (traverse h <$> traverse g (Quotient.mk (isSetoid \u03b1) a))\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 H \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 traverse (Comp.mk \u2218 Functor.map h \u2218 g) (Quotient.mk (isSetoid \u03b1) a\u271d) =\n Comp.mk (traverse h <$> traverse g (Quotient.mk (isSetoid \u03b1) a\u271d))\n[PROOFSTEP]\nsimp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map, functor_norm]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 H \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 Comp.mk (((fun x => ofList <$> x) \u2218 Traversable.traverse h) <$> Traversable.traverse g a\u271d) =\n Comp.mk\n ((Quotient.lift (Functor.map ofList \u2218 Traversable.traverse h)\n (_ :\n \u2200 (a b : List \u03b2),\n a \u2248 b \u2192\n (Functor.map Coe.coe \u2218 Traversable.traverse h) a = (Functor.map Coe.coe \u2218 Traversable.traverse h) b) \u2218\n ofList) <$>\n Traversable.traverse g a\u271d)\n[PROOFSTEP]\nsimp only [Function.comp, lift_coe]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\nx : Multiset \u03b1\n\u22a2 Functor.map h <$> traverse g x = traverse (Functor.map h \u2218 g) x\n[PROOFSTEP]\nrefine' Quotient.inductionOn x _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\nx : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n Functor.map h <$> traverse g (Quotient.mk (isSetoid \u03b1) a) =\n traverse (Functor.map h \u2218 g) (Quotient.mk (isSetoid \u03b1) a)\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 Functor.map h <$> traverse g (Quotient.mk (isSetoid \u03b1) a\u271d) =\n traverse (Functor.map h \u2218 g) (Quotient.mk (isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp only [traverse, quot_mk_to_coe, lift_coe, Function.comp_apply, Functor.map_map, map_comp_coe]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 (Coe.coe \u2218 Functor.map h) <$> Traversable.traverse g a\u271d = Coe.coe <$> Traversable.traverse (Functor.map h \u2218 g) a\u271d\n[PROOFSTEP]\nrw [LawfulFunctor.comp_map, Traversable.map_traverse']\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 Coe.coe <$> Functor.map h <$> Traversable.traverse g a\u271d =\n Coe.coe <$> (Functor.map (Functor.map h) \u2218 Traversable.traverse g) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 G \u03b3\nx : Multiset \u03b1\n\u22a2 traverse h (map g x) = traverse (h \u2218 g) x\n[PROOFSTEP]\nrefine' Quotient.inductionOn x _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 G \u03b3\nx : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), traverse h (map g (Quotient.mk (isSetoid \u03b1) a)) = traverse (h \u2218 g) (Quotient.mk (isSetoid \u03b1) a)\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 G \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 traverse h (map g (Quotient.mk (isSetoid \u03b1) a\u271d)) = traverse (h \u2218 g) (Quotient.mk (isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp only [traverse, quot_mk_to_coe, coe_map, lift_coe, Function.comp_apply]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\nG : Type u_1 \u2192 Type u_1\ninst\u271d\u00b9 : Applicative G\ninst\u271d : CommApplicative G\n\u03b1 \u03b2 \u03b3 : Type u_1\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 G \u03b3\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 Coe.coe <$> Traversable.traverse h (List.map g a\u271d) = Coe.coe <$> Traversable.traverse (h \u2218 g) a\u271d\n[PROOFSTEP]\nrw [\u2190 Traversable.traverse_map h g, List.map_eq_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\neta : ApplicativeTransformation G H\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 G \u03b2\nx : Multiset \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) (traverse f x) =\n traverse ((fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) \u2218 f) x\n[PROOFSTEP]\nrefine' Quotient.inductionOn x _\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\neta : ApplicativeTransformation G H\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 G \u03b2\nx : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n (fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) (traverse f (Quotient.mk (isSetoid \u03b1) a)) =\n traverse ((fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) \u2218 f) (Quotient.mk (isSetoid \u03b1) a)\n[PROOFSTEP]\nintro\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u2075 : Applicative F\ninst\u271d\u2074 : CommApplicative F\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\nG H : Type u_1 \u2192 Type u_1\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : Applicative H\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : CommApplicative H\neta : ApplicativeTransformation G H\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 G \u03b2\nx : Multiset \u03b1\na\u271d : List \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) (traverse f (Quotient.mk (isSetoid \u03b1) a\u271d)) =\n traverse ((fun {\u03b1} => ApplicativeTransformation.app eta \u03b1) \u2218 f) (Quotient.mk (isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply, ApplicativeTransformation.preserves_map,\n LawfulTraversable.naturality]\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Functor", "llama_tokens": 11292, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.285246173780828}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u2074 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d\u00b3 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesFiniteLimits F\nJ : Type w\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\napply preservesLimitsOfShapeOfEquiv (FinCategory.equivAsType J)\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\nJ\u271d : Type w\ninst\u271d\u00b9 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d : PreservesLimitsOfSize.{w, w\u2082, v\u2081, v\u2082, u\u2081, u\u2082} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\nhaveI := preservesSmallestLimitsOfPreservesLimits 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\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d\u00b9 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d : PreservesLimitsOfSize.{w, w\u2082, v\u2081, v\u2082, u\u2081, u\u2082} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2082, u\u2081, u\u2082} F\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\nexact preservesLimitsOfShapeOfEquiv (FinCategory.equivAsType J) 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\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\nletI : Category (ULiftHom (ULift J)) := ULiftHom.category\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\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\nthis : Category.{?u.3382, ?u.3380} (ULiftHom (ULift J)) := ULiftHom.category\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\nhaveI := h (ULiftHom (ULift J)) CategoryTheory.finCategoryUlift\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\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\nthis\u271d : Category.{?u.3382, ?u.3380} (ULiftHom (ULift J)) := ULiftHom.category\nthis : PreservesLimitsOfShape (ULiftHom (ULift J)) F\n\u22a2 PreservesLimitsOfShape J F\n[PROOFSTEP]\nexact preservesLimitsOfShapeOfEquiv (ULiftHomULiftCategory.equiv J).symm 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\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nJ : Type w\ninst\u271d : SmallCategory J\nK : J \u2964 C\nx\u271d\u00b2 : Type\nx\u271d\u00b9 : SmallCategory x\u271d\u00b2\nx\u271d : FinCategory x\u271d\u00b2\n\u22a2 PreservesLimitsOfShape x\u271d\u00b2 (\ud835\udfed C)\n[PROOFSTEP]\ninfer_instance\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\nE : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} E\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\nK : J \u2964 C\nF : C \u2964 D\nG : D \u2964 E\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteLimits G\nx\u271d\u00b2 : Type\nx\u271d\u00b9 : SmallCategory x\u271d\u00b2\nx\u271d : FinCategory x\u271d\u00b2\n\u22a2 PreservesLimitsOfShape x\u271d\u00b2 (F \u22d9 G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u2074 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d\u00b3 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d\u00b2 : PreservesFiniteColimits F\nJ : Type w\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : FinCategory J\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\napply preservesColimitsOfShapeOfEquiv (FinCategory.equivAsType J)\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\nJ\u271d : Type w\ninst\u271d\u00b9 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d : PreservesColimitsOfSize.{w, w\u2082, v\u2081, v\u2082, u\u2081, u\u2082} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\nhaveI := preservesSmallestColimitsOfPreservesColimits 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\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d\u00b9 : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\ninst\u271d : PreservesColimitsOfSize.{w, w\u2082, v\u2081, v\u2082, u\u2081, u\u2082} F\nJ : Type\nsJ : SmallCategory J\nfJ : FinCategory J\nthis : PreservesColimitsOfSize.{0, 0, v\u2081, v\u2082, u\u2081, u\u2082} F\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\nexact preservesColimitsOfShapeOfEquiv (FinCategory.equivAsType J) 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\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesColimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\nletI : Category (ULiftHom (ULift J)) := ULiftHom.category\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\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesColimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\nthis : Category.{?u.13228, ?u.13226} (ULiftHom (ULift J)) := ULiftHom.category\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\nhaveI := h (ULiftHom (ULift J)) CategoryTheory.finCategoryUlift\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\nJ\u271d : Type w\ninst\u271d : SmallCategory J\u271d\nK : J\u271d \u2964 C\nF : C \u2964 D\nh : (J : Type w) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesColimitsOfShape J F\nJ : Type\nx\u271d\u00b9 : SmallCategory J\nx\u271d : FinCategory J\nthis\u271d : Category.{?u.13228, ?u.13226} (ULiftHom (ULift J)) := ULiftHom.category\nthis : PreservesColimitsOfShape (ULiftHom (ULift J)) F\n\u22a2 PreservesColimitsOfShape J F\n[PROOFSTEP]\nexact preservesColimitsOfShapeOfEquiv (ULiftHomULiftCategory.equiv J).symm F\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\nE : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} E\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\nK : J \u2964 C\nF : C \u2964 D\nG : D \u2964 E\ninst\u271d\u00b9 : PreservesFiniteColimits F\ninst\u271d : PreservesFiniteColimits G\nx\u271d\u00b2 : Type\nx\u271d\u00b9 : SmallCategory x\u271d\u00b2\nx\u271d : FinCategory x\u271d\u00b2\n\u22a2 PreservesColimitsOfShape x\u271d\u00b2 (F \u22d9 G)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Finite", "llama_tokens": 3287, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.4263215925474903, "lm_q1q2_score": 0.28515809782494833}} {"text": "[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : List \u0393\n\u22a2 l = l ++ List.replicate 0 default\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\n\u22a2 BlankExtends l\u2081 l\u2082 \u2192 BlankExtends l\u2082 l\u2083 \u2192 BlankExtends l\u2081 l\u2083\n[PROOFSTEP]\nrintro \u27e8i, rfl\u27e9 \u27e8j, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 : List \u0393\ni j : \u2115\n\u22a2 BlankExtends l\u2081 (l\u2081 ++ List.replicate i default ++ List.replicate j default)\n[PROOFSTEP]\nexact \u27e8i + j, by simp [List.replicate_add]\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 : List \u0393\ni j : \u2115\n\u22a2 l\u2081 ++ List.replicate i default ++ List.replicate j default = l\u2081 ++ List.replicate (i + j) default\n[PROOFSTEP]\nsimp [List.replicate_add]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl l\u2081 l\u2082 : List \u0393\n\u22a2 BlankExtends l l\u2081 \u2192 BlankExtends l l\u2082 \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 BlankExtends l\u2081 l\u2082\n[PROOFSTEP]\nrintro \u27e8i, rfl\u27e9 \u27e8j, rfl\u27e9 h\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : List \u0393\ni j : \u2115\nh : List.length (l ++ List.replicate i default) \u2264 List.length (l ++ List.replicate j default)\n\u22a2 BlankExtends (l ++ List.replicate i default) (l ++ List.replicate j default)\n[PROOFSTEP]\nuse j - i\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : List \u0393\ni j : \u2115\nh : List.length (l ++ List.replicate i default) \u2264 List.length (l ++ List.replicate j default)\n\u22a2 l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default\n[PROOFSTEP]\nsimp only [List.length_append, add_le_add_iff_left, List.length_replicate] at h \n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : List \u0393\ni j : \u2115\nh : i \u2264 j\n\u22a2 l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default\n[PROOFSTEP]\nsimp only [\u2190 List.replicate_add, add_tsub_cancel_of_le h, List.append_assoc]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl l\u2081 l\u2082 : List \u0393\n\u22a2 BlankExtends l\u2081 l \u2192 BlankExtends l\u2082 l \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 BlankExtends l\u2081 l\u2082\n[PROOFSTEP]\nrintro \u27e8i, rfl\u27e9 \u27e8j, e\u27e9 h\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\ne : l\u2081 ++ List.replicate i default = l\u2082 ++ List.replicate j default\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 BlankExtends l\u2081 l\u2082\n[PROOFSTEP]\nuse i - j\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\ne : l\u2081 ++ List.replicate i default = l\u2082 ++ List.replicate j default\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 l\u2082 = l\u2081 ++ List.replicate (i - j) default\n[PROOFSTEP]\nrefine' List.append_right_cancel (e.symm.trans _)\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\ne : l\u2081 ++ List.replicate i default = l\u2082 ++ List.replicate j default\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 l\u2081 ++ List.replicate i default = l\u2081 ++ List.replicate (i - j) default ++ List.replicate j default\n[PROOFSTEP]\nrw [List.append_assoc, \u2190 List.replicate_add, tsub_add_cancel_of_le]\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\ne : l\u2081 ++ List.replicate i default = l\u2082 ++ List.replicate j default\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 j \u2264 i\n[PROOFSTEP]\napply_fun List.length at e \n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\nh : List.length l\u2081 \u2264 List.length l\u2082\ne : List.length (l\u2081 ++ List.replicate i default) = List.length (l\u2082 ++ List.replicate j default)\n\u22a2 j \u2264 i\n[PROOFSTEP]\nsimp only [List.length_append, List.length_replicate] at e \n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\ni j : \u2115\nh : List.length l\u2081 \u2264 List.length l\u2082\ne : List.length l\u2081 + i = List.length l\u2082 + j\n\u22a2 j \u2264 i\n[PROOFSTEP]\nrwa [\u2190 add_le_add_iff_left, e, add_le_add_iff_right]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\n\u22a2 BlankRel l\u2081 l\u2082 \u2192 BlankRel l\u2082 l\u2083 \u2192 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nrintro (h\u2081 | h\u2081) (h\u2082 | h\u2082)\n[GOAL]\ncase inl.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2081 l\u2082\nh\u2082 : BlankExtends l\u2082 l\u2083\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inl (h\u2081.trans h\u2082)\n[GOAL]\ncase inl.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2081 l\u2082\nh\u2082 : BlankExtends l\u2083 l\u2082\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\ncases' le_total l\u2081.length l\u2083.length with h h\n[GOAL]\ncase inl.inr.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2081 l\u2082\nh\u2082 : BlankExtends l\u2083 l\u2082\nh : List.length l\u2081 \u2264 List.length l\u2083\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inl (h\u2081.above_of_le h\u2082 h)\n[GOAL]\ncase inl.inr.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2081 l\u2082\nh\u2082 : BlankExtends l\u2083 l\u2082\nh : List.length l\u2083 \u2264 List.length l\u2081\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inr (h\u2082.above_of_le h\u2081 h)\n[GOAL]\ncase inr.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2082 l\u2081\nh\u2082 : BlankExtends l\u2082 l\u2083\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\ncases' le_total l\u2081.length l\u2083.length with h h\n[GOAL]\ncase inr.inl.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2082 l\u2081\nh\u2082 : BlankExtends l\u2082 l\u2083\nh : List.length l\u2081 \u2264 List.length l\u2083\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inl (h\u2081.below_of_le h\u2082 h)\n[GOAL]\ncase inr.inl.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2082 l\u2081\nh\u2082 : BlankExtends l\u2082 l\u2083\nh : List.length l\u2083 \u2264 List.length l\u2081\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inr (h\u2082.below_of_le h\u2081 h)\n[GOAL]\ncase inr.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 l\u2083 : List \u0393\nh\u2081 : BlankExtends l\u2082 l\u2081\nh\u2082 : BlankExtends l\u2083 l\u2082\n\u22a2 BlankRel l\u2081 l\u2083\n[PROOFSTEP]\nexact Or.inr (h\u2082.trans h\u2081)\n[GOAL]\n\u0393 : Type ?u.9261\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\n\u22a2 { l // BlankExtends l\u2081 l \u2227 BlankExtends l\u2082 l }\n[PROOFSTEP]\nrefine'\n if hl : l\u2081.length \u2264 l\u2082.length then \u27e8l\u2082, Or.elim h id fun h' \u21a6 _, BlankExtends.refl _\u27e9\n else \u27e8l\u2081, BlankExtends.refl _, Or.elim h (fun h' \u21a6 _) id\u27e9\n[GOAL]\ncase refine'_1\n\u0393 : Type ?u.9261\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : List.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2082 l\u2081\n\u22a2 BlankExtends l\u2081 l\u2082\ncase refine'_2\n\u0393 : Type ?u.9261\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2081 l\u2082\n\u22a2 BlankExtends l\u2082 l\u2081\n[PROOFSTEP]\nexact (BlankExtends.refl _).above_of_le h' hl\n[GOAL]\ncase refine'_2\n\u0393 : Type ?u.9261\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2081 l\u2082\n\u22a2 BlankExtends l\u2082 l\u2081\n[PROOFSTEP]\nexact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)\n[GOAL]\n\u0393 : Type ?u.10721\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\n\u22a2 { l // BlankExtends l l\u2081 \u2227 BlankExtends l l\u2082 }\n[PROOFSTEP]\nrefine'\n if hl : l\u2081.length \u2264 l\u2082.length then \u27e8l\u2081, BlankExtends.refl _, Or.elim h id fun h' \u21a6 _\u27e9\n else \u27e8l\u2082, Or.elim h (fun h' \u21a6 _) id, BlankExtends.refl _\u27e9\n[GOAL]\ncase refine'_1\n\u0393 : Type ?u.10721\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : List.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2082 l\u2081\n\u22a2 BlankExtends l\u2081 l\u2082\ncase refine'_2\n\u0393 : Type ?u.10721\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2081 l\u2082\n\u22a2 BlankExtends l\u2082 l\u2081\n[PROOFSTEP]\nexact (BlankExtends.refl _).above_of_le h' hl\n[GOAL]\ncase refine'_2\n\u0393 : Type ?u.10721\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nh : BlankRel l\u2081 l\u2082\nhl : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh' : BlankExtends l\u2081 l\u2082\n\u22a2 BlankExtends l\u2082 l\u2081\n[PROOFSTEP]\nexact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)\n[GOAL]\n\u0393 : Type ?u.13118\ninst\u271d : Inhabited \u0393\n\u03b1 : Sort ?u.13130\nl : ListBlank \u0393\nf : List \u0393 \u2192 \u03b1\nH : \u2200 (a b : List \u0393), BlankExtends a b \u2192 f a = f b\n\u22a2 \u2200 (a b : List \u0393), Setoid.r a b \u2192 f a = f b\n[PROOFSTEP]\nrintro a b (h | h) <;> [exact H _ _ h; exact (H _ _ h).symm]\n[GOAL]\n\u0393 : Type ?u.13118\ninst\u271d : Inhabited \u0393\n\u03b1 : Sort ?u.13130\nl : ListBlank \u0393\nf : List \u0393 \u2192 \u03b1\nH : \u2200 (a b : List \u0393), BlankExtends a b \u2192 f a = f b\n\u22a2 \u2200 (a b : List \u0393), Setoid.r a b \u2192 f a = f b\n[PROOFSTEP]\nrintro a b (h | h)\n[GOAL]\ncase inl\n\u0393 : Type ?u.13118\ninst\u271d : Inhabited \u0393\n\u03b1 : Sort ?u.13130\nl : ListBlank \u0393\nf : List \u0393 \u2192 \u03b1\nH : \u2200 (a b : List \u0393), BlankExtends a b \u2192 f a = f b\na b : List \u0393\nh : BlankExtends a b\n\u22a2 f a = f b\n[PROOFSTEP]\nexact H _ _ h\n[GOAL]\ncase inr\n\u0393 : Type ?u.13118\ninst\u271d : Inhabited \u0393\n\u03b1 : Sort ?u.13130\nl : ListBlank \u0393\nf : List \u0393 \u2192 \u03b1\nH : \u2200 (a b : List \u0393), BlankExtends a b \u2192 f a = f b\na b : List \u0393\nh : BlankExtends b a\n\u22a2 f a = f b\n[PROOFSTEP]\nexact (H _ _ h).symm\n[GOAL]\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 \u0393\n[PROOFSTEP]\napply l.liftOn List.headI\n[GOAL]\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 \u2200 (a b : List \u0393), BlankExtends a b \u2192 List.headI a = List.headI b\n[PROOFSTEP]\nrintro a _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\na : List \u0393\ni : \u2115\n\u22a2 List.headI a = List.headI (a ++ List.replicate i default)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase intro.nil\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\n\u22a2 List.headI [] = List.headI ([] ++ List.replicate i default)\n[PROOFSTEP]\ncases i\n[GOAL]\ncase intro.nil.zero\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 List.headI [] = List.headI ([] ++ List.replicate Nat.zero default)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.nil.succ\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn\u271d : \u2115\n\u22a2 List.headI [] = List.headI ([] ++ List.replicate (Nat.succ n\u271d) default)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.cons\n\u0393 : Type ?u.14152\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\nhead\u271d : \u0393\ntail\u271d : List \u0393\n\u22a2 List.headI (head\u271d :: tail\u271d) = List.headI (head\u271d :: tail\u271d ++ List.replicate i default)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\napply l.liftOn (fun l \u21a6 ListBlank.mk l.tail)\n[GOAL]\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 \u2200 (a b : List \u0393), BlankExtends a b \u2192 mk (List.tail a) = mk (List.tail b)\n[PROOFSTEP]\nrintro a _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\na : List \u0393\ni : \u2115\n\u22a2 mk (List.tail a) = mk (List.tail (a ++ List.replicate i default))\n[PROOFSTEP]\nrefine' Quotient.sound' (Or.inl _)\n[GOAL]\ncase intro\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\na : List \u0393\ni : \u2115\n\u22a2 BlankExtends (List.tail a) (List.tail (a ++ List.replicate i default))\n[PROOFSTEP]\ncases a\n[GOAL]\ncase intro.nil\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\n\u22a2 BlankExtends (List.tail []) (List.tail ([] ++ List.replicate i default))\n[PROOFSTEP]\ncases' i with i <;> [exact \u27e80, rfl\u27e9; exact \u27e8i, rfl\u27e9]\n[GOAL]\ncase intro.nil\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\n\u22a2 BlankExtends (List.tail []) (List.tail ([] ++ List.replicate i default))\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase intro.nil.zero\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 BlankExtends (List.tail []) (List.tail ([] ++ List.replicate Nat.zero default))\n[PROOFSTEP]\nexact \u27e80, rfl\u27e9\n[GOAL]\ncase intro.nil.succ\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\n\u22a2 BlankExtends (List.tail []) (List.tail ([] ++ List.replicate (Nat.succ i) default))\n[PROOFSTEP]\nexact \u27e8i, rfl\u27e9\n[GOAL]\ncase intro.cons\n\u0393 : Type ?u.14936\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\ni : \u2115\nhead\u271d : \u0393\ntail\u271d : List \u0393\n\u22a2 BlankExtends (List.tail (head\u271d :: tail\u271d)) (List.tail (head\u271d :: tail\u271d ++ List.replicate i default))\n[PROOFSTEP]\nexact \u27e8i, rfl\u27e9\n[GOAL]\n\u0393 : Type ?u.15941\ninst\u271d : Inhabited \u0393\na : \u0393\nl : ListBlank \u0393\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\napply l.liftOn (fun l \u21a6 ListBlank.mk (List.cons a l))\n[GOAL]\n\u0393 : Type ?u.15941\ninst\u271d : Inhabited \u0393\na : \u0393\nl : ListBlank \u0393\n\u22a2 \u2200 (a_1 b : List \u0393), BlankExtends a_1 b \u2192 mk (a :: a_1) = mk (a :: b)\n[PROOFSTEP]\nrintro _ _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.15941\ninst\u271d : Inhabited \u0393\na : \u0393\nl : ListBlank \u0393\na\u271d : List \u0393\ni : \u2115\n\u22a2 mk (a :: a\u271d) = mk (a :: (a\u271d ++ List.replicate i default))\n[PROOFSTEP]\nexact Quotient.sound' (Or.inl \u27e8i, rfl\u27e9)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u22a2 \u2200 (l : ListBlank \u0393), cons (head l) (tail l) = l\n[PROOFSTEP]\napply Quotient.ind'\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u22a2 \u2200 (a : List \u0393), cons (head (Quotient.mk'' a)) (tail (Quotient.mk'' a)) = Quotient.mk'' a\n[PROOFSTEP]\nrefine' fun l \u21a6 Quotient.sound' (Or.inr _)\n[GOAL]\ncase h\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : List \u0393\n\u22a2 BlankExtends l (head (Quotient.mk'' l) :: List.tail l)\n[PROOFSTEP]\ncases l\n[GOAL]\ncase h.nil\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u22a2 BlankExtends [] (head (Quotient.mk'' []) :: List.tail [])\n[PROOFSTEP]\nexact \u27e81, rfl\u27e9\n[GOAL]\ncase h.cons\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nhead\u271d : \u0393\ntail\u271d : List \u0393\n\u22a2 BlankExtends (head\u271d :: tail\u271d) (head (Quotient.mk'' (head\u271d :: tail\u271d)) :: List.tail (head\u271d :: tail\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n\u22a2 \u0393\n[PROOFSTEP]\napply l.liftOn (fun l \u21a6 List.getI l n)\n[GOAL]\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n\u22a2 \u2200 (a b : List \u0393), BlankExtends a b \u2192 List.getI a n = List.getI b n\n[PROOFSTEP]\nrintro l _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\n\u22a2 List.getI l n = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\ncases' lt_or_le n _ with h h\n[GOAL]\ncase intro.inl\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\nh : n < ?m.17919\n\u22a2 List.getI l n = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\nrw [List.getI_append _ _ _ h]\n[GOAL]\ncase intro.inr\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\nh : List.length l \u2264 n\n\u22a2 List.getI l n = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\nrw [List.getI_eq_default _ h]\n[GOAL]\ncase intro.inr\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\nh : List.length l \u2264 n\n\u22a2 default = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\ncases' le_or_lt _ n with h\u2082 h\u2082\n[GOAL]\ncase intro.inr.inl\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\nh : List.length l \u2264 n\nh\u2082 : ?m.18386 \u2264 n\n\u22a2 default = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\nrw [List.getI_eq_default _ h\u2082]\n[GOAL]\ncase intro.inr.inr\n\u0393 : Type ?u.17656\ninst\u271d : Inhabited \u0393\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\ni : \u2115\nh : List.length l \u2264 n\nh\u2082 : n < List.length (l ++ List.replicate i default)\n\u22a2 default = List.getI (l ++ List.replicate i default) n\n[PROOFSTEP]\nrw [List.getI_eq_get _ h\u2082, List.get_append_right' h, List.get_replicate]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 nth l 0 = head l\n[PROOFSTEP]\nconv => lhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n| nth l 0 = head l\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n| nth l 0 = head l\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n| nth l 0 = head l\n[PROOFSTEP]\nlhs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n| nth l 0\n[PROOFSTEP]\nrw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\n\u22a2 nth (cons (head l) (tail l)) 0 = head l\n[PROOFSTEP]\nexact Quotient.inductionOn' l.tail fun l \u21a6 rfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n\u22a2 nth l (n + 1) = nth (tail l) n\n[PROOFSTEP]\nconv => lhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n| nth l (n + 1) = nth (tail l) n\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n| nth l (n + 1) = nth (tail l) n\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n| nth l (n + 1) = nth (tail l) n\n[PROOFSTEP]\nlhs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n| nth l (n + 1)\n[PROOFSTEP]\nrw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl : ListBlank \u0393\nn : \u2115\n\u22a2 nth (cons (head l) (tail l)) (n + 1) = nth (tail l) n\n[PROOFSTEP]\nexact Quotient.inductionOn' l.tail fun l \u21a6 rfl\n[GOAL]\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\n\u22a2 (\u2200 (i_1 : \u2115), nth L\u2081 i_1 = nth L\u2082 i_1) \u2192 L\u2081 = L\u2082\n[PROOFSTEP]\nrefine' ListBlank.induction_on L\u2081 fun l\u2081 \u21a6 ListBlank.induction_on L\u2082 fun l\u2082 H \u21a6 _\n[GOAL]\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\nwlog h : l\u2081.length \u2264 l\u2082.length\n[GOAL]\ncase inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\ncases le_total l\u2081.length l\u2082.length <;> [skip; symm]\n[GOAL]\ncase inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\ncases le_total l\u2081.length l\u2082.length\n[GOAL]\ncase inr.inl\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\nskip\n[GOAL]\ncase inr.inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\nsymm\n[GOAL]\ncase inr.inl\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\napply this\n[GOAL]\ncase inr.inr\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 mk l\u2082 = mk l\u2081\n[PROOFSTEP]\napply this\n[GOAL]\ncase inr.inl.L\u2081\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inl.L\u2081\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inl.L\u2082\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inl.L\u2082\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inl.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inl.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inl.h\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 List.length l\u2081 \u2264 List.length l\u2082\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inl.h\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 List.length l\u2081 \u2264 List.length l\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.L\u2081\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inr.L\u2081\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.L\u2082\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inr.L\u2082\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 \u2200 (i_1 : \u2115), nth (mk l\u2082) i_1 = nth (mk l\u2081) i_1\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 \u2200 (i_1 : \u2115), nth (mk l\u2082) i_1 = nth (mk l\u2081) i_1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.h\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 List.length l\u2082 \u2264 List.length l\u2081\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase inr.inr.h\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 List.length l\u2082 \u2264 List.length l\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\n\u22a2 \u2200 (i_1 : \u2115), nth (mk l\u2082) i_1 = nth (mk l\u2081) i_1\n[PROOFSTEP]\nintro\n[GOAL]\ncase inr.inr.H\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nthis :\n \u2200 {\u0393 : Type u_1} [i : Inhabited \u0393] {L\u2081 L\u2082 : ListBlank \u0393} (l\u2081 l\u2082 : List \u0393),\n (\u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1) \u2192 List.length l\u2081 \u2264 List.length l\u2082 \u2192 mk l\u2081 = mk l\u2082\nh : \u00acList.length l\u2081 \u2264 List.length l\u2082\nh\u271d : List.length l\u2082 \u2264 List.length l\u2081\ni\u271d : \u2115\n\u22a2 nth (mk l\u2082) i\u271d = nth (mk l\u2081) i\u271d\n[PROOFSTEP]\nrw [H]\n[GOAL]\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 mk l\u2081 = mk l\u2082\n[PROOFSTEP]\nrefine' Quotient.sound' (Or.inl \u27e8l\u2082.length - l\u2081.length, _\u27e9)\n[GOAL]\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 l\u2082 = l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default\n[PROOFSTEP]\nrefine' List.ext_get _ fun i h h\u2082 \u21a6 Eq.symm _\n[GOAL]\ncase refine'_1\n\u0393 : Type u_1\ni : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i_1 : \u2115), nth (mk l\u2081) i_1 = nth (mk l\u2082) i_1\nh : List.length l\u2081 \u2264 List.length l\u2082\n\u22a2 List.length l\u2082 = List.length (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default)\n[PROOFSTEP]\nsimp only [add_tsub_cancel_of_le h, List.length_append, List.length_replicate]\n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), nth (mk l\u2081) i = nth (mk l\u2082) i\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\ni : \u2115\nh : i < List.length l\u2082\nh\u2082 : i < List.length (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default)\n\u22a2 List.get (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default) { val := i, isLt := h\u2082 } =\n List.get l\u2082 { val := i, isLt := h }\n[PROOFSTEP]\nsimp only [ListBlank.nth_mk] at H \n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), List.getI l\u2081 i = List.getI l\u2082 i\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\ni : \u2115\nh : i < List.length l\u2082\nh\u2082 : i < List.length (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default)\n\u22a2 List.get (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default) { val := i, isLt := h\u2082 } =\n List.get l\u2082 { val := i, isLt := h }\n[PROOFSTEP]\ncases' lt_or_le i l\u2081.length with h' h'\n[GOAL]\ncase refine'_2.inl\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), List.getI l\u2081 i = List.getI l\u2082 i\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\ni : \u2115\nh : i < List.length l\u2082\nh\u2082 : i < List.length (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default)\nh' : i < List.length l\u2081\n\u22a2 List.get (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default) { val := i, isLt := h\u2082 } =\n List.get l\u2082 { val := i, isLt := h }\n[PROOFSTEP]\nsimp only [List.get_append _ h', List.get?_eq_get h, List.get?_eq_get h', \u2190 List.getI_eq_get _ h, \u2190\n List.getI_eq_get _ h', H]\n[GOAL]\ncase refine'_2.inr\n\u0393 : Type u_1\ni\u271d : Inhabited \u0393\nL\u2081 L\u2082 : ListBlank \u0393\nl\u2081 l\u2082 : List \u0393\nH : \u2200 (i : \u2115), List.getI l\u2081 i = List.getI l\u2082 i\nh\u271d : List.length l\u2081 \u2264 List.length l\u2082\ni : \u2115\nh : i < List.length l\u2082\nh\u2082 : i < List.length (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default)\nh' : List.length l\u2081 \u2264 i\n\u22a2 List.get (l\u2081 ++ List.replicate (List.length l\u2082 - List.length l\u2081) default) { val := i, isLt := h\u2082 } =\n List.get l\u2082 { val := i, isLt := h }\n[PROOFSTEP]\nsimp only [List.get_append_right' h', List.get_replicate, List.get?_eq_get h, List.get?_len_le h', \u2190\n List.getI_eq_default _ h', H, List.getI_eq_get _ h]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\nn i : \u2115\nL : ListBlank \u0393\n\u22a2 nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\n[PROOFSTEP]\ninduction' n with n IH generalizing i L\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni\u271d : \u2115\nL\u271d : ListBlank \u0393\ni : \u2115\nL : ListBlank \u0393\n\u22a2 nth (modifyNth f Nat.zero L) i = if i = Nat.zero then f (nth L i) else nth L i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero.zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni : \u2115\nL\u271d L : ListBlank \u0393\n\u22a2 nth (modifyNth f Nat.zero L) Nat.zero = if Nat.zero = Nat.zero then f (nth L Nat.zero) else nth L Nat.zero\n[PROOFSTEP]\nsimp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.nth_succ, if_false,\n ListBlank.tail_cons, Nat.zero_eq]\n[GOAL]\ncase zero.succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni : \u2115\nL\u271d L : ListBlank \u0393\nn\u271d : \u2115\n\u22a2 nth (modifyNth f Nat.zero L) (Nat.succ n\u271d) =\n if Nat.succ n\u271d = Nat.zero then f (nth L (Nat.succ n\u271d)) else nth L (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.nth_succ, if_false,\n ListBlank.tail_cons, Nat.zero_eq]\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni\u271d : \u2115\nL\u271d : ListBlank \u0393\nn : \u2115\nIH : \u2200 (i : \u2115) (L : ListBlank \u0393), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\ni : \u2115\nL : ListBlank \u0393\n\u22a2 nth (modifyNth f (Nat.succ n) L) i = if i = Nat.succ n then f (nth L i) else nth L i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase succ.zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni : \u2115\nL\u271d : ListBlank \u0393\nn : \u2115\nIH : \u2200 (i : \u2115) (L : ListBlank \u0393), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank \u0393\n\u22a2 nth (modifyNth f (Nat.succ n) L) Nat.zero = if Nat.zero = Nat.succ n then f (nth L Nat.zero) else nth L Nat.zero\n[PROOFSTEP]\nrw [if_neg (Nat.succ_ne_zero _).symm]\n[GOAL]\ncase succ.zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni : \u2115\nL\u271d : ListBlank \u0393\nn : \u2115\nIH : \u2200 (i : \u2115) (L : ListBlank \u0393), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank \u0393\n\u22a2 nth (modifyNth f (Nat.succ n) L) Nat.zero = nth L Nat.zero\n[PROOFSTEP]\nsimp only [ListBlank.nth_zero, ListBlank.head_cons, ListBlank.modifyNth, Nat.zero_eq]\n[GOAL]\ncase succ.succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\ni : \u2115\nL\u271d : ListBlank \u0393\nn : \u2115\nIH : \u2200 (i : \u2115) (L : ListBlank \u0393), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank \u0393\nn\u271d : \u2115\n\u22a2 nth (modifyNth f (Nat.succ n) L) (Nat.succ n\u271d) =\n if Nat.succ n\u271d = Nat.succ n then f (nth L (Nat.succ n\u271d)) else nth L (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [IH, ListBlank.modifyNth, ListBlank.nth_succ, ListBlank.tail_cons, Nat.succ.injEq]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : List \u0393\n\u22a2 List.headI (List.map f.f l) = Turing.PointedMap.f f (List.headI l)\n[PROOFSTEP]\ncases l <;> [exact (PointedMap.map_pt f).symm; rfl]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : List \u0393\n\u22a2 List.headI (List.map f.f l) = Turing.PointedMap.f f (List.headI l)\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\n\u22a2 List.headI (List.map f.f []) = Turing.PointedMap.f f (List.headI [])\n[PROOFSTEP]\nexact (PointedMap.map_pt f).symm\n[GOAL]\ncase cons\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nhead\u271d : \u0393\ntail\u271d : List \u0393\n\u22a2 List.headI (List.map f.f (head\u271d :: tail\u271d)) = Turing.PointedMap.f f (List.headI (head\u271d :: tail\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type ?u.27995\n\u0393' : Type ?u.27994\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 ListBlank \u0393'\n[PROOFSTEP]\napply l.liftOn (fun l \u21a6 ListBlank.mk (List.map f l))\n[GOAL]\n\u0393 : Type ?u.27995\n\u0393' : Type ?u.27994\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 \u2200 (a b : List \u0393), BlankExtends a b \u2192 mk (List.map f.f a) = mk (List.map f.f b)\n[PROOFSTEP]\nrintro l _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.27995\n\u0393' : Type ?u.27994\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nl : List \u0393\ni : \u2115\n\u22a2 mk (List.map f.f l) = mk (List.map f.f (l ++ List.replicate i default))\n[PROOFSTEP]\nrefine' Quotient.sound' (Or.inl \u27e8i, _\u27e9)\n[GOAL]\ncase intro\n\u0393 : Type ?u.27995\n\u0393' : Type ?u.27994\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nl : List \u0393\ni : \u2115\n\u22a2 List.map f.f (l ++ List.replicate i default) = List.map f.f l ++ List.replicate i default\n[PROOFSTEP]\nsimp only [PointedMap.map_pt, List.map_append, List.map_replicate]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 head (map f l) = PointedMap.f f (head l)\n[PROOFSTEP]\nconv => lhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| head (map f l) = PointedMap.f f (head l)\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| head (map f l) = PointedMap.f f (head l)\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| head (map f l) = PointedMap.f f (head l)\n[PROOFSTEP]\nlhs\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| head (map f l)\n[PROOFSTEP]\nrw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 head (map f (cons (head l) (tail l))) = PointedMap.f f (head l)\n[PROOFSTEP]\nexact Quotient.inductionOn' l fun a \u21a6 rfl\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 tail (map f l) = map f (tail l)\n[PROOFSTEP]\nconv => lhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| tail (map f l) = map f (tail l)\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| tail (map f l) = map f (tail l)\n[PROOFSTEP]\nlhs; rw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| tail (map f l) = map f (tail l)\n[PROOFSTEP]\nlhs\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n| tail (map f l)\n[PROOFSTEP]\nrw [\u2190 ListBlank.cons_head_tail l]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 tail (map f (cons (head l) (tail l))) = map f (tail l)\n[PROOFSTEP]\nexact Quotient.inductionOn' l fun a \u21a6 rfl\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\na : \u0393\n\u22a2 map f (cons a l) = cons (PointedMap.f f a) (map f l)\n[PROOFSTEP]\nrefine' (ListBlank.cons_head_tail _).symm.trans _\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\na : \u0393\n\u22a2 cons (head (map f (cons a l))) (tail (map f (cons a l))) = cons (PointedMap.f f a) (map f l)\n[PROOFSTEP]\nsimp only [ListBlank.head_map, ListBlank.head_cons, ListBlank.tail_map, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\nn : \u2115\n\u22a2 nth (map f l) n = PointedMap.f f (nth l n)\n[PROOFSTEP]\nrefine'\n l.inductionOn fun l \u21a6\n _\n -- Porting note: Added `suffices` to get `simp` to work.\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\n\u22a2 nth (map f (Quotient.mk (BlankRel.setoid \u0393) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid \u0393) l) n)\n[PROOFSTEP]\nsuffices ((mk l).map f).nth n = f ((mk l).nth n) by exact this\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\nthis : nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n)\n\u22a2 nth (map f (Quotient.mk (BlankRel.setoid \u0393) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid \u0393) l) n)\n[PROOFSTEP]\nexact this\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\n\u22a2 nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n)\n[PROOFSTEP]\nsimp only [List.get?_map, ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\n\u22a2 Option.iget (Option.map f.f (List.get? l n)) = PointedMap.f f (Option.iget (List.get? l n))\n[PROOFSTEP]\ncases l.get? n\n[GOAL]\ncase none\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\n\u22a2 Option.iget (Option.map f.f none) = PointedMap.f f (Option.iget none)\n[PROOFSTEP]\nexact f.2.symm\n[GOAL]\ncase some\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl\u271d : ListBlank \u0393\nn : \u2115\nl : List \u0393\nval\u271d : \u0393\n\u22a2 Option.iget (Option.map f.f (some val\u271d)) = PointedMap.f f (Option.iget (some val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u0393 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 Inhabited (\u0393 i)\ni : \u03b9\nL : ListBlank ((i : \u03b9) \u2192 \u0393 i)\nn : \u2115\n\u22a2 ListBlank.nth (ListBlank.map (proj i) L) n = ListBlank.nth L n i\n[PROOFSTEP]\nrw [ListBlank.nth_map]\n[GOAL]\n\u03b9 : Type u_1\n\u0393 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 Inhabited (\u0393 i)\ni : \u03b9\nL : ListBlank ((i : \u03b9) \u2192 \u0393 i)\nn : \u2115\n\u22a2 PointedMap.f (proj i) (ListBlank.nth L n) = ListBlank.nth L n i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nF : PointedMap \u0393 \u0393'\nf : \u0393 \u2192 \u0393\nf' : \u0393' \u2192 \u0393'\nH : \u2200 (x : \u0393), PointedMap.f F (f x) = f' (PointedMap.f F x)\nn : \u2115\nL : ListBlank \u0393\n\u22a2 map F (modifyNth f n L) = modifyNth f' n (map F L)\n[PROOFSTEP]\ninduction' n with n IH generalizing L\n[GOAL]\ncase zero\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nF : PointedMap \u0393 \u0393'\nf : \u0393 \u2192 \u0393\nf' : \u0393' \u2192 \u0393'\nH : \u2200 (x : \u0393), PointedMap.f F (f x) = f' (PointedMap.f F x)\nL\u271d L : ListBlank \u0393\n\u22a2 map F (modifyNth f Nat.zero L) = modifyNth f' Nat.zero (map F L)\n[PROOFSTEP]\nsimp only [*, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map]\n[GOAL]\ncase succ\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nF : PointedMap \u0393 \u0393'\nf : \u0393 \u2192 \u0393\nf' : \u0393' \u2192 \u0393'\nH : \u2200 (x : \u0393), PointedMap.f F (f x) = f' (PointedMap.f F x)\nL\u271d : ListBlank \u0393\nn : \u2115\nIH : \u2200 (L : ListBlank \u0393), map F (modifyNth f n L) = modifyNth f' n (map F L)\nL : ListBlank \u0393\n\u22a2 map F (modifyNth f (Nat.succ n) L) = modifyNth f' (Nat.succ n) (map F L)\n[PROOFSTEP]\nsimp only [*, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\n\u22a2 append l\u2081 (mk l\u2082) = mk (l\u2081 ++ l\u2082)\n[PROOFSTEP]\ninduction l\u2081\n[GOAL]\ncase nil\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2082 : List \u0393\n\u22a2 append [] (mk l\u2082) = mk ([] ++ l\u2082)\n[PROOFSTEP]\nsimp only [*, ListBlank.append, List.nil_append, List.cons_append, ListBlank.cons_mk]\n[GOAL]\ncase cons\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2082 : List \u0393\nhead\u271d : \u0393\ntail\u271d : List \u0393\ntail_ih\u271d : append tail\u271d (mk l\u2082) = mk (tail\u271d ++ l\u2082)\n\u22a2 append (head\u271d :: tail\u271d) (mk l\u2082) = mk (head\u271d :: tail\u271d ++ l\u2082)\n[PROOFSTEP]\nsimp only [*, ListBlank.append, List.nil_append, List.cons_append, ListBlank.cons_mk]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nl\u2083 : ListBlank \u0393\n\u22a2 append (l\u2081 ++ l\u2082) l\u2083 = append l\u2081 (append l\u2082 l\u2083)\n[PROOFSTEP]\nrefine'\n l\u2083.inductionOn fun l \u21a6\n _\n -- Porting note: Added `suffices` to get `simp` to work.\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nl\u2083 : ListBlank \u0393\nl : List \u0393\n\u22a2 append (l\u2081 ++ l\u2082) (Quotient.mk (BlankRel.setoid \u0393) l) = append l\u2081 (append l\u2082 (Quotient.mk (BlankRel.setoid \u0393) l))\n[PROOFSTEP]\nsuffices append (l\u2081 ++ l\u2082) (mk l) = append l\u2081 (append l\u2082 (mk l)) by exact this\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nl\u2083 : ListBlank \u0393\nl : List \u0393\nthis : append (l\u2081 ++ l\u2082) (mk l) = append l\u2081 (append l\u2082 (mk l))\n\u22a2 append (l\u2081 ++ l\u2082) (Quotient.mk (BlankRel.setoid \u0393) l) = append l\u2081 (append l\u2082 (Quotient.mk (BlankRel.setoid \u0393) l))\n[PROOFSTEP]\nexact this\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nl\u2081 l\u2082 : List \u0393\nl\u2083 : ListBlank \u0393\nl : List \u0393\n\u22a2 append (l\u2081 ++ l\u2082) (mk l) = append l\u2081 (append l\u2082 (mk l))\n[PROOFSTEP]\nsimp only [ListBlank.append_mk, List.append_assoc]\n[GOAL]\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\n\u22a2 ListBlank \u0393'\n[PROOFSTEP]\napply l.liftOn (fun l \u21a6 ListBlank.mk (List.bind l f))\n[GOAL]\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\n\u22a2 \u2200 (a b : List \u0393), BlankExtends a b \u2192 mk (List.bind a f) = mk (List.bind b f)\n[PROOFSTEP]\nrintro l _ \u27e8i, rfl\u27e9\n[GOAL]\ncase intro\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\ni : \u2115\n\u22a2 mk (List.bind l f) = mk (List.bind (l ++ List.replicate i default) f)\n[PROOFSTEP]\ncases' hf with n e\n[GOAL]\ncase intro.intro\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\ni n : \u2115\ne : f default = List.replicate n default\n\u22a2 mk (List.bind l f) = mk (List.bind (l ++ List.replicate i default) f)\n[PROOFSTEP]\nrefine' Quotient.sound' (Or.inl \u27e8i * n, _\u27e9)\n[GOAL]\ncase intro.intro\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\ni n : \u2115\ne : f default = List.replicate n default\n\u22a2 List.bind (l ++ List.replicate i default) f = List.bind l f ++ List.replicate (i * n) default\n[PROOFSTEP]\nrw [List.append_bind, mul_comm]\n[GOAL]\ncase intro.intro\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\ni n : \u2115\ne : f default = List.replicate n default\n\u22a2 List.bind l f ++ List.bind (List.replicate i default) f = List.bind l f ++ List.replicate (n * i) default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.e_a\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\ni n : \u2115\ne : f default = List.replicate n default\n\u22a2 List.bind (List.replicate i default) f = List.replicate (n * i) default\n[PROOFSTEP]\ninduction' i with i IH\n[GOAL]\ncase intro.intro.e_a.zero\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\nn : \u2115\ne : f default = List.replicate n default\n\u22a2 List.bind (List.replicate Nat.zero default) f = List.replicate (n * Nat.zero) default\ncase intro.intro.e_a.succ\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\nn : \u2115\ne : f default = List.replicate n default\ni : \u2115\nIH : List.bind (List.replicate i default) f = List.replicate (n * i) default\n\u22a2 List.bind (List.replicate (Nat.succ i) default) f = List.replicate (n * Nat.succ i) default\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.e_a.succ\n\u0393 : Type ?u.36140\n\u0393' : Type ?u.36152\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nl : List \u0393\nn : \u2115\ne : f default = List.replicate n default\ni : \u2115\nIH : List.bind (List.replicate i default) f = List.replicate (n * i) default\n\u22a2 List.bind (List.replicate (Nat.succ i) default) f = List.replicate (n * Nat.succ i) default\n[PROOFSTEP]\nsimp only [IH, e, List.replicate_add, Nat.mul_succ, add_comm, List.replicate_succ, List.cons_bind]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\n\u22a2 bind (cons a l) f hf = append (f a) (bind l f hf)\n[PROOFSTEP]\nrefine'\n l.inductionOn fun l \u21a6\n _\n -- Porting note: Added `suffices` to get `simp` to work.\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf)\n[PROOFSTEP]\nsuffices ((mk l).cons a).bind f hf = ((mk l).bind f hf).append (f a) by exact this\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\nthis : bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf)\n\u22a2 bind (cons a (Quotient.mk (BlankRel.setoid \u0393) l)) f hf = append (f a) (bind (Quotient.mk (BlankRel.setoid \u0393) l) f hf)\n[PROOFSTEP]\nexact this\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\na : \u0393\nl\u271d : ListBlank \u0393\nf : \u0393 \u2192 List \u0393'\nhf : \u2203 n, f default = List.replicate n default\nl : List \u0393\n\u22a2 bind (cons a (mk l)) f hf = append (f a) (bind (mk l) f hf)\n[PROOFSTEP]\nsimp only [ListBlank.append_mk, ListBlank.bind_mk, ListBlank.cons_mk, List.cons_bind]\n[GOAL]\n\u0393 : Type ?u.39827\ninst\u271d : Inhabited \u0393\n\u22a2 Tape \u0393\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase head\n\u0393 : Type ?u.39827\ninst\u271d : Inhabited \u0393\n\u22a2 \u0393\n[PROOFSTEP]\napply default\n[GOAL]\ncase left\n\u0393 : Type ?u.39827\ninst\u271d : Inhabited \u0393\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\napply default\n[GOAL]\ncase right\n\u0393 : Type ?u.39827\ninst\u271d : Inhabited \u0393\n\u22a2 ListBlank \u0393\n[PROOFSTEP]\napply default\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 move Dir.right (move Dir.left T) = T\n[PROOFSTEP]\ncases T\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 move Dir.right (move Dir.left { head := head\u271d, left := left\u271d, right := right\u271d }) =\n { head := head\u271d, left := left\u271d, right := right\u271d }\n[PROOFSTEP]\nsimp [Tape.move]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 move Dir.left (move Dir.right T) = T\n[PROOFSTEP]\ncases T\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 move Dir.left (move Dir.right { head := head\u271d, left := left\u271d, right := right\u271d }) =\n { head := head\u271d, left := left\u271d, right := right\u271d }\n[PROOFSTEP]\nsimp [Tape.move]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 mk' T.left (right\u2080 T) = T\n[PROOFSTEP]\ncases T\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 mk' { head := head\u271d, left := left\u271d, right := right\u271d }.left\n (right\u2080 { head := head\u271d, left := left\u271d, right := right\u271d }) =\n { head := head\u271d, left := left\u271d, right := right\u271d }\n[PROOFSTEP]\nsimp only [Tape.right\u2080, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nL R : ListBlank \u0393\n\u22a2 move Dir.left (mk' L R) = mk' (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R)\n[PROOFSTEP]\nsimp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff,\n ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nL R : ListBlank \u0393\n\u22a2 move Dir.right (mk' L R) = mk' (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)\n[PROOFSTEP]\nsimp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff,\n ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\nn : \u2115\n\u22a2 ListBlank.nth (right\u2080 T) n = nth T \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 ListBlank.nth (right\u2080 T) Nat.zero = nth T \u2191Nat.zero\n[PROOFSTEP]\nsimp only [Tape.nth, Tape.right\u2080, Int.ofNat_zero, ListBlank.nth_zero, ListBlank.nth_succ, ListBlank.head_cons,\n ListBlank.tail_cons, Nat.zero_eq]\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\nn\u271d : \u2115\n\u22a2 ListBlank.nth (right\u2080 T) (Nat.succ n\u271d) = nth T \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [Tape.nth, Tape.right\u2080, Int.ofNat_zero, ListBlank.nth_zero, ListBlank.nth_succ, ListBlank.head_cons,\n ListBlank.tail_cons, Nat.zero_eq]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 nth (mk' L R) \u2191n = ListBlank.nth R n\n[PROOFSTEP]\nrw [\u2190 Tape.right\u2080_nth, Tape.mk'_right\u2080]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\na : \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 nth (move Dir.left { head := a, left := L, right := R }) (\u2191(n + 1) + 1) =\n nth { head := a, left := L, right := R } (\u2191(n + 1) + 1 - 1)\n[PROOFSTEP]\nrw [add_sub_cancel]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\na : \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 nth (move Dir.left { head := a, left := L, right := R }) (\u2191(n + 1) + 1) =\n nth { head := a, left := L, right := R } \u2191(n + 1)\n[PROOFSTEP]\nchange (R.cons a).nth (n + 1) = R.nth n\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\na : \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 ListBlank.nth (ListBlank.cons a R) (n + 1) = ListBlank.nth R n\n[PROOFSTEP]\nrw [ListBlank.nth_succ, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n\u22a2 nth (move Dir.right T) i = nth T (i + 1)\n[PROOFSTEP]\nconv => rhs; rw [\u2190 T.move_right_left]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n| nth (move Dir.right T) i = nth T (i + 1)\n[PROOFSTEP]\nrhs; rw [\u2190 T.move_right_left]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n| nth (move Dir.right T) i = nth T (i + 1)\n[PROOFSTEP]\nrhs; rw [\u2190 T.move_right_left]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n| nth (move Dir.right T) i = nth T (i + 1)\n[PROOFSTEP]\nrhs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n| nth T (i + 1)\n[PROOFSTEP]\nrw [\u2190 T.move_right_left]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2124\n\u22a2 nth (move Dir.right T) i = nth (move Dir.left (move Dir.right T)) (i + 1)\n[PROOFSTEP]\nrw [Tape.move_left_nth, add_sub_cancel]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\ni : \u2115\n\u22a2 ((move Dir.right)^[i] T).head = nth T \u2191i\n[PROOFSTEP]\ninduction i generalizing T\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.zero] T).head = nth T \u2191Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nn\u271d : \u2115\nn_ih\u271d : \u2200 (T : Tape \u0393), ((move Dir.right)^[n\u271d] T).head = nth T \u2191n\u271d\nT : Tape \u0393\n\u22a2 ((move Dir.right)^[Nat.succ n\u271d] T).head = nth T \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [*, Tape.move_right_nth, Int.ofNat_succ, iterate_succ, Function.comp_apply]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u22a2 \u2200 (T : Tape \u0393), write T.head T = T\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 write { head := head\u271d, left := left\u271d, right := right\u271d }.head { head := head\u271d, left := left\u271d, right := right\u271d } =\n { head := head\u271d, left := left\u271d, right := right\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\na b : \u0393\nL R : ListBlank \u0393\n\u22a2 write b (mk' L (ListBlank.cons a R)) = mk' L (ListBlank.cons b R)\n[PROOFSTEP]\nsimp only [Tape.write, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\n\u22a2 \u2200 (T : Tape \u0393), (map f T).head = PointedMap.f f T.head\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mk\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 (map f { head := head\u271d, left := left\u271d, right := right\u271d }).head =\n PointedMap.f f { head := head\u271d, left := left\u271d, right := right\u271d }.head\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nb : \u0393\n\u22a2 \u2200 (T : Tape \u0393), map f (write b T) = write (PointedMap.f f b) (map f T)\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mk\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nb head\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 map f (write b { head := head\u271d, left := left\u271d, right := right\u271d }) =\n write (PointedMap.f f b) (map f { head := head\u271d, left := left\u271d, right := right\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\nL R : ListBlank \u0393\nn : \u2115\n\u22a2 write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) =\n (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R))\n[PROOFSTEP]\ninduction' n with n IH generalizing L R\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\nL\u271d R\u271d L R : ListBlank \u0393\n\u22a2 write (f (ListBlank.nth R Nat.zero)) ((move Dir.right)^[Nat.zero] (mk' L R)) =\n (move Dir.right)^[Nat.zero] (mk' L (ListBlank.modifyNth f Nat.zero R))\n[PROOFSTEP]\nsimp only [ListBlank.nth_zero, ListBlank.modifyNth, iterate_zero_apply, Nat.zero_eq]\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\nL\u271d R\u271d L R : ListBlank \u0393\n\u22a2 write (f (ListBlank.head R)) (mk' L R) = mk' L (ListBlank.cons (f (ListBlank.head R)) (ListBlank.tail R))\n[PROOFSTEP]\nrw [\u2190 Tape.write_mk', ListBlank.cons_head_tail]\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\nf : \u0393 \u2192 \u0393\nL\u271d R\u271d : ListBlank \u0393\nn : \u2115\nIH :\n \u2200 (L R : ListBlank \u0393),\n write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) =\n (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R))\nL R : ListBlank \u0393\n\u22a2 write (f (ListBlank.nth R (Nat.succ n))) ((move Dir.right)^[Nat.succ n] (mk' L R)) =\n (move Dir.right)^[Nat.succ n] (mk' L (ListBlank.modifyNth f (Nat.succ n) R))\n[PROOFSTEP]\nsimp only [ListBlank.head_cons, ListBlank.nth_succ, ListBlank.modifyNth, Tape.move_right_mk', ListBlank.tail_cons,\n iterate_succ_apply, IH]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nT : Tape \u0393\nd : Dir\n\u22a2 map f (move d T) = move d (map f T)\n[PROOFSTEP]\ncases T\n[GOAL]\ncase mk\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nd : Dir\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 map f (move d { head := head\u271d, left := left\u271d, right := right\u271d }) =\n move d (map f { head := head\u271d, left := left\u271d, right := right\u271d })\n[PROOFSTEP]\ncases d\n[GOAL]\ncase mk.left\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 map f (move Dir.left { head := head\u271d, left := left\u271d, right := right\u271d }) =\n move Dir.left (map f { head := head\u271d, left := left\u271d, right := right\u271d })\n[PROOFSTEP]\nsimp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true, ListBlank.map_cons, and_self_iff,\n ListBlank.tail_map]\n[GOAL]\ncase mk.right\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nhead\u271d : \u0393\nleft\u271d right\u271d : ListBlank \u0393\n\u22a2 map f (move Dir.right { head := head\u271d, left := left\u271d, right := right\u271d }) =\n move Dir.right (map f { head := head\u271d, left := left\u271d, right := right\u271d })\n[PROOFSTEP]\nsimp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true, ListBlank.map_cons, and_self_iff,\n ListBlank.tail_map]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nL R : ListBlank \u0393\n\u22a2 map f (mk' L R) = mk' (ListBlank.map f L) (ListBlank.map f R)\n[PROOFSTEP]\nsimp only [Tape.mk', Tape.map, ListBlank.head_map, eq_self_iff_true, and_self_iff, ListBlank.tail_map]\n[GOAL]\n\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nL R : List \u0393\n\u22a2 map f (mk\u2082 L R) = mk\u2082 (List.map f.f L) (List.map f.f R)\n[PROOFSTEP]\nsimp only [Tape.mk\u2082, Tape.map_mk', ListBlank.map_mk]\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b c : \u03c3\nh : f a = f b\n\u22a2 (\u2203 b_1, b_1 \u2208 f b \u2227 ReflTransGen (fun a b => b \u2208 f a) b_1 c) \u2194 \u2203 b, b \u2208 f a \u2227 ReflTransGen (fun a b => b \u2208 f a) b c\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b c : \u03c3\nh\u2081 : Reaches\u2081 f a c\nh\u2082 : b \u2208 f a\n\u22a2 Reaches f b c\n[PROOFSTEP]\nrcases TransGen.head'_iff.1 h\u2081 with \u27e8b', hab, hbc\u27e9\n[GOAL]\ncase intro.intro\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b c : \u03c3\nh\u2081 : Reaches\u2081 f a c\nh\u2082 : b \u2208 f a\nb' : \u03c3\nhab : b' \u2208 f a\nhbc : ReflTransGen (fun a b => b \u2208 f a) b' c\n\u22a2 Reaches f b c\n[PROOFSTEP]\ncases Option.mem_unique hab h\u2082\n[GOAL]\ncase intro.intro.refl\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b c : \u03c3\nh\u2081 : Reaches\u2081 f a c\nh\u2082 hab : b \u2208 f a\nhbc : ReflTransGen (fun a b => b \u2208 f a) b c\n\u22a2 Reaches f b c\n[PROOFSTEP]\nexact hbc\n[GOAL]\n\u03c3 : Type ?u.63381\nf : \u03c3 \u2192 Option \u03c3\nb : \u03c3\nC : \u03c3 \u2192 Sort u_1\na : \u03c3\nh : b \u2208 eval f a\nH : (a : \u03c3) \u2192 b \u2208 eval f a \u2192 ((a' : \u03c3) \u2192 f a = some a' \u2192 C a') \u2192 C a\na' : \u03c3\nha' : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a'\nh' : (a'' : \u03c3) \u2192 Sum.inr a'' \u2208 Part.some (Option.elim (f a') (Sum.inl a') Sum.inr) \u2192 C a''\nb' : \u03c3\ne : f a' = some b'\n\u22a2 Sum.inr b' = Option.elim (f a') (Sum.inl a') Sum.inr\n[PROOFSTEP]\nrw [e]\n[GOAL]\n\u03c3 : Type ?u.63381\nf : \u03c3 \u2192 Option \u03c3\nb : \u03c3\nC : \u03c3 \u2192 Sort u_1\na : \u03c3\nh : b \u2208 eval f a\nH : (a : \u03c3) \u2192 b \u2208 eval f a \u2192 ((a' : \u03c3) \u2192 f a = some a' \u2192 C a') \u2192 C a\na' : \u03c3\nha' : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a'\nh' : (a'' : \u03c3) \u2192 Sum.inr a'' \u2208 Part.some (Option.elim (f a') (Sum.inl a') Sum.inr) \u2192 C a''\nb' : \u03c3\ne : f a' = some b'\n\u22a2 Sum.inr b' = Option.elim (some b') (Sum.inl a') Sum.inr\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\n\u22a2 b \u2208 eval f a \u2194 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 _\u27e9\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nh : b \u2208 eval f a\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrefine' @evalInduction _ _ _ (fun a \u21a6 Reaches f a b \u2227 f b = none) _ h fun a h IH \u21a6 _\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\n\u22a2 (fun a => Reaches f a b \u2227 f b = none) a\n[PROOFSTEP]\ncases' e : f a with a'\n[GOAL]\ncase refine'_1.none\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\ne : f a = none\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrw [Part.mem_unique h (PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e] <;> rfl)]\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\ne : f a = none\n\u22a2 Sum.inl ?m.64287 = Option.elim (f a) (Sum.inl a) Sum.inr\n[PROOFSTEP]\nrw [e]\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\ne : f a = none\n\u22a2 Sum.inl ?m.64287 = Option.elim none (Sum.inl a) Sum.inr\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.none\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\ne : f a = none\n\u22a2 Reaches f a a \u2227 f a = none\n[PROOFSTEP]\nexact \u27e8ReflTransGen.refl, e\u27e9\n[GOAL]\ncase refine'_1.some\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d : b \u2208 eval f a\u271d\na : \u03c3\nh : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrcases PFun.mem_fix_iff.1 h with (h | \u27e8_, h, _\u27e9)\n[GOAL]\ncase refine'_1.some.inl\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nh : Sum.inl b \u2208 Part.some (Option.elim (f a) (Sum.inl a) Sum.inr)\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\ncase refine'_1.some.inr.intro.intro\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nw\u271d : \u03c3\nh : Sum.inr w\u271d \u2208 Part.some (Option.elim (f a) (Sum.inl a) Sum.inr)\nright\u271d : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) w\u271d\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\ncase refine'_1.some.inl\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nh : Sum.inl b \u2208 Part.some (Option.elim (some a') (Sum.inl a) Sum.inr)\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\ncases Part.mem_some_iff.1 h\n[GOAL]\ncase refine'_1.some.inr.intro.intro\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nw\u271d : \u03c3\nh : Sum.inr w\u271d \u2208 Part.some (Option.elim (some a') (Sum.inl a) Sum.inr)\nright\u271d : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) w\u271d\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\ncases Part.mem_some_iff.1 h\n[GOAL]\ncase refine'_1.some.inr.intro.intro.refl\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nh : Sum.inr a' \u2208 Part.some (Option.elim (some a') (Sum.inl a) Sum.inr)\nright\u271d : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a'\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\ncases' IH a' e with h\u2081 h\u2082\n[GOAL]\ncase refine'_1.some.inr.intro.intro.refl.intro\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na\u271d b : \u03c3\nh\u271d\u00b9 : b \u2208 eval f a\u271d\na : \u03c3\nh\u271d : b \u2208 eval f a\nIH : \u2200 (a' : \u03c3), f a = some a' \u2192 (fun a => Reaches f a b \u2227 f b = none) a'\na' : \u03c3\ne : f a = some a'\nh : Sum.inr a' \u2208 Part.some (Option.elim (some a') (Sum.inl a) Sum.inr)\nright\u271d : b \u2208 PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a'\nh\u2081 : Reaches f a' b\nh\u2082 : f b = none\n\u22a2 Reaches f a b \u2227 f b = none\n[PROOFSTEP]\nexact \u27e8ReflTransGen.head e h\u2081, h\u2082\u27e9\n[GOAL]\ncase refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\n\u22a2 b \u2208 eval f a\n[PROOFSTEP]\nrefine' ReflTransGen.head_induction_on h\u2081 _ fun h _ IH \u21a6 _\n[GOAL]\ncase refine'_2.refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\n\u22a2 b \u2208 eval f b\n[PROOFSTEP]\nrefine' PFun.mem_fix_iff.2 (Or.inl _)\n[GOAL]\ncase refine'_2.refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\n\u22a2 Sum.inl b \u2208 Part.some (Option.elim (f b) (Sum.inl b) Sum.inr)\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\ncase refine'_2.refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\n\u22a2 Sum.inl b \u2208 Part.some (Option.elim none (Sum.inl b) Sum.inr)\n[PROOFSTEP]\napply Part.mem_some\n[GOAL]\ncase refine'_2.refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d\u00b9 : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\na\u271d c\u271d : \u03c3\nh : c\u271d \u2208 f a\u271d\nx\u271d : ReflTransGen (fun a b => b \u2208 f a) c\u271d b\nIH : b \u2208 eval f c\u271d\n\u22a2 b \u2208 eval f a\u271d\n[PROOFSTEP]\nrefine' PFun.mem_fix_iff.2 (Or.inr \u27e8_, _, IH\u27e9)\n[GOAL]\ncase refine'_2.refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d\u00b9 : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\na\u271d c\u271d : \u03c3\nh : c\u271d \u2208 f a\u271d\nx\u271d : ReflTransGen (fun a b => b \u2208 f a) c\u271d b\nIH : b \u2208 eval f c\u271d\n\u22a2 Sum.inr c\u271d \u2208 Part.some (Option.elim (f a\u271d) (Sum.inl a\u271d) Sum.inr)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase refine'_2.refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nx\u271d\u00b9 : Reaches f a b \u2227 f b = none\nh\u2081 : Reaches f a b\nh\u2082 : f b = none\na\u271d c\u271d : \u03c3\nh : c\u271d \u2208 f a\u271d\nx\u271d : ReflTransGen (fun a b => b \u2208 f a) c\u271d b\nIH : b \u2208 eval f c\u271d\n\u22a2 Sum.inr c\u271d \u2208 Part.some (Option.elim (some c\u271d) (Sum.inl a\u271d) Sum.inr)\n[PROOFSTEP]\napply Part.mem_some\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nh : b \u2208 eval f a\nc : \u03c3\nx\u271d : Reaches\u2081 f b c\nbc : Reaches\u2081 f b c := x\u271d\n\u22a2 False\n[PROOFSTEP]\nlet \u27e8_, b0\u27e9 := mem_eval.1 h\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nh : b \u2208 eval f a\nc : \u03c3\nx\u271d : Reaches\u2081 f b c\nbc : Reaches\u2081 f b c := x\u271d\nleft\u271d : Reaches f a b\nb0 : f b = none\n\u22a2 False\n[PROOFSTEP]\nlet \u27e8b', h', _\u27e9 := TransGen.head'_iff.1 bc\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nh : b \u2208 eval f a\nc : \u03c3\nx\u271d : Reaches\u2081 f b c\nbc : Reaches\u2081 f b c := x\u271d\nleft\u271d : Reaches f a b\nb0 : f b = none\nb' : \u03c3\nh' : b' \u2208 f b\nright\u271d : ReflTransGen (fun a b => b \u2208 f a) b' c\n\u22a2 False\n[PROOFSTEP]\ncases b0.symm.trans h'\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nh : b \u2208 eval f a\nc : \u03c3\nleft\u271d : Reaches f a b\nb0 : f b = none\nb' : \u03c3\nh' : b' \u2208 f b\n\u22a2 False\n[PROOFSTEP]\ncases b0.symm.trans h'\n[GOAL]\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\n\u22a2 eval f a = eval f b\n[PROOFSTEP]\nrefine' Part.ext fun _ \u21a6 \u27e8fun h \u21a6 _, fun h \u21a6 _\u27e9\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\n\u22a2 x\u271d \u2208 eval f b\n[PROOFSTEP]\nhave \u27e8ac, c0\u27e9 := mem_eval.1 h\n[GOAL]\ncase refine'_1\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f a\nac : Reaches f a x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f b\n[PROOFSTEP]\nexact mem_eval.2 \u27e8(or_iff_left_of_imp fun cb \u21a6 (eval_maximal h).1 cb \u25b8 ReflTransGen.refl).1 (reaches_total ab ac), c0\u27e9\n[GOAL]\ncase refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\n\u22a2 x\u271d \u2208 eval f a\n[PROOFSTEP]\nhave \u27e8bc, c0\u27e9 := mem_eval.1 h\n[GOAL]\ncase refine'_2\n\u03c3 : Type u_1\nf : \u03c3 \u2192 Option \u03c3\na b : \u03c3\nab : Reaches f a b\nx\u271d : \u03c3\nh : x\u271d \u2208 eval f b\nbc : Reaches f b x\u271d\nc0 : f x\u271d = none\n\u22a2 x\u271d \u2208 eval f a\n[PROOFSTEP]\nexact mem_eval.2 \u27e8ab.trans bc, c0\u27e9\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab : Reaches\u2081 f\u2081 a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\ninduction' ab with c\u2081 ac c\u2081 d\u2081 _ cd IH\n[GOAL]\ncase single\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 : \u03c3\u2081\nac : c\u2081 \u2208 f\u2081 a\u2081\n\u22a2 \u2203 b\u2082, tr c\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nhave := H aa\n[GOAL]\ncase single\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 : \u03c3\u2081\nac : c\u2081 \u2208 f\u2081 a\u2081\nthis :\n match f\u2081 a\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n | none => f\u2082 a\u2082 = none\n\u22a2 \u2203 b\u2082, tr c\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nrwa [show f\u2081 a\u2081 = _ from ac] at this \n[GOAL]\ncase tail\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 d\u2081 : \u03c3\u2081\na\u271d : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 c\u2081\ncd : d\u2081 \u2208 f\u2081 c\u2081\nIH : \u2203 b\u2082, tr c\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr d\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nrcases IH with \u27e8c\u2082, cc, ac\u2082\u27e9\n[GOAL]\ncase tail.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 d\u2081 : \u03c3\u2081\na\u271d : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 c\u2081\ncd : d\u2081 \u2208 f\u2081 c\u2081\nc\u2082 : \u03c3\u2082\ncc : tr c\u2081 c\u2082\nac\u2082 : Reaches\u2081 f\u2082 a\u2082 c\u2082\n\u22a2 \u2203 b\u2082, tr d\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nhave := H cc\n[GOAL]\ncase tail.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 d\u2081 : \u03c3\u2081\na\u271d : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 c\u2081\ncd : d\u2081 \u2208 f\u2081 c\u2081\nc\u2082 : \u03c3\u2082\ncc : tr c\u2081 c\u2082\nac\u2082 : Reaches\u2081 f\u2082 a\u2082 c\u2082\nthis :\n match f\u2081 c\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none\n\u22a2 \u2203 b\u2082, tr d\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nrw [show f\u2081 c\u2081 = _ from cd] at this \n[GOAL]\ncase tail.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 d\u2081 : \u03c3\u2081\na\u271d : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 c\u2081\ncd : d\u2081 \u2208 f\u2081 c\u2081\nc\u2082 : \u03c3\u2082\ncc : tr c\u2081 c\u2082\nac\u2082 : Reaches\u2081 f\u2082 a\u2082 c\u2082\nthis :\n match some d\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none\n\u22a2 \u2203 b\u2082, tr d\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nrcases this with \u27e8d\u2082, dd, cd\u2082\u27e9\n[GOAL]\ncase tail.intro.intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 c\u2081 d\u2081 : \u03c3\u2081\na\u271d : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 c\u2081\ncd : d\u2081 \u2208 f\u2081 c\u2081\nc\u2082 : \u03c3\u2082\ncc : tr c\u2081 c\u2082\nac\u2082 : Reaches\u2081 f\u2082 a\u2082 c\u2082\nd\u2082 : \u03c3\u2082\ndd : tr d\u2081 d\u2082\ncd\u2082 : Reaches\u2081 f\u2082 c\u2082 d\u2082\n\u22a2 \u2203 b\u2082, tr d\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nexact \u27e8_, dd, ac\u2082.trans cd\u2082\u27e9\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab : Reaches f\u2081 a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nrcases reflTransGen_iff_eq_or_transGen.1 ab with (rfl | ab)\n[GOAL]\ncase inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2082 : \u03c3\u2082\nb\u2081 : \u03c3\u2081\naa : tr b\u2081 a\u2082\nab : Reaches f\u2081 b\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nexact \u27e8_, aa, ReflTransGen.refl\u27e9\n[GOAL]\ncase inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nhave \u27e8b\u2082, bb, h\u27e9 := tr_reaches\u2081 H aa ab\n[GOAL]\ncase inr\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2081 : \u03c3\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nab : TransGen (fun a b => b \u2208 f\u2081 a) a\u2081 b\u2081\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nh : Reaches\u2081 f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches f\u2082 a\u2082 b\u2082\n[PROOFSTEP]\nexact \u27e8b\u2082, bb, h.to_reflTransGen\u27e9\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 : \u03c3\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 b\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\ninduction' ab with c\u2082 d\u2082 _ cd IH\n[GOAL]\ncase refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 : \u03c3\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 a\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nexact \u27e8_, _, ReflTransGen.refl, aa, ReflTransGen.refl\u27e9\n[GOAL]\ncase tail\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\nIH : \u2203 c\u2081 c\u2082_1, Reaches f\u2082 c\u2082 c\u2082_1 \u2227 tr c\u2081 c\u2082_1 \u2227 Reaches f\u2081 a\u2081 c\u2081\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nrcases IH with \u27e8e\u2081, e\u2082, ce, ee, ae\u27e9\n[GOAL]\ncase tail.intro.intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\ne\u2082 : \u03c3\u2082\nce : Reaches f\u2082 c\u2082 e\u2082\nee : tr e\u2081 e\u2082\nae : Reaches f\u2081 a\u2081 e\u2081\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nrcases ReflTransGen.cases_head ce with (rfl | \u27e8d', cd', de\u27e9)\n[GOAL]\ncase tail.intro.intro.intro.intro.inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nhave := H ee\n[GOAL]\ncase tail.intro.intro.intro.intro.inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\nthis :\n match f\u2081 e\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase tail.intro.intro.intro.intro.inl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\n\u22a2 (match f\u2081 e\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none) \u2192\n \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\ncases' eg : f\u2081 e\u2081 with g\u2081\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.none\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\neg : f\u2081 e\u2081 = none\n\u22a2 (match none with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none) \u2192\n \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nsimp only [Respects, and_imp, exists_imp]\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\ng\u2081 : \u03c3\u2081\neg : f\u2081 e\u2081 = some g\u2081\n\u22a2 (match some g\u2081 with\n | some b\u2081 => \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 c\u2082 b\u2082\n | none => f\u2082 c\u2082 = none) \u2192\n \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nsimp only [Respects, and_imp, exists_imp]\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.none\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\neg : f\u2081 e\u2081 = none\n\u22a2 f\u2082 c\u2082 = none \u2192 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nintro c0\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.none\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\neg : f\u2081 e\u2081 = none\nc0 : f\u2082 c\u2082 = none\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\ncases cd.symm.trans c0\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\ng\u2081 : \u03c3\u2081\neg : f\u2081 e\u2081 = some g\u2081\n\u22a2 \u2200 (x : \u03c3\u2082), tr g\u2081 x \u2192 Reaches\u2081 f\u2082 c\u2082 x \u2192 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nintro g\u2082 gg cg\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\ng\u2081 : \u03c3\u2081\neg : f\u2081 e\u2081 = some g\u2081\ng\u2082 : \u03c3\u2082\ngg : tr g\u2081 g\u2082\ncg : Reaches\u2081 f\u2082 c\u2082 g\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nrcases TransGen.head'_iff.1 cg with \u27e8d', cd', dg\u27e9\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.some.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\ng\u2081 : \u03c3\u2081\neg : f\u2081 e\u2081 = some g\u2081\ng\u2082 : \u03c3\u2082\ngg : tr g\u2081 g\u2082\ncg : Reaches\u2081 f\u2082 c\u2082 g\u2082\nd' : \u03c3\u2082\ncd' : d' \u2208 f\u2082 c\u2082\ndg : ReflTransGen (fun a b => b \u2208 f\u2082 a) d' g\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\ncases Option.mem_unique cd cd'\n[GOAL]\ncase tail.intro.intro.intro.intro.inl.some.intro.intro.refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\nae : Reaches f\u2081 a\u2081 e\u2081\nce : Reaches f\u2082 c\u2082 c\u2082\nee : tr e\u2081 c\u2082\ng\u2081 : \u03c3\u2081\neg : f\u2081 e\u2081 = some g\u2081\ng\u2082 : \u03c3\u2082\ngg : tr g\u2081 g\u2082\ncg : Reaches\u2081 f\u2082 c\u2082 g\u2082\ncd' : d\u2082 \u2208 f\u2082 c\u2082\ndg : ReflTransGen (fun a b => b \u2208 f\u2082 a) d\u2082 g\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nexact \u27e8_, _, dg, gg, ae.tail eg\u27e9\n[GOAL]\ncase tail.intro.intro.intro.intro.inr.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\ne\u2082 : \u03c3\u2082\nce : Reaches f\u2082 c\u2082 e\u2082\nee : tr e\u2081 e\u2082\nae : Reaches f\u2081 a\u2081 e\u2081\nd' : \u03c3\u2082\ncd' : d' \u2208 f\u2082 c\u2082\nde : ReflTransGen (fun a b => b \u2208 f\u2082 a) d' e\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\ncases Option.mem_unique cd cd'\n[GOAL]\ncase tail.intro.intro.intro.intro.inr.intro.intro.refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nb\u2082 c\u2082 d\u2082 : \u03c3\u2082\na\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) a\u2082 c\u2082\ncd : d\u2082 \u2208 f\u2082 c\u2082\ne\u2081 : \u03c3\u2081\ne\u2082 : \u03c3\u2082\nce : Reaches f\u2082 c\u2082 e\u2082\nee : tr e\u2081 e\u2082\nae : Reaches f\u2081 a\u2081 e\u2081\ncd' : d\u2082 \u2208 f\u2082 c\u2082\nde : ReflTransGen (fun a b => b \u2208 f\u2082 a) d\u2082 e\u2082\n\u22a2 \u2203 c\u2081 c\u2082, Reaches f\u2082 d\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081\n[PROOFSTEP]\nexact \u27e8_, _, de, ee, ae\u27e9\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab : b\u2081 \u2208 eval f\u2081 a\u2081\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082\n[PROOFSTEP]\ncases' mem_eval.1 ab with ab b0\n[GOAL]\ncase intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2081 \u2208 eval f\u2081 a\u2081\nab : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082\n[PROOFSTEP]\nrcases tr_reaches H aa ab with \u27e8b\u2082, bb, ab\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 \u2203 b\u2082, tr b\u2081 b\u2082 \u2227 b\u2082 \u2208 eval f\u2082 a\u2082\n[PROOFSTEP]\nrefine' \u27e8_, bb, mem_eval.2 \u27e8ab, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\n\u22a2 f\u2082 b\u2082 = none\n[PROOFSTEP]\nhave := H bb\n[GOAL]\ncase intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 b\u2081 : \u03c3\u2081\na\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d\u00b9 : b\u2081 \u2208 eval f\u2081 a\u2081\nab\u271d : Reaches f\u2081 a\u2081 b\u2081\nb0 : f\u2081 b\u2081 = none\nb\u2082 : \u03c3\u2082\nbb : tr b\u2081 b\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nthis :\n match f\u2081 b\u2081 with\n | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n | none => f\u2082 b\u2082 = none\n\u22a2 f\u2082 b\u2082 = none\n[PROOFSTEP]\nrwa [b0] at this \n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab : b\u2082 \u2208 eval f\u2082 a\u2082\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081\n[PROOFSTEP]\ncases' mem_eval.1 ab with ab b0\n[GOAL]\ncase intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081\n[PROOFSTEP]\nrcases tr_reaches_rev H aa ab with \u27e8c\u2081, c\u2082, bc, cc, ac\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nc\u2082 : \u03c3\u2082\nbc : Reaches f\u2082 b\u2082 c\u2082\ncc : tr c\u2081 c\u2082\nac : Reaches f\u2081 a\u2081 c\u2081\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081\n[PROOFSTEP]\ncases (reflTransGen_iff_eq (Option.eq_none_iff_forall_not_mem.1 b0)).1 bc\n[GOAL]\ncase intro.intro.intro.intro.intro.refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081\n[PROOFSTEP]\nrefine' \u27e8_, cc, mem_eval.2 \u27e8ac, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\n\u22a2 f\u2081 c\u2081 = none\n[PROOFSTEP]\nhave := H cc\n[GOAL]\ncase intro.intro.intro.intro.intro.refl\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n match f\u2081 c\u2081 with\n | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n | none => f\u2082 b\u2082 = none\n\u22a2 f\u2081 c\u2081 = none\n[PROOFSTEP]\ncases' hfc : f\u2081 c\u2081 with d\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.refl.none\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n match f\u2081 c\u2081 with\n | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = none\n\u22a2 none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro.intro.refl.some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nthis :\n match f\u2081 c\u2081 with\n | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n | none => f\u2082 b\u2082 = none\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none\n[PROOFSTEP]\nrw [hfc] at this \n[GOAL]\ncase intro.intro.intro.intro.intro.refl.some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nthis :\n match some d\u2081 with\n | some b\u2081 => \u2203 b\u2082_1, tr b\u2081 b\u2082_1 \u2227 Reaches\u2081 f\u2082 b\u2082 b\u2082_1\n | none => f\u2082 b\u2082 = none\nhfc : f\u2081 c\u2081 = some d\u2081\n\u22a2 some d\u2081 = none\n[PROOFSTEP]\nrcases this with \u27e8d\u2082, _, bd\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refl.some.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\n\u22a2 some d\u2081 = none\n[PROOFSTEP]\nrcases TransGen.head'_iff.1 bd with \u27e8e, h, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refl.some.intro.intro.intro.intro\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop\nH : Respects f\u2081 f\u2082 tr\na\u2081 : \u03c3\u2081\nb\u2082 a\u2082 : \u03c3\u2082\naa : tr a\u2081 a\u2082\nab\u271d : b\u2082 \u2208 eval f\u2082 a\u2082\nab : Reaches f\u2082 a\u2082 b\u2082\nb0 : f\u2082 b\u2082 = none\nc\u2081 : \u03c3\u2081\nac : Reaches f\u2081 a\u2081 c\u2081\nbc : Reaches f\u2082 b\u2082 b\u2082\ncc : tr c\u2081 b\u2082\nd\u2081 : \u03c3\u2081\nhfc : f\u2081 c\u2081 = some d\u2081\nd\u2082 : \u03c3\u2082\nleft\u271d : tr d\u2081 d\u2082\nbd : Reaches\u2081 f\u2082 b\u2082 d\u2082\ne : \u03c3\u2082\nh : e \u2208 f\u2082 b\u2082\nright\u271d : ReflTransGen (fun a b => b \u2208 f\u2082 a) e d\u2082\n\u22a2 some d\u2081 = none\n[PROOFSTEP]\ncases b0.symm.trans h\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\na\u2082 b\u2082 : \u03c3\u2082\nh : f\u2082 a\u2082 = f\u2082 b\u2082\n\u22a2 FRespects f\u2082 tr a\u2082 none \u2194 FRespects f\u2082 tr b\u2082 none\n[PROOFSTEP]\nunfold FRespects\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\na\u2082 b\u2082 : \u03c3\u2082\nh : f\u2082 a\u2082 = f\u2082 b\u2082\n\u22a2 (match none with\n | some b\u2081 => Reaches\u2081 f\u2082 a\u2082 (tr b\u2081)\n | none => f\u2082 a\u2082 = none) \u2194\n match none with\n | some b\u2081 => Reaches\u2081 f\u2082 b\u2082 (tr b\u2081)\n | none => f\u2082 b\u2082 = none\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\na\u2081 : \u03c3\u2081\n\u22a2 (\u2200 \u2983a\u2082 : \u03c3\u2082\u2984,\n (fun a b => tr a = b) a\u2081 a\u2082 \u2192\n match f\u2081 a\u2081 with\n | some b\u2081 => \u2203 b\u2082, (fun a b => tr a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n | none => f\u2082 a\u2082 = none) \u2194\n FRespects f\u2082 tr (tr a\u2081) (f\u2081 a\u2081)\n[PROOFSTEP]\ncases f\u2081 a\u2081\n[GOAL]\ncase none\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\na\u2081 : \u03c3\u2081\n\u22a2 (\u2200 \u2983a\u2082 : \u03c3\u2082\u2984,\n (fun a b => tr a = b) a\u2081 a\u2082 \u2192\n match none with\n | some b\u2081 => \u2203 b\u2082, (fun a b => tr a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n | none => f\u2082 a\u2082 = none) \u2194\n FRespects f\u2082 tr (tr a\u2081) none\n[PROOFSTEP]\nsimp only [FRespects, Respects, exists_eq_left', forall_eq']\n[GOAL]\ncase some\n\u03c3\u2081 : Type u_1\n\u03c3\u2082 : Type u_2\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\na\u2081 val\u271d : \u03c3\u2081\n\u22a2 (\u2200 \u2983a\u2082 : \u03c3\u2082\u2984,\n (fun a b => tr a = b) a\u2081 a\u2082 \u2192\n match some val\u271d with\n | some b\u2081 => \u2203 b\u2082, (fun a b => tr a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 f\u2082 a\u2082 b\u2082\n | none => f\u2082 a\u2082 = none) \u2194\n FRespects f\u2082 tr (tr a\u2081) (some val\u271d)\n[PROOFSTEP]\nsimp only [FRespects, Respects, exists_eq_left', forall_eq']\n[GOAL]\n\u03c3\u2081 \u03c3\u2082 : Type u_1\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\nH : Respects f\u2081 f\u2082 fun a b => tr a = b\na\u2081 : \u03c3\u2081\nb\u2082 : \u03c3\u2082\nh : b\u2082 \u2208 tr <$> eval f\u2081 a\u2081\n\u22a2 b\u2082 \u2208 eval f\u2082 (tr a\u2081)\n[PROOFSTEP]\nrcases(Part.mem_map_iff _).1 h with \u27e8b\u2081, ab, bb\u27e9\n[GOAL]\ncase intro.intro\n\u03c3\u2081 \u03c3\u2082 : Type u_1\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\nH : Respects f\u2081 f\u2082 fun a b => tr a = b\na\u2081 : \u03c3\u2081\nb\u2082 : \u03c3\u2082\nh : b\u2082 \u2208 tr <$> eval f\u2081 a\u2081\nb\u2081 : \u03c3\u2081\nab : b\u2081 \u2208 eval f\u2081 a\u2081\nbb : tr b\u2081 = b\u2082\n\u22a2 b\u2082 \u2208 eval f\u2082 (tr a\u2081)\n[PROOFSTEP]\nrcases tr_eval H rfl ab with \u27e8_, rfl, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03c3\u2081 \u03c3\u2082 : Type u_1\nf\u2081 : \u03c3\u2081 \u2192 Option \u03c3\u2081\nf\u2082 : \u03c3\u2082 \u2192 Option \u03c3\u2082\ntr : \u03c3\u2081 \u2192 \u03c3\u2082\nH : Respects f\u2081 f\u2082 fun a b => tr a = b\na\u2081 : \u03c3\u2081\nb\u2082 : \u03c3\u2082\nh\u271d : b\u2082 \u2208 tr <$> eval f\u2081 a\u2081\nb\u2081 : \u03c3\u2081\nab : b\u2081 \u2208 eval f\u2081 a\u2081\nbb : tr b\u2081 = b\u2082\nh : tr b\u2081 \u2208 eval f\u2082 (tr a\u2081)\n\u22a2 b\u2082 \u2208 eval f\u2082 (tr a\u2081)\n[PROOFSTEP]\nrwa [bb] at h \n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\n\u22a2 Inhabited Machine\u2080\n[PROOFSTEP]\nunfold Machine\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\n\u22a2 Inhabited (\u039b \u2192 \u0393 \u2192 Option (\u039b \u00d7 Stmt\u2080))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\nS : Set \u039b\nss : Supports M S\n\u22a2 \u2200 {c c' : Cfg\u2080}, c' \u2208 step M c \u2192 c.q \u2208 S \u2192 c'.q \u2208 S\n[PROOFSTEP]\nintro \u27e8q, T\u27e9 c' h\u2081 h\u2082\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\nS : Set \u039b\nss : Supports M S\nq : \u039b\nT : Tape \u0393\nc' : Cfg\u2080\nh\u2081 : c' \u2208 step M { q := q, Tape := T }\nh\u2082 : { q := q, Tape := T }.q \u2208 S\n\u22a2 c'.q \u2208 S\n[PROOFSTEP]\nrcases Option.map_eq_some'.1 h\u2081 with \u27e8\u27e8q', a\u27e9, h, rfl\u27e9\n[GOAL]\ncase intro.mk.intro\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\nS : Set \u039b\nss : Supports M S\nq : \u039b\nT : Tape \u0393\nh\u2082 : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : Stmt\u2080\nh : M q T.head = some (q', a)\nh\u2081 :\n (fun x =>\n match x with\n | (q', a) =>\n { q := q',\n Tape :=\n match a with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (q', a) \u2208\n step M { q := q, Tape := T }\n\u22a2 ((fun x =>\n match x with\n | (q', a) =>\n { q := q',\n Tape :=\n match a with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (q', a)).q \u2208\n S\n[PROOFSTEP]\nexact ss.2 h h\u2082\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\n\u22a2 Supports M Set.univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\n\u22a2 default \u2208 Set.univ\n[PROOFSTEP]\nintros\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\n\u22a2 \u2200 {q : \u039b} {a : \u0393} {q' : \u039b} {s : Stmt\u2080}, (q', s) \u2208 M q a \u2192 q \u2208 Set.univ \u2192 q' \u2208 Set.univ\n[PROOFSTEP]\nintros\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\n\u22a2 default \u2208 Set.univ\n[PROOFSTEP]\napply Set.mem_univ\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : Machine\u2080\nq\u271d : \u039b\na\u271d\u00b2 : \u0393\nq'\u271d : \u039b\ns\u271d : Stmt\u2080\na\u271d\u00b9 : (q'\u271d, s\u271d) \u2208 M q\u271d a\u271d\u00b2\na\u271d : q\u271d \u2208 Set.univ\n\u22a2 q'\u271d \u2208 Set.univ\n[PROOFSTEP]\napply Set.mem_univ\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (Cfg.map f\u2081 g\u2081) (step M { q := q, Tape := T }) =\n step (map M f\u2081 f\u2082 g\u2081 g\u2082) (Cfg.map f\u2081 g\u2081 { q := q, Tape := T })\n[PROOFSTEP]\nunfold step Machine.map Cfg.map\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map\n (fun x =>\n match x with\n | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T })\n (match { q := q, Tape := T } with\n | { q := q, Tape := T } =>\n Option.map\n (fun x =>\n match x with\n | (q', a) =>\n { q := q',\n Tape :=\n match a with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (M q T.head)) =\n match\n match { q := q, Tape := T } with\n | { q := q, Tape := T } => { q := g\u2081 q, Tape := Tape.map f\u2081 T } with\n | { q := q, Tape := T } =>\n Option.map\n (fun x =>\n match x with\n | (q', a) =>\n { q := q',\n Tape :=\n match a with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (match q, T.head with\n | q, l => Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M (g\u2082 q) (PointedMap.f f\u2082 l)))\n[PROOFSTEP]\nsimp only [Turing.Tape.map_fst, g\u2082\u2081 q h, f\u2082\u2081 _]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n (Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (M q T.head)) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (M q T.head))\n[PROOFSTEP]\nrcases M q T.1 with (_ | \u27e8q', d | a\u27e9)\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n (Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n none) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) none)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n (Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (some (q', Stmt.move d))) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.move d)))\n[PROOFSTEP]\nsimp only [step, Cfg.map, Option.map_some', Tape.map_move f\u2081]\n[GOAL]\ncase some.mk.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\nd : Dir\n\u22a2 some { q := g\u2081 q', Tape := Tape.move d (Tape.map f\u2081 T) } =\n some\n { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).fst,\n Tape :=\n match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.move d)).snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 Option.map (fun x => { q := g\u2081 x.q, Tape := Tape.map f\u2081 x.Tape })\n (Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d T\n | Stmt.write a => Tape.write a T })\n (some (q', Stmt.write a))) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) })\n (Option.map (Prod.map g\u2081 (Stmt.map f\u2081)) (some (q', Stmt.write a)))\n[PROOFSTEP]\nsimp only [step, Cfg.map, Option.map_some', Tape.map_write]\n[GOAL]\ncase some.mk.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081 : \u039b \u2192 \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (g\u2081 q) = q\nq : \u039b\nT : Tape \u0393\nh : { q := q, Tape := T }.q \u2208 S\nq' : \u039b\na : \u0393\n\u22a2 some { q := g\u2081 q', Tape := Tape.write (PointedMap.f f\u2081 a) (Tape.map f\u2081 T) } =\n some\n { q := (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).fst,\n Tape :=\n match (Prod.map g\u2081 (Stmt.map f\u2081) (q', Stmt.write a)).snd with\n | Stmt.move d => Tape.move d (Tape.map f\u2081 T)\n | Stmt.write a => Tape.write a (Tape.map f\u2081 T) }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\n\u22a2 Respects (step M) (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b\n[PROOFSTEP]\nintro c _ \u27e8cs, rfl\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\n\u22a2 match step M c with\n | some b\u2081 =>\n \u2203 b\u2082,\n (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none\n[PROOFSTEP]\ncases e : step M c\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n | some b\u2081 =>\n \u2203 b\u2082,\n (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none\n[PROOFSTEP]\nrw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\ne : step M c = none\n\u22a2 match none with\n | some b\u2081 =>\n \u2203 b\u2082,\n (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n | none => Option.map (Cfg.map f\u2081 g\u2081.f) none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 match some val\u271d with\n | some b\u2081 =>\n \u2203 b\u2082,\n (fun a b => a.q \u2208 S \u2227 Cfg.map f\u2081 g\u2081.f a = b) b\u2081 b\u2082 \u2227 Reaches\u2081 (step (map M f\u2081 f\u2082 g\u2081.f g\u2082)) (Cfg.map f\u2081 g\u2081.f c) b\u2082\n | none => step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c) = none\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8step_supports M ss e cs, rfl\u27e9, TransGen.single _\u27e9\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 step (map M f\u2081 f\u2082 g\u2081.f g\u2082) (Cfg.map f\u2081 g\u2081.f c)\n[PROOFSTEP]\nrw [\u2190 M.map_step f\u2081 f\u2082 g\u2081 g\u2082 f\u2082\u2081 g\u2082\u2081 _ cs, e]\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u0393' : Type u_2\ninst\u271d\u00b2 : Inhabited \u0393'\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u039b' : Type u_4\ninst\u271d : Inhabited \u039b'\nM : Machine \u0393 \u039b\nf\u2081 : PointedMap \u0393 \u0393'\nf\u2082 : PointedMap \u0393' \u0393\ng\u2081\u271d : \u039b \u2192 \u039b'\ng\u2082\u271d : \u039b' \u2192 \u039b\ng\u2081 : PointedMap \u039b \u039b'\ng\u2082 : \u039b' \u2192 \u039b\nS : Set \u039b\nss : Supports M S\nf\u2082\u2081 : Function.RightInverse f\u2081.f f\u2082.f\ng\u2082\u2081 : \u2200 (q : \u039b), q \u2208 S \u2192 g\u2082 (PointedMap.f g\u2081 q) = q\nc : Cfg \u0393 \u039b\na\u2082\u271d : Cfg \u0393' \u039b'\ncs : c.q \u2208 S\nval\u271d : Cfg \u0393 \u039b\ne : step M c = some val\u271d\n\u22a2 Cfg.map f\u2081 g\u2081.f val\u271d \u2208 Option.map (Cfg.map f\u2081 g\u2081.f) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq : Stmt\u2081\n\u22a2 q \u2208 stmts\u2081 q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\n\u22a2 move a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (move a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\n\u22a2 write a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (write a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\n\u22a2 load a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (load a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\n\u22a2 branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\n\u22a2 goto a\u271d \u2208 stmts\u2081 (goto a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\n\u22a2 halt \u2208 stmts\u2081 halt\n[PROOFSTEP]\nsimp only [stmts\u2081, Finset.mem_insert_self, Finset.mem_singleton_self]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\n\u22a2 q\u2081 \u2208 stmts\u2081 q\u2082 \u2192 stmts\u2081 q\u2081 \u2286 stmts\u2081 q\u2082\n[PROOFSTEP]\nintro h\u2081\u2082 q\u2080 h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\n\u22a2 q\u2080 \u2208 stmts\u2081 q\u2082\n[PROOFSTEP]\ninduction' q\u2082 with _ q IH _ q IH _ q IH\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (move a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (move a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (write a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (write a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (load a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (load a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 q\u2080 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (goto a\u271d)\n\u22a2 q\u2080 \u2208 stmts\u2081 (goto a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 halt\n\u22a2 q\u2080 \u2208 stmts\u2081 halt\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 \u2208 {goto a\u271d}\n\u22a2 q\u2080 \u2208 {goto a\u271d}\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 \u2208 {halt}\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h\u2081\u2082 \n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = move a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\niterate 3 \n rcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = move a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase move.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : Dir\nq : Stmt\u2081\nh\u2081\u2082 : move a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (move a\u271d q)\nIH : move a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase move.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : Dir\nq : Stmt\u2081\nh\u2081\u2082 : move a\u271d q \u2208 stmts\u2081 q\u2082\nIH : move a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase move.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (move a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase write.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nh\u2081\u2082 : write a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (write a\u271d q)\nIH : write a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase write.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nh\u2081\u2082 : write a\u271d q \u2208 stmts\u2081 q\u2082\nIH : write a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase write.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (write a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase load.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nh\u2081\u2082 : load a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (load a\u271d q)\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase load.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nh\u2081\u2082 : load a\u271d q \u2208 stmts\u2081 q\u2082\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase load.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n rcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH\u2081 h\u2081\u2082)\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH\u2082 h\u2081\u2082)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d = branch p q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n rcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH\u2081 h\u2081\u2082)\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH\u2082 h\u2081\u2082)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d = branch p q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n[GOAL]\ncase inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082\u271d q\u2080 : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (branch p q\u2081 q\u2082)\nIH\u2081 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082\u271d q\u2080 : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase inr.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH\u2081 h\u2081\u2082)\n[GOAL]\ncase inr.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch p q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH\u2082 h\u2081\u2082)\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase goto l => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto l\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\ncase goto l => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto l\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\nsubst h\u2081\u2082\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2081\u2082 : goto l \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (goto l)\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase halt => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase halt => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2081\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nsubst h\u2081\u2082\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nq\u2082 q\u2080 : Stmt\u2081\nh\u2081\u2082 : halt \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh : q\u2081 \u2208 stmts\u2081 q\u2082\nhs : SupportsStmt S q\u2082\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ninduction' q\u2082 with _ q IH _ q IH _ q IH\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (move a\u271d q)\nhs : SupportsStmt S (move a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (write a\u271d q)\nhs : SupportsStmt S (write a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (load a\u271d q)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nh : q\u2081 \u2208 stmts\u2081 (goto a\u271d)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nh : q\u2081 \u2208 stmts\u2081 halt\nhs : SupportsStmt S halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = move a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\niterate 3 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = move a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = move a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase move.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082 : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nhs : SupportsStmt S q\nh : move a\u271d q \u2208 stmts\u2081 q\u2082\nIH : move a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (move a\u271d q)\n\u22a2 SupportsStmt S (move a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase move.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : Dir\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = write a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase write.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082 : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S q\nh : write a\u271d q \u2208 stmts\u2081 q\u2082\nIH : write a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (write a\u271d q)\n\u22a2 SupportsStmt S (write a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase write.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase load.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082 : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nhs : SupportsStmt S q\nh : load a\u271d q \u2208 stmts\u2081 q\u2082\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (load a\u271d q)\n\u22a2 SupportsStmt S (load a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase load.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 => rcases h with (rfl | h | h); exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d = branch p q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 => rcases h with (rfl | h | h); exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d = branch p q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\nrcases h with (rfl | h | h)\n[GOAL]\ncase inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082\u271d : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S (branch p q\u2081 q\u2082)\nIH\u2082 : branch p q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S (branch p q\u2081 q\u2082)\n\u22a2 SupportsStmt S (branch p q\u2081 q\u2082)\ncase inr.inl\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 SupportsStmt S q\u2081\u271d\ncase inr.inr\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2081\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\nexacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), a\u271d a v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase goto l => subst h; exact hs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), l a v \u2208 S\nh : q\u2081 = goto l\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase goto l => subst h; exact hs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), l a v \u2208 S\nh : q\u2081 = goto l\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082 : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : \u2200 (a : \u0393) (v : \u03c3), l a v \u2208 S\nh : goto l \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S (goto l)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase halt => subst h; trivial\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase halt => subst h; trivial\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nS : Finset \u039b\nq\u2082 : Stmt\u2081\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : halt \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S halt\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 some q\u2082 \u2208 stmts M S \u2192 some q\u2081 \u2208 stmts M S\n[PROOFSTEP]\nsimp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq, forall_eq', exists_imp,\n and_imp]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2081\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q\u2082 \u2208 stmts\u2081 (M x) \u2192 \u2203 a, a \u2208 S \u2227 q\u2081 \u2208 stmts\u2081 (M a)\n[PROOFSTEP]\nexact fun l ls h\u2082 \u21a6 \u27e8_, ls, stmts\u2081_trans h\u2082 h\u2081\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q\n[PROOFSTEP]\nsimp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq, forall_eq', exists_imp,\n and_imp]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q\n[PROOFSTEP]\nexact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\nc' : Cfg\u2081\nh\u2081 : c' \u2208 step M { l := some l\u2081, var := v, Tape := T }\nh\u2082 : { l := some l\u2081, var := v, Tape := T }.l \u2208 \u2191Finset.insertNone S\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nreplace h\u2082 := ss.2 _ (Finset.some_mem_insertNone.1 h\u2082)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\nc' : Cfg\u2081\nh\u2081 : c' \u2208 step M { l := some l\u2081, var := v, Tape := T }\nh\u2082 : SupportsStmt S (M l\u2081)\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nsimp only [step, Option.mem_def, Option.some.injEq] at h\u2081 \n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\nc' : Cfg\u2081\nh\u2082 : SupportsStmt S (M l\u2081)\nh\u2081 : stepAux (M l\u2081) v T = c'\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nsubst c'\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\nh\u2082 : SupportsStmt S (M l\u2081)\n\u22a2 (stepAux (M l\u2081) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nrevert h\u2082\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (M l\u2081) \u2192 (stepAux (M l\u2081) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ninduction' M l\u2081 with _ q IH _ q IH _ q IH generalizing v T\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : Dir\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (move a\u271d q) \u2192 (stepAux (move a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (write a\u271d q) \u2192 (stepAux (write a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (load a\u271d q) \u2192 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S (goto a\u271d) \u2192 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\n\u22a2 SupportsStmt S halt \u2192 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : Dir\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (move a\u271d q)\n\u22a2 (stepAux (move a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (write a\u271d q)\n\u22a2 (stepAux (write a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\niterate 3 exact IH _ _ hs\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : Dir\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (move a\u271d q)\n\u22a2 (stepAux (move a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (write a\u271d q)\n\u22a2 (stepAux (write a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (write a\u271d q)\n\u22a2 (stepAux (write a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase branch p q\u2081' q\u2082' IH\u2081 IH\u2082 =>\n unfold stepAux; cases p T.1 v\n \u00b7 exact IH\u2082 _ _ hs.2\n \u00b7 exact IH\u2081 _ _ hs.1\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2081\nIH\u2081 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (stepAux (branch p q\u2081' q\u2082') v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase branch p q\u2081' q\u2082' IH\u2081 IH\u2082 =>\n unfold stepAux; cases p T.1 v\n \u00b7 exact IH\u2082 _ _ hs.2\n \u00b7 exact IH\u2081 _ _ hs.1\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2081\nIH\u2081 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (stepAux (branch p q\u2081' q\u2082') v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nunfold stepAux\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2081\nIH\u2081 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif p T.head v then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncases p T.1 v\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2081\nIH\u2081 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif false then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH\u2082 _ _ hs.2\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2081\nIH\u2081 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : Tape \u0393), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif true then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH\u2081 _ _ hs.1\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase goto => exact Finset.some_mem_insertNone.2 (hs _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase goto => exact Finset.some_mem_insertNone.2 (hs _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact Finset.some_mem_insertNone.2 (hs _ _)\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase halt => apply Multiset.mem_cons_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase halt => apply Multiset.mem_cons_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nv : \u03c3\nT : Tape \u0393\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\napply Multiset.mem_cons_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := l\u2081, var := v, Tape := T })\n (TM1.step M { l := l\u2081, var := v, Tape := T })\n[PROOFSTEP]\ncases' l\u2081 with l\u2081\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := none, var := v, Tape := T })\n (TM1.step M { l := none, var := v, Tape := T })\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (TM0.step (tr M)) (fun c\u2081 => trCfg M c\u2081) (trCfg M { l := some l\u2081, var := v, Tape := T })\n (TM1.step M { l := some l\u2081, var := v, Tape := T })\n[PROOFSTEP]\nsimp only [trCfg, TM1.step, FRespects, Option.map]\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv : \u03c3\nT : Tape \u0393\nl\u2081 : \u039b\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (M l\u2081), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (M l\u2081) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (M l\u2081) v T).var),\n Tape := (TM1.stepAux (M l\u2081) v T).Tape }\n[PROOFSTEP]\ninduction' M l\u2081 with _ q IH _ q IH _ q IH generalizing v T\n[GOAL]\ncase some.move\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : Dir\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.move a\u271d q) v T).Tape }\ncase some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\ncase some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\ncase some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d\u00b9 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d\u00b9 v T).var),\n Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d v T).var),\n Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\ncase move d q IH => exact TransGen.head rfl (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nd : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : Dir\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).Tape }\n[PROOFSTEP]\ncase move d q IH => exact TransGen.head rfl (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nd : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : Dir\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.move IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.move IH\u271d q) v T).Tape }\n[PROOFSTEP]\nexact TransGen.head rfl (IH _ _)\n[GOAL]\ncase some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.write a\u271d q) v T).Tape }\ncase some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\ncase some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d\u00b9 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d\u00b9 v T).var),\n Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d v T).var),\n Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\ncase write a q IH => exact TransGen.head rfl (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).Tape }\n[PROOFSTEP]\ncase write a q IH => exact TransGen.head rfl (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.write IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.write IH\u271d q) v T).Tape }\n[PROOFSTEP]\nexact TransGen.head rfl (IH _ _)\n[GOAL]\ncase some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load a\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.load a\u271d q) v T).Tape }\ncase some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d\u00b9 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d\u00b9 v T).var),\n Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d v T).var),\n Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\ncase load a q IH => exact (reaches\u2081_eq (by rfl)).2 (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).Tape }\n[PROOFSTEP]\ncase load a q IH => exact (reaches\u2081_eq (by rfl)).2 (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.load IH\u271d q) v T).Tape }\n[PROOFSTEP]\nexact (reaches\u2081_eq (by rfl)).2 (IH _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\na : \u03c3\nq\u271d : Tape \u0393\nl\u2081 : \u039b\nIH\u271d : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q v T).var),\n Tape := (TM1.stepAux q v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.load IH\u271d q), v), Tape := T } =\n TM0.step (tr M) { q := (some q, IH\u271d T.head v), Tape := T }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d\u00b9, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d\u00b9 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d\u00b9 v T).var),\n Tape := (TM1.stepAux a\u271d\u00b9 v T).Tape }\na_ih\u271d :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some a\u271d, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux a\u271d v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux a\u271d v T).var),\n Tape := (TM1.stepAux a\u271d v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).Tape }\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold TM1.stepAux; cases e : p T.1 v\n \u00b7 exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2082 _ _)\n \u00b7 exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2081 _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).Tape }\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold TM1.stepAux; cases e : p T.1 v\n \u00b7 exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2082 _ _)\n \u00b7 exact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2081 _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.branch p q\u2081 q\u2082) v T).Tape }\n[PROOFSTEP]\nunfold TM1.stepAux\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n {\n q :=\n (match (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n Tape := (bif p T.head v then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }\n[PROOFSTEP]\ncases e : p T.1 v\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n {\n q :=\n (match (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n Tape := (bif false then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }\n[PROOFSTEP]\nexact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2082 _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =\n TM0.step (tr M) { q := (some q\u2082, v), Tape := T }\n[PROOFSTEP]\nsimp only [TM0.step, tr, trAux, e]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = false\n\u22a2 Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | move d => Tape.move d T\n | write a => Tape.write a T })\n (some (bif false then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | move d => Tape.move d T\n | write a => Tape.write a T })\n (some (trAux M T.head q\u2082 v))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T }\n {\n q :=\n (match (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).var),\n Tape := (bif true then TM1.stepAux q\u2081 v T else TM1.stepAux q\u2082 v T).Tape }\n[PROOFSTEP]\nexact (reaches\u2081_eq (by simp only [TM0.step, tr, trAux, e]; rfl)).2 (IH\u2081 _ _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 TM0.step (tr M) { q := (some (TM1.Stmt.branch p q\u2081 q\u2082), v), Tape := T } =\n TM0.step (tr M) { q := (some q\u2081, v), Tape := T }\n[PROOFSTEP]\nsimp only [TM0.step, tr, trAux, e]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2081, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2081 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2081 v T).var),\n Tape := (TM1.stepAux q\u2081 v T).Tape }\nIH\u2082 :\n \u2200 (v : \u03c3) (T : Tape \u0393),\n Reaches\u2081 (TM0.step (tr M)) { q := (some q\u2082, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux q\u2082 v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux q\u2082 v T).var),\n Tape := (TM1.stepAux q\u2082 v T).Tape }\nv : \u03c3\nT : Tape \u0393\ne : p T.head v = true\n\u22a2 Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | move d => Tape.move d T\n | write a => Tape.write a T })\n (some (bif true then trAux M T.head q\u2081 v else trAux M T.head q\u2082 v)) =\n Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | move d => Tape.move d T\n | write a => Tape.write a T })\n (some (trAux M T.head q\u2081 v))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\niterate 2 exact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T)))\n[GOAL]\ncase some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some (TM1.Stmt.goto a\u271d), v), Tape := T }\n {\n q :=\n (match (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).var),\n Tape := (TM1.stepAux (TM1.Stmt.goto a\u271d) v T).Tape }\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\nexact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T)))\n[GOAL]\ncase some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nT\u271d : Tape \u0393\nl\u2081 : \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 Reaches\u2081 (TM0.step (tr M)) { q := (some TM1.Stmt.halt, v), Tape := T }\n {\n q :=\n (match (TM1.stepAux TM1.Stmt.halt v T).l with\n | some x => some (M x)\n | none => none,\n (TM1.stepAux TM1.Stmt.halt v T).var),\n Tape := (TM1.stepAux TM1.Stmt.halt v T).Tape }\n[PROOFSTEP]\nexact TransGen.single (congr_arg some (congr (congr_arg TM0.Cfg.mk rfl) (Tape.write_self T)))\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nl : List \u0393\n\u22a2 Part.map (fun c => Tape.right\u2080 c.Tape)\n ((fun a => trCfg M a) <$> eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =\n TM1.eval M l\n[PROOFSTEP]\nrw [Part.map_eq_map, Part.map_map, TM1.eval]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\nl : List \u0393\n\u22a2 Part.map ((fun c => Tape.right\u2080 c.Tape) \u2218 fun a => trCfg M a)\n (eval (TM1.step M) { l := some default, var := default, Tape := Tape.mk\u2081 l }) =\n Part.map (fun c => Tape.right\u2080 c.Tape) (eval (TM1.step M) (TM1.init l))\n[PROOFSTEP]\ncongr with \u27e8\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 TM0.Supports (tr M) \u2191(trStmts M S)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\napply Finset.mem_product.2\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.fst \u2208 TM1.stmts M S \u2227 default.snd \u2208 Finset.univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.fst \u2208 TM1.stmts M S\n[PROOFSTEP]\nsimp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj, forall_eq', Finset.mem_biUnion]\n[GOAL]\ncase left.left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2203 a, a \u2208 S \u2227 M default \u2208 TM1.stmts\u2081 (M a)\n[PROOFSTEP]\nexact \u27e8_, ss.1, TM1.stmts\u2081_self\u27e9\n[GOAL]\ncase left.right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 default.snd \u2208 Finset.univ\n[PROOFSTEP]\napply Finset.mem_univ\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\n\u22a2 \u2200 {q : \u039b'\u2081\u2080} {a : \u0393} {q' : \u039b'\u2081\u2080} {s : Stmt\u2080}, (q', s) \u2208 tr M q a \u2192 q \u2208 \u2191(trStmts M S) \u2192 q' \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\nintro q a q' s h\u2081 h\u2082\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\nq : \u039b'\u2081\u2080\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nh\u2081 : (q', s) \u2208 tr M q a\nh\u2082 : q \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\nrcases q with \u27e8_ | q, v\u27e9\n[GOAL]\ncase right.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nh\u2081 : (q', s) \u2208 tr M (none, v) a\nh\u2082 : (none, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase right.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nq' : \u039b'\u2081\u2080\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2081 : (q', s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\n\u22a2 q' \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\ncases' q' with q' v'\n[GOAL]\ncase right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nh\u2082 : (some q, v) \u2208 \u2191(trStmts M S)\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\n\u22a2 (q', v') \u2208 \u2191(trStmts M S)\n[PROOFSTEP]\nsimp only [trStmts, Finset.mem_coe] at h\u2082 \u22a2\n[GOAL]\ncase right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v) \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ\n\u22a2 (q', v') \u2208 TM1.stmts M S \u00d7\u02e2 Finset.univ\n[PROOFSTEP]\nrw [Finset.mem_product] at h\u2082 \u22a2\n[GOAL]\ncase right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : (some q, v).fst \u2208 TM1.stmts M S \u2227 (some q, v).snd \u2208 Finset.univ\n\u22a2 (q', v').fst \u2208 TM1.stmts M S \u2227 (q', v').snd \u2208 Finset.univ\n[PROOFSTEP]\nsimp only [Finset.mem_univ, and_true_iff] at h\u2082 \u22a2\n[GOAL]\ncase right.mk.some.mk\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nq' : Option Stmt\u2081\nv' : \u03c3\nh\u2081 : ((q', v'), s) \u2208 tr M (some q, v) a\nh\u2082 : some q \u2208 TM1.stmts M S\n\u22a2 q' \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases q'\n[GOAL]\ncase right.mk.some.mk.none\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nh\u2081 : ((none, v'), s) \u2208 tr M (some q, v) a\n\u22a2 none \u2208 TM1.stmts M S\n[PROOFSTEP]\nexact Multiset.mem_cons_self _ _\n[GOAL]\ncase right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : ((some val\u271d, v'), s) \u2208 tr M (some q, v) a\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nsimp only [tr, Option.mem_def] at h\u2081 \n[GOAL]\ncase right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nhave := TM1.stmts_supportsStmt ss h\u2082\n[GOAL]\ncase right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\nthis : TM1.SupportsStmt S q\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase right.mk.some.mk.some\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv : \u03c3\nq : Stmt\u2081\nv' : \u03c3\nh\u2082 : some q \u2208 TM1.stmts M S\nval\u271d : Stmt\u2081\nh\u2081 : some (trAux M a q v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ninduction q generalizing v\n[GOAL]\ncase right.mk.some.mk.some.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3),\n some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S (TM1.Stmt.goto a\u271d) \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\n\u22a2 TM1.SupportsStmt S TM1.Stmt.halt \u2192 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase right.mk.some.mk.some.move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3),\n some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase move d q =>\n cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nd : Stmt\u2081\nq :\n \u2200 (v : \u03c3),\n some d \u2208 TM1.stmts M S \u2192\n some (trAux M a d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase move d q =>\n cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nd : Stmt\u2081\nq :\n \u2200 (v : \u03c3),\n some d \u2208 TM1.stmts M S \u2192\n some (trAux M a d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.move a\u271d d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.move a\u271d d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrefine' TM1.stmts_trans _ h\u2082\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.move a\u271d val\u271d)\n[PROOFSTEP]\nunfold TM1.stmts\u2081\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : Dir\nh\u2082 : some (TM1.Stmt.move a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.move a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), move a\u271d) \u2192 TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.move a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\ncase right.mk.some.mk.some.write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3),\n some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase write b q =>\n cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nb : Stmt\u2081\nq :\n \u2200 (v : \u03c3),\n some b \u2208 TM1.stmts M S \u2192\n some (trAux M a b v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S b \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d b) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d b) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d b)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase write b q =>\n cases h\u2081; refine' TM1.stmts_trans _ h\u2082\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nb : Stmt\u2081\nq :\n \u2200 (v : \u03c3),\n some b \u2208 TM1.stmts M S \u2192\n some (trAux M a b v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S b \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.write a\u271d b) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.write a\u271d b) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d b)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrefine' TM1.stmts_trans _ h\u2082\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 TM1.stmts\u2081 (TM1.Stmt.write a\u271d val\u271d)\n[PROOFSTEP]\nunfold TM1.stmts\u2081\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u0393\nh\u2082 : some (TM1.Stmt.write a\u271d val\u271d) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.write a\u271d val\u271d)\nq :\n \u2200 (v : \u03c3),\n some val\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a val\u271d v) = some ((some val\u271d, v'), write (a\u271d a v')) \u2192\n TM1.SupportsStmt S val\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\n\u22a2 val\u271d \u2208 insert (TM1.Stmt.write a\u271d val\u271d) (TM1.stmts\u2081 val\u271d)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\ncase right.mk.some.mk.some.load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3),\n some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase load b q IH =>\n refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3),\n some q \u2208 TM1.stmts M S \u2192\n some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase load b q IH =>\n refine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3),\n some q \u2208 TM1.stmts M S \u2192\n some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrefine' IH _ (TM1.stmts_trans _ h\u2082) h\u2081 hs\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3),\n some q \u2208 TM1.stmts M S \u2192\n some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 TM1.stmts\u2081 (TM1.Stmt.load b q)\n[PROOFSTEP]\nunfold TM1.stmts\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nb : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3),\n some q \u2208 TM1.stmts M S \u2192\n some (trAux M a q v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.load b q) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.load b q)\n\u22a2 q \u2208 insert (TM1.Stmt.load b q) (TM1.stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem TM1.stmts\u2081_self\n[GOAL]\ncase right.mk.some.mk.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3),\n some a\u271d\u00b9 \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d\u00b9 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d\u00b9 \u2192 some val\u271d \u2208 TM1.stmts M S\na_ih\u271d :\n \u2200 (v : \u03c3),\n some a\u271d \u2208 TM1.stmts M S \u2192\n some (trAux M a a\u271d v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S a\u271d \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n cases h : p a v <;> rw [trAux, h] at h\u2081 \n \u00b7 refine' IH\u2082 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.2\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts\u2081_self)\n \u00b7 refine' IH\u2081 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.1\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts\u2081_self)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n cases h : p a v <;> rw [trAux, h] at h\u2081 \n \u00b7 refine' IH\u2082 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.2\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts\u2081_self)\n \u00b7 refine' IH\u2081 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.1\n unfold TM1.stmts\u2081\n exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts\u2081_self)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases h : p a v\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrw [trAux, h] at h\u2081 \n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.branch p q\u2081 q\u2082) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrw [trAux, h] at h\u2081 \n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrefine' IH\u2082 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.2\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)\n[PROOFSTEP]\nunfold TM1.stmts\u2081\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif false then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = false\n\u22a2 q\u2082 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts\u2081_self)\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\nrefine' IH\u2081 _ (TM1.stmts_trans _ h\u2082) h\u2081 hs.1\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 TM1.stmts\u2081 (TM1.Stmt.branch p q\u2081 q\u2082)\n[PROOFSTEP]\nunfold TM1.stmts\u2081\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3),\n some q\u2081 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2081 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2081 \u2192 some val\u271d \u2208 TM1.stmts M S\nIH\u2082 :\n \u2200 (v : \u03c3),\n some q\u2082 \u2208 TM1.stmts M S \u2192\n some (trAux M a q\u2082 v) = some ((some val\u271d, v'), s) \u2192 TM1.SupportsStmt S q\u2082 \u2192 some val\u271d \u2208 TM1.stmts M S\nv : \u03c3\nh\u2082 : some (TM1.Stmt.branch p q\u2081 q\u2082) \u2208 TM1.stmts M S\nh\u2081 : some (bif true then trAux M a q\u2081 v else trAux M a q\u2082 v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.branch p q\u2081 q\u2082)\nh : p a v = true\n\u22a2 q\u2081 \u2208 insert (TM1.Stmt.branch p q\u2081 q\u2082) (TM1.stmts\u2081 q\u2081 \u222a TM1.stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts\u2081_self)\n[GOAL]\ncase right.mk.some.mk.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto a\u271d) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto a\u271d) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto a\u271d)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase goto l =>\n cases h\u2081\n exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto l) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase goto l =>\n cases h\u2081\n exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a (TM1.Stmt.goto l) v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\nv' : \u03c3\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nh\u2082 : some (TM1.Stmt.goto l) \u2208 TM1.stmts M S\nhs : TM1.SupportsStmt S (TM1.Stmt.goto l)\n\u22a2 some (M (l a v')) \u2208 TM1.stmts M S\n[PROOFSTEP]\nexact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 \u27e8_, hs _ _, TM1.stmts\u2081_self\u27e9)\n[GOAL]\ncase right.mk.some.mk.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase halt => cases h\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncase halt => cases h\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2081\ninst\u271d : Fintype \u03c3\nS : Finset \u039b\nss : TM1.Supports M S\na : \u0393\ns : Stmt\u2080\nv' : \u03c3\nval\u271d : Stmt\u2081\nv : \u03c3\nh\u2082 : some TM1.Stmt.halt \u2208 TM1.stmts M S\nh\u2081 : some (trAux M a TM1.Stmt.halt v) = some ((some val\u271d, v'), s)\nhs : TM1.SupportsStmt S TM1.Stmt.halt\n\u22a2 some val\u271d \u2208 TM1.stmts M S\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nletI := Classical.decEq \u0393\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nlet n := Fintype.card \u0393\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nobtain \u27e8F\u27e9 := Fintype.truncEquivFin \u0393\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nlet G : Fin n \u21aa Fin n \u2192 Bool :=\n \u27e8fun a b \u21a6 a = b, fun a b h \u21a6 Bool.of_decide_true <| (congr_fun h b).trans <| Bool.decide_true rfl\u27e9\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n { toFun := fun a b => decide (a = b),\n inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nlet H := (F.toEmbedding.trans G).trans (Equiv.vectorEquivFin _ _).symm.toEmbedding\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n { toFun := fun a b => decide (a = b),\n inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nclassical\nlet enc := H.setValue default (Vector.replicate n false)\nexact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n { toFun := fun a b => decide (a = b),\n inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nlet enc := H.setValue default (Vector.replicate n false)\n[GOAL]\ncase mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Fintype \u0393\nthis : DecidableEq \u0393 := Classical.decEq \u0393\nn : \u2115 := Fintype.card \u0393\nx\u271d : Trunc (\u0393 \u2243 Fin (Fintype.card \u0393))\nF : \u0393 \u2243 Fin (Fintype.card \u0393)\nG : Fin n \u21aa Fin n \u2192 Bool :=\n { toFun := fun a b => decide (a = b),\n inj' := (_ : \u2200 (a b : Fin n), (fun a b => decide (a = b)) a = (fun a b => decide (a = b)) b \u2192 a = b) }\nH : \u0393 \u21aa Vector Bool n :=\n Function.Embedding.trans (Function.Embedding.trans (Equiv.toEmbedding F) G)\n (Equiv.toEmbedding (Equiv.vectorEquivFin Bool n).symm)\nenc : \u0393 \u21aa Vector Bool n := Function.Embedding.setValue H default (Vector.replicate n false)\n\u22a2 \u2203 n enc dec, enc default = Vector.replicate n false \u2227 \u2200 (a : \u0393), dec (enc a) = a\n[PROOFSTEP]\nexact \u27e8_, enc, Function.invFun enc, H.setValue_eq _ _, Function.leftInverse_invFun enc.2\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT : Tape Bool\n\u22a2 stepAux (move d q) v T = stepAux q v ((Tape.move d)^[n] T)\n[PROOFSTEP]\nsuffices : \u2200 i, stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT : Tape Bool\nthis : \u2200 (i : \u2115), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\n\u22a2 stepAux (move d q) v T = stepAux q v ((Tape.move d)^[n] T)\ncase this\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT : Tape Bool\n\u22a2 \u2200 (i : \u2115), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\n[PROOFSTEP]\nexact this n\n[GOAL]\ncase this\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT : Tape Bool\n\u22a2 \u2200 (i : \u2115), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase this\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT : Tape Bool\ni : \u2115\n\u22a2 stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\n[PROOFSTEP]\ninduction' i with i IH generalizing T\n[GOAL]\ncase this.zero\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT\u271d T : Tape Bool\n\u22a2 stepAux ((Stmt.move d)^[Nat.zero] q) v T = stepAux q v ((Tape.move d)^[Nat.zero] T)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase this.succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT\u271d : Tape Bool\ni : \u2115\nIH : \u2200 (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\nT : Tape Bool\n\u22a2 stepAux ((Stmt.move d)^[Nat.succ i] q) v T = stepAux q v ((Tape.move d)^[Nat.succ i] T)\n[PROOFSTEP]\nrw [iterate_succ', iterate_succ]\n[GOAL]\ncase this.succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT\u271d : Tape Bool\ni : \u2115\nIH : \u2200 (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\nT : Tape Bool\n\u22a2 stepAux ((Stmt.move d \u2218 (Stmt.move d)^[i]) q) v T = stepAux q v (((Tape.move d)^[i] \u2218 Tape.move d) T)\n[PROOFSTEP]\nsimp only [stepAux, Function.comp_apply]\n[GOAL]\ncase this.succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nd : Dir\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nT\u271d : Tape Bool\ni : \u2115\nIH : \u2200 (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T)\nT : Tape Bool\n\u22a2 stepAux ((Stmt.move d)^[i] q) v (Tape.move d T) = stepAux q v ((Tape.move d)^[i] (Tape.move d T))\n[PROOFSTEP]\nrw [IH]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S (move d q) = SupportsStmt S q\n[PROOFSTEP]\nsuffices \u2200 {i}, SupportsStmt S ((Stmt.move d)^[i] q) = _ from this\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nq : Stmt Bool \u039b' \u03c3\n\u22a2 \u2200 {i : \u2115}, SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q\n[PROOFSTEP]\nintro i\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nq : Stmt Bool \u039b' \u03c3\ni : \u2115\n\u22a2 SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q\n[PROOFSTEP]\ninduction i generalizing q\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S ((Stmt.move d)^[Nat.zero] q) = SupportsStmt S q\n[PROOFSTEP]\nsimp only [*, iterate]\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nn\u271d : \u2115\nn_ih\u271d : \u2200 {q : Stmt Bool \u039b' \u03c3}, SupportsStmt S ((Stmt.move d)^[n\u271d] q) = SupportsStmt S q\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S ((Stmt.move d)^[Nat.succ n\u271d] q) = SupportsStmt S q\n[PROOFSTEP]\nsimp only [*, iterate]\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nd : Dir\nn\u271d : \u2115\nn_ih\u271d : \u2200 {q : Stmt Bool \u039b' \u03c3}, SupportsStmt S ((Stmt.move d)^[n\u271d] q) = SupportsStmt S q\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S (Stmt.move d q) = SupportsStmt S q\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nl : List Bool\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S (write l q) = SupportsStmt S q\n[PROOFSTEP]\ninduction' l with _ l IH\n[GOAL]\ncase nil\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nq : Stmt Bool \u039b' \u03c3\n\u22a2 SupportsStmt S (write [] q) = SupportsStmt S q\n[PROOFSTEP]\nsimp only [write, SupportsStmt, *]\n[GOAL]\ncase cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nq : Stmt Bool \u039b' \u03c3\nhead\u271d : Bool\nl : List Bool\nIH : SupportsStmt S (write l q) = SupportsStmt S q\n\u22a2 SupportsStmt S (write (head\u271d :: l) q) = SupportsStmt S q\n[PROOFSTEP]\nsimp only [write, SupportsStmt, *]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni : \u2115\nf : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool i), SupportsStmt S (f v)\n\u22a2 SupportsStmt S (readAux i f)\n[PROOFSTEP]\ninduction' i with i IH\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni : \u2115\nf\u271d : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i), SupportsStmt S (f\u271d v)\nf : Vector Bool Nat.zero \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool Nat.zero), SupportsStmt S (f v)\n\u22a2 SupportsStmt S (readAux Nat.zero f)\n[PROOFSTEP]\nexact hf _\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\n\u22a2 SupportsStmt S (readAux (Nat.succ i) f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.left\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\n\u22a2 SupportsStmt S (Stmt.move Dir.right (readAux i fun v => f (true ::\u1d65 v)))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase succ.right\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\n\u22a2 SupportsStmt S (Stmt.move Dir.right (readAux i fun v => f (false ::\u1d65 v)))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase succ.left.hf\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\n\u22a2 \u2200 (v : Vector Bool i), SupportsStmt S (f (true ::\u1d65 v))\n[PROOFSTEP]\nintro\n[GOAL]\ncase succ.right.hf\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\n\u22a2 \u2200 (v : Vector Bool i), SupportsStmt S (f (false ::\u1d65 v))\n[PROOFSTEP]\nintro\n[GOAL]\ncase succ.left.hf\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\nv\u271d : Vector Bool i\n\u22a2 SupportsStmt S (f (true ::\u1d65 v\u271d))\n[PROOFSTEP]\napply hf\n[GOAL]\ncase succ.right.hf\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d\u00b9 : \u0393 \u2192 Stmt Bool \u039b' \u03c3\ni\u271d : \u2115\nf\u271d : Vector Bool i\u271d \u2192 Stmt Bool \u039b' \u03c3\nhf\u271d : \u2200 (v : Vector Bool i\u271d), SupportsStmt S (f\u271d v)\ni : \u2115\nIH :\n \u2200 (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3), (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nf : Vector Bool (Nat.succ i) \u2192 Stmt Bool \u039b' \u03c3\nhf : \u2200 (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v)\nv\u271d : Vector Bool i\n\u22a2 SupportsStmt S (f (false ::\u1d65 v\u271d))\n[PROOFSTEP]\napply hf\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nthis :\n \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3),\n (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nhf : \u2200 (a : \u0393), SupportsStmt S (f\u271d a)\n\u22a2 \u2200 (v : Vector Bool n), SupportsStmt S (move Dir.left (f\u271d (dec v)))\n[PROOFSTEP]\nintro\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nS : Finset \u039b'\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nthis :\n \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3),\n (\u2200 (v : Vector Bool i), SupportsStmt S (f v)) \u2192 SupportsStmt S (readAux i f)\nhf : \u2200 (a : \u0393), SupportsStmt S (f\u271d a)\nv\u271d : Vector Bool n\n\u22a2 SupportsStmt S (move Dir.left (f\u271d (dec v\u271d)))\n[PROOFSTEP]\nsimp only [supportsStmt_move, hf]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nL R : ListBlank \u0393\n\u22a2 Tape Bool\n[PROOFSTEP]\nrefine' Tape.mk' (L.bind (fun x \u21a6 (enc x).toList.reverse) \u27e8n, _\u27e9) (R.bind (fun x \u21a6 (enc x).toList) \u27e8n, _\u27e9)\n[GOAL]\ncase refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nL R : ListBlank \u0393\n\u22a2 (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n default\n[PROOFSTEP]\nsimp only [enc0, Vector.replicate, List.reverse_replicate, Bool.default_bool, Vector.toList_mk]\n[GOAL]\ncase refine'_2\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nL R : ListBlank \u0393\n\u22a2 (fun x => Vector.toList (enc x)) default = List.replicate n default\n[PROOFSTEP]\nsimp only [enc0, Vector.replicate, List.reverse_replicate, Bool.default_bool, Vector.toList_mk]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nL R : ListBlank \u0393\n\u22a2 trTape enc0 (Tape.mk' L R) = trTape' enc0 L R\n[PROOFSTEP]\nsimp only [trTape, Tape.mk'_left, Tape.mk'_right\u2080]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\n\u22a2 (Tape.move Dir.left)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R)\n[PROOFSTEP]\nobtain \u27e8a, L, rfl\u27e9 := L.exists_cons\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\n\u22a2 (Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons a L) R) =\n trTape' enc0 (ListBlank.tail (ListBlank.cons a L)) (ListBlank.cons (ListBlank.head (ListBlank.cons a L)) R)\n[PROOFSTEP]\nsimp only [trTape', ListBlank.cons_bind, ListBlank.head_cons, ListBlank.tail_cons]\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\n\u22a2 (Tape.move Dir.left)^[n]\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) =\n Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nsuffices\n \u2200 {L' R' l\u2081 l\u2082} (_ : Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082),\n (Tape.move Dir.left)^[l\u2081.length] (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R')\n by simpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nthis :\n \u2200 {L' R' : ListBlank Bool} {l\u2081 l\u2082 : List Bool},\n Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082 \u2192\n (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R')\n\u22a2 (Tape.move Dir.left)^[n]\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) =\n Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nsimpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\n\u22a2 \u2200 {L' R' : ListBlank Bool} {l\u2081 l\u2082 : List Bool},\n Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082 \u2192\n (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R')\n[PROOFSTEP]\nintro _ _ l\u2081 l\u2082 e\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081 l\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082\n\u22a2 (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\n[PROOFSTEP]\ninduction' l\u2081 with b l\u2081 IH generalizing l\u2082\n[GOAL]\ncase intro.intro.nil\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081 l\u2082\u271d : List Bool\ne\u271d : Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082\u271d\nl\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux [] l\u2082\n\u22a2 (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\n[PROOFSTEP]\ncases e\n[GOAL]\ncase intro.intro.nil.refl\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081 l\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082\n\u22a2 (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'\u271d) (ListBlank.append (enc a).1 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\ne\u271d : Vector.toList (enc a) = List.reverseAux l\u2081\u271d l\u2082\u271d\nb : Bool\nl\u2081 : List Bool\nIH :\n \u2200 {l\u2082 : List Bool},\n Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082 \u2192\n (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\nl\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux (b :: l\u2081) l\u2082\n\u22a2 (Tape.move Dir.left)^[List.length (b :: l\u2081)] (Tape.mk' (ListBlank.append (b :: l\u2081) L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\n[PROOFSTEP]\nsimp only [List.length, List.cons_append, iterate_succ_apply]\n[GOAL]\ncase intro.intro.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\ne\u271d : Vector.toList (enc a) = List.reverseAux l\u2081\u271d l\u2082\u271d\nb : Bool\nl\u2081 : List Bool\nIH :\n \u2200 {l\u2082 : List Bool},\n Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082 \u2192\n (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\nl\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux (b :: l\u2081) l\u2082\n\u22a2 (Tape.move Dir.left)^[List.length l\u2081]\n (Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l\u2081) L'\u271d) (ListBlank.append l\u2082 R'\u271d))) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\n[PROOFSTEP]\nconvert IH e\n[GOAL]\ncase h.e'_2.h.e'_4\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nR : ListBlank \u0393\na : \u0393\nL : ListBlank \u0393\nL'\u271d R'\u271d : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\ne\u271d : Vector.toList (enc a) = List.reverseAux l\u2081\u271d l\u2082\u271d\nb : Bool\nl\u2081 : List Bool\nIH :\n \u2200 {l\u2082 : List Bool},\n Vector.toList (enc a) = List.reverseAux l\u2081 l\u2082 \u2192\n (Tape.move Dir.left)^[List.length l\u2081] (Tape.mk' (ListBlank.append l\u2081 L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' L'\u271d (ListBlank.append (Vector.toList (enc a)) R'\u271d)\nl\u2082 : List Bool\ne : Vector.toList (enc a) = List.reverseAux (b :: l\u2081) l\u2082\n\u22a2 Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l\u2081) L'\u271d) (ListBlank.append l\u2082 R'\u271d)) =\n Tape.mk' (ListBlank.append l\u2081 L'\u271d) (ListBlank.append (b :: l\u2082) R'\u271d)\n[PROOFSTEP]\nsimp only [ListBlank.tail_cons, ListBlank.append, Tape.move_left_mk', ListBlank.head_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\n\u22a2 (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)\n[PROOFSTEP]\nsuffices \u2200 i L, (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L\n by\n refine' (Eq.symm _).trans (this n _)\n simp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\nthis : \u2200 (i : \u2115) (L : Tape (?m.366455 i)), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L\n\u22a2 (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)\n[PROOFSTEP]\nrefine' (Eq.symm _).trans (this n _)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\nthis : \u2200 (i : \u2115) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L\n\u22a2 (Tape.move Dir.right)^[n]\n ((Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))) =\n (Tape.move Dir.right)^[n] (trTape' enc0 L R)\n[PROOFSTEP]\nsimp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\n\u22a2 \u2200 (i : \u2115) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L\n[PROOFSTEP]\nintro i _\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\ni : \u2115\nL\u271d : Tape Bool\n\u22a2 (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L\u271d) = L\u271d\n[PROOFSTEP]\ninduction' i with i IH\n[GOAL]\ncase zero\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\nL\u271d : Tape Bool\n\u22a2 (Tape.move Dir.right)^[Nat.zero] ((Tape.move Dir.left)^[Nat.zero] L\u271d) = L\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nL R : ListBlank \u0393\nL\u271d : Tape Bool\ni : \u2115\nIH : (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L\u271d) = L\u271d\n\u22a2 (Tape.move Dir.right)^[Nat.succ i] ((Tape.move Dir.left)^[Nat.succ i] L\u271d) = L\u271d\n[PROOFSTEP]\nrw [iterate_succ_apply, iterate_succ_apply', Tape.move_left_right, IH]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\na b : \u0393\nL R : ListBlank \u0393\n\u22a2 stepAux (write (Vector.toList (enc a)) q) v (trTape' enc0 L (ListBlank.cons b R)) =\n stepAux q v (trTape' enc0 (ListBlank.cons a L) R)\n[PROOFSTEP]\nsimp only [trTape', ListBlank.cons_bind]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\na b : \u0393\nL R : ListBlank \u0393\n\u22a2 stepAux (write (Vector.toList (enc a)) q) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc b))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux q v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nsuffices\n \u2200 {L' R'} (l\u2081 l\u2082 l\u2082' : List Bool) (_ : l\u2082'.length = l\u2082.length),\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (L'.append (List.reverseAux l\u2082 l\u2081)) R')\n by refine' this [] _ _ ((enc b).2.trans (enc a).2.symm)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\na b : \u0393\nL R : ListBlank \u0393\nthis :\n \u2200 {L' R' : ListBlank Bool} (l\u2081 l\u2082 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n\u22a2 stepAux (write (Vector.toList (enc a)) q) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc b))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux q v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nrefine' this [] _ _ ((enc b).2.trans (enc a).2.symm)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\na b : \u0393\nL R : ListBlank \u0393\n\u22a2 \u2200 {L' R' : ListBlank Bool} (l\u2081 l\u2082 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\nclear a b L R\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\n\u22a2 \u2200 {L' R' : ListBlank Bool} (l\u2081 l\u2082 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\nintro L' R' l\u2081 l\u2082 l\u2082' e\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 l\u2082' : List Bool\ne : List.length l\u2082' = List.length l\u2082\n\u22a2 stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\ninduction' l\u2082 with a l\u2082 IH generalizing l\u2081 l\u2082'\n[GOAL]\ncase nil\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082 l\u2082'\u271d : List Bool\ne\u271d : List.length l\u2082'\u271d = List.length l\u2082\nl\u2081 l\u2082' : List Bool\ne : List.length l\u2082' = List.length []\n\u22a2 stepAux (write [] q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R')\n[PROOFSTEP]\ncases List.length_eq_zero.1 e\n[GOAL]\ncase nil.refl\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082 l\u2082' : List Bool\ne\u271d : List.length l\u2082' = List.length l\u2082\nl\u2081 : List Bool\ne : List.length [] = List.length []\n\u22a2 stepAux (write [] q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R')\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d l\u2082'\u271d : List Bool\ne\u271d : List.length l\u2082'\u271d = List.length l\u2082\u271d\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 l\u2082' : List Bool\ne : List.length l\u2082' = List.length (a :: l\u2082)\n\u22a2 stepAux (write (a :: l\u2082) q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\ncases' l\u2082' with b l\u2082'\n[GOAL]\ncase cons.nil\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d l\u2082' : List Bool\ne\u271d : List.length l\u2082' = List.length l\u2082\u271d\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\ne : List.length [] = List.length (a :: l\u2082)\n\u22a2 stepAux (write (a :: l\u2082) q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\nsimp only [List.length_nil, List.length_cons, Nat.succ_inj'] at e \n[GOAL]\ncase cons.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d l\u2082'\u271d : List Bool\ne\u271d : List.length l\u2082'\u271d = List.length l\u2082\u271d\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nb : Bool\nl\u2082' : List Bool\ne : List.length (b :: l\u2082') = List.length (a :: l\u2082)\n\u22a2 stepAux (write (a :: l\u2082) q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (b :: l\u2082') R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\nsimp only [List.length_nil, List.length_cons, Nat.succ_inj'] at e \n[GOAL]\ncase cons.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d l\u2082'\u271d : List Bool\ne\u271d : List.length l\u2082'\u271d = List.length l\u2082\u271d\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nb : Bool\nl\u2082' : List Bool\ne : List.length l\u2082' = List.length l\u2082\n\u22a2 stepAux (write (a :: l\u2082) q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (b :: l\u2082') R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\nrw [List.reverseAux, \u2190 IH (a :: l\u2081) l\u2082' e]\n[GOAL]\ncase cons.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nq : Stmt Bool \u039b' \u03c3\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d l\u2082'\u271d : List Bool\ne\u271d : List.length l\u2082'\u271d = List.length l\u2082\u271d\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 l\u2082' : List Bool),\n List.length l\u2082' = List.length l\u2082 \u2192\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082' R')) =\n stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nb : Bool\nl\u2082' : List Bool\ne : List.length l\u2082' = List.length l\u2082\n\u22a2 stepAux (write (a :: l\u2082) q) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (b :: l\u2082') R')) =\n stepAux (write l\u2082 q) v (Tape.mk' (ListBlank.append (a :: l\u2081) L') (ListBlank.append l\u2082' R'))\n[PROOFSTEP]\nsimp only [stepAux, ListBlank.append, Tape.write_mk', Tape.move_right_mk', ListBlank.head_cons, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)\n[PROOFSTEP]\nsuffices\n \u2200 f, stepAux (readAux n f) v (trTape' enc0 L R) = stepAux (f (enc R.head)) v (trTape' enc0 (L.cons R.head) R.tail)\n by\n rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]\n simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\nthis :\n \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n\u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)\n[PROOFSTEP]\nrw [read, this, stepAux_move, encdec, trTape'_move_left enc0]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\nthis :\n \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n\u22a2 stepAux (f (ListBlank.head R)) v\n (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))\n (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =\n stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)\n[PROOFSTEP]\nsimp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n[PROOFSTEP]\nobtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =\n stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v\n (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (ListBlank.cons a R)))\n[PROOFSTEP]\nsimp only [ListBlank.head_cons, ListBlank.tail_cons, trTape', ListBlank.cons_bind, ListBlank.append_assoc]\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nsuffices\n \u2200 i f L' R' l\u2081 l\u2082 h,\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f \u27e8l\u2082, h\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\n by\n intro f\n exact this n f (L.bind (fun x => (enc x).1.reverse) _) (R.bind (fun x => (enc x).1) _) [] _ (enc a).2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\nthis :\n \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nintro f\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf\u271d : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\nthis :\n \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nf : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : \u2203 n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : \u2203 n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))\n[PROOFSTEP]\nexact this n f (L.bind (fun x => (enc x).1.reverse) _) (R.bind (fun x => (enc x).1) _) [] _ (enc a).2\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nf : \u0393 \u2192 Stmt Bool \u039b' \u03c3\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\nclear f L a R\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\n\u22a2 \u2200 (i : \u2115) (f : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3) (L' R' : ListBlank Bool) (l\u2081 l\u2082 : List Bool) (h : List.length l\u2082 = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\nintro i f L' R' l\u2081 l\u2082 _\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\ni : \u2115\nf : Vector Bool i \u2192 Stmt Bool \u039b' \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nh\u271d : List.length l\u2082 = i\n\u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := h\u271d }) v (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\nsubst i\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081 l\u2082 : List Bool\nf : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\n[PROOFSTEP]\ninduction' l\u2082 with a l\u2082 IH generalizing l\u2081\n[GOAL]\ncase intro.intro.nil\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082 : List Bool\nf\u271d : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3\nl\u2081 : List Bool\nf : Vector Bool (List.length []) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux [] l\u2081) L') R')\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.cons\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\ntrans stepAux (readAux l\u2082.length fun v \u21a6 f (a ::\u1d65 v)) v (Tape.mk' ((L'.append l\u2081).cons a) (R'.append l\u2082))\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length (a :: l\u2082)) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n[PROOFSTEP]\ndsimp [readAux, stepAux]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif ListBlank.head (ListBlank.cons a (ListBlank.append l\u2082 R')) then\n stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))\n else\n stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.cons a (ListBlank.append l\u2082 R'))))) =\n stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif a then\n stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n else\n stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =\n stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (false :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif false then\n stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons false (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n else\n stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons false (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =\n stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons false (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (true :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif true then\n stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons true (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n else\n stepAux (readAux (List.length l\u2082) fun v => f (false ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons true (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))) =\n stepAux (readAux (List.length l\u2082) fun v => f (true ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons true (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R'))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (List.length l\u2082) fun v => f (a ::\u1d65 v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l\u2081 L')) (ListBlank.append l\u2082 R')) =\n stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\nrw [\u2190 ListBlank.append, IH]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nv : \u03c3\nL' R' : ListBlank Bool\nl\u2081\u271d l\u2082\u271d : List Bool\nf\u271d : Vector Bool (List.length l\u2082\u271d) \u2192 Stmt Bool \u039b' \u03c3\na : Bool\nl\u2082 : List Bool\nIH :\n \u2200 (l\u2081 : List Bool) (f : Vector Bool (List.length l\u2082) \u2192 Stmt Bool \u039b' \u03c3),\n stepAux (readAux (List.length l\u2082) f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n stepAux (f { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (List.length (a :: l\u2082)) \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (f (a ::\u1d65 { val := l\u2082, property := (_ : List.length l\u2082 = List.length l\u2082) })) v\n (Tape.mk' (ListBlank.append (List.reverseAux l\u2082 (a :: l\u2081)) L') R') =\n stepAux (f { val := a :: l\u2082, property := (_ : List.length (a :: l\u2082) = List.length (a :: l\u2082)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l\u2082) l\u2081) L') R')\n[PROOFSTEP]\nrfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nT : Tape \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081) (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := T })\n (step M { l := l\u2081, var := v, Tape := T })\n[PROOFSTEP]\nobtain \u27e8L, R, rfl\u27e9 := T.exists_mk'\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nl\u2081 : Option \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n (trCfg enc enc\u2080 { l := l\u2081, var := v, Tape := Tape.mk' L R }) (step M { l := l\u2081, var := v, Tape := Tape.mk' L R })\n[PROOFSTEP]\ncases' l\u2081 with l\u2081\n[GOAL]\ncase intro.intro.none\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n (trCfg enc enc\u2080 { l := none, var := v, Tape := Tape.mk' L R })\n (step M { l := none, var := v, Tape := Tape.mk' L R })\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n[PROOFSTEP]\nsuffices\n \u2200 q R,\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n by\n refine' TransGen.head' rfl _\n rw [trTape_mk']\n exact this _ R\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 FRespects (step (tr enc dec M)) (fun c\u2081 => trCfg enc enc\u2080 c\u2081)\n (trCfg enc enc\u2080 { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n (step M { l := some l\u2081, var := v, Tape := Tape.mk' L R })\n[PROOFSTEP]\nrefine' TransGen.head' rfl _\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape enc\u2080 (Tape.mk' L R)))\n ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))\n[PROOFSTEP]\nrw [trTape_mk']\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\nthis :\n \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a) (stepAux (tr enc dec M (\u039b'.normal l\u2081)) v (trTape' enc\u2080 L R))\n ((fun c\u2081 => trCfg enc enc\u2080 c\u2081) (stepAux (M l\u2081) v (Tape.mk' L R)))\n[PROOFSTEP]\nexact this _ R\n[GOAL]\ncase intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL R : ListBlank \u0393\nl\u2081 : \u039b\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n[PROOFSTEP]\nclear R l\u2081\n[GOAL]\ncase intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\n\u22a2 \u2200 (q : Stmt\u2081) (R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n[PROOFSTEP]\nintro q R\n[GOAL]\ncase intro.intro.some\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv : \u03c3\nL : ListBlank \u0393\nq : Stmt\u2081\nR : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\n[PROOFSTEP]\ninduction' q generalizing v L R\n[GOAL]\ncase intro.intro.some.move\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.move a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase move d q IH =>\n cases d <;>\n simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons, Tape.move_left_mk',\n ListBlank.cons_head_tail, ListBlank.tail_cons, trTape'_move_left enc0, trTape'_move_right enc0] <;>\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nd : Dir\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move d q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.move d q) v (Tape.mk' L R)))\n[PROOFSTEP]\ncase move d q IH =>\n cases d <;>\n simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons, Tape.move_left_mk',\n ListBlank.cons_head_tail, ListBlank.tail_cons, trTape'_move_left enc0, trTape'_move_right enc0] <;>\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nd : Dir\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move d q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.move d q) v (Tape.mk' L R)))\n[PROOFSTEP]\ncases d\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move Dir.left q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.move Dir.left q) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons, Tape.move_left_mk', ListBlank.cons_head_tail,\n ListBlank.tail_cons, trTape'_move_left enc0, trTape'_move_right enc0]\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.move Dir.right q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.move Dir.right q) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons, Tape.move_left_mk', ListBlank.cons_head_tail,\n ListBlank.tail_cons, trTape'_move_left enc0, trTape'_move_right enc0]\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (stepAux (trNormal dec q) v (trTape' enc0 (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R)))\n (trCfg enc enc0 (stepAux q v (Tape.mk' (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R))))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (stepAux (trNormal dec q) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R)))\n (trCfg enc enc0 (stepAux q v (Tape.move Dir.right (Tape.mk' L R))))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase intro.intro.some.write\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.write a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase write f q IH =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n refine' ReflTransGen.head rfl _\n obtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move, trTape'_move_left enc0, ListBlank.head_cons,\n ListBlank.tail_cons, Tape.write_mk']\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write f q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.write f q) v (Tape.mk' L R)))\n[PROOFSTEP]\ncase write f q IH =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n refine' ReflTransGen.head rfl _\n obtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move, trTape'_move_left enc0, ListBlank.head_cons,\n ListBlank.tail_cons, Tape.write_mk']\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write f q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.write f q) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.write (f (ListBlank.head R) v) q), var := v, Tape := trTape' enc0 L R }\n (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))\n[PROOFSTEP]\nrefine' ReflTransGen.head rfl _\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head R) v) q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.write (f (Tape.mk' L R).head v) (Tape.mk' L R))))\n[PROOFSTEP]\nobtain \u27e8a, R, rfl\u27e9 := R.exists_cons\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n (stepAux (tr enc dec M (\u039b'.write (f (ListBlank.head (ListBlank.cons a R)) v) q)) v\n (trTape' enc0 L (ListBlank.cons a R)))\n (trCfg enc enc0\n (stepAux q v (Tape.write (f (Tape.mk' L (ListBlank.cons a R)).head v) (Tape.mk' L (ListBlank.cons a R)))))\n[PROOFSTEP]\nrw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move, trTape'_move_left enc0, ListBlank.head_cons,\n ListBlank.tail_cons, Tape.write_mk']\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL : ListBlank \u0393\na : \u0393\nR : ListBlank \u0393\n\u22a2 ReflTransGen (fun a b => b \u2208 step (tr enc dec M) a)\n (stepAux (trNormal dec q) v (trTape' enc0 L (ListBlank.cons (f a v) R)))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L (ListBlank.cons (f a v) R))))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase intro.intro.some.load\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase load a q IH =>\n simp only [trNormal, stepAux_read dec enc0 encdec]\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))\n[PROOFSTEP]\ncase load a q IH =>\n simp only [trNormal, stepAux_read dec enc0 encdec]\n apply IH\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, stepAux_read dec enc0 encdec]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (stepAux (Stmt.load (fun x s => a (ListBlank.head R) s) (trNormal dec q)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.load a q) v (Tape.mk' L R)))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase intro.intro.some.branch\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d\u00b9) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d\u00b9 v (Tape.mk' L R)))\na_ih\u271d :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec a\u271d) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux a\u271d v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n cases p R.head v <;> [apply IH\u2082; apply IH\u2081]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch p q\u2081 q\u2082)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch p q\u2081 q\u2082) v (Tape.mk' L R)))\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n cases p R.head v <;> [apply IH\u2082; apply IH\u2081]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch p q\u2081 q\u2082)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.branch p q\u2081 q\u2082) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, stepAux_read dec enc0 encdec, stepAux]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (bif p (ListBlank.head R) v then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R)\n else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (bif p (Tape.mk' L R).head v then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R)))\n[PROOFSTEP]\ncases p R.head v <;> [apply IH\u2082; apply IH\u2081]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (bif p (ListBlank.head R) v then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R)\n else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (bif p (Tape.mk' L R).head v then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R)))\n[PROOFSTEP]\ncases p R.head v\n[GOAL]\ncase false\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (bif false then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R) else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (bif false then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R)))\n[PROOFSTEP]\napply IH\u2082\n[GOAL]\ncase true\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2081) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2081 v (Tape.mk' L R)))\nIH\u2082 :\n \u2200 (v : \u03c3) (L R : ListBlank \u0393),\n Reaches (step (tr enc dec M)) (stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux q\u2082 v (Tape.mk' L R)))\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M))\n (bif true then stepAux (trNormal dec q\u2081) v (trTape' enc0 L R) else stepAux (trNormal dec q\u2082) v (trTape' enc0 L R))\n (trCfg enc enc0 (bif true then stepAux q\u2081 v (Tape.mk' L R) else stepAux q\u2082 v (Tape.mk' L R)))\n[PROOFSTEP]\napply IH\u2081\n[GOAL]\ncase intro.intro.some.goto\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a\u271d)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto a\u271d) v (Tape.mk' L R)))\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase goto l =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']\n apply ReflTransGen.refl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto l)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto l) v (Tape.mk' L R)))\n[PROOFSTEP]\ncase goto l =>\n simp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']\n apply ReflTransGen.refl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto l)) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux (Stmt.goto l) v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, stepAux_read dec enc0 encdec, stepAux, trCfg, trTape_mk']\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) { l := some (\u039b'.normal (l (ListBlank.head R) v)), var := v, Tape := trTape' enc0 L R }\n { l := Option.map \u039b'.normal (some (l (Tape.mk' L R).head v)), var := v, Tape := trTape' enc0 L R }\n[PROOFSTEP]\napply ReflTransGen.refl\n[GOAL]\ncase intro.intro.some.halt\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase\n halt =>\n simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0, trTape'_move_right enc0, trTape_mk']\n apply ReflTransGen.refl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\ncase\n halt =>\n simp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0, trTape'_move_right enc0, trTape_mk']\n apply ReflTransGen.refl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))\n (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))\n[PROOFSTEP]\nsimp only [trNormal, stepAux, trCfg, stepAux_move, trTape'_move_left enc0, trTape'_move_right enc0, trTape_mk']\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b2 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\nenc\u2080 : enc default = Vector.replicate n false\nx\u271d : Cfg\u2081\nv\u271d : \u03c3\nL\u271d R\u271d : ListBlank \u0393\nv : \u03c3\nL R : ListBlank \u0393\n\u22a2 Reaches (step (tr enc dec M)) { l := none, var := v, Tape := trTape' enc0 L R }\n { l := Option.map \u039b'.normal none, var := v, Tape := trTape' enc0 L R }\n[PROOFSTEP]\napply ReflTransGen.refl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)\n[PROOFSTEP]\nsuffices\n \u2200 q,\n SupportsStmt S q \u2192\n (\u2200 q' \u2208 writes q, q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 q' \u2208 writes q, SupportsStmt (trSupp M S) (tr enc dec M q')\n by\n rcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9\n have := this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9\n rcases Finset.mem_insert.1 h with (rfl | h)\n exacts [this.1, this.2 _ h]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nthis :\n \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)\n[PROOFSTEP]\nrcases Finset.mem_biUnion.1 h with \u27e8l, hl, h\u27e9\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis :\n \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)\n[PROOFSTEP]\nhave := this _ (ss.2 _ hl) fun q' hq \u21a6 Finset.mem_biUnion.2 \u27e8_, hl, Finset.mem_insert_of_mem hq\u27e9\n[GOAL]\ncase intro.intro\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d : q \u2208 trSupp M S\nthis\u271d :\n \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)\n[PROOFSTEP]\nrcases Finset.mem_insert.1 h with (rfl | h)\n[GOAL]\ncase intro.intro.inl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nthis\u271d :\n \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nthis :\n SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh\u271d : \u039b'.normal l \u2208 trSupp M S\nh : \u039b'.normal l \u2208 insert (\u039b'.normal l) (writes (M l))\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.normal l))\ncase intro.intro.inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh\u271d\u00b9 : q \u2208 trSupp M S\nthis\u271d :\n \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nl : \u039b\nhl : l \u2208 S\nh\u271d : q \u2208 insert (\u039b'.normal l) (writes (M l))\nthis :\n SupportsStmt (trSupp M S) (trNormal dec (M l)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (M l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nh : q \u2208 writes (M l)\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q)\n[PROOFSTEP]\nexacts [this.1, this.2 _ h]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2081),\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nintro q hs hw\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nq : Stmt\u2081\nhs : SupportsStmt S q\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ninduction q\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : Dir\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n SupportsStmt S a\u271d\u00b9 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase move d q IH =>\n unfold writes at hw \u22a2\n replace IH := IH hs hw; refine' \u27e8_, IH.2\u27e9\n cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase move d q IH =>\n unfold writes at hw \u22a2\n replace IH := IH hs hw; refine' \u27e8_, IH.2\u27e9\n cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.move d q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nunfold writes at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nreplace IH := IH hs hw\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrefine' \u27e8_, IH.2\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nd : Dir\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.move d q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move d q))\n[PROOFSTEP]\ncases d\n[GOAL]\ncase left\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nq : Stmt\u2081\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move Dir.left q)\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move Dir.left q))\n[PROOFSTEP]\nsimp only [trNormal, iterate, supportsStmt_move, IH]\n[GOAL]\ncase right\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nq : Stmt\u2081\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.move Dir.right q)\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.move Dir.right q))\n[PROOFSTEP]\nsimp only [trNormal, iterate, supportsStmt_move, IH]\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u0393\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n SupportsStmt S a\u271d\u00b9 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase write f q IH =>\n unfold writes at hw \u22a2\n simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff] at hw \u22a2\n replace IH := IH hs fun q hq \u21a6 hw q (Or.inr hq)\n refine' \u27e8supportsStmt_read _ fun a _ s \u21a6 hw _ (Or.inl \u27e8_, rfl\u27e9), fun q' hq \u21a6 _\u27e9\n rcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq)\n \u00b7 simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]\n \u00b7 exact IH.2 _ hq\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase write f q IH =>\n unfold writes at hw \u22a2\n simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff] at hw \u22a2\n replace IH := IH hs fun q hq \u21a6 hw q (Or.inr hq)\n refine' \u27e8supportsStmt_read _ fun a _ s \u21a6 hw _ (Or.inl \u27e8_, rfl\u27e9), fun q' hq \u21a6 _\u27e9\n rcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq)\n \u00b7 simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]\n \u00b7 exact IH.2 _ hq\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.write f q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nunfold writes at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n \u2200 (q' : \u039b'),\n q' \u2208 Finset.image (fun a => \u039b'.write a q) Finset.univ \u222a writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nsimp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff] at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nreplace IH := IH hs fun q hq \u21a6 hw q (Or.inr hq)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.write f q)) \u2227\n \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrefine' \u27e8supportsStmt_read _ fun a _ s \u21a6 hw _ (Or.inl \u27e8_, rfl\u27e9), fun q' hq \u21a6 _\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrcases hq with (\u27e8a, q\u2082, rfl\u27e9 | hq)\n[GOAL]\ncase inl.intro.refl\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na : \u0393\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M (\u039b'.write a q))\n[PROOFSTEP]\nsimp only [tr, supportsStmt_write, supportsStmt_move, IH.1]\n[GOAL]\ncase inr\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\nf : \u0393 \u2192 \u03c3 \u2192 \u0393\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.write f q)\nhw : \u2200 (q' : \u039b'), (\u2203 a, \u039b'.write a q = q') \u2228 q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nq' : \u039b'\nhq : q' \u2208 writes q\n\u22a2 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nexact IH.2 _ hq\n[GOAL]\ncase load\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b9 : \u0393 \u2192 \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2081\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n SupportsStmt S a\u271d\u00b9 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase load a q IH =>\n unfold writes at hw \u22a2\n replace IH := IH hs hw\n refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase load a q IH =>\n unfold writes at hw \u22a2\n replace IH := IH hs hw\n refine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.load a q) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nunfold writes at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nIH :\n SupportsStmt S q \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nreplace IH := IH hs hw\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq\u271d : \u039b'\nh : q\u271d \u2208 trSupp M S\na : \u0393 \u2192 \u03c3 \u2192 \u03c3\nq : Stmt\u2081\nhs : SupportsStmt S (Stmt.load a q)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 q' \u2208 trSupp M S\nIH :\n SupportsStmt (trSupp M S) (trNormal dec q) \u2227 \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.load a q)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrefine' \u27e8supportsStmt_read _ fun _ \u21a6 IH.1, IH.2\u27e9\n[GOAL]\ncase branch\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d\u00b2 : \u0393 \u2192 \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2081\na_ih\u271d\u00b9 :\n SupportsStmt S a\u271d\u00b9 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d\u00b9) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d\u00b9 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\na_ih\u271d :\n SupportsStmt S a\u271d \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec a\u271d) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes a\u271d \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold writes at hw \u22a2\n simp only [Finset.mem_union] at hw \u22a2\n replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)\n replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)\n exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n SupportsStmt S q\u2081 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n SupportsStmt S q\u2082 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase branch p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold writes at hw \u22a2\n simp only [Finset.mem_union] at hw \u22a2\n replace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)\n replace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)\n exact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n SupportsStmt S q\u2081 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n SupportsStmt S q\u2082 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.branch p q\u2081 q\u2082) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nunfold writes at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n SupportsStmt S q\u2081 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n SupportsStmt S q\u2082 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u222a writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nsimp only [Finset.mem_union] at hw \u22a2\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2081 :\n SupportsStmt S q\u2081 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n SupportsStmt S q\u2082 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nreplace IH\u2081 := IH\u2081 hs.1 fun q hq \u21a6 hw q (Or.inl hq)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nIH\u2082 :\n SupportsStmt S q\u2082 \u2192\n (\u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S) \u2192\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nreplace IH\u2082 := IH\u2082 hs.2 fun q hq \u21a6 hw q (Or.inr hq)\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\np : \u0393 \u2192 \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2081\nhs : SupportsStmt S (Stmt.branch p q\u2081 q\u2082)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 q' \u2208 trSupp M S\nIH\u2081 :\n SupportsStmt (trSupp M S) (trNormal dec q\u2081) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\nIH\u2082 :\n SupportsStmt (trSupp M S) (trNormal dec q\u2082) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes q\u2081 \u2228 q' \u2208 writes q\u2082 \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nexact \u27e8supportsStmt_read _ fun _ \u21a6 \u27e8IH\u2081.1, IH\u2082.1\u27e9, fun q \u21a6 Or.rec (IH\u2081.2 _) (IH\u2082.2 _)\u27e9\n[GOAL]\ncase goto\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\na\u271d : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto a\u271d)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto a\u271d)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto a\u271d) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase goto l =>\n simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n refine' supportsStmt_read _ fun a _ s \u21a6 _\n exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase goto l =>\n simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n refine' supportsStmt_read _ fun a _ s \u21a6 _\n exact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nsimp only [writes, Finset.not_mem_empty]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l)) \u2227\n \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrefine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec (Stmt.goto l))\n[PROOFSTEP]\nrefine' supportsStmt_read _ fun a _ s \u21a6 _\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nl : \u0393 \u2192 \u03c3 \u2192 \u039b\nhs : SupportsStmt S (Stmt.goto l)\nhw : \u2200 (q' : \u039b'), q' \u2208 writes (Stmt.goto l) \u2192 q' \u2208 trSupp M S\na : \u0393\nx\u271d : Bool\ns : \u03c3\n\u22a2 (fun x s => \u039b'.normal (l a s)) x\u271d s \u2208 trSupp M S\n[PROOFSTEP]\nexact Finset.mem_biUnion.2 \u27e8_, hs _ _, Finset.mem_insert_self _ _\u27e9\n[GOAL]\ncase halt\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase halt =>\n simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n simp only [SupportsStmt, supportsStmt_move, trNormal]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\ncase halt =>\n simp only [writes, Finset.not_mem_empty]; refine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n simp only [SupportsStmt, supportsStmt_move, trNormal]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227\n \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nsimp only [writes, Finset.not_mem_empty]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt) \u2227 \u2200 (q' : \u039b'), False \u2192 SupportsStmt (trSupp M S) (tr enc dec M q')\n[PROOFSTEP]\nrefine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b3 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d\u00b2 : Inhabited \u039b\n\u03c3 : Type u_3\ninst\u271d\u00b9 : Inhabited \u03c3\nn : \u2115\nenc : \u0393 \u2192 Vector Bool n\ndec : Vector Bool n \u2192 \u0393\nenc0 : enc default = Vector.replicate n false\nM : \u039b \u2192 Stmt\u2081\nencdec : \u2200 (a : \u0393), dec (enc a) = a\ninst\u271d : Fintype \u0393\nS : Finset \u039b\nss : Supports M S\nq : \u039b'\nh : q \u2208 trSupp M S\nhs : SupportsStmt S Stmt.halt\nhw : \u2200 (q' : \u039b'), q' \u2208 writes Stmt.halt \u2192 q' \u2208 trSupp M S\n\u22a2 SupportsStmt (trSupp M S) (trNormal dec Stmt.halt)\n[PROOFSTEP]\nsimp only [SupportsStmt, supportsStmt_move, trNormal]\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n[PROOFSTEP]\ncases' e : M q T.1 with val\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n[PROOFSTEP]\nsimp only [TM0.step, trCfg, e]\n[GOAL]\ncase none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\ne : M q T.head = none\n\u22a2 FRespects (TM1.step (tr M))\n (fun a =>\n { l := bif Option.isSome (M a.q a.Tape.head) then some (\u039b'.normal a.q) else none, var := (), Tape := a.Tape })\n { l := bif Option.isSome none then some (\u039b'.normal q) else none, var := (), Tape := T }\n (Option.map\n (fun x =>\n { q := x.fst,\n Tape :=\n match x.snd with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T })\n none)\n[PROOFSTEP]\nexact Eq.refl none\n[GOAL]\ncase some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nval : \u039b \u00d7 TM0.Stmt \u0393\ne : M q T.head = some val\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n[PROOFSTEP]\ncases' val with q' s\n[GOAL]\ncase some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 FRespects (TM1.step (tr M)) (fun a => trCfg M a) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })\n[PROOFSTEP]\nsimp only [FRespects, TM0.step, trCfg, e, Option.isSome, cond, Option.map_some']\n[GOAL]\ncase some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n {\n l :=\n match\n match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nrevert e\n[GOAL]\ncase some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\n\u22a2 M q T.head = some (q', s) \u2192\n Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n {\n l :=\n match\n match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nhave :\n TM1.step (tr M) \u27e8some (\u039b'.act s q'), (), T\u27e9 =\n some\n \u27e8some (\u039b'.normal q'), (),\n match s with\n | TM0.Stmt.move d => T.move d\n | TM0.Stmt.write a => T.write a\u27e9 :=\n by cases' s with d a <;> rfl\n[GOAL]\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\n\u22a2 TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\ncases' s with d a\n[GOAL]\ncase move\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\nd : Dir\n\u22a2 TM1.step (tr M) { l := some (\u039b'.act (TM0.Stmt.move d) q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match TM0.Stmt.move d with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase write\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\na : \u0393\n\u22a2 TM1.step (tr M) { l := some (\u039b'.act (TM0.Stmt.write a) q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match TM0.Stmt.write a with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n\u22a2 M q T.head = some (q', s) \u2192\n Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n {\n l :=\n match\n match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nintro e\n[GOAL]\ncase some.mk\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 Reaches\u2081 (TM1.step (tr M)) { l := some (\u039b'.normal q), var := (), Tape := T }\n {\n l :=\n match\n match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nrefine' TransGen.head _ (TransGen.head' this _)\n[GOAL]\ncase some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208 TM1.step (tr M) { l := some (\u039b'.normal q), var := (), Tape := T }\n[PROOFSTEP]\nsimp only [TM1.step, TM1.stepAux]\n[GOAL]\ncase some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n some\n (bif Option.isNone (M q T.head) then { l := none, var := (), Tape := T }\n else\n {\n l :=\n some\n (match M q T.head with\n | none => default\n | some (q', s) => \u039b'.act s q'),\n var := (), Tape := T })\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase some.mk.refine'_1\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 { l := some (\u039b'.act s q'), var := (), Tape := T } \u2208\n some\n (bif Option.isNone (some (q', s)) then { l := none, var := (), Tape := T }\n else\n {\n l :=\n some\n (match some (q', s) with\n | none => default\n | some (q', s) => \u039b'.act s q'),\n var := (), Tape := T })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.mk.refine'_2\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n {\n l :=\n match\n match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\ncases e' : M q' _\n[GOAL]\ncase some.mk.refine'_2.none\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head =\n none\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n {\n l :=\n match\n match none with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\napply ReflTransGen.single\n[GOAL]\ncase some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head =\n none\n\u22a2 {\n l :=\n match\n match none with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T } \u2208\n TM1.step (tr M)\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nsimp only [TM1.step, TM1.stepAux]\n[GOAL]\ncase some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head =\n none\n\u22a2 { l := none, var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T } \u2208\n some\n (bif\n Option.isNone\n (M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head) then\n { l := none, var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n else\n {\n l :=\n some\n (match\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head with\n | none => default\n | some (q', s) => \u039b'.act s q'),\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T })\n[PROOFSTEP]\nrw [e']\n[GOAL]\ncase some.mk.refine'_2.none.hab\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\ne' :\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head =\n none\n\u22a2 { l := none, var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T } \u2208\n some\n (bif Option.isNone none then\n { l := none, var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n else\n {\n l :=\n some\n (match none with\n | none => default\n | some (q', s) => \u039b'.act s q'),\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.mk.refine'_2.some\n\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\ninst\u271d : Inhabited \u039b\nM : TM0.Machine \u0393 \u039b\nx\u271d : Cfg\u2080\nq : \u039b\nT : Tape \u0393\nq' : \u039b\ns : TM0.Stmt \u0393\nthis :\n TM1.step (tr M) { l := some (\u039b'.act s q'), var := (), Tape := T } =\n some\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\nval\u271d : \u039b \u00d7 TM0.Stmt \u0393\ne' :\n M q'\n (match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T).head =\n some val\u271d\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n { l := some (\u039b'.normal q'), var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n {\n l :=\n match\n match some val\u271d with\n | some val => true\n | none => false with\n | true => some (\u039b'.normal q')\n | false => none,\n var := (),\n Tape :=\n match s with\n | TM0.Stmt.move d => Tape.move d T\n | TM0.Stmt.write a => Tape.write a T }\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq : Stmt\u2082\n\u22a2 q \u2208 stmts\u2081 q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nk\u271d : K\na\u271d\u00b9 : \u03c3 \u2192 \u0393 k\u271d\na\u271d : Stmt\u2082\n\u22a2 push k\u271d a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (push k\u271d a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nk\u271d : K\na\u271d\u00b9 : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\na\u271d : Stmt\u2082\n\u22a2 peek k\u271d a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (peek k\u271d a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nk\u271d : K\na\u271d\u00b9 : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\na\u271d : Stmt\u2082\n\u22a2 pop k\u271d a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (pop k\u271d a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\na\u271d\u00b9 : \u03c3 \u2192 \u03c3\na\u271d : Stmt\u2082\n\u22a2 load a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (load a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\n\u22a2 branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\na\u271d : \u03c3 \u2192 \u039b\n\u22a2 goto a\u271d \u2208 stmts\u2081 (goto a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\n\u22a2 halt \u2208 stmts\u2081 halt\n[PROOFSTEP]\nsimp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts\u2081]\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\n\u22a2 q\u2081 \u2208 stmts\u2081 q\u2082 \u2192 stmts\u2081 q\u2081 \u2286 stmts\u2081 q\u2082\n[PROOFSTEP]\nintro h\u2081\u2082 q\u2080 h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\n\u22a2 q\u2080 \u2208 stmts\u2081 q\u2082\n[PROOFSTEP]\ninduction' q\u2082 with _ _ q IH _ _ q IH _ _ q IH _ q IH\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (push k\u271d a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (push k\u271d a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (peek k\u271d a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (peek k\u271d a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (pop k\u271d a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (pop k\u271d a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (load a\u271d q)\n\u22a2 q\u2080 \u2208 stmts\u2081 (load a\u271d q)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 q\u2080 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 (goto a\u271d)\n\u22a2 q\u2080 \u2208 stmts\u2081 (goto a\u271d)\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 halt\n\u22a2 q\u2080 \u2208 stmts\u2081 halt\n[PROOFSTEP]\nsimp only [stmts\u2081] at h\u2081\u2082 \u22a2\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 \u2208 {goto a\u271d}\n\u22a2 q\u2080 \u2208 {goto a\u271d}\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 \u2208 {halt}\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nsimp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h\u2081\u2082 \n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\niterate 4 \n rcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase push.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nh\u2081\u2082 : push k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (push k\u271d a\u271d q)\nIH : push k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase push.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nh\u2081\u2082 : push k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : push k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase push.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (push k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase peek.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : peek k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (peek k\u271d a\u271d q)\nIH : peek k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase peek.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : peek k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : peek k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase peek.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (peek k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase pop.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : pop k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (pop k\u271d a\u271d q)\nIH : pop k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase pop.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : pop k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : pop k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase pop.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (pop k\u271d a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082)\n[GOAL]\ncase load.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : load a\u271d q \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (load a\u271d q)\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase load.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nh\u2081\u2082 : load a\u271d q \u2208 stmts\u2081 q\u2082\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2080\u2081 : q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase load.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 q\u2080 \u2208 stmts\u2081 q\nh\u2081\u2082 : q\u2081 \u2208 stmts\u2081 q\n\u22a2 q\u2080 \u2208 insert (load a\u271d q) (stmts\u2081 q)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (IH h\u2081\u2082)\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\u00b9\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 q\u2080 \u2208 stmts\u2081 a\u271d\nh\u2081\u2082 : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 q\u2080 \u2208 insert (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) (stmts\u2081 a\u271d\u00b9 \u222a stmts\u2081 a\u271d)\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase branch f q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n rcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_left _ (IH\u2081 h\u2081\u2082))\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_right _ (IH\u2082 h\u2081\u2082))\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d = branch f q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\ncase branch f q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n rcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n \u00b7 unfold stmts\u2081 at h\u2080\u2081 \n exact h\u2080\u2081\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_left _ (IH\u2081 h\u2081\u2082))\n \u00b7 exact Finset.mem_insert_of_mem (Finset.mem_union_right _ (IH\u2082 h\u2081\u2082))\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d = branch f q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nrcases h\u2081\u2082 with (rfl | h\u2081\u2082 | h\u2081\u2082)\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082\u271d q\u2080 : Stmt\u2082\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (branch f q\u2081 q\u2082)\nIH\u2081 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nunfold stmts\u2081 at h\u2080\u2081 \n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082\u271d q\u2080 : Stmt\u2082\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase inr.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_left _ (IH\u2081 h\u2081\u2082))\n[GOAL]\ncase inr.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 q\u2080 \u2208 stmts\u2081 q\u2081\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 q\u2080 \u2208 stmts\u2081 q\u2082\nh\u2081\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 q\u2080 \u2208 insert (branch f q\u2081 q\u2082) (stmts\u2081 q\u2081 \u222a stmts\u2081 q\u2082)\n[PROOFSTEP]\nexact Finset.mem_insert_of_mem (Finset.mem_union_right _ (IH\u2082 h\u2081\u2082))\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\na\u271d : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto a\u271d\n\u22a2 q\u2080 \u2208 {goto a\u271d}\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase goto l => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nl : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto l\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\ncase goto l => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nl : \u03c3 \u2192 \u039b\nh\u2081\u2082 : q\u2081 = goto l\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\nsubst h\u2081\u2082\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nl : \u03c3 \u2192 \u039b\nh\u2081\u2082 : goto l \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 (goto l)\n\u22a2 q\u2080 \u2208 {goto l}\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase halt => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\ncase halt => subst h\u2081\u2082; exact h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081\u2082\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nq\u2080 : Stmt\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 q\u2081\nh\u2081\u2082 : q\u2081 = halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nsubst h\u2081\u2082\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nq\u2082 q\u2080 : Stmt\u2082\nh\u2081\u2082 : halt \u2208 stmts\u2081 q\u2082\nh\u2080\u2081 : q\u2080 \u2208 stmts\u2081 halt\n\u22a2 q\u2080 \u2208 {halt}\n[PROOFSTEP]\nexact h\u2080\u2081\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh : q\u2081 \u2208 stmts\u2081 q\u2082\nhs : SupportsStmt S q\u2082\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ninduction' q\u2082 with _ _ q IH _ _ q IH _ _ q IH _ q IH\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (push k\u271d a\u271d q)\nhs : SupportsStmt S (push k\u271d a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (peek k\u271d a\u271d q)\nhs : SupportsStmt S (peek k\u271d a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (pop k\u271d a\u271d q)\nhs : SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (load a\u271d q)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nh : q\u2081 \u2208 stmts\u2081 (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nh : q\u2081 \u2208 stmts\u2081 (goto a\u271d)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nh : q\u2081 \u2208 stmts\u2081 halt\nhs : SupportsStmt S halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsimp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs \n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\niterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase push.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nhs : SupportsStmt S q\nh : push k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : push k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (push k\u271d a\u271d q)\n\u22a2 SupportsStmt S (push k\u271d a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase push.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase peek.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nhs : SupportsStmt S q\nh : peek k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : peek k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (peek k\u271d a\u271d q)\n\u22a2 SupportsStmt S (peek k\u271d a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase peek.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase pop.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nhs : SupportsStmt S q\nh : pop k\u271d a\u271d q \u2208 stmts\u2081 q\u2082\nIH : pop k\u271d a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 SupportsStmt S (pop k\u271d a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase pop.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h) <;> [exact hs; exact IH h hs]\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase load.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nhs : SupportsStmt S q\nh : load a\u271d q \u2208 stmts\u2081 q\u2082\nIH : load a\u271d q \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S (load a\u271d q)\n\u22a2 SupportsStmt S (load a\u271d q)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase load.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nexact IH h hs\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase branch f q\u2081 q\u2082 IH\u2081 IH\u2082 => rcases h with (rfl | h | h); exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d = branch f q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\ncase branch f q\u2081 q\u2082 IH\u2081 IH\u2082 => rcases h with (rfl | h | h); exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d = branch f q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\nrcases h with (rfl | h | h)\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082\u271d : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\nIH\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\n\u22a2 SupportsStmt S (branch f q\u2081 q\u2082)\ncase inr.inl\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 SupportsStmt S q\u2081\u271d\ncase inr.inr\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d\n[PROOFSTEP]\nexacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase goto l => subst h; exact hs\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : q\u2081 = goto l\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase goto l => subst h; exact hs\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : q\u2081 = goto l\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : goto l \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S (goto l)\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase halt => subst h; trivial\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\ncase halt => subst h; trivial\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : halt \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S halt\n[PROOFSTEP]\ntrivial\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 some q\u2082 \u2208 stmts M S \u2192 some q\u2081 \u2208 stmts M S\n[PROOFSTEP]\nsimp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq, forall_eq', exists_imp,\n and_imp]\n[GOAL]\nK : Type u_1\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u2081 : q\u2081 \u2208 stmts\u2081 q\u2082\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q\u2082 \u2208 stmts\u2081 (M x) \u2192 \u2203 a, a \u2208 S \u2227 q\u2081 \u2208 stmts\u2081 (M a)\n[PROOFSTEP]\nexact fun l ls h\u2082 \u21a6 \u27e8_, ls, stmts\u2081_trans h\u2082 h\u2081\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq : Stmt\u2082\nss : Supports M S\n\u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q\n[PROOFSTEP]\nsimp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq, forall_eq', exists_imp,\n and_imp]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nq : Stmt\u2082\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q\n[PROOFSTEP]\nexact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nc' : Cfg\u2082\nh\u2081 : c' \u2208 step M { l := some l\u2081, var := v, stk := T }\nh\u2082 : { l := some l\u2081, var := v, stk := T }.l \u2208 \u2191Finset.insertNone S\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nreplace h\u2082 := ss.2 _ (Finset.some_mem_insertNone.1 h\u2082)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nc' : Cfg\u2082\nh\u2081 : c' \u2208 step M { l := some l\u2081, var := v, stk := T }\nh\u2082 : SupportsStmt S (M l\u2081)\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nsimp only [step, Option.mem_def, Option.some.injEq] at h\u2081 \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nc' : Cfg\u2082\nh\u2082 : SupportsStmt S (M l\u2081)\nh\u2081 : stepAux (M l\u2081) v T = c'\n\u22a2 c'.l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nsubst c'\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nh\u2082 : SupportsStmt S (M l\u2081)\n\u22a2 (stepAux (M l\u2081) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nrevert h\u2082\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (M l\u2081) \u2192 (stepAux (M l\u2081) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ninduction' M l\u2081 with _ _ q IH _ _ q IH _ _ q IH _ q IH generalizing v T\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (push k\u271d a\u271d q) \u2192 (stepAux (push k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (peek k\u271d a\u271d q) \u2192 (stepAux (peek k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (pop k\u271d a\u271d q) \u2192 (stepAux (pop k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (load a\u271d q) \u2192 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) \u2192 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S (goto a\u271d) \u2192 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\n\u22a2 SupportsStmt S halt \u2192 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (push k\u271d a\u271d q)\n\u22a2 (stepAux (push k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase peek\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (peek k\u271d a\u271d q)\n\u22a2 (stepAux (peek k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase pop\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 (stepAux (pop k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\niterate 4 exact IH _ _ hs\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (push k\u271d a\u271d q)\n\u22a2 (stepAux (push k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase peek\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (peek k\u271d a\u271d q)\n\u22a2 (stepAux (peek k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase pop\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 (stepAux (pop k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (peek k\u271d a\u271d q)\n\u22a2 (stepAux (peek k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase pop\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 (stepAux (pop k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (pop k\u271d a\u271d q)\n\u22a2 (stepAux (pop k\u271d a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase load\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q \u2192 (stepAux q v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (load a\u271d q)\n\u22a2 (stepAux (load a\u271d q) v T).l \u2208 \u2191Finset.insertNone S\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH _ _ hs\n[GOAL]\ncase branch\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d\u00b9 \u2192 (stepAux a\u271d\u00b9 v T).l \u2208 \u2191Finset.insertNone S\na_ih\u271d : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S a\u271d \u2192 (stepAux a\u271d v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d)\n\u22a2 (stepAux (branch a\u271d\u00b2 a\u271d\u00b9 a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase branch p q\u2081' q\u2082' IH\u2081 IH\u2082 =>\n unfold stepAux; cases p v\n \u00b7 exact IH\u2082 _ _ hs.2\n \u00b7 exact IH\u2081 _ _ hs.1\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\np : \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2082\nIH\u2081 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (stepAux (branch p q\u2081' q\u2082') v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase branch p q\u2081' q\u2082' IH\u2081 IH\u2082 =>\n unfold stepAux; cases p v\n \u00b7 exact IH\u2082 _ _ hs.2\n \u00b7 exact IH\u2081 _ _ hs.1\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\np : \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2082\nIH\u2081 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (stepAux (branch p q\u2081' q\u2082') v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nunfold stepAux\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\np : \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2082\nIH\u2081 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif p v then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncases p v\n[GOAL]\ncase false\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\np : \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2082\nIH\u2081 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif false then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH\u2082 _ _ hs.2\n[GOAL]\ncase true\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\np : \u03c3 \u2192 Bool\nq\u2081' q\u2082' : Stmt\u2082\nIH\u2081 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2081' \u2192 (stepAux q\u2081' v T).l \u2208 \u2191Finset.insertNone S\nIH\u2082 : \u2200 (v : \u03c3) (T : (k : K) \u2192 List (\u0393 k)), SupportsStmt S q\u2082' \u2192 (stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (branch p q\u2081' q\u2082')\n\u22a2 (bif true then stepAux q\u2081' v T else stepAux q\u2082' v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact IH\u2081 _ _ hs.1\n[GOAL]\ncase goto\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase goto => exact Finset.some_mem_insertNone.2 (hs _)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase goto => exact Finset.some_mem_insertNone.2 (hs _)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\na\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S (goto a\u271d)\n\u22a2 (stepAux (goto a\u271d) v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\nexact Finset.some_mem_insertNone.2 (hs _)\n[GOAL]\ncase halt\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase halt => apply Multiset.mem_cons_self\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\ncase halt => apply Multiset.mem_cons_self\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : Supports M S\nl\u2081 : \u039b\nv\u271d : \u03c3\nT\u271d : (k : K) \u2192 List (\u0393 k)\nv : \u03c3\nT : (k : K) \u2192 List (\u0393 k)\nhs : SupportsStmt S halt\n\u22a2 (stepAux halt v T).l \u2208 \u2191Finset.insertNone S\n[PROOFSTEP]\napply Multiset.mem_cons_self\n[GOAL]\nK : Type u_1\n\u0393 : K \u2192 Type u_2\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nk : K\nS : List (\u0393 k)\nn : \u2115\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\n\u22a2 ListBlank.nth L n k = List.get? (List.reverse S) n\n[PROOFSTEP]\nrw [\u2190 proj_map_nth, hL, \u2190 List.map_reverse, ListBlank.nth_mk, List.getI_eq_iget_get?, List.get?_map]\n[GOAL]\nK : Type u_1\n\u0393 : K \u2192 Type u_2\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nk : K\nS : List (\u0393 k)\nn : \u2115\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\n\u22a2 Option.iget (Option.map some (List.get? (List.reverse S) n)) = List.get? (List.reverse S) n\n[PROOFSTEP]\ncases S.reverse.get? n\n[GOAL]\ncase none\nK : Type u_1\n\u0393 : K \u2192 Type u_2\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nk : K\nS : List (\u0393 k)\nn : \u2115\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\n\u22a2 Option.iget (Option.map some none) = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nK : Type u_1\n\u0393 : K \u2192 Type u_2\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nk : K\nS : List (\u0393 k)\nn : \u2115\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nval\u271d : \u0393 k\n\u22a2 Option.iget (Option.map some (some val\u271d)) = some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 default.snd = default\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) } (addBottom L) = L\n[PROOFSTEP]\nsimp only [addBottom, ListBlank.map_cons]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.cons (ListBlank.head L)\n (ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.tail L))) =\n L\n[PROOFSTEP]\nconvert ListBlank.cons_head_tail L\n[GOAL]\ncase h.e'_2.h.e'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n ListBlank.tail L\n[PROOFSTEP]\ngeneralize ListBlank.tail L = L'\n[GOAL]\ncase h.e'_2.h.e'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL L' : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } L') =\n L'\n[PROOFSTEP]\nrefine' L'.induction_on fun l \u21a6 _\n[GOAL]\ncase h.e'_2.h.e'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL L' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nl : List ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.mk l)) =\n ListBlank.mk l\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.fst, f a.snd)) n (addBottom L) = addBottom (ListBlank.modifyNth f n L)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 ListBlank.modifyNth (fun a => (a.fst, f a.snd)) Nat.zero (addBottom L) = addBottom (ListBlank.modifyNth f Nat.zero L)\n[PROOFSTEP]\nsimp only [addBottom, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.tail_cons]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.fst, f a.snd)) (Nat.succ n\u271d) (addBottom L) =\n addBottom (ListBlank.modifyNth f (Nat.succ n\u271d) L)\n[PROOFSTEP]\nsimp only [addBottom, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.tail_cons]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.cons (true, ListBlank.head L)\n (ListBlank.modifyNth (fun a => (a.fst, f a.snd)) n\u271d\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.tail L))) =\n ListBlank.cons (true, ListBlank.head L)\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.modifyNth f n\u271d (ListBlank.tail L)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.modifyNth (fun a => (a.fst, f a.snd)) n\u271d\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.modifyNth f n\u271d (ListBlank.tail L))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase succ.e_l\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.modifyNth f n\u271d (ListBlank.tail L)) =\n ListBlank.modifyNth (fun a => (a.fst, f a.snd)) n\u271d\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L))\n[PROOFSTEP]\napply ListBlank.map_modifyNth\n[GOAL]\ncase succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\n\u22a2 \u2200 (x : (k : K) \u2192 Option (\u0393 k)),\n PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x) =\n ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).fst,\n f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x).snd)\n[PROOFSTEP]\nintro\n[GOAL]\ncase succ.e_l.H\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nf : ((k : K) \u2192 Option (\u0393 k)) \u2192 (k : K) \u2192 Option (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn\u271d : \u2115\nx\u271d : (k : K) \u2192 Option (\u0393 k)\n\u22a2 PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (f x\u271d) =\n ((PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).fst,\n f (PointedMap.f { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } x\u271d).snd)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 (ListBlank.nth (addBottom L) n).snd = ListBlank.nth L n\n[PROOFSTEP]\nconv => rhs; rw [\u2190 addBottom_map L, ListBlank.nth_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n| (ListBlank.nth (addBottom L) n).snd = ListBlank.nth L n\n[PROOFSTEP]\nrhs; rw [\u2190 addBottom_map L, ListBlank.nth_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n| (ListBlank.nth (addBottom L) n).snd = ListBlank.nth L n\n[PROOFSTEP]\nrhs; rw [\u2190 addBottom_map L, ListBlank.nth_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n| (ListBlank.nth (addBottom L) n).snd = ListBlank.nth L n\n[PROOFSTEP]\nrhs\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n| ListBlank.nth L n\n[PROOFSTEP]\nrw [\u2190 addBottom_map L, ListBlank.nth_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 (ListBlank.nth (addBottom L) (n + 1)).fst = false\n[PROOFSTEP]\nrw [ListBlank.nth_succ, addBottom, ListBlank.tail_cons, ListBlank.nth_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 (ListBlank.head (addBottom L)).fst = true\n[PROOFSTEP]\nrw [addBottom, ListBlank.head_cons]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nS : Finset \u039b\nk : K\ns : StAct k\nq : Stmt\u2082\n\u22a2 TM2.SupportsStmt S (stRun s q) \u2194 TM2.SupportsStmt S q\n[PROOFSTEP]\ncases s\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nS : Finset \u039b\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 \u0393 k\n\u22a2 TM2.SupportsStmt S (stRun (push a\u271d) q) \u2194 TM2.SupportsStmt S q\n[PROOFSTEP]\nrfl\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nS : Finset \u039b\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.SupportsStmt S (stRun (peek a\u271d) q) \u2194 TM2.SupportsStmt S q\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nS : Finset \u039b\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.SupportsStmt S (stRun (pop a\u271d) q) \u2194 TM2.SupportsStmt S q\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =\n TM2.stepAux q (stVar v (S k) (StAct.peek f)) (update S k (stWrite v (S k) (StAct.peek f)))\n[PROOFSTEP]\nunfold stWrite\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S =\n TM2.stepAux q (stVar v (S k) (StAct.peek f))\n (update S k\n (match StAct.peek f with\n | StAct.push f => f v :: S k\n | StAct.peek a => S k\n | StAct.pop a => List.tail (S k)))\n[PROOFSTEP]\nrw [Function.update_eq_self]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM2.stepAux (stRun (StAct.peek f) q) v S = TM2.stepAux q (stVar v (S k) (StAct.peek f)) S\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\ns : StAct k\nq : Stmt\u2082\n\u22a2 trNormal (stRun s q) = goto fun x x => go k s q\n[PROOFSTEP]\ncases s\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 \u0393 k\n\u22a2 trNormal (stRun (StAct.push a\u271d) q) = goto fun x x => go k (StAct.push a\u271d) q\n[PROOFSTEP]\nrfl\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 trNormal (stRun (StAct.peek a\u271d) q) = goto fun x x => go k (StAct.peek a\u271d) q\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 trNormal (stRun (StAct.pop a\u271d) q) = goto fun x x => go k (StAct.pop a\u271d) q\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\ns : StAct k\nq : Stmt\u2082\n\u22a2 trStmts\u2081 (stRun s q) = {go k s q, ret q} \u222a trStmts\u2081 q\n[PROOFSTEP]\ncases s\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 \u0393 k\n\u22a2 trStmts\u2081 (stRun (StAct.push a\u271d) q) = {go k (StAct.push a\u271d) q, ret q} \u222a trStmts\u2081 q\n[PROOFSTEP]\nsimp only [trStmts\u2081]\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 trStmts\u2081 (stRun (StAct.peek a\u271d) q) = {go k (StAct.peek a\u271d) q, ret q} \u222a trStmts\u2081 q\n[PROOFSTEP]\nsimp only [trStmts\u2081]\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : Stmt\u2082\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 trStmts\u2081 (stRun (StAct.pop a\u271d) q) = {go k (StAct.pop a\u271d) q, ret q} \u222a trStmts\u2081 q\n[PROOFSTEP]\nsimp only [trStmts\u2081]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\n\u22a2 let v' := stVar v (S k) o;\n let Sk' := stWrite v (S k) o;\n let S' := update S k Sk';\n \u2203 L',\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S' k)))) \u2227\n TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q v' ((Tape.move Dir.right)^[List.length (S' k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ndsimp only\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) \u2227\n TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update S k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) \u2227\n TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (stWrite v (S k) o)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncases o\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 \u0393 k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) (StAct.push a\u271d)) k_1)))) \u2227\n TM1.stepAux (trStAct q (StAct.push a\u271d)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (stVar v (S k) (StAct.push a\u271d))\n ((Tape.move Dir.right)^[List.length (stWrite v (S k) (StAct.push a\u271d))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nsimp only [stWrite, stVar, trStAct, TM1.stepAux]\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) (StAct.peek a\u271d)) k_1)))) \u2227\n TM1.stepAux (trStAct q (StAct.peek a\u271d)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (stVar v (S k) (StAct.peek a\u271d))\n ((Tape.move Dir.right)^[List.length (stWrite v (S k) (StAct.peek a\u271d))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nsimp only [stWrite, stVar, trStAct, TM1.stepAux]\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) (StAct.pop a\u271d)) k_1)))) \u2227\n TM1.stepAux (trStAct q (StAct.pop a\u271d)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (stVar v (S k) (StAct.pop a\u271d))\n ((Tape.move Dir.right)^[List.length (stWrite v (S k) (StAct.pop a\u271d))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nsimp only [stWrite, stVar, trStAct, TM1.stepAux]\n[GOAL]\ncase push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 \u0393 k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (a\u271d v :: S k) k_1)))) \u2227\n TM1.stepAux q v\n (Tape.move Dir.right\n (Tape.write\n (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst,\n update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.snd k (some (a\u271d v)))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q v ((Tape.move Dir.right)^[List.length (a\u271d v :: S k)] (Tape.mk' \u2205 (addBottom L')))\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) \u2227\n TM1.stepAux q\n (a\u271d v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (a\u271d v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L')))\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (a\u271d v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (a\u271d v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.snd k\n none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (a\u271d v (List.head? (S k)))\n ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase push f =>\n have := Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k (some (f v)))\n refine'\n \u27e8_, fun k' \u21a6 _, by\n -- Porting note: `rw [...]` to `erw [...]; rfl`.\n -- https://github.com/leanprover-community/mathlib4/issues/5164\n erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this,\n addBottom_modifyNth fun a \u21a6 update a k (some (f v)), Nat.add_one, iterate_succ']\n rfl\u27e9\n refine' ListBlank.ext fun i \u21a6 _\n rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n by_cases h' : k' = k\n \u00b7 subst k'\n split_ifs with h <;>\n simp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map]\n -- Porting note: `le_refl` is required.\n \u00b7\n rw [List.getI_eq_get, List.get_append_right'] <;>\n simp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append,\n lt_add_iff_pos_right, List.length, le_refl]\n rw [\u2190 proj_map_nth, hL, ListBlank.nth_mk]\n cases' lt_or_gt_of_ne h with h h\n \u00b7 rw [List.getI_append]\n simpa only [List.length_map, List.length_reverse] using h\n \u00b7 rw [gt_iff_lt] at h \n rw [List.getI_eq_default, List.getI_eq_default] <;>\n simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append,\n List.length_map]\n \u00b7 split_ifs <;> rw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n rw [Function.update_noteq h']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) \u2227\n TM1.stepAux q v\n (Tape.move Dir.right\n (Tape.write\n (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst,\n update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.snd k (some (f v)))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase push f =>\n have := Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k (some (f v)))\n refine'\n \u27e8_, fun k' \u21a6 _, by\n -- Porting note: `rw [...]` to `erw [...]; rfl`.\n -- https://github.com/leanprover-community/mathlib4/issues/5164\n erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this,\n addBottom_modifyNth fun a \u21a6 update a k (some (f v)), Nat.add_one, iterate_succ']\n rfl\u27e9\n refine' ListBlank.ext fun i \u21a6 _\n rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n by_cases h' : k' = k\n \u00b7 subst k'\n split_ifs with h <;>\n simp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map]\n -- Porting note: `le_refl` is required.\n \u00b7\n rw [List.getI_eq_get, List.get_append_right'] <;>\n simp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append,\n lt_add_iff_pos_right, List.length, le_refl]\n rw [\u2190 proj_map_nth, hL, ListBlank.nth_mk]\n cases' lt_or_gt_of_ne h with h h\n \u00b7 rw [List.getI_append]\n simpa only [List.length_map, List.length_reverse] using h\n \u00b7 rw [gt_iff_lt] at h \n rw [List.getI_eq_default, List.getI_eq_default] <;>\n simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append,\n List.length_map]\n \u00b7 split_ifs <;> rw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n rw [Function.update_noteq h']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) \u2227\n TM1.stepAux q v\n (Tape.move Dir.right\n (Tape.write\n (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst,\n update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.snd k (some (f v)))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nhave := Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k (some (f v)))\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) \u2227\n TM1.stepAux q v\n (Tape.move Dir.right\n (Tape.write\n (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst,\n update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.snd k (some (f v)))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nrefine'\n \u27e8_, fun k' \u21a6 _, by\n -- Porting note: `rw [...]` to `erw [...]; rfl`.\n -- https://github.com/leanprover-community/mathlib4/issues/5164\n erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this,\n addBottom_modifyNth fun a \u21a6 update a k (some (f v)), Nat.add_one, iterate_succ']\n rfl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 TM1.stepAux q v\n (Tape.move Dir.right\n (Tape.write\n (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst,\n update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.snd k (some (f v)))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' \u2205 (addBottom ?m.655566)))\n[PROOFSTEP]\nerw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this, addBottom_modifyNth fun a \u21a6 update a k (some (f v)),\n Nat.add_one, iterate_succ']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 TM1.stepAux q v\n (Tape.move Dir.right\n ((Tape.move Dir.right)^[List.length (S k)]\n (Tape.mk' \u2205 (addBottom (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L))))) =\n TM1.stepAux q v\n ((Tape.move Dir.right \u2218 (Tape.move Dir.right)^[List.length (S k)]) (Tape.mk' \u2205 (addBottom ?m.655566)))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\n\u22a2 ListBlank ((k : K) \u2192 Option (\u0393 k))\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\n\u22a2 ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L) =\n ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))\n[PROOFSTEP]\nrefine' ListBlank.ext fun i \u21a6 _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\n\u22a2 ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L))\n i =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nrw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\n\u22a2 ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nby_cases h' : k' = k\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\nh' : k' = k\n\u22a2 ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nsubst k'\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\n\u22a2 ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k)))) i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : i = List.length (S k)\n\u22a2 update (ListBlank.nth L i) k (some (f v)) k =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k)))) i\n[PROOFSTEP]\nsimp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map]\n -- Porting note: `le_refl` is required.\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : \u00aci = List.length (S k)\n\u22a2 ListBlank.nth L i k = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k)))) i\n[PROOFSTEP]\nsimp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map]\n -- Porting note: `le_refl` is required.\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : i = List.length (S k)\n\u22a2 some (f v) = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\nrw [List.getI_eq_get, List.get_append_right']\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : i = List.length (S k)\n\u22a2 some (f v) =\n List.get [some (f v)]\n { val := i - List.length (List.reverse (List.map some (S k))),\n isLt := (_ : i - List.length (List.reverse (List.map some (S k))) < List.length [some (f v)]) }\n[PROOFSTEP]\nsimp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append,\n lt_add_iff_pos_right, List.length, le_refl]\n[GOAL]\ncase pos.h\u2081\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : i = List.length (S k)\n\u22a2 List.length (List.reverse (List.map some (S k))) \u2264 i\n[PROOFSTEP]\nsimp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append,\n lt_add_iff_pos_right, List.length, le_refl]\n[GOAL]\ncase pos.h\u2082\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : i = List.length (S k)\n\u22a2 i < List.length (List.reverse (List.map some (S k)) ++ [some (f v)])\n[PROOFSTEP]\nsimp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append,\n lt_add_iff_pos_right, List.length, le_refl]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : \u00aci = List.length (S k)\n\u22a2 ListBlank.nth L i k = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\nrw [\u2190 proj_map_nth, hL, ListBlank.nth_mk]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh : \u00aci = List.length (S k)\n\u22a2 List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h with h h\n[GOAL]\ncase neg.inl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : i < List.length (S k)\n\u22a2 List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\nrw [List.getI_append]\n[GOAL]\ncase neg.inl.h\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : i < List.length (S k)\n\u22a2 i < List.length (List.reverse (List.map some (S k)))\n[PROOFSTEP]\nsimpa only [List.length_map, List.length_reverse] using h\n[GOAL]\ncase neg.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : i > List.length (S k)\n\u22a2 List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\nrw [gt_iff_lt] at h \n[GOAL]\ncase neg.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : List.length (S k) < i\n\u22a2 List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i\n[PROOFSTEP]\nrw [List.getI_eq_default, List.getI_eq_default]\n[GOAL]\ncase neg.inr.hn\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : List.length (S k) < i\n\u22a2 List.length (List.reverse (List.map some (S k)) ++ [some (f v)]) \u2264 i\n[PROOFSTEP]\nsimp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map]\n[GOAL]\ncase neg.inr.hn\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\ni : \u2115\nh\u271d : \u00aci = List.length (S k)\nh : List.length (S k) < i\n\u22a2 List.length (List.reverse (List.map some (S k))) \u2264 i\n[PROOFSTEP]\nsimp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\nh' : \u00ack' = k\n\u22a2 ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : i = List.length (S k)\n\u22a2 update (ListBlank.nth L i) k (some (f v)) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : \u00aci = List.length (S k)\n\u22a2 ListBlank.nth L i k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 \u0393 k\nthis :\n \u2200 (L R : ListBlank \u0393') (n : \u2115),\n Tape.write ((ListBlank.nth R n).fst, update (ListBlank.nth R n).snd k (some (f v)))\n ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =\n (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.fst, update a.snd k (some (f v)))) n R))\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : i = List.length (S k)\n\u22a2 ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h']\n[GOAL]\ncase peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) \u2227\n TM1.stepAux q\n (a\u271d v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (a\u271d v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L')))\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (a\u271d v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (a\u271d v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.snd k\n none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (a\u271d v (List.head? (S k)))\n ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase peek f =>\n rw [Function.update_eq_self]\n use L, hL; rw [Tape.move_left_right]; congr\n cases e : S k; \u00b7 rfl\n rw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n addBottom_nth_snd, stk_nth_val _ (hL k), e, List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length]\n rfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) \u2227\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase peek f =>\n rw [Function.update_eq_self]\n use L, hL; rw [Tape.move_left_right]; congr\n cases e : S k; \u00b7 rfl\n rw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n addBottom_nth_snd, stk_nth_val _ (hL k), e, List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length]\n rfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) \u2227\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nrw [Function.update_eq_self]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nuse L, hL\n[GOAL]\ncase right\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head k))\n (Tape.move Dir.right\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))))) =\n TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n[PROOFSTEP]\nrw [Tape.move_left_right]\n[GOAL]\ncase right\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head k))\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))) =\n TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase right.e_a.e_a\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head k =\n List.head? (S k)\n[PROOFSTEP]\ncases e : S k\n[GOAL]\ncase right.e_a.e_a.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\ne : S k = []\n\u22a2 Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))).head k =\n List.head? []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right.e_a.e_a.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\ne : S k = head\u271d :: tail\u271d\n\u22a2 Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (head\u271d :: tail\u271d)] (Tape.mk' \u2205 (addBottom L)))).head\n k =\n List.head? (head\u271d :: tail\u271d)\n[PROOFSTEP]\nrw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n addBottom_nth_snd, stk_nth_val _ (hL k), e, List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length]\n[GOAL]\ncase right.e_a.e_a.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\ne : S k = head\u271d :: tail\u271d\n\u22a2 some head\u271d = List.head? (head\u271d :: tail\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\na\u271d : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (a\u271d v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (a\u271d v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.snd k\n none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (a\u271d v (List.head? (S k)))\n ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase pop f =>\n cases' e : S k with hd tl\n \u00b7 simp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length, Tape.write_mk', List.head?,\n iterate_zero_apply, List.tail_nil]\n rw [\u2190 e, Function.update_eq_self]\n exact \u27e8L, hL, by rw [addBottom_head_fst, cond]\u27e9\n \u00b7 refine'\n \u27e8_, fun k' \u21a6 _, by\n erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ',\n Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k none), addBottom_modifyNth fun a \u21a6 update a k none,\n addBottom_nth_snd, stk_nth_val _ (hL k), e,\n show (List.cons hd tl).reverse.get? tl.length = some hd by\n rw [List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length],\n List.head?, List.tail]\u27e9\n refine' ListBlank.ext fun i \u21a6 _\n rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n by_cases h' : k' = k\n \u00b7 subst k'\n split_ifs with h <;> simp only [Function.update_same, ListBlank.nth_mk, List.tail]\n \u00b7 rw [List.getI_eq_default]\n \u00b7 rfl\n rw [h, List.length_reverse, List.length_map]\n rw [\u2190 proj_map_nth, hL, ListBlank.nth_mk, e, List.map, List.reverse_cons]\n cases' lt_or_gt_of_ne h with h h\n \u00b7 rw [List.getI_append]\n simpa only [List.length_map, List.length_reverse] using h\n \u00b7 rw [gt_iff_lt] at h \n rw [List.getI_eq_default, List.getI_eq_default] <;>\n simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append,\n List.length_map]\n \u00b7 split_ifs <;> rw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n rw [Function.update_noteq h']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.snd k\n none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (f v (List.head? (S k)))\n ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncase pop f =>\n cases' e : S k with hd tl\n \u00b7 simp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length, Tape.write_mk', List.head?,\n iterate_zero_apply, List.tail_nil]\n rw [\u2190 e, Function.update_eq_self]\n exact \u27e8L, hL, by rw [addBottom_head_fst, cond]\u27e9\n \u00b7 refine'\n \u27e8_, fun k' \u21a6 _, by\n erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ',\n Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k none), addBottom_modifyNth fun a \u21a6 update a k none,\n addBottom_nth_snd, stk_nth_val _ (hL k), e,\n show (List.cons hd tl).reverse.get? tl.length = some hd by\n rw [List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length],\n List.head?, List.tail]\u27e9\n refine' ListBlank.ext fun i \u21a6 _\n rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n by_cases h' : k' = k\n \u00b7 subst k'\n split_ifs with h <;> simp only [Function.update_same, ListBlank.nth_mk, List.tail]\n \u00b7 rw [List.getI_eq_default]\n \u00b7 rfl\n rw [h, List.length_reverse, List.length_map]\n rw [\u2190 proj_map_nth, hL, ListBlank.nth_mk, e, List.map, List.reverse_cons]\n cases' lt_or_gt_of_ne h with h h\n \u00b7 rw [List.getI_append]\n simpa only [List.length_map, List.length_reverse] using h\n \u00b7 rw [gt_iff_lt] at h \n rw [List.getI_eq_default, List.getI_eq_default] <;>\n simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append,\n List.length_map]\n \u00b7 split_ifs <;> rw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n rw [Function.update_noteq h']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))).head.snd k\n none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (f v (List.head? (S k)))\n ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\ncases' e : S k with hd tl\n[GOAL]\ncase nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\ne : S k = []\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail []) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (f v\n (Prod.snd (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))).head.snd\n k none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (f v (List.head? []))\n ((Tape.move Dir.right)^[List.length (List.tail [])] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nsimp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length, Tape.write_mk', List.head?,\n iterate_zero_apply, List.tail_nil]\n[GOAL]\ncase nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\ne : S k = []\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k [] k_1)))) \u2227\n (bif (ListBlank.head (addBottom L)).fst then TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L))\n else\n TM1.stepAux q (f v (Prod.snd (ListBlank.head \u2205) k))\n (Tape.mk' (ListBlank.tail \u2205)\n (ListBlank.cons ((ListBlank.head \u2205).fst, update (ListBlank.head \u2205).snd k none) (addBottom L)))) =\n TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L'))\n[PROOFSTEP]\nrw [\u2190 e, Function.update_eq_self]\n[GOAL]\ncase nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\ne : S k = []\n\u22a2 \u2203 L',\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227\n (bif (ListBlank.head (addBottom L)).fst then TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L))\n else\n TM1.stepAux q (f v (Prod.snd (ListBlank.head \u2205) k))\n (Tape.mk' (ListBlank.tail \u2205)\n (ListBlank.cons ((ListBlank.head \u2205).fst, update (ListBlank.head \u2205).snd k none) (addBottom L)))) =\n TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L'))\n[PROOFSTEP]\nexact \u27e8L, hL, by rw [addBottom_head_fst, cond]\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\ne : S k = []\n\u22a2 (bif (ListBlank.head (addBottom L)).fst then TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L))\n else\n TM1.stepAux q (f v (Prod.snd (ListBlank.head \u2205) k))\n (Tape.mk' (ListBlank.tail \u2205)\n (ListBlank.cons ((ListBlank.head \u2205).fst, update (ListBlank.head \u2205).snd k none) (addBottom L)))) =\n TM1.stepAux q (f v none) (Tape.mk' \u2205 (addBottom L))\n[PROOFSTEP]\nrw [addBottom_head_fst, cond]\n[GOAL]\ncase cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\n\u22a2 \u2203 L',\n (\u2200 (k_1 : K),\n ListBlank.map (proj k_1) L' =\n ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k_1)))) \u2227\n (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (f v\n (Prod.snd\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head\n k))\n (Tape.write\n ((Tape.move Dir.left\n ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left\n ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head.snd\n k none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (f v (List.head? (hd :: tl)))\n ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' \u2205 (addBottom L')))\n[PROOFSTEP]\nrefine'\n \u27e8_, fun k' \u21a6 _, by\n erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ',\n Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k none), addBottom_modifyNth fun a \u21a6 update a k none,\n addBottom_nth_snd, stk_nth_val _ (hL k), e,\n show (List.cons hd tl).reverse.get? tl.length = some hd by\n rw [List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length],\n List.head?, List.tail]\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\n\u22a2 (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L))).head.fst then\n TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))\n else\n TM1.stepAux q\n (f v\n (Prod.snd\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head k))\n (Tape.write\n ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head.fst,\n update\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))).head.snd\n k none)\n (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' \u2205 (addBottom L)))))) =\n TM1.stepAux q (f v (List.head? (hd :: tl)))\n ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' \u2205 (addBottom ?m.672274)))\n[PROOFSTEP]\nerw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ',\n Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat,\n Tape.write_move_right_n fun a : \u0393' \u21a6 (a.1, update a.2 k none), addBottom_modifyNth fun a \u21a6 update a k none,\n addBottom_nth_snd, stk_nth_val _ (hL k), e,\n show (List.cons hd tl).reverse.get? tl.length = some hd by\n rw [List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length],\n List.head?, List.tail]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\n\u22a2 List.get? (List.reverse (hd :: tl)) (List.length tl) = some hd\n[PROOFSTEP]\nrw [List.reverse_cons, \u2190 List.length_reverse, List.get?_concat_length]\n[GOAL]\ncase cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\n\u22a2 ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L) =\n ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))\n[PROOFSTEP]\nrefine' ListBlank.ext fun i \u21a6 _\n[GOAL]\ncase cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\n\u22a2 ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L)) i =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nrw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]\n[GOAL]\ncase cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\n\u22a2 ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nby_cases h' : k' = k\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\nh' : k' = k\n\u22a2 ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nsubst k'\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\n\u22a2 ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k)))) i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : i = List.length tl\n\u22a2 update (ListBlank.nth L i) k none k =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k)))) i\n[PROOFSTEP]\nsimp only [Function.update_same, ListBlank.nth_mk, List.tail]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : \u00aci = List.length tl\n\u22a2 ListBlank.nth L i k =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k)))) i\n[PROOFSTEP]\nsimp only [Function.update_same, ListBlank.nth_mk, List.tail]\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : i = List.length tl\n\u22a2 none = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\nrw [List.getI_eq_default]\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : i = List.length tl\n\u22a2 none = default\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.hn\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : i = List.length tl\n\u22a2 List.length (List.reverse (List.map some tl)) \u2264 i\n[PROOFSTEP]\nrw [h, List.length_reverse, List.length_map]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : \u00aci = List.length tl\n\u22a2 ListBlank.nth L i k = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\nrw [\u2190 proj_map_nth, hL, ListBlank.nth_mk, e, List.map, List.reverse_cons]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh : \u00aci = List.length tl\n\u22a2 List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h with h h\n[GOAL]\ncase neg.inl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : i < List.length tl\n\u22a2 List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\nrw [List.getI_append]\n[GOAL]\ncase neg.inl.h\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : i < List.length tl\n\u22a2 i < List.length (List.reverse (List.map some tl))\n[PROOFSTEP]\nsimpa only [List.length_map, List.length_reverse] using h\n[GOAL]\ncase neg.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : i > List.length tl\n\u22a2 List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\nrw [gt_iff_lt] at h \n[GOAL]\ncase neg.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : List.length tl < i\n\u22a2 List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i\n[PROOFSTEP]\nrw [List.getI_eq_default, List.getI_eq_default]\n[GOAL]\ncase neg.inr.hn\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : List.length tl < i\n\u22a2 List.length (List.reverse (List.map some tl)) \u2264 i\n[PROOFSTEP]\nsimp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map]\n[GOAL]\ncase neg.inr.hn\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\ni : \u2115\nh\u271d : \u00aci = List.length tl\nh : List.length tl < i\n\u22a2 List.length (List.reverse (List.map some tl) ++ [some hd]) \u2264 i\n[PROOFSTEP]\nsimp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\nh' : \u00ack' = k\n\u22a2 ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : i = List.length tl\n\u22a2 update (ListBlank.nth L i) k none k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : \u00aci = List.length tl\n\u22a2 ListBlank.nth L i k' =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h', \u2190 proj_map_nth, hL]\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nk : K\nq : TM1.Stmt \u0393' \u039b' \u03c3\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nf : \u03c3 \u2192 Option (\u0393 k) \u2192 \u03c3\nhd : \u0393 k\ntl : List (\u0393 k)\ne : S k = hd :: tl\nk' : K\ni : \u2115\nh' : \u00ack' = k\nh\u271d : i = List.length tl\n\u22a2 ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i\n[PROOFSTEP]\nrw [Function.update_noteq h']\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn : \u2115\nH : n \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn : \u2115\nH\u271d : n \u2264 List.length S\nH : Nat.zero \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\napply (IH (le_of_lt H)).tail\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nrw [iterate_succ_apply']\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 { l := some (go k o q), var := v,\n Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) } \u2208\n TM1.step (tr M) { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nsimp only [TM1.step, TM1.stepAux, tr, Tape.mk'_nth_nat, Tape.move_right_n_head, addBottom_nth_snd, Option.mem_def]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n (bif Option.isNone (ListBlank.nth L n k) then\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n else\n { l := some (go k o q), var := v,\n Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n some\n { l := some (go k o q), var := v,\n Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }\n[PROOFSTEP]\nrw [stk_nth_val _ hL, List.get?_eq_get]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 some\n (bif Option.isNone (some (List.get (List.reverse S) { val := n, isLt := ?succ })) then\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))\n else\n { l := some (go k o q), var := v,\n Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }) =\n some\n { l := some (go k o q), var := v,\n Tape := Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))) }\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\no : StAct k\nq : Stmt\u2082\nv : \u03c3\nS : List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S))\nn\u271d : \u2115\nH\u271d : n\u271d \u2264 List.length S\nn : \u2115\nIH :\n n \u2264 List.length S \u2192\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\nH : Nat.succ n \u2264 List.length S\n\u22a2 n < List.length (List.reverse S)\n[PROOFSTEP]\nrwa [List.length_reverse]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.zero] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n[PROOFSTEP]\nrefine' Reaches\u2080.head _ IH\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) } \u2208\n TM1.step (tr M)\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nsimp only [Option.mem_def, TM1.step]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 some (TM1.stepAux (tr M (ret q)) v ((Tape.move Dir.right)^[Nat.succ n] (Tape.mk' \u2205 (addBottom L)))) =\n some { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nrw [Option.some_inj, tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, TM1.stepAux,\n iterate_succ', Function.comp_apply, Tape.move_right_left]\n[GOAL]\ncase succ\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : Stmt\u2082\nv : \u03c3\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nn : \u2115\nIH :\n Reaches\u2080 (TM1.step (tr M))\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n { l := some (ret q), var := v, Tape := Tape.mk' \u2205 (addBottom L) }\n\u22a2 (bif false then\n TM1.stepAux (trNormal q) v (Tape.move Dir.right ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L))))\n else TM1.stepAux (goto fun x x => ret q) v ((Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)))) =\n { l := some (ret q), var := v, Tape := (Tape.move Dir.right)^[n] (Tape.mk' \u2205 (addBottom L)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (stRun o q) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nsimp only [trNormal_run, step_run]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nhave hgo := tr_respects_aux\u2081 M o q v (hT k) _ le_rfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nobtain \u27e8T', hT', hrun\u27e9 := tr_respects_aux\u2082 hT o\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.705248 (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct ?m.705247 o) ?m.705248 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux ?m.705247 (stVar ?m.705248 (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.705248 (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nhave := hgo.tail' rfl\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite ?m.705248 (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct ?m.705247 o) ?m.705248 ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux ?m.705247 (stVar ?m.705248 (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite ?m.705248 (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\nthis :\n Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nrw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd, stk_nth_val _ (hT k),\n List.get?_len_le (le_of_eq (List.length_reverse _)), Option.isNone, cond, hrun, TM1.stepAux] at this \n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n (match\n match none with\n | some val => false\n | none => true with\n | true =>\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\n | false =>\n TM1.stepAux (goto fun x x => go k o q) v\n (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nobtain \u27e8c, gc, rc\u27e9 := IH hT'\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n (match\n match none with\n | some val => false\n | none => true with\n | true =>\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\n | false =>\n TM1.stepAux (goto fun x x => go k o q) v\n (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q ?m.709134 fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.709134 (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' \u2205 (addBottom T))) b\n[PROOFSTEP]\nrefine' \u27e8c, gc, (this.to\u2080.trans (tr_respects_aux\u2083 M _) c (TransGen.head' rfl _)).to_reflTransGen\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n (match\n match none with\n | some val => false\n | none => true with\n | true =>\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\n | false =>\n TM1.stepAux (goto fun x x => go k o q) v\n (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n[PROOFSTEP]\nrw [tr, TM1.stepAux, Tape.mk'_head, addBottom_head_fst]\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nq : TM2.Stmt (fun k => \u0393 k) \u039b \u03c3\nv : \u03c3\nT : ListBlank ((i : K) \u2192 Option (\u0393 i))\nk : K\nS : (k : K) \u2192 List (\u0393 k)\nhT : \u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))\no : StAct k\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} {T : ListBlank ((k : K) \u2192 Option (\u0393 k))},\n (\u2200 (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom T))) b\nhgo :\n Reaches\u2080 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)) }\nT' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT' :\n \u2200 (k_1 : K),\n ListBlank.map (proj k_1) T' =\n ListBlank.mk (List.reverse (List.map some (update (fun k => S k) k (stWrite v (S k) o) k_1)))\nhrun :\n TM1.stepAux (trStAct (goto fun x x => ret q) o) v\n ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T))) =\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)] (Tape.mk' \u2205 (addBottom T')))\nthis :\n Reaches\u2081 (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' \u2205 (addBottom T) }\n (match\n match none with\n | some val => false\n | none => true with\n | true =>\n TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n ((Tape.move Dir.right)^[List.length (update (fun k => S k) k (stWrite v (S k) o) k)]\n (Tape.mk' \u2205 (addBottom T')))\n | false =>\n TM1.stepAux (goto fun x x => go k o q) v\n (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' \u2205 (addBottom T)))))\nc : TM1.Cfg \u0393' \u039b' \u03c3\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) fun k_1 => update (fun k => S k) k (stWrite v (S k) o) k_1) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))) c\n\u22a2 ReflTransGen (fun a b => b \u2208 TM1.step (tr M) a)\n (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T'))\n else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' \u2205 (addBottom T')))\n c\n[PROOFSTEP]\nexact rc\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\n\u22a2 Respects (TM2.step M) (TM1.step (tr M)) TrCfg\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nc\u2081 : Cfg\u2082\nc\u2082 : TM1.Cfg \u0393' \u039b' \u03c3\nh : TrCfg c\u2081 c\u2082\n\u22a2 match TM2.step M c\u2081 with\n | some b\u2081 => \u2203 b\u2082, TrCfg b\u2081 b\u2082 \u2227 Reaches\u2081 (TM1.step (tr M)) c\u2082 b\u2082\n | none => TM1.step (tr M) c\u2082 = none\n[PROOFSTEP]\ncases' h with l v S L hT\n[GOAL]\ncase mk\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : Option \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := l, var := v, stk := S } with\n | some b\u2081 =>\n \u2203 b\u2082,\n TrCfg b\u2081 b\u2082 \u2227\n Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n | none => TM1.step (tr M) { l := Option.map normal l, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none\n[PROOFSTEP]\ncases' l with l\n[GOAL]\ncase mk.none\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 match TM2.step M { l := none, var := v, stk := S } with\n | some b\u2081 =>\n \u2203 b\u2082,\n TrCfg b\u2081 b\u2082 \u2227\n Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n | none => TM1.step (tr M) { l := Option.map normal none, var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.some\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 match TM2.step M { l := some l, var := v, stk := S } with\n | some b\u2081 =>\n \u2203 b\u2082,\n TrCfg b\u2081 b\u2082 \u2227\n Reaches\u2081 (TM1.step (tr M)) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n | none => TM1.step (tr M) { l := Option.map normal (some l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } = none\n[PROOFSTEP]\nsimp only [TM2.step, Respects, Option.map_some']\n[GOAL]\ncase mk.some\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b\u2082,\n TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n[PROOFSTEP]\nrsuffices \u27e8b, c, r\u27e9 : \u2203 b, _ \u2227 Reaches (TM1.step (tr M)) _ _\n[GOAL]\ncase mk.some.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nb : ?m.714525\nc : ?m.714852 b\nr : Reaches (TM1.step (tr M)) (?m.714853 b) (?m.714854 b)\n\u22a2 \u2203 b\u2082,\n TrCfg (TM2.stepAux (M l) v S) b\u2082 \u2227\n Reaches\u2081 (TM1.step (tr M)) { l := some (normal l), var := v, Tape := Tape.mk' \u2205 (addBottom L) } b\u2082\n[PROOFSTEP]\nexact \u27e8b, c, TransGen.head' rfl r\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (M l) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (tr M (normal l)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nsimp only [tr]\n -- Porting note: `refine'` failed because of implicit lambda, so `induction` is used.\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (M l) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (M l)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\ngeneralize M l = N\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nN : Stmt\u2082\n\u22a2 \u2203 b, TrCfg (TM2.stepAux N v S) b \u2227 Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\ninduction N using stmtStRec generalizing v S L hT with\n| H\u2081 k s q IH => exact tr_respects_aux M hT s @IH\n| H\u2082 a _ IH => exact IH _ hT\n| H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold TM2.stepAux trNormal TM1.stepAux\n simp only []\n cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n| H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n| H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\nl : \u039b\nN : Stmt\u2082\n\u22a2 \u2203 b, TrCfg (TM2.stepAux N v S) b \u2227 Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal N) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\ninduction N using stmtStRec generalizing v S L hT with\n| H\u2081 k s q IH => exact tr_respects_aux M hT s @IH\n| H\u2082 a _ IH => exact IH _ hT\n| H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold TM2.stepAux trNormal TM1.stepAux\n simp only []\n cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n| H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n| H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\ncase H\u2081\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nk : K\ns : StAct k\nq : Stmt\u2082\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (stRun s q) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun s q)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\n\n| H\u2081 k s q IH => exact tr_respects_aux M hT s @IH\n[GOAL]\ncase H\u2081\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nk : K\ns : StAct k\nq : Stmt\u2082\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (stRun s q) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun s q)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nexact tr_respects_aux M hT s @IH\n[GOAL]\ncase H\u2082\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\na : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u271d v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u271d) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.load a q\u271d) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.load a q\u271d)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\n\n| H\u2082 a _ IH => exact IH _ hT\n[GOAL]\ncase H\u2082\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\na : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u271d v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u271d) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.load a q\u271d) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.load a q\u271d)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nexact IH _ hT\n[GOAL]\ncase H\u2083\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.branch p q\u2081 q\u2082) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\n\n| H\u2083 p q\u2081 q\u2082 IH\u2081 IH\u2082 =>\n unfold TM2.stepAux trNormal TM1.stepAux\n simp only []\n cases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n[GOAL]\ncase H\u2083\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.branch p q\u2081 q\u2082) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nunfold TM2.stepAux trNormal TM1.stepAux\n[GOAL]\ncase H\u2083\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M))\n (bif (fun x => p) (Tape.mk' \u2205 (addBottom L)).head v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n b\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase H\u2083\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M))\n (bif p v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n b\n[PROOFSTEP]\ncases p v <;> [exact IH\u2082 _ hT; exact IH\u2081 _ hT]\n[GOAL]\ncase H\u2083\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (bif p v then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M))\n (bif p v then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n b\n[PROOFSTEP]\ncases p v\n[GOAL]\ncase H\u2083.false\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (bif false then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M))\n (bif false then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n b\n[PROOFSTEP]\nexact IH\u2082 _ hT\n[GOAL]\ncase H\u2083.true\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\np : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2081 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))) b\nIH\u2082 :\n \u2200 {v : \u03c3} {S : (k : K) \u2192 List (\u0393 k)} (L : ListBlank ((k : K) \u2192 Option (\u0393 k))),\n (\u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))) \u2192\n \u2203 b,\n TrCfg (TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L))) b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (bif true then TM2.stepAux q\u2081 v S else TM2.stepAux q\u2082 v S) b \u2227\n Reaches (TM1.step (tr M))\n (bif true then TM1.stepAux (trNormal q\u2081) v (Tape.mk' \u2205 (addBottom L))\n else TM1.stepAux (trNormal q\u2082) v (Tape.mk' \u2205 (addBottom L)))\n b\n[PROOFSTEP]\nexact IH\u2081 _ hT\n[GOAL]\ncase H\u2084\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nl\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.goto l\u271d) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.goto l\u271d)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\n\n| H\u2084 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\ncase H\u2084\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nl\u271d : \u03c3 \u2192 \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux (TM2.Stmt.goto l\u271d) v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (TM2.Stmt.goto l\u271d)) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\ncase H\u2085\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux TM2.Stmt.halt v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal TM2.Stmt.halt) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\n\n| H\u2085 => exact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\ncase H\u2085\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nl : \u039b\nv : \u03c3\nS : (k : K) \u2192 List (\u0393 k)\nL : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))\n\u22a2 \u2203 b,\n TrCfg (TM2.stepAux TM2.Stmt.halt v S) b \u2227\n Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal TM2.Stmt.halt) v (Tape.mk' \u2205 (addBottom L))) b\n[PROOFSTEP]\nexact \u27e8_, \u27e8_, hT\u27e9, ReflTransGen.refl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TrCfg (TM2.init k L) (TM1.init (trInit k L))\n[PROOFSTEP]\nrw [(_ : TM1.init _ = _)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TrCfg (TM2.init k L) ?m.724426\n[PROOFSTEP]\nrefine' \u27e8ListBlank.mk (L.reverse.map fun a \u21a6 update default k (some a)), fun k' \u21a6 _\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\n\u22a2 ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L))) =\n ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))\n[PROOFSTEP]\nrefine' ListBlank.ext fun i \u21a6 _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 ListBlank.nth\n (ListBlank.map (proj k') (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) i =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nrw [ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?, List.map_map]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nhave :\n ((proj k').f \u2218 fun a => update (\u03b2 := fun k => Option (\u0393 k)) default k (some a)) = fun a =>\n (proj k').f (update (\u03b2 := fun k => Option (\u0393 k)) default k (some a)) :=\n rfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (List.get? (List.map ((proj k').f \u2218 fun a => update default k (some a)) (List.reverse L)) i) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nrw [this, List.get?_map, proj, PointedMap.mk_val]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nsimp only []\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nby_cases h : k' = k\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : k' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nsubst k'\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k) (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k)))) i\n[PROOFSTEP]\nsimp only [Function.update_same]\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\ni : \u2115\nthis : ((proj k).f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k) (update default k (some a))\n\u22a2 Option.iget (Option.map (fun a => some a) (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some L))) i\n[PROOFSTEP]\nrw [ListBlank.nth_mk, List.getI_eq_iget_get?, \u2190 List.map_reverse, List.get?_map]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => update default k (some a) k') (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update (fun x => []) k L k')))) i\n[PROOFSTEP]\nsimp only [Function.update_noteq h]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) =\n ListBlank.nth (ListBlank.mk (List.reverse (List.map some []))) i\n[PROOFSTEP]\nrw [ListBlank.nth_mk, List.getI_eq_iget_get?, List.map, List.reverse_nil]\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') (List.get? (List.reverse L) i)) = Option.iget (List.get? [] i)\n[PROOFSTEP]\ncases L.reverse.get? i\n[GOAL]\ncase neg.none\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\n\u22a2 Option.iget (Option.map (fun a => default k') none) = Option.iget (List.get? [] i)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.some\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nk' : K\ni : \u2115\nthis : ((proj k').f \u2218 fun a => update default k (some a)) = fun a => PointedMap.f (proj k') (update default k (some a))\nh : \u00ack' = k\nval\u271d : \u0393 k\n\u22a2 Option.iget (Option.map (fun a => default k') (some val\u271d)) = Option.iget (List.get? [] i)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 TM1.init (trInit k L) =\n { l := Option.map normal (some default), var := default,\n Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }\n[PROOFSTEP]\nrw [trInit, TM1.init]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n Tape :=\n Tape.mk\u2081\n (let L' := List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L);\n (true, (List.headI L').snd) :: List.tail L') } =\n { l := Option.map normal (some default), var := default,\n Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 { l := some default, var := default,\n Tape :=\n Tape.mk\u2081\n ((true, (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))).snd) ::\n List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))) } =\n { l := Option.map normal (some default), var := default,\n Tape := Tape.mk' \u2205 (addBottom (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_Tape.e_l.e_head.e_snd\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L))).snd =\n ListBlank.head (ListBlank.mk (List.map (fun a => update default k (some a)) (List.reverse L)))\n[PROOFSTEP]\ncases L.reverse\n[GOAL]\ncase e_Tape.e_l.e_tail\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (List.reverse L)) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) (List.reverse L)))\n[PROOFSTEP]\ncases L.reverse\n[GOAL]\ncase e_Tape.e_l.e_head.e_snd.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) [])).snd =\n ListBlank.head (ListBlank.mk (List.map (fun a => update default k (some a)) []))\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase e_Tape.e_l.e_head.e_snd.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) [])).snd =\n ListBlank.head (ListBlank.mk (List.map (fun a => update default k (some a)) []))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_Tape.e_l.e_head.e_snd.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d))).snd =\n ListBlank.head (ListBlank.mk (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase e_Tape.e_l.e_head.e_snd.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 (List.headI (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d))).snd =\n ListBlank.head (ListBlank.mk (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_Tape.e_l.e_tail.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) []) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) []))\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase e_Tape.e_l.e_tail.nil\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) []) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) []))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.tail (List.map (fun a => (false, update (fun x => none) k (some a))) (head\u271d :: tail\u271d)) =\n List.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }.f\n (List.tail (List.map (fun a => update default k (some a)) (head\u271d :: tail\u271d)))\n[PROOFSTEP]\nsimp only [List.map_map, List.tail_cons, List.map]\n[GOAL]\ncase e_Tape.e_l.e_tail.cons\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nhead\u271d : \u0393 k\ntail\u271d : List (\u0393 k)\n\u22a2 List.map (fun a => (false, update (fun x => none) k (some a))) tail\u271d =\n List.map (Prod.mk false \u2218 fun a => update default k (some a)) tail\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nL\u2081 : ListBlank \u0393'\nL\u2082 : List (\u0393 k)\nH\u2081 : L\u2081 \u2208 TM1.eval (tr M) (trInit k L)\nH\u2082 : L\u2082 \u2208 TM2.eval M k L\n\u22a2 \u2203 S L',\n addBottom L' = L\u2081 \u2227\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227 S k = L\u2082\n[PROOFSTEP]\nobtain \u27e8c\u2081, h\u2081, rfl\u27e9 := (Part.mem_map_iff _).1 H\u2081\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL L\u2082 : List (\u0393 k)\nH\u2082 : L\u2082 \u2208 TM2.eval M k L\nc\u2081 : TM1.Cfg \u0393' \u039b' \u03c3\nh\u2081 : c\u2081 \u2208 eval (TM1.step (tr M)) (TM1.init (trInit k L))\nH\u2081 : Tape.right\u2080 c\u2081.Tape \u2208 TM1.eval (tr M) (trInit k L)\n\u22a2 \u2203 S L',\n addBottom L' = Tape.right\u2080 c\u2081.Tape \u2227\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227 S k = L\u2082\n[PROOFSTEP]\nobtain \u27e8c\u2082, h\u2082, rfl\u27e9 := (Part.mem_map_iff _).1 H\u2082\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nc\u2081 : TM1.Cfg \u0393' \u039b' \u03c3\nh\u2081 : c\u2081 \u2208 eval (TM1.step (tr M)) (TM1.init (trInit k L))\nH\u2081 : Tape.right\u2080 c\u2081.Tape \u2208 TM1.eval (tr M) (trInit k L)\nc\u2082 : Cfg\u2082\nh\u2082 : c\u2082 \u2208 eval (TM2.step M) (TM2.init k L)\nH\u2082 : TM2.Cfg.stk c\u2082 k \u2208 TM2.eval M k L\n\u22a2 \u2203 S L',\n addBottom L' = Tape.right\u2080 c\u2081.Tape \u2227\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227\n S k = TM2.Cfg.stk c\u2082 k\n[PROOFSTEP]\nobtain \u27e8_, \u27e8L', hT\u27e9, h\u2083\u27e9 := Turing.tr_eval (tr_respects M) (trCfg_init k L) h\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nc\u2081 : TM1.Cfg \u0393' \u039b' \u03c3\nh\u2081 : c\u2081 \u2208 eval (TM1.step (tr M)) (TM1.init (trInit k L))\nH\u2081 : Tape.right\u2080 c\u2081.Tape \u2208 TM1.eval (tr M) (trInit k L)\nq\u271d : Option \u039b\nv\u271d : \u03c3\nS\u271d : (k : K) \u2192 List (\u0393 k)\nL' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S\u271d k)))\nh\u2082 : { l := q\u271d, var := v\u271d, stk := S\u271d } \u2208 eval (TM2.step M) (TM2.init k L)\nH\u2082 : TM2.Cfg.stk { l := q\u271d, var := v\u271d, stk := S\u271d } k \u2208 TM2.eval M k L\nh\u2083 :\n { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') } \u2208\n eval (TM1.step (tr M)) (TM1.init (trInit k L))\n\u22a2 \u2203 S L',\n addBottom L' = Tape.right\u2080 c\u2081.Tape \u2227\n (\u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227\n S k = TM2.Cfg.stk { l := q\u271d, var := v\u271d, stk := S\u271d } k\n[PROOFSTEP]\ncases Part.mem_unique h\u2081 h\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.mk.refl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nq\u271d : Option \u039b\nv\u271d : \u03c3\nS\u271d : (k : K) \u2192 List (\u0393 k)\nL' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S\u271d k)))\nh\u2082 : { l := q\u271d, var := v\u271d, stk := S\u271d } \u2208 eval (TM2.step M) (TM2.init k L)\nH\u2082 : TM2.Cfg.stk { l := q\u271d, var := v\u271d, stk := S\u271d } k \u2208 TM2.eval M k L\nh\u2083 :\n { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') } \u2208\n eval (TM1.step (tr M)) (TM1.init (trInit k L))\nh\u2081 :\n { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') } \u2208\n eval (TM1.step (tr M)) (TM1.init (trInit k L))\nH\u2081 :\n Tape.right\u2080 { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') }.Tape \u2208\n TM1.eval (tr M) (trInit k L)\n\u22a2 \u2203 S L'_1,\n addBottom L'_1 = Tape.right\u2080 { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') }.Tape \u2227\n (\u2200 (k : K), ListBlank.map (proj k) L'_1 = ListBlank.mk (List.reverse (List.map some (S k)))) \u2227\n S k = TM2.Cfg.stk { l := q\u271d, var := v\u271d, stk := S\u271d } k\n[PROOFSTEP]\nexact \u27e8_, L', by simp only [Tape.mk'_right\u2080], hT, rfl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nk : K\nL : List (\u0393 k)\nq\u271d : Option \u039b\nv\u271d : \u03c3\nS\u271d : (k : K) \u2192 List (\u0393 k)\nL' : ListBlank ((k : K) \u2192 Option (\u0393 k))\nhT : \u2200 (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S\u271d k)))\nh\u2082 : { l := q\u271d, var := v\u271d, stk := S\u271d } \u2208 eval (TM2.step M) (TM2.init k L)\nH\u2082 : TM2.Cfg.stk { l := q\u271d, var := v\u271d, stk := S\u271d } k \u2208 TM2.eval M k L\nh\u2083 :\n { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') } \u2208\n eval (TM1.step (tr M)) (TM1.init (trInit k L))\nh\u2081 :\n { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') } \u2208\n eval (TM1.step (tr M)) (TM1.init (trInit k L))\nH\u2081 :\n Tape.right\u2080 { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') }.Tape \u2208\n TM1.eval (tr M) (trInit k L)\n\u22a2 addBottom L' = Tape.right\u2080 { l := Option.map normal q\u271d, var := v\u271d, Tape := Tape.mk' \u2205 (addBottom L') }.Tape\n[PROOFSTEP]\nsimp only [Tape.mk'_right\u2080]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nsuffices\n \u2200 (q) (_ : TM2.SupportsStmt S q) (_ : \u2200 x \u2208 trStmts\u2081 q, x \u2208 trSupp M S),\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227 \u2200 l' \u2208 trStmts\u2081 q, TM1.SupportsStmt (trSupp M S) (tr M l')\n by\n rcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9\n have := this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9\n rcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\nthis :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrcases Finset.mem_biUnion.1 h with \u27e8l, lS, h\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nhave := this _ (ss.2 l lS) fun x hx \u21a6 Finset.mem_biUnion.2 \u27e8_, lS, Finset.mem_insert_of_mem hx\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis\u271d :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrcases Finset.mem_insert.1 h with (rfl | h) <;> [exact this.1; exact this.2 _ h]\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d : l' \u2208 trSupp M S\nthis\u271d :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrcases Finset.mem_insert.1 h with (rfl | h)\n[GOAL]\ncase intro.intro.inl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nthis\u271d :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nh\u271d : normal l \u2208 trSupp M S\nh : normal l \u2208 insert (normal l) (trStmts\u2081 (M l))\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (normal l))\n[PROOFSTEP]\nexact this.1\n[GOAL]\ncase intro.intro.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh\u271d\u00b9 : l' \u2208 trSupp M S\nthis\u271d :\n \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b\nlS : l \u2208 S\nh\u271d : l' \u2208 insert (normal l) (trStmts\u2081 (M l))\nthis :\n TM1.SupportsStmt (trSupp M S) (trNormal (M l)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (M l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nh : l' \u2208 trStmts\u2081 (M l)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nexact this.2 _ h\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl' : \u039b'\nh : l' \u2208 trSupp M S\n\u22a2 \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nclear h l'\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (q : Stmt\u2082),\n TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrefine' stmtStRec _ _ _ _ _\n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (k : K) (s : StAct k) (q : Stmt\u2082),\n (TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n TM2.SupportsStmt S (stRun s q) \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q) \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nintro _ s _ IH ss' sub\n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (stRun s q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrw [TM2to1.supports_run] at ss' \n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nsimp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton] at sub \n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nhave hgo := sub _ (Or.inl <| Or.inl rfl)\n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nhave hret := sub _ (Or.inl <| Or.inr rfl)\n[GOAL]\ncase refine'_1\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\ncases' IH ss' fun x hx \u21a6 sub x <| Or.inr hx with IH\u2081 IH\u2082\n[GOAL]\ncase refine'_1.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (stRun s q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrefine' \u27e8by simp only [trNormal_run, TM1.SupportsStmt]; intros; exact hgo, fun l h \u21a6 _\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (stRun s q\u271d))\n[PROOFSTEP]\nsimp only [trNormal_run, TM1.SupportsStmt]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 \u0393' \u2192 \u03c3 \u2192 go k\u271d s q\u271d \u2208 trSupp M S\n[PROOFSTEP]\nintros\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u0393'\nv\u271d : \u03c3\n\u22a2 go k\u271d s q\u271d \u2208 trSupp M S\n[PROOFSTEP]\nexact hgo\n[GOAL]\ncase refine'_1.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (stRun s q\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nrw [trStmts\u2081_run] at h \n[GOAL]\ncase refine'_1.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 {go k\u271d s q\u271d, ret q\u271d} \u222a trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nsimp only [TM2to1.trStmts\u2081_run, Finset.mem_union, Finset.mem_insert, Finset.mem_singleton] at h \n[GOAL]\ncase refine'_1.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : (l = go k\u271d s q\u271d \u2228 l = ret q\u271d) \u2228 l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nrcases h with (\u27e8rfl | rfl\u27e9 | h)\n[GOAL]\ncase refine'_1.intro.inl.inl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d s q\u271d))\n[PROOFSTEP]\ncases s\n[GOAL]\ncase refine'_1.intro.inl.inl.push\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.push a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.push a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.push a\u271d) q\u271d))\n[PROOFSTEP]\nexact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9\n[GOAL]\ncase refine'_1.intro.inl.inl.peek\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.peek a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.peek a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.peek a\u271d) q\u271d))\n[PROOFSTEP]\nexact \u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hgo\u27e9\n[GOAL]\ncase refine'_1.intro.inl.inl.pop\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nsub : \u2200 (x : \u039b'), (x = go k\u271d (StAct.pop a\u271d) q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d (StAct.pop a\u271d) q\u271d \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (go k\u271d (StAct.pop a\u271d) q\u271d))\n[PROOFSTEP]\nexact \u27e8\u27e8fun _ _ \u21a6 hret, fun _ _ \u21a6 hret\u27e9, fun _ _ \u21a6 hgo\u27e9\n[GOAL]\ncase refine'_1.intro.inl.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M (ret q\u271d))\n[PROOFSTEP]\nunfold TM1.SupportsStmt TM2to1.tr\n[GOAL]\ncase refine'_1.intro.inl.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 match\n match ret q\u271d with\n | normal q => trNormal (M q)\n | go k s q =>\n branch (fun a x => Option.isNone (Prod.snd a k)) (trStAct (goto fun x x => ret q) s)\n (move Dir.right (goto fun x x => go k s q))\n | ret q => branch (fun a x => a.fst) (trNormal q) (move Dir.left (goto fun x x => ret q)) with\n | move a q => TM1.SupportsStmt (trSupp M S) q\n | write a q => TM1.SupportsStmt (trSupp M S) q\n | load a q => TM1.SupportsStmt (trSupp M S) q\n | branch a q\u2081 q\u2082 => TM1.SupportsStmt (trSupp M S) q\u2081 \u2227 TM1.SupportsStmt (trSupp M S) q\u2082\n | goto l => \u2200 (a : \u0393') (v : \u03c3), l a v \u2208 trSupp M S\n | halt => True\n[PROOFSTEP]\nexact \u27e8IH\u2081, fun _ _ \u21a6 hret\u27e9\n[GOAL]\ncase refine'_1.intro.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nk\u271d : K\ns : StAct k\u271d\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S q\u271d\nsub : \u2200 (x : \u039b'), (x = go k\u271d s q\u271d \u2228 x = ret q\u271d) \u2228 x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\nhgo : go k\u271d s q\u271d \u2208 trSupp M S\nhret : ret q\u271d \u2208 trSupp M S\nIH\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u271d)\nIH\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nexact IH\u2082 _ h\n[GOAL]\ncase refine'_2\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (a : \u03c3 \u2192 \u03c3) (q : Stmt\u2082),\n (TM2.SupportsStmt S q \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n TM2.SupportsStmt S (TM2.Stmt.load a q) \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a q)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a q) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nintro _ _ IH ss' sub\n[GOAL]\ncase refine'_2\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.load a\u271d q\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nunfold TM2to1.trStmts\u2081 at ss' sub \u22a2\n[GOAL]\ncase refine'_2\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\na\u271d : \u03c3 \u2192 \u03c3\nq\u271d : Stmt\u2082\nIH :\n TM2.SupportsStmt S q\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.load a\u271d q\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.load a\u271d q\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nexact IH ss' sub\n[GOAL]\ncase refine'_3\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (p : \u03c3 \u2192 Bool) (q\u2081 q\u2082 : Stmt\u2082),\n (TM2.SupportsStmt S q\u2081 \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081 \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n (TM2.SupportsStmt S q\u2082 \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082 \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082 \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')) \u2192\n TM2.SupportsStmt S (TM2.Stmt.branch p q\u2081 q\u2082) \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p q\u2081 q\u2082)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p q\u2081 q\u2082) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nintro _ _ _ IH\u2081 IH\u2082 ss' sub\n[GOAL]\ncase refine'_3\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nunfold TM2to1.trStmts\u2081 at sub \n[GOAL]\ncase refine'_3\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\ncases' IH\u2081 ss'.1 fun x hx \u21a6 sub x <| Finset.mem_union_left _ hx with IH\u2081\u2081 IH\u2081\u2082\n[GOAL]\ncase refine'_3.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\ncases' IH\u2082 ss'.2 fun x hx \u21a6 sub x <| Finset.mem_union_right _ hx with IH\u2082\u2081 IH\u2082\u2082\n[GOAL]\ncase refine'_3.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrefine' \u27e8\u27e8IH\u2081\u2081, IH\u2082\u2081\u27e9, fun l h \u21a6 _\u27e9\n[GOAL]\ncase refine'_3.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nrw [trStmts\u2081] at h \n[GOAL]\ncase refine'_3.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nrcases Finset.mem_union.1 h with (h | h) <;> [exact IH\u2081\u2082 _ h; exact IH\u2082\u2082 _ h]\n[GOAL]\ncase refine'_3.intro.intro\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nrcases Finset.mem_union.1 h with (h | h)\n[GOAL]\ncase refine'_3.intro.intro.inl\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh\u271d : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\nh : l \u2208 trStmts\u2081 q\u2081\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nexact IH\u2081\u2082 _ h\n[GOAL]\ncase refine'_3.intro.intro.inr\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\np\u271d : \u03c3 \u2192 Bool\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nIH\u2081 :\n TM2.SupportsStmt S q\u2081\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082 :\n TM2.SupportsStmt S q\u2082\u271d \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nss' : TM2.SupportsStmt S (TM2.Stmt.branch p\u271d q\u2081\u271d q\u2082\u271d)\nsub : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d \u2192 x \u2208 trSupp M S\nIH\u2081\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2081\u271d)\nIH\u2081\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2081\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nIH\u2082\u2081 : TM1.SupportsStmt (trSupp M S) (trNormal q\u2082\u271d)\nIH\u2082\u2082 : \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 q\u2082\u271d \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\nl : \u039b'\nh\u271d : l \u2208 trStmts\u2081 q\u2081\u271d \u222a trStmts\u2081 q\u2082\u271d\nh : l \u2208 trStmts\u2081 q\u2082\u271d\n\u22a2 TM1.SupportsStmt (trSupp M S) (tr M l)\n[PROOFSTEP]\nexact IH\u2082\u2082 _ h\n[GOAL]\ncase refine'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 \u2200 (l : \u03c3 \u2192 \u039b),\n TM2.SupportsStmt S (TM2.Stmt.goto l) \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nintro _ ss'\n _\n -- goto\n[GOAL]\ncase refine'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nsimp only [trStmts\u2081, Finset.not_mem_empty]\n[GOAL]\ncase refine'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d)) \u2227\n \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nrefine' \u27e8_, fun _ \u21a6 False.elim\u27e9\n[GOAL]\ncase refine'_4\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nl\u271d : \u03c3 \u2192 \u039b\nss' : TM2.SupportsStmt S (TM2.Stmt.goto l\u271d)\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 (TM2.Stmt.goto l\u271d) \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal (TM2.Stmt.goto l\u271d))\n[PROOFSTEP]\nexact fun _ v \u21a6 Finset.mem_biUnion.2 \u27e8_, ss' v, Finset.mem_insert_self _ _\u27e9\n[GOAL]\ncase refine'_5\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\n\u22a2 TM2.SupportsStmt S TM2.Stmt.halt \u2192\n (\u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S) \u2192\n TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nintro _\n _\n -- halt\n[GOAL]\ncase refine'_5\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227\n \u2200 (l' : \u039b'), l' \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nsimp only [trStmts\u2081, Finset.not_mem_empty]\n[GOAL]\ncase refine'_5\nK : Type u_1\ninst\u271d\u00b2 : DecidableEq K\n\u0393 : K \u2192 Type u_2\n\u039b : Type u_3\ninst\u271d\u00b9 : Inhabited \u039b\n\u03c3 : Type u_4\ninst\u271d : Inhabited \u03c3\nM : \u039b \u2192 Stmt\u2082\nS : Finset \u039b\nss : TM2.Supports M S\nx\u271d\u00b9 : TM2.SupportsStmt S TM2.Stmt.halt\nx\u271d : \u2200 (x : \u039b'), x \u2208 trStmts\u2081 TM2.Stmt.halt \u2192 x \u2208 trSupp M S\n\u22a2 TM1.SupportsStmt (trSupp M S) (trNormal TM2.Stmt.halt) \u2227 \u2200 (l' : \u039b'), False \u2192 TM1.SupportsStmt (trSupp M S) (tr M l')\n[PROOFSTEP]\nexact \u27e8trivial, fun _ \u21a6 False.elim\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Computability.TuringMachine", "llama_tokens": 358268, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.5, "lm_q1q2_score": 0.2849263330122419}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\na\u2082 : \u03b1\nf\u2082 : \u03b2 a\u2082 \u2192 WType fun a => \u03b2 a\nh : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2082 f\u2082)\n\u22a2 mk a\u2081 f\u2081 = mk a\u2082 f\u2082\n[PROOFSTEP]\nobtain \u27e8rfl, h\u27e9 := Sigma.mk.inj_iff.mp (f\u03b3_injective h)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\n\u22a2 mk a\u2081 f\u2081 = mk a\u2081 f\u2082\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase intro.e_f.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\n\u03b3 : Type u_3\nf\u03b3 : (a : \u03b1) \u00d7 (\u03b2 a \u2192 \u03b3) \u2192 \u03b3\nf\u03b3_injective : Function.Injective f\u03b3\na\u2081 : \u03b1\nf\u2081 f\u2082 : \u03b2 a\u2081 \u2192 WType fun a => \u03b2 a\nh\u271d : elim \u03b3 f\u03b3 (mk a\u2081 f\u2081) = elim \u03b3 f\u03b3 (mk a\u2081 f\u2082)\nh : HEq (fun b => elim \u03b3 f\u03b3 (f\u2081 b)) fun b => elim \u03b3 f\u03b3 (f\u2082 b)\nx : \u03b2 a\u2081\n\u22a2 f\u2081 x = f\u2082 x\n[PROOFSTEP]\nexact elim_injective \u03b3 f\u03b3 f\u03b3_injective (congr_fun (eq_of_heq h) x : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\n\u22a2 \u00acFinite (WType \u03b2)\n[PROOFSTEP]\nintro hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\n\u22a2 False\n[PROOFSTEP]\nhave hba : b \u2260 a := fun h => ha.elim (IsEmpty.elim' (show IsEmpty (\u03b2 a) from h \u25b8 he))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\n\u22a2 False\n[PROOFSTEP]\nrefine'\n not_injective_infinite_finite\n (fun n : \u2115 => show WType \u03b2 from Nat.recOn n \u27e8b, IsEmpty.elim' he\u27e9 fun _ ih => \u27e8a, fun _ => ih\u27e9) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\n\u22a2 Function.Injective fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this\n[PROOFSTEP]\nintro n m h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn m : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\n\u22a2 n = m\n[PROOFSTEP]\ninduction' n with n ih generalizing m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn m\u271d : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\u271d\nm : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n Nat.zero =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\n\u22a2 Nat.zero = m\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn m : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n Nat.zero =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n Nat.zero\n\u22a2 Nat.zero = Nat.zero\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase zero.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn m\u271d : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\u271d\nm : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n Nat.zero =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ m)\n\u22a2 Nat.zero = Nat.succ m\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn\u271d m\u271d : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n\u271d =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\u271d\nn : \u2115\nih :\n \u2200 \u2983m : \u2115\u2984,\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m \u2192\n n = m\nm : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ n) =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\n\u22a2 Nat.succ n = m\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn\u271d m : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n\u271d =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\nn : \u2115\nih :\n \u2200 \u2983m : \u2115\u2984,\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m \u2192\n n = m\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ n) =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n Nat.zero\n\u22a2 Nat.succ n = Nat.zero\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn\u271d m\u271d : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n\u271d =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\u271d\nn : \u2115\nih :\n \u2200 \u2983m : \u2115\u2984,\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m \u2192\n n = m\nm : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ n) =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ m)\n\u22a2 Nat.succ n = Nat.succ m\n[PROOFSTEP]\nrefine' congr_arg Nat.succ (ih _)\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\na b : \u03b1\nha : Nonempty (\u03b2 a)\nhe : IsEmpty (\u03b2 b)\nhf : Finite (WType \u03b2)\nhba : b \u2260 a\nn\u271d m\u271d : \u2115\nh\u271d :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n\u271d =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\u271d\nn : \u2115\nih :\n \u2200 \u2983m : \u2115\u2984,\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m \u2192\n n = m\nm : \u2115\nh :\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ n) =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n (Nat.succ m)\n\u22a2 (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n n =\n (fun n =>\n let_fun this := Nat.recOn n (mk b (IsEmpty.elim' he)) fun x ih => mk a fun x => ih;\n this)\n m\n[PROOFSTEP]\nsimp_all [Function.funext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\nt : WType \u03b2\n\u22a2 0 < depth t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\na\u271d : \u03b1\nf\u271d : \u03b2 a\u271d \u2192 WType \u03b2\n\u22a2 0 < depth (mk a\u271d f\u271d)\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b2 a)\nf : WType.WType' \u03b2 0 \u2192 Empty :=\n fun x =>\n match x with\n | { val := x, property := h } => False.elim (_ : False)\n\u22a2 Empty \u2192 WType.WType' \u03b2 0\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b2 a)\nf : WType.WType' \u03b2 0 \u2192 Empty :=\n fun x =>\n match x with\n | { val := x, property := h } => False.elim (_ : False)\nx : Empty\n\u22a2 WType.WType' \u03b2 0\n[PROOFSTEP]\ncases x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b2 a)\nn : \u2115\nt : WType \u03b2\nh : depth t \u2264 n + 1\n\u22a2 (a : \u03b1) \u00d7 (\u03b2 a \u2192 WType.WType' \u03b2 n)\n[PROOFSTEP]\ncases' t with a f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b2 a)\nn : \u2115\na : \u03b1\nf : \u03b2 a \u2192 WType \u03b2\nh : depth (mk a f) \u2264 n + 1\n\u22a2 (a : \u03b1) \u00d7 (\u03b2 a \u2192 WType.WType' \u03b2 n)\n[PROOFSTEP]\nhave h\u2080 : \u2200 i : \u03b2 a, WType.depth (f i) \u2264 n := fun i =>\n Nat.le_of_lt_succ (lt_of_lt_of_le (WType.depth_lt_depth_mk a f i) h)\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d : (a : \u03b1) \u2192 Encodable (\u03b2 a)\nn : \u2115\na : \u03b1\nf : \u03b2 a \u2192 WType \u03b2\nh : depth (mk a f) \u2264 n + 1\nh\u2080 : \u2200 (i : \u03b2 a), depth (f i) \u2264 n\n\u22a2 (a : \u03b1) \u00d7 (\u03b2 a \u2192 WType.WType' \u03b2 n)\n[PROOFSTEP]\nexact \u27e8a, fun i : \u03b2 a => \u27e8f i, h\u2080 i\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nn : \u2115\nh : Encodable (WType.WType' \u03b2 n)\n\u22a2 \u2200 (b : WType.WType' \u03b2 (n + 1)), WType.finv n (WType.f n b) = b\n[PROOFSTEP]\nrintro \u27e8\u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nn : \u2115\nh : Encodable (WType.WType' \u03b2 n)\na\u271d : \u03b1\nf\u271d : \u03b2 a\u271d \u2192 WType \u03b2\nproperty\u271d : depth (mk a\u271d f\u271d) \u2264 n + 1\n\u22a2 WType.finv n (WType.f n { val := mk a\u271d f\u271d, property := property\u271d }) = { val := mk a\u271d f\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\n\u22a2 Encodable (WType \u03b2)\n[PROOFSTEP]\nhaveI h' : \u2200 n, Encodable (WType' \u03b2 n) := fun n => Nat.rec encodable_zero encodable_succ n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nh' : (n : \u2115) \u2192 Encodable (WType.WType' \u03b2 n)\n\u22a2 Encodable (WType \u03b2)\n[PROOFSTEP]\nlet f : WType \u03b2 \u2192 \u03a3 n, WType' \u03b2 n := fun t => \u27e8t.depth, \u27e8t, le_rfl\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nh' : (n : \u2115) \u2192 Encodable (WType.WType' \u03b2 n)\nf : WType \u03b2 \u2192 (n : \u2115) \u00d7 WType.WType' \u03b2 n :=\n fun t => { fst := depth t, snd := { val := t, property := (_ : depth t \u2264 depth t) } }\n\u22a2 Encodable (WType \u03b2)\n[PROOFSTEP]\nlet finv : (\u03a3 n, WType' \u03b2 n) \u2192 WType \u03b2 := fun p => p.2.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nh' : (n : \u2115) \u2192 Encodable (WType.WType' \u03b2 n)\nf : WType \u03b2 \u2192 (n : \u2115) \u00d7 WType.WType' \u03b2 n :=\n fun t => { fst := depth t, snd := { val := t, property := (_ : depth t \u2264 depth t) } }\nfinv : (n : \u2115) \u00d7 WType.WType' \u03b2 n \u2192 WType \u03b2 := fun p => \u2191p.snd\n\u22a2 Encodable (WType \u03b2)\n[PROOFSTEP]\nhave : \u2200 t, finv (f t) = t := fun t => rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Fintype (\u03b2 a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Encodable (\u03b2 a)\ninst\u271d : Encodable \u03b1\nh' : (n : \u2115) \u2192 Encodable (WType.WType' \u03b2 n)\nf : WType \u03b2 \u2192 (n : \u2115) \u00d7 WType.WType' \u03b2 n :=\n fun t => { fst := depth t, snd := { val := t, property := (_ : depth t \u2264 depth t) } }\nfinv : (n : \u2115) \u00d7 WType.WType' \u03b2 n \u2192 WType \u03b2 := fun p => \u2191p.snd\nthis : \u2200 (t : WType \u03b2), finv (f t) = t\n\u22a2 Encodable (WType \u03b2)\n[PROOFSTEP]\nexact Encodable.ofLeftInverse f finv this\n", "meta": {"mathlib_filename": "Mathlib.Data.W.Basic", "llama_tokens": 6270, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.28423636717820466}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nhb : card t \u2022 b < \u2211 x in s, w x\n\u22a2 \u2211 i in t, b < \u2211 i in t, \u2211 x in filter (fun x => f x = i) s, w x\n[PROOFSTEP]\nsimpa only [sum_fiberwise_of_maps_to hf, sum_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nht : \u2200 (y : \u03b2), \u00acy \u2208 t \u2192 \u2211 x in filter (fun x => f x = y) s, w x \u2264 0\nhb : card t \u2022 b < \u2211 x in s, w x\n\u22a2 \u2211 _y in t, b < \u2211 x in s, w x\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : Finset.Nonempty t\nhb : card t \u2022 b \u2264 \u2211 x in s, w x\n\u22a2 \u2211 i in t, b \u2264 \u2211 i in t, \u2211 x in filter (fun x => f x = i) s, w x\n[PROOFSTEP]\nsimpa only [sum_fiberwise_of_maps_to hf, sum_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nhf : \u2200 (y : \u03b2), \u00acy \u2208 t \u2192 \u2211 x in filter (fun x => f x = y) s, w x \u2264 0\nht : Finset.Nonempty t\nhb : card t \u2022 b \u2264 \u2211 x in s, w x\n\u22a2 \u2211 _y in t, b \u2264 \u2211 x in s, w x\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : card t \u2022 b < \u2191(card s)\n\u22a2 \u2203 y, y \u2208 t \u2227 b < \u2191(card (filter (fun x => f x = y) s))\n[PROOFSTEP]\nsimp_rw [cast_card] at ht \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : card t \u2022 b < \u2211 a in s, 1\n\u22a2 \u2203 y, y \u2208 t \u2227 b < \u2211 a in filter (fun x => f x = y) s, 1\n[PROOFSTEP]\nexact exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum hf ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nht : \u2191(card s) < card t \u2022 b\n\u22a2 \u2203 y, y \u2208 t \u2227 \u2191(card (filter (fun x => f x = y) s)) < b\n[PROOFSTEP]\nsimp_rw [cast_card] at ht \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nht : \u2211 a in s, 1 < card t \u2022 b\n\u22a2 \u2203 y, y \u2208 t \u2227 \u2211 a in filter (fun x => f x = y) s, 1 < b\n[PROOFSTEP]\nexact exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul (fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : Finset.Nonempty t\nhb : card t \u2022 b \u2264 \u2191(card s)\n\u22a2 \u2203 y, y \u2208 t \u2227 b \u2264 \u2191(card (filter (fun x => f x = y) s))\n[PROOFSTEP]\nsimp_rw [cast_card] at hb \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : Finset.Nonempty t\nhb : card t \u2022 b \u2264 \u2211 a in s, 1\n\u22a2 \u2203 y, y \u2208 t \u2227 b \u2264 \u2211 a in filter (fun x => f x = y) s, 1\n[PROOFSTEP]\nexact exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum hf ht hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nht : Finset.Nonempty t\nhb : \u2191(card s) \u2264 card t \u2022 b\n\u22a2 \u2203 y, y \u2208 t \u2227 \u2191(card (filter (fun x => f x = y) s)) \u2264 b\n[PROOFSTEP]\nsimp_rw [cast_card] at hb \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nht : Finset.Nonempty t\nhb : \u2211 a in s, 1 \u2264 card t \u2022 b\n\u22a2 \u2203 y, y \u2208 t \u2227 \u2211 a in filter (fun x => f x = y) s, 1 \u2264 b\n[PROOFSTEP]\nrefine' exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul (fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht hb\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Pigeonhole", "llama_tokens": 2170, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.28423243351001287}} {"text": "[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\n\u22a2 IsClique G s \u2194 induce s G = \u22a4\n[PROOFSTEP]\nrw [isClique_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\n\u22a2 Set.Pairwise s G.Adj \u2194 induce s G = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\n\u22a2 Set.Pairwise s G.Adj \u2192 induce s G = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : Set.Pairwise s G.Adj\n\u22a2 induce s G = \u22a4\n[PROOFSTEP]\next \u27e8v, hv\u27e9 \u27e8w, hw\u27e9\n[GOAL]\ncase mp.Adj.h.mk.h.mk.a\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : Set.Pairwise s G.Adj\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\n\u22a2 Adj (induce s G) { val := v, property := hv } { val := w, property := hw } \u2194\n Adj \u22a4 { val := v, property := hv } { val := w, property := hw }\n[PROOFSTEP]\nsimp only [comap_Adj, Subtype.coe_mk, top_adj, Ne.def, Subtype.mk_eq_mk]\n[GOAL]\ncase mp.Adj.h.mk.h.mk.a\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : Set.Pairwise s G.Adj\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\n\u22a2 Adj G (\u2191(Function.Embedding.subtype fun x => x \u2208 s) { val := v, property := hv })\n (\u2191(Function.Embedding.subtype fun x => x \u2208 s) { val := w, property := hw }) \u2194\n \u00acv = w\n[PROOFSTEP]\nexact \u27e8Adj.ne, h hv hw\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\n\u22a2 induce s G = \u22a4 \u2192 Set.Pairwise s G.Adj\n[PROOFSTEP]\nintro h v hv w hw hne\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\n\u22a2 Adj G v w\n[PROOFSTEP]\nhave h2 : (G.induce s).Adj \u27e8v, hv\u27e9 \u27e8w, hw\u27e9 = _ := rfl\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 :\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw } =\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n\u22a2 Adj G v w\n[PROOFSTEP]\nconv_lhs at h2 => rw [h]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 :\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw } =\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n| Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 :\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw } =\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n| Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 :\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw } =\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n| Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 :\n Adj \u22a4 { val := v, property := hv } { val := w, property := hw } =\n Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n\u22a2 Adj G v w\n[PROOFSTEP]\nsimp only [top_adj, ne_eq, Subtype.mk.injEq, eq_iff_iff] at h2 \n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : induce s G = \u22a4\nv : \u03b1\nhv : v \u2208 s\nw : \u03b1\nhw : w \u2208 s\nhne : v \u2260 w\nh2 : \u00acv = w \u2194 Adj (induce s G) { val := v, property := hv } { val := w, property := hw }\n\u22a2 Adj G v w\n[PROOFSTEP]\nexact h2.1 hne\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : G \u2264 H\n\u22a2 IsClique G s \u2192 IsClique H s\n[PROOFSTEP]\nsimp_rw [isClique_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : G \u2264 H\n\u22a2 Set.Pairwise s G.Adj \u2192 Set.Pairwise s H.Adj\n[PROOFSTEP]\nexact Set.Pairwise.mono' h\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : t \u2286 s\n\u22a2 IsClique G s \u2192 IsClique G t\n[PROOFSTEP]\nsimp_rw [isClique_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : t \u2286 s\n\u22a2 Set.Pairwise s G.Adj \u2192 Set.Pairwise t G.Adj\n[PROOFSTEP]\nexact Set.Pairwise.mono h\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\nh : G \u2264 H\n\u22a2 IsNClique G n s \u2192 IsNClique H n s\n[PROOFSTEP]\nsimp_rw [isNClique_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\nh : G \u2264 H\n\u22a2 IsClique G \u2191s \u2227 Finset.card s = n \u2192 IsClique H \u2191s \u2227 Finset.card s = n\n[PROOFSTEP]\nexact And.imp_left (IsClique.mono h)\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\n\u22a2 IsNClique \u22a5 n s \u2194 n \u2264 1 \u2227 Finset.card s = n\n[PROOFSTEP]\nrw [isNClique_iff, isClique_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\n\u22a2 Set.Subsingleton \u2191s \u2227 Finset.card s = n \u2194 n \u2264 1 \u2227 Finset.card s = n\n[PROOFSTEP]\nrefine' and_congr_left _\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\n\u22a2 Finset.card s = n \u2192 (Set.Subsingleton \u2191s \u2194 n \u2264 1)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns : Finset \u03b1\n\u22a2 Set.Subsingleton \u2191s \u2194 Finset.card s \u2264 1\n[PROOFSTEP]\nexact card_le_one.symm\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 IsNClique G 3 {a, b, c} \u2194 Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsimp only [isNClique_iff, isClique_iff, Set.pairwise_insert_of_symmetric G.symm, coe_insert]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hbc : b = c\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hbc : b = c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : b = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hac : a = c\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : \u00acb = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hac : a = c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : b = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hac : a = c\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nby_cases hac : a = c\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : b = c\nhac : a = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : b = c\nhac : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : \u00acb = c\nhac : a = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : a = b\nhbc : \u00acb = c\nhac : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : b = c\nhac : a = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : b = c\nhac : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : \u00acb = c\nhac : a = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : \u00acb = c\nhac : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nc : \u03b1\nhac : c = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b : \u03b1), b \u2208 \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n \u2200 (b : \u03b1), b \u2208 insert c \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n Finset.card {c, c, c} = 3 \u2194\n Adj G c c \u2227 Adj G c c \u2227 Adj G c c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nc : \u03b1\nhac : \u00acc = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b : \u03b1), b \u2208 \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n \u2200 (b : \u03b1), b \u2208 insert c \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n Finset.card {c, c, c} = 3 \u2194\n Adj G c c \u2227 Adj G c c \u2227 Adj G c c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nc : \u03b1\nhbc : \u00acc = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b : \u03b1), b \u2208 \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n \u2200 (b : \u03b1), b \u2208 insert c \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n Finset.card {c, c, c} = 3 \u2194\n Adj G c c \u2227 Adj G c c \u2227 Adj G c c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nb c : \u03b1\nhbc hac : \u00acb = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n Finset.card {b, b, c} = 3 \u2194\n Adj G b b \u2227 Adj G b c \u2227 Adj G b c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nc : \u03b1\nhab : \u00acc = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b : \u03b1), b \u2208 \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n \u2200 (b : \u03b1), b \u2208 insert c \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n Finset.card {c, c, c} = 3 \u2194\n Adj G c c \u2227 Adj G c c \u2227 Adj G c c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na c : \u03b1\nhac hab : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b : \u03b1), b \u2208 \u2191{c} \u2192 c \u2260 b \u2192 Adj G c b) \u2227\n \u2200 (b : \u03b1), b \u2208 insert c \u2191{c} \u2192 a \u2260 b \u2192 Adj G a b) \u2227\n Finset.card {a, c, c} = 3 \u2194\n Adj G a c \u2227 Adj G a c \u2227 Adj G c c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nb c : \u03b1\nhbc : \u00acb = c\nhab : \u00acc = b\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 c \u2260 b_1 \u2192 Adj G c b_1) \u2227\n Finset.card {c, b, c} = 3 \u2194\n Adj G c b \u2227 Adj G c c \u2227 Adj G b c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhab : \u00aca = b\nhbc : \u00acb = c\nhac : \u00aca = c\n\u22a2 ((Set.Pairwise (\u2191{c}) G.Adj \u2227 \u2200 (b_1 : \u03b1), b_1 \u2208 \u2191{c} \u2192 b \u2260 b_1 \u2192 Adj G b b_1) \u2227\n \u2200 (b_1 : \u03b1), b_1 \u2208 insert b \u2191{c} \u2192 a \u2260 b_1 \u2192 Adj G a b_1) \u2227\n Finset.card {a, b, c} = 3 \u2194\n Adj G a b \u2227 Adj G a c \u2227 Adj G b c\n[PROOFSTEP]\nsimp [G.ne_of_adj, and_rotate, *]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 IsNClique G 3 s \u2194 \u2203 a b c, Adj G a b \u2227 Adj G a c \u2227 Adj G b c \u2227 s = {a, b, c}\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nh : IsNClique G 3 s\n\u22a2 \u2203 a b c, Adj G a b \u2227 Adj G a c \u2227 Adj G b c \u2227 s = {a, b, c}\n[PROOFSTEP]\nobtain \u27e8a, b, c, -, -, -, hs\u27e9 := card_eq_three.1 h.card_eq\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na\u271d b\u271d c\u271d : \u03b1\nh : IsNClique G 3 s\na b c : \u03b1\nhs : s = {a, b, c}\n\u22a2 \u2203 a b c, Adj G a b \u2227 Adj G a c \u2227 Adj G b c \u2227 s = {a, b, c}\n[PROOFSTEP]\nrefine' \u27e8a, b, c, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na\u271d b\u271d c\u271d : \u03b1\nh : IsNClique G 3 s\na b c : \u03b1\nhs : s = {a, b, c}\n\u22a2 Adj G a b \u2227 Adj G a c \u2227 Adj G b c \u2227 s = {a, b, c}\n[PROOFSTEP]\nrwa [hs, eq_self_iff_true, and_true, is3Clique_triple_iff.symm, \u2190 hs]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ns : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 (\u2203 a b c, Adj G a b \u2227 Adj G a c \u2227 Adj G b c \u2227 s = {a, b, c}) \u2192 IsNClique G 3 s\n[PROOFSTEP]\nrintro \u27e8a, b, c, hab, hbc, hca, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\nhab : Adj G a b\nhbc : Adj G a c\nhca : Adj G b c\n\u22a2 IsNClique G 3 {a, b, c}\n[PROOFSTEP]\nexact is3Clique_triple_iff.2 \u27e8hab, hbc, hca\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\n\u22a2 \u00acCliqueFree G n\n[PROOFSTEP]\nsimp only [CliqueFree, isNClique_iff, isClique_iff_induce_eq, not_forall, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\n\u22a2 \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\n[PROOFSTEP]\nuse Finset.univ.map f.toEmbedding\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\n\u22a2 induce (\u2191(map f.toEmbedding univ)) G = \u22a4 \u2227 Finset.card (map f.toEmbedding univ) = n\n[PROOFSTEP]\nsimp only [card_map, Finset.card_fin, eq_self_iff_true, and_true_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\n\u22a2 induce (\u2191(map f.toEmbedding univ)) G = \u22a4\n[PROOFSTEP]\next \u27e8v, hv\u27e9 \u27e8w, hw\u27e9\n[GOAL]\ncase h.Adj.h.mk.h.mk.a\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\nv : \u03b1\nhv : v \u2208 \u2191(map f.toEmbedding univ)\nw : \u03b1\nhw : w \u2208 \u2191(map f.toEmbedding univ)\n\u22a2 Adj (induce (\u2191(map f.toEmbedding univ)) G) { val := v, property := hv } { val := w, property := hw } \u2194\n Adj \u22a4 { val := v, property := hv } { val := w, property := hw }\n[PROOFSTEP]\nsimp only [coe_map, Set.mem_image, coe_univ, Set.mem_univ, true_and_iff] at hv hw \n[GOAL]\ncase h.Adj.h.mk.h.mk.a\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\nv : \u03b1\nhv\u271d : v \u2208 \u2191(map f.toEmbedding univ)\nw : \u03b1\nhw\u271d : w \u2208 \u2191(map f.toEmbedding univ)\nhv : \u2203 x, \u2191f.toEmbedding x = v\nhw : \u2203 x, \u2191f.toEmbedding x = w\n\u22a2 Adj (induce (\u2191(map f.toEmbedding univ)) G) { val := v, property := hv\u271d } { val := w, property := hw\u271d } \u2194\n Adj \u22a4 { val := v, property := hv\u271d } { val := w, property := hw\u271d }\n[PROOFSTEP]\nobtain \u27e8v', rfl\u27e9 := hv\n[GOAL]\ncase h.Adj.h.mk.h.mk.a.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\nw : \u03b1\nhw\u271d : w \u2208 \u2191(map f.toEmbedding univ)\nhw : \u2203 x, \u2191f.toEmbedding x = w\nv' : Fin n\nhv : \u2191f.toEmbedding v' \u2208 \u2191(map f.toEmbedding univ)\n\u22a2 Adj (induce (\u2191(map f.toEmbedding univ)) G) { val := \u2191f.toEmbedding v', property := hv }\n { val := w, property := hw\u271d } \u2194\n Adj \u22a4 { val := \u2191f.toEmbedding v', property := hv } { val := w, property := hw\u271d }\n[PROOFSTEP]\nobtain \u27e8w', rfl\u27e9 := hw\n[GOAL]\ncase h.Adj.h.mk.h.mk.a.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nf : \u22a4 \u21aag G\nv' : Fin n\nhv : \u2191f.toEmbedding v' \u2208 \u2191(map f.toEmbedding univ)\nw' : Fin n\nhw : \u2191f.toEmbedding w' \u2208 \u2191(map f.toEmbedding univ)\n\u22a2 Adj (induce (\u2191(map f.toEmbedding univ)) G) { val := \u2191f.toEmbedding v', property := hv }\n { val := \u2191f.toEmbedding w', property := hw } \u2194\n Adj \u22a4 { val := \u2191f.toEmbedding v', property := hv } { val := \u2191f.toEmbedding w', property := hw }\n[PROOFSTEP]\nsimp only [coe_sort_coe, RelEmbedding.coe_toEmbedding, comap_Adj, Function.Embedding.coe_subtype, f.map_adj_iff,\n top_adj, ne_eq, Subtype.mk.injEq, RelEmbedding.inj]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u00acCliqueFree G n\n\u22a2 \u22a4 \u21aag G\n[PROOFSTEP]\nsimp only [CliqueFree, isNClique_iff, isClique_iff_induce_eq, not_forall, Classical.not_not] at h \n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\n\u22a2 \u22a4 \u21aag G\n[PROOFSTEP]\nobtain \u27e8ha, hb\u27e9 := h.choose_spec\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\n\u22a2 \u22a4 \u21aag G\n[PROOFSTEP]\nhave : (\u22a4 : SimpleGraph (Fin h.choose.card)) \u2243g (\u22a4 : SimpleGraph h.choose) :=\n by\n apply Iso.completeGraph\n simpa using (Fintype.equivFin h.choose).symm\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\n\u22a2 \u22a4 \u2243g \u22a4\n[PROOFSTEP]\napply Iso.completeGraph\n[GOAL]\ncase f\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\n\u22a2 Fin (Finset.card (Exists.choose h)) \u2243 { x // x \u2208 Exists.choose h }\n[PROOFSTEP]\nsimpa using (Fintype.equivFin h.choose).symm\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\nthis : \u22a4 \u2243g \u22a4\n\u22a2 \u22a4 \u21aag G\n[PROOFSTEP]\nrw [\u2190 ha] at this \n[GOAL]\ncase intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\nthis : \u22a4 \u2243g induce (\u2191(Exists.choose h)) G\n\u22a2 \u22a4 \u21aag G\n[PROOFSTEP]\nconvert (Embedding.induce \u2191h.choose.toSet).comp this.toEmbedding\n[GOAL]\ncase h.e'_1.h.e'_1\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\nh : \u2203 x, induce (\u2191x) G = \u22a4 \u2227 Finset.card x = n\nha : induce (\u2191(Exists.choose h)) G = \u22a4\nhb : Finset.card (Exists.choose h) = n\nthis : \u22a4 \u2243g induce (\u2191(Exists.choose h)) G\n\u22a2 n = Finset.card (Exists.choose h)\n[PROOFSTEP]\nexact hb.symm\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n\u271d : \u2115\ns : Finset \u03b1\nn : \u2115\n\u22a2 CliqueFree G n \u2194 IsEmpty (\u22a4 \u21aag G)\n[PROOFSTEP]\nrw [\u2190 not_iff_not, not_cliqueFree_iff, not_isEmpty_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nf : \u22a4 \u21aag G\n\u22a2 \u00acCliqueFree G (Fintype.card \u03b1)\n[PROOFSTEP]\nrw [not_cliqueFree_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nf : \u22a4 \u21aag G\n\u22a2 Nonempty (\u22a4 \u21aag G)\n[PROOFSTEP]\nexact \u27e8(Iso.completeGraph (Fintype.equivFin \u03b1)).symm.toEmbedding.trans f\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\nh : 2 \u2264 n\n\u22a2 CliqueFree \u22a5 n\n[PROOFSTEP]\nintro t ht\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\nh : 2 \u2264 n\nt : Finset \u03b1\nht : IsNClique \u22a5 n t\n\u22a2 False\n[PROOFSTEP]\nhave := le_trans h (isNClique_bot_iff.1 ht).1\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\nh : 2 \u2264 n\nt : Finset \u03b1\nht : IsNClique \u22a5 n t\nthis : 2 \u2264 1\n\u22a2 False\n[PROOFSTEP]\nsimp only at this \n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\nh : m \u2264 n\n\u22a2 CliqueFree G m \u2192 CliqueFree G n\n[PROOFSTEP]\nintro hG s hs\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns\u271d : Finset \u03b1\nh : m \u2264 n\nhG : CliqueFree G m\ns : Finset \u03b1\nhs : IsNClique G n s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8t, hts, ht\u27e9 := s.exists_smaller_set _ (h.trans hs.card_eq.ge)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns\u271d : Finset \u03b1\nh : m \u2264 n\nhG : CliqueFree G m\ns : Finset \u03b1\nhs : IsNClique G n s\nt : Finset \u03b1\nhts : t \u2286 s\nht : Finset.card t = m\n\u22a2 False\n[PROOFSTEP]\nexact hG _ \u27e8hs.clique.subset hts, ht\u27e9\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\n\u22a2 CliqueFree G n\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 False\n[PROOFSTEP]\nrefine' Nat.lt_le_antisymm hc _\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 n \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nrw [cliqueFree_iff, not_isEmpty_iff] at h \n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : Nonempty (\u22a4 \u21aag G)\n\u22a2 n \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nsimpa only [Fintype.card_fin] using Fintype.card_le_of_embedding h.some.toEmbedding\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nn : \u2115\na b c : \u03b1\ns : Finset \u03b1\n\u22a2 cliqueSet G n = \u2205 \u2194 CliqueFree G n\n[PROOFSTEP]\nsimp_rw [CliqueFree, Set.eq_empty_iff_forall_not_mem, mem_cliqueSet_iff]\n[GOAL]\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\nn : \u2115\na b c : \u03b1\ns : Finset \u03b1\n\u22a2 cliqueFinset G n = \u2205 \u2194 CliqueFree G n\n[PROOFSTEP]\nsimp_rw [CliqueFree, eq_empty_iff_forall_not_mem, mem_cliqueFinset_iff]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Clique", "llama_tokens": 12391, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.28390412341375226}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f)) (cons (a ::\u2098 m) a' b' (cons m a b f))\n[PROOFSTEP]\napply hfunext rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\n\u22a2 \u2200 (a_1 a'_1 : \u03b1),\n HEq a_1 a'_1 \u2192 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a_1) (cons (a ::\u2098 m) a' b' (cons m a b f) a'_1)\n[PROOFSTEP]\nsimp only [heq_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\n\u22a2 \u2200 (a_1 a'_1 : \u03b1),\n a_1 = a'_1 \u2192 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a_1) (cons (a ::\u2098 m) a' b' (cons m a b f) a'_1)\n[PROOFSTEP]\nrintro a'' _ rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a'') (cons (a ::\u2098 m) a' b' (cons m a b f) a'')\n[PROOFSTEP]\nrefine' hfunext (by rw [Multiset.cons_swap]) fun ha\u2081 ha\u2082 _ => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\n\u22a2 (a'' \u2208 a ::\u2098 a' ::\u2098 m) = (a'' \u2208 a' ::\u2098 a ::\u2098 m)\n[PROOFSTEP]\nrw [Multiset.cons_swap]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\nha\u2081 : a'' \u2208 a ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a' b' (cons m a b f) a'' ha\u2082)\n[PROOFSTEP]\nrcases ne_or_eq a'' a with (h\u2081 | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\nha\u2081 : a'' \u2208 a ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\nh\u2081 : a'' \u2260 a\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a' b' (cons m a b f) a'' ha\u2082)\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na' : \u03b1\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\na'' : \u03b1\nb : \u03b4 a''\nh : a'' \u2260 a'\nha\u2081 : a'' \u2208 a'' ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a'' ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a' ::\u2098 m) a'' b (cons m a' b' f) a'' ha\u2081) (cons (a'' ::\u2098 m) a' b' (cons m a'' b f) a'' ha\u2082)\n[PROOFSTEP]\nrcases eq_or_ne a'' a' with (rfl | h\u2082)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na : \u03b1\nb : \u03b4 a\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\na'' : \u03b1\nh\u2081 : a'' \u2260 a\nb' : \u03b4 a''\nh : a \u2260 a''\nha\u2081 : a'' \u2208 a ::\u2098 a'' ::\u2098 m\nha\u2082 : a'' \u2208 a'' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a'' ::\u2098 m) a b (cons m a'' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a'' b' (cons m a b f) a'' ha\u2082)\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\nha\u2081 : a'' \u2208 a ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\nh\u2081 : a'' \u2260 a\nh\u2082 : a'' \u2260 a'\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a' b' (cons m a b f) a'' ha\u2082)\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na' : \u03b1\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\na'' : \u03b1\nb : \u03b4 a''\nh : a'' \u2260 a'\nha\u2081 : a'' \u2208 a'' ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a'' ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a' ::\u2098 m) a'' b (cons m a' b' f) a'' ha\u2081) (cons (a'' ::\u2098 m) a' b' (cons m a'' b f) a'' ha\u2082)\n[PROOFSTEP]\nall_goals simp [*, Pi.cons_same, Pi.cons_ne]\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na : \u03b1\nb : \u03b4 a\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\na'' : \u03b1\nh\u2081 : a'' \u2260 a\nb' : \u03b4 a''\nh : a \u2260 a''\nha\u2081 : a'' \u2208 a ::\u2098 a'' ::\u2098 m\nha\u2082 : a'' \u2208 a'' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a'' ::\u2098 m) a b (cons m a'' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a'' b' (cons m a b f) a'' ha\u2082)\n[PROOFSTEP]\nsimp [*, Pi.cons_same, Pi.cons_ne]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na a' : \u03b1\nb : \u03b4 a\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\nh : a \u2260 a'\na'' : \u03b1\nha\u2081 : a'' \u2208 a ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\nh\u2081 : a'' \u2260 a\nh\u2082 : a'' \u2260 a'\n\u22a2 HEq (cons (a' ::\u2098 m) a b (cons m a' b' f) a'' ha\u2081) (cons (a ::\u2098 m) a' b' (cons m a b f) a'' ha\u2082)\n[PROOFSTEP]\nsimp [*, Pi.cons_same, Pi.cons_ne]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na' : \u03b1\nb' : \u03b4 a'\nm : Multiset \u03b1\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a\na'' : \u03b1\nb : \u03b4 a''\nh : a'' \u2260 a'\nha\u2081 : a'' \u2208 a'' ::\u2098 a' ::\u2098 m\nha\u2082 : a'' \u2208 a' ::\u2098 a'' ::\u2098 m\nx\u271d : HEq ha\u2081 ha\u2082\n\u22a2 HEq (cons (a' ::\u2098 m) a'' b (cons m a' b' f) a'' ha\u2081) (cons (a'' ::\u2098 m) a' b' (cons m a'' b f) a'' ha\u2082)\n[PROOFSTEP]\nsimp [*, Pi.cons_same, Pi.cons_ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 a ::\u2098 m \u2192 \u03b4 a'\n\u22a2 (Pi.cons m a (f a (_ : a \u2208 a ::\u2098 m)) fun a' ha' => f a' (_ : a' \u2208 a ::\u2098 m)) = f\n[PROOFSTEP]\next a' h'\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 a ::\u2098 m \u2192 \u03b4 a'\na' : \u03b1\nh' : a' \u2208 a ::\u2098 m\n\u22a2 Pi.cons m a (f a (_ : a \u2208 a ::\u2098 m)) (fun a' ha' => f a' (_ : a' \u2208 a ::\u2098 m)) a' h' = f a' h'\n[PROOFSTEP]\nby_cases h : a' = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 a ::\u2098 m \u2192 \u03b4 a'\na' : \u03b1\nh' : a' \u2208 a ::\u2098 m\nh : a' = a\n\u22a2 Pi.cons m a (f a (_ : a \u2208 a ::\u2098 m)) (fun a' ha' => f a' (_ : a' \u2208 a ::\u2098 m)) a' h' = f a' h'\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\na' : \u03b1\nf : (a'_1 : \u03b1) \u2192 a'_1 \u2208 a' ::\u2098 m \u2192 \u03b4 a'_1\nh' : a' \u2208 a' ::\u2098 m\n\u22a2 Pi.cons m a' (f a' (_ : a' \u2208 a' ::\u2098 m)) (fun a'_1 ha' => f a'_1 (_ : a'_1 \u2208 a' ::\u2098 m)) a' h' = f a' h'\n[PROOFSTEP]\nrw [Pi.cons_same]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\na : \u03b1\nf : (a' : \u03b1) \u2192 a' \u2208 a ::\u2098 m \u2192 \u03b4 a'\na' : \u03b1\nh' : a' \u2208 a ::\u2098 m\nh : \u00aca' = a\n\u22a2 Pi.cons m a (f a (_ : a \u2208 a ::\u2098 m)) (fun a' ha' => f a' (_ : a' \u2208 a ::\u2098 m)) a' h' = f a' h'\n[PROOFSTEP]\nrw [Pi.cons_ne _ h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na : \u03b1\nb : \u03b4 a\ns : Multiset \u03b1\nhs : \u00aca \u2208 s\nf\u2081 f\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b f\u2081 = cons s a b f\u2082\na' : \u03b1\nh' : a' \u2208 s\nne : a \u2260 a'\nthis : a' \u2208 a ::\u2098 s\n\u22a2 f\u2081 a' h' = cons s a b f\u2081 a' this\n[PROOFSTEP]\nrw [Pi.cons_ne this ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na : \u03b1\nb : \u03b4 a\ns : Multiset \u03b1\nhs : \u00aca \u2208 s\nf\u2081 f\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b f\u2081 = cons s a b f\u2082\na' : \u03b1\nh' : a' \u2208 s\nne : a \u2260 a'\nthis : a' \u2208 a ::\u2098 s\n\u22a2 cons s a b f\u2081 a' this = cons s a b f\u2082 a' this\n[PROOFSTEP]\nrw [eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\na : \u03b1\nb : \u03b4 a\ns : Multiset \u03b1\nhs : \u00aca \u2208 s\nf\u2081 f\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b f\u2081 = cons s a b f\u2082\na' : \u03b1\nh' : a' \u2208 s\nne : a \u2260 a'\nthis : a' \u2208 a ::\u2098 s\n\u22a2 cons s a b f\u2082 a' this = f\u2082 a' h'\n[PROOFSTEP]\nrw [Pi.cons_ne this ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\n\u22a2 \u2200 (a a' : \u03b1) (m : Multiset \u03b1) (b : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)),\n HEq (bind (t a) fun b_1 => map (Pi.cons (a' ::\u2098 m) a b_1) (bind (t a') fun b_2 => map (Pi.cons m a' b_2) b))\n (bind (t a') fun b_1 => map (Pi.cons (a ::\u2098 m) a' b_1) (bind (t a) fun b_2 => map (Pi.cons m a b_2) b))\n[PROOFSTEP]\nintro a a' m n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\n\u22a2 HEq (bind (t a) fun b => map (Pi.cons (a' ::\u2098 m) a b) (bind (t a') fun b => map (Pi.cons m a' b) n))\n (bind (t a') fun b => map (Pi.cons (a ::\u2098 m) a' b) (bind (t a) fun b => map (Pi.cons m a b) n))\n[PROOFSTEP]\nby_cases eq : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : a = a'\n\u22a2 HEq (bind (t a) fun b => map (Pi.cons (a' ::\u2098 m) a b) (bind (t a') fun b => map (Pi.cons m a' b) n))\n (bind (t a') fun b => map (Pi.cons (a ::\u2098 m) a' b) (bind (t a) fun b => map (Pi.cons m a b) n))\n[PROOFSTEP]\nsubst eq\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\n\u22a2 HEq (bind (t a) fun b => map (Pi.cons (a ::\u2098 m) a b) (bind (t a) fun b => map (Pi.cons m a b) n))\n (bind (t a) fun b => map (Pi.cons (a ::\u2098 m) a b) (bind (t a) fun b => map (Pi.cons m a b) n))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\n\u22a2 HEq (bind (t a) fun b => map (Pi.cons (a' ::\u2098 m) a b) (bind (t a') fun b => map (Pi.cons m a' b) n))\n (bind (t a') fun b => map (Pi.cons (a ::\u2098 m) a' b) (bind (t a) fun b => map (Pi.cons m a b) n))\n[PROOFSTEP]\nsimp [map_bind, bind_bind (t a') (t a)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\n\u22a2 HEq (bind (t a) fun b => bind (t a') fun a_1 => map (fun x => Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' a_1 x)) n)\n (bind (t a) fun b => bind (t a') fun a_1 => map (fun x => Pi.cons (a ::\u2098 m) a' a_1 (Pi.cons m a b x)) n)\n[PROOFSTEP]\napply bind_hcongr\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\n\u22a2 ((a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 a' ::\u2098 m \u2192 \u03b2 a_1) = ((a_1 : \u03b1) \u2192 a_1 \u2208 a' ::\u2098 a ::\u2098 m \u2192 \u03b2 a_1)\n[PROOFSTEP]\nrw [cons_swap a a']\n[GOAL]\ncase neg.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\n\u22a2 \u2200 (a_1 : \u03b2 a),\n a_1 \u2208 t a \u2192\n HEq (bind (t a') fun a_3 => map (fun x => Pi.cons (a' ::\u2098 m) a a_1 (Pi.cons m a' a_3 x)) n)\n (bind (t a') fun a_3 => map (fun x => Pi.cons (a ::\u2098 m) a' a_3 (Pi.cons m a a_1 x)) n)\n[PROOFSTEP]\nintro b _\n[GOAL]\ncase neg.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d : b \u2208 t a\n\u22a2 HEq (bind (t a') fun a_1 => map (fun x => Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' a_1 x)) n)\n (bind (t a') fun a_1 => map (fun x => Pi.cons (a ::\u2098 m) a' a_1 (Pi.cons m a b x)) n)\n[PROOFSTEP]\napply bind_hcongr\n[GOAL]\ncase neg.hf.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d : b \u2208 t a\n\u22a2 ((a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 a' ::\u2098 m \u2192 \u03b2 a_1) = ((a_1 : \u03b1) \u2192 a_1 \u2208 a' ::\u2098 a ::\u2098 m \u2192 \u03b2 a_1)\n[PROOFSTEP]\nrw [cons_swap a a']\n[GOAL]\ncase neg.hf.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d : b \u2208 t a\n\u22a2 \u2200 (a_1 : \u03b2 a'),\n a_1 \u2208 t a' \u2192\n HEq (map (fun x => Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' a_1 x)) n)\n (map (fun x => Pi.cons (a ::\u2098 m) a' a_1 (Pi.cons m a b x)) n)\n[PROOFSTEP]\nintro b' _\n[GOAL]\ncase neg.hf.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d\u00b9 : b \u2208 t a\nb' : \u03b2 a'\na\u271d : b' \u2208 t a'\n\u22a2 HEq (map (fun x => Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' b' x)) n)\n (map (fun x => Pi.cons (a ::\u2098 m) a' b' (Pi.cons m a b x)) n)\n[PROOFSTEP]\napply map_hcongr\n[GOAL]\ncase neg.hf.hf.h\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d\u00b9 : b \u2208 t a\nb' : \u03b2 a'\na\u271d : b' \u2208 t a'\n\u22a2 ((a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 a' ::\u2098 m \u2192 \u03b2 a_1) = ((a_1 : \u03b1) \u2192 a_1 \u2208 a' ::\u2098 a ::\u2098 m \u2192 \u03b2 a_1)\n[PROOFSTEP]\nrw [cons_swap a a']\n[GOAL]\ncase neg.hf.hf.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d\u00b9 : b \u2208 t a\nb' : \u03b2 a'\na\u271d : b' \u2208 t a'\n\u22a2 \u2200 (a_1 : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a),\n a_1 \u2208 n \u2192 HEq (Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' b' a_1)) (Pi.cons (a ::\u2098 m) a' b' (Pi.cons m a b a_1))\n[PROOFSTEP]\nintro f _\n[GOAL]\ncase neg.hf.hf.hf\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na a' : \u03b1\nm : Multiset \u03b1\nn : Multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a)\neq : \u00aca = a'\nb : \u03b2 a\na\u271d\u00b2 : b \u2208 t a\nb' : \u03b2 a'\na\u271d\u00b9 : b' \u2208 t a'\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\na\u271d : f \u2208 n\n\u22a2 HEq (Pi.cons (a' ::\u2098 m) a b (Pi.cons m a' b' f)) (Pi.cons (a ::\u2098 m) a' b' (Pi.cons m a b f))\n[PROOFSTEP]\nexact Pi.cons_swap eq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\n\u22a2 \u2191card (pi 0 t) = prod (map (fun a => \u2191card (t a)) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n \u2191card (pi s t) = prod (map (fun a => \u2191card (t a)) s) \u2192\n \u2191card (pi (a ::\u2098 s) t) = prod (map (fun a => \u2191card (t a)) (a ::\u2098 s))\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1},\n (Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)) \u2192\n Nodup (a ::\u2098 s) \u2192 (\u2200 (a_3 : \u03b1), a_3 \u2208 a ::\u2098 s \u2192 Nodup (t a_3)) \u2192 Nodup (pi (a ::\u2098 s) t)\n[PROOFSTEP]\nintro a s ih hs ht\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\n\u22a2 Nodup (pi (a ::\u2098 s) t)\n[PROOFSTEP]\nhave has : a \u2209 s := by simp at hs ; exact hs.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\n\u22a2 \u00aca \u2208 s\n[PROOFSTEP]\nsimp at hs \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhs : \u00aca \u2208 s \u2227 Nodup s\n\u22a2 \u00aca \u2208 s\n[PROOFSTEP]\nexact hs.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\n\u22a2 Nodup (pi (a ::\u2098 s) t)\n[PROOFSTEP]\nhave hs : Nodup s := by simp at hs ; exact hs.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\n\u22a2 Nodup s\n[PROOFSTEP]\nsimp at hs \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : \u00aca \u2208 s \u2227 Nodup s\n\u22a2 Nodup s\n[PROOFSTEP]\nexact hs.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\n\u22a2 Nodup (pi (a ::\u2098 s) t)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\n\u22a2 (\u2200 (a_1 : \u03b2 a), a_1 \u2208 t a \u2192 Nodup (Multiset.map (Pi.cons s a a_1) (pi s t))) \u2227\n Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)\n[PROOFSTEP]\nrefine' \u27e8fun b _ => ((ih hs) fun a' h' => ht a' <| mem_cons_of_mem h').map (Pi.cons_injective has), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\n\u22a2 Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)\n[PROOFSTEP]\nrefine' (ht a <| mem_cons_self _ _).pairwise _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\n\u22a2 \u2200 (a_1 : \u03b2 a),\n a_1 \u2208 t a \u2192\n \u2200 (b : \u03b2 a),\n b \u2208 t a \u2192 a_1 \u2260 b \u2192 Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t))\n[PROOFSTEP]\nexact fun b\u2081 _ b\u2082 _ neb =>\n disjoint_map_map.2 fun f _ g _ eq =>\n have : Pi.cons s a b\u2081 f a (mem_cons_self _ _) = Pi.cons s a b\u2082 g a (mem_cons_self _ _) := by rw [eq]\n neb <| show b\u2081 = b\u2082 by rwa [Pi.cons_same, Pi.cons_same] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\nb\u2081 : \u03b2 a\nx\u271d\u00b3 : b\u2081 \u2208 t a\nb\u2082 : \u03b2 a\nx\u271d\u00b2 : b\u2082 \u2208 t a\nneb : b\u2081 \u2260 b\u2082\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2 a\nx\u271d\u00b9 : f \u2208 pi s t\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2 a\nx\u271d : g \u2208 pi s t\neq : Pi.cons s a b\u2081 f = Pi.cons s a b\u2082 g\n\u22a2 Pi.cons s a b\u2081 f a (_ : a \u2208 a ::\u2098 s) = Pi.cons s a b\u2082 g a (_ : a \u2208 a ::\u2098 s)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ns\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\na : \u03b1\ns : Multiset \u03b1\nih : Nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 Nodup (t a)) \u2192 Nodup (pi s t)\nhs\u271d : Nodup (a ::\u2098 s)\nht : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Nodup (t a_1)\nhas : \u00aca \u2208 s\nhs : Nodup s\nb\u2081 : \u03b2 a\nx\u271d\u00b3 : b\u2081 \u2208 t a\nb\u2082 : \u03b2 a\nx\u271d\u00b2 : b\u2082 \u2208 t a\nneb : b\u2081 \u2260 b\u2082\nf : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2 a\nx\u271d\u00b9 : f \u2208 pi s t\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2 a\nx\u271d : g \u2208 pi s t\neq : Pi.cons s a b\u2081 f = Pi.cons s a b\u2082 g\nthis : Pi.cons s a b\u2081 f a (_ : a \u2208 a ::\u2098 s) = Pi.cons s a b\u2082 g a (_ : a \u2208 a ::\u2098 s)\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nrwa [Pi.cons_same, Pi.cons_same] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\n\u22a2 \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\n\u22a2 f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\n[PROOFSTEP]\ninduction' m using Multiset.induction_on with a m ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 0 \u2192 \u03b2 a\n\u22a2 f \u2208 pi 0 t \u2194 \u2200 (a : \u03b1) (h : a \u2208 0), f a h \u2208 t a\n[PROOFSTEP]\nhave : f = Pi.empty \u03b2 := funext (fun _ => funext fun h => (not_mem_zero _ h).elim)\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 0 \u2192 \u03b2 a\nthis : f = Pi.empty \u03b2\n\u22a2 f \u2208 pi 0 t \u2194 \u2200 (a : \u03b1) (h : a \u2208 0), f a h \u2208 t a\n[PROOFSTEP]\nsimp only [this, pi_zero, mem_singleton, true_iff]\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 0 \u2192 \u03b2 a\nthis : f = Pi.empty \u03b2\n\u22a2 \u2200 (a : \u03b1) (h : a \u2208 0), Pi.empty \u03b2 a h \u2208 t a\n[PROOFSTEP]\nintro _ h\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nf : (a : \u03b1) \u2192 a \u2208 0 \u2192 \u03b2 a\nthis : f = Pi.empty \u03b2\na\u271d : \u03b1\nh : a\u271d \u2208 0\n\u22a2 Pi.empty \u03b2 a\u271d h \u2208 t a\u271d\n[PROOFSTEP]\nexact (not_mem_zero _ h).elim\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\n\u22a2 f \u2208 pi (a ::\u2098 m) t \u2194 \u2200 (a_1 : \u03b1) (h : a_1 \u2208 a ::\u2098 m), f a_1 h \u2208 t a_1\n[PROOFSTEP]\nsimp_rw [pi_cons, mem_bind, mem_map, ih]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\n\u22a2 (\u2203 a_1, a_1 \u2208 t a \u2227 \u2203 a_2, (\u2200 (a : \u03b1) (h : a \u2208 m), a_2 a h \u2208 t a) \u2227 Pi.cons m a a_1 a_2 = f) \u2194\n \u2200 (a_1 : \u03b1) (h : a_1 \u2208 a ::\u2098 m), f a_1 h \u2208 t a_1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\n\u22a2 (\u2203 a_1, a_1 \u2208 t a \u2227 \u2203 a_2, (\u2200 (a : \u03b1) (h : a \u2208 m), a_2 a h \u2208 t a) \u2227 Pi.cons m a a_1 a_2 = f) \u2192\n \u2200 (a_2 : \u03b1) (h : a_2 \u2208 a ::\u2098 m), f a_2 h \u2208 t a_2\n[PROOFSTEP]\nrintro \u27e8b, hb, f', hf', rfl\u27e9 a' ha'\n[GOAL]\ncase cons.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nb : \u03b2 a\nhb : b \u2208 t a\nf' : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nhf' : \u2200 (a : \u03b1) (h : a \u2208 m), f' a h \u2208 t a\na' : \u03b1\nha' : a' \u2208 a ::\u2098 m\n\u22a2 Pi.cons m a b f' a' ha' \u2208 t a'\n[PROOFSTEP]\nby_cases h : a' = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nb : \u03b2 a\nhb : b \u2208 t a\nf' : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nhf' : \u2200 (a : \u03b1) (h : a \u2208 m), f' a h \u2208 t a\na' : \u03b1\nha' : a' \u2208 a ::\u2098 m\nh : a' = a\n\u22a2 Pi.cons m a b f' a' ha' \u2208 t a'\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf' : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nhf' : \u2200 (a : \u03b1) (h : a \u2208 m), f' a h \u2208 t a\na' : \u03b1\nb : \u03b2 a'\nhb : b \u2208 t a'\nha' : a' \u2208 a' ::\u2098 m\n\u22a2 Pi.cons m a' b f' a' ha' \u2208 t a'\n[PROOFSTEP]\nrwa [Pi.cons_same]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nb : \u03b2 a\nhb : b \u2208 t a\nf' : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nhf' : \u2200 (a : \u03b1) (h : a \u2208 m), f' a h \u2208 t a\na' : \u03b1\nha' : a' \u2208 a ::\u2098 m\nh : \u00aca' = a\n\u22a2 Pi.cons m a b f' a' ha' \u2208 t a'\n[PROOFSTEP]\nrw [Pi.cons_ne _ h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nb : \u03b2 a\nhb : b \u2208 t a\nf' : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a\nhf' : \u2200 (a : \u03b1) (h : a \u2208 m), f' a h \u2208 t a\na' : \u03b1\nha' : a' \u2208 a ::\u2098 m\nh : \u00aca' = a\n\u22a2 f' a' (_ : a' \u2208 m) \u2208 t a'\n[PROOFSTEP]\napply hf'\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\n\u22a2 (\u2200 (a_1 : \u03b1) (h : a_1 \u2208 a ::\u2098 m), f a_1 h \u2208 t a_1) \u2192\n \u2203 a_2, a_2 \u2208 t a \u2227 \u2203 a_3, (\u2200 (a : \u03b1) (h : a \u2208 m), a_3 a h \u2208 t a) \u2227 Pi.cons m a a_2 a_3 = f\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\nhf : \u2200 (a_1 : \u03b1) (h : a_1 \u2208 a ::\u2098 m), f a_1 h \u2208 t a_1\n\u22a2 \u2203 a_1, a_1 \u2208 t a \u2227 \u2203 a_2, (\u2200 (a : \u03b1) (h : a \u2208 m), a_2 a h \u2208 t a) \u2227 Pi.cons m a a_1 a_2 = f\n[PROOFSTEP]\nrefine' \u27e8_, hf a (mem_cons_self _ _), _, fun a ha => hf a (mem_cons_of_mem ha), _\u27e9\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\nm\u271d : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03b2 a)\nf\u271d : (a : \u03b1) \u2192 a \u2208 m\u271d \u2192 \u03b2 a\na : \u03b1\nm : Multiset \u03b1\nih : \u2200 (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a), f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a\nf : (a_1 : \u03b1) \u2192 a_1 \u2208 a ::\u2098 m \u2192 \u03b2 a_1\nhf : \u2200 (a_1 : \u03b1) (h : a_1 \u2208 a ::\u2098 m), f a_1 h \u2208 t a_1\n\u22a2 (Pi.cons m a (f a (_ : a \u2208 a ::\u2098 m)) fun a_1 ha => f a_1 (_ : a_1 \u2208 a ::\u2098 m)) = f\n[PROOFSTEP]\nrw [pi.cons_eta]\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Pi", "llama_tokens": 15306, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.28352507933486504}} {"text": "[GOAL]\n\u22a2 Set.Nonempty (interior { carrier := Icc 0 1, isCompact' := (_ : IsCompact (Icc 0 1)) }.carrier)\n[PROOFSTEP]\nsimp_rw [interior_Icc, nonempty_Ioo, zero_lt_one]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 Set.Nonempty\n (interior\n { carrier := Set.pi univ fun x => Icc 0 1, isCompact' := (_ : IsCompact (Set.pi univ fun x => Icc 0 1)) }.carrier)\n[PROOFSTEP]\nsimp only [interior_pi_set, Set.toFinite, interior_Icc, univ_pi_nonempty_iff, nonempty_Ioo, imp_true_iff, zero_lt_one]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = \u2191(PositiveCompacts.piIcc01 \u03b9)\n[PROOFSTEP]\nrefine' Eq.trans _ ((uIcc_of_le _).trans (Set.pi_univ_Icc _ _).symm)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1\n[PROOFSTEP]\nclassical convert parallelepiped_single (\u03b9 := \u03b9) 1\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191(parallelepiped (Pi.basisFun \u211d \u03b9)) = uIcc (fun i => 0) fun i => 1\n[PROOFSTEP]\nconvert parallelepiped_single (\u03b9 := \u03b9) 1\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 (fun i => 0) \u2264 fun i => 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\n\u22a2 addHaarMeasure Icc01 = volume\n[PROOFSTEP]\nconvert (addHaarMeasure_unique volume Icc01).symm\n[GOAL]\ncase h.e'_2\n\u22a2 addHaarMeasure Icc01 = \u2191\u2191volume \u2191Icc01 \u2022 addHaarMeasure Icc01\n[PROOFSTEP]\nsimp [Icc01]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 addHaarMeasure (piIcc01 \u03b9) = volume\n[PROOFSTEP]\nconvert (addHaarMeasure_unique volume (piIcc01 \u03b9)).symm\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 addHaarMeasure (piIcc01 \u03b9) = \u2191\u2191volume \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)\n[PROOFSTEP]\nsimp only [piIcc01, volume_pi_pi fun _ => Icc (0 : \u211d) 1, PositiveCompacts.coe_mk, Compacts.coe_mk,\n Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl \u221e\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u22a4 < \u22a4\n[PROOFSTEP]\ncalc\n \u221e = \u2211' _ : \u2115, \u03bc s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm\n _ = \u2211' n : \u2115, \u03bc ({u n} + s) := by congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]\n _ = \u03bc (\u22c3 n, {u n} + s) :=\n (Eq.symm <|\n measure_iUnion hs fun n => by simpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's)\n _ = \u03bc (range u + s) := by rw [\u2190 iUnion_add, iUnion_singleton_eq_range]\n _ < \u221e := Bounded.measure_lt_top (hu.add sb)\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2211' (x : \u2115), \u2191\u2191\u03bc s = \u2211' (n : \u2115), \u2191\u2191\u03bc ({u n} + s)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 (fun x => \u2191\u2191\u03bc s) = fun n => \u2191\u2191\u03bc ({u n} + s)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase e_f.h\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\nn : \u2115\n\u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc ({u n} + s)\n[PROOFSTEP]\nsimp only [image_add_left, measure_preimage_add, singleton_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\nn : \u2115\n\u22a2 MeasurableSet ({u n} + s)\n[PROOFSTEP]\nsimpa only [image_add_left, singleton_add] using measurable_id.const_add _ h's\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nsb : Metric.Bounded s\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nh : \u00ac\u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), {u n} + s) = \u2191\u2191\u03bc (range u + s)\n[PROOFSTEP]\nrw [\u2190 iUnion_add, iUnion_singleton_eq_range]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\nsuffices H : \u2200 R, \u03bc (s \u2229 closedBall 0 R) = 0\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s = 0\n[PROOFSTEP]\napply le_antisymm _ (zero_le _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s \u2264 0\n[PROOFSTEP]\ncalc\n \u03bc s \u2264 \u2211' n : \u2115, \u03bc (s \u2229 closedBall 0 n) :=\n by\n conv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0]\n exact measure_iUnion_le _\n _ = 0 := by simp only [H, tsum_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc s \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 iUnion_inter_closedBall_nat s 0]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 iUnion_inter_closedBall_nat s 0]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 iUnion_inter_closedBall_nat s 0]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n| \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 iUnion_inter_closedBall_nat s 0]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n)\n[PROOFSTEP]\nexact measure_iUnion_le _\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (s \u2229 closedBall 0 \u2191n) = 0\n[PROOFSTEP]\nsimp only [H, tsum_zero]\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n[PROOFSTEP]\nintro R\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall 0 R) = 0\n[PROOFSTEP]\napply\n addHaar_eq_zero_of_disjoint_translates_aux \u03bc u (bounded_closedBall.mono (inter_subset_right _ _)) hu _\n (h's.inter measurableSet_closedBall)\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\n\u22a2 Pairwise (Disjoint on fun n => {u n} + s \u2229 closedBall 0 R)\n[PROOFSTEP]\nrefine pairwise_disjoint_mono hs fun n => ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set E\nu : \u2115 \u2192 E\nhu : Metric.Bounded (range u)\nhs : Pairwise (Disjoint on fun n => {u n} + s)\nh's : MeasurableSet s\nR : \u211d\nn : \u2115\n\u22a2 {u n} + s \u2229 closedBall 0 R \u2264 {u n} + s\n[PROOFSTEP]\nexact add_subset_add Subset.rfl (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x, x \u2209 s := by simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2203 x, \u00acx \u2208 s\n[PROOFSTEP]\nsimpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nobtain \u27e8c, cpos, cone\u27e9 : \u2203 c : \u211d, 0 < c \u2227 c < 1 := \u27e81 / 2, by norm_num, by norm_num\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nhave A : Bounded (range fun n : \u2115 => c ^ n \u2022 x) :=\n haveI : Tendsto (fun n : \u2115 => c ^ n \u2022 x) atTop (\ud835\udcdd ((0 : \u211d) \u2022 x)) :=\n (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x\n bounded_range_of_tendsto _ this\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\napply addHaar_eq_zero_of_disjoint_translates \u03bc _ A _ (Submodule.closed_of_finiteDimensional s).measurableSet\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\n\u22a2 Pairwise (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s)\n[PROOFSTEP]\nintro m n hmn\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 (Disjoint on fun n => {c ^ n \u2022 x} + \u2191s) m n\n[PROOFSTEP]\nsimp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage, SetLike.mem_coe]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\n\u22a2 \u2200 \u2983a : E\u2984, -(c ^ m \u2022 x) + a \u2208 s \u2192 \u00ac-(c ^ n \u2022 x) + a \u2208 s\n[PROOFSTEP]\nintro y hym hyn\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nhave A : (c ^ n - c ^ m) \u2022 x \u2208 s := by\n convert s.sub_mem hym hyn using 1\n simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 (c ^ n - c ^ m) \u2022 x \u2208 s\n[PROOFSTEP]\nconvert s.sub_mem hym hyn using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\n\u22a2 (c ^ n - c ^ m) \u2022 x = -(c ^ m \u2022 x) + y - (-(c ^ n \u2022 x) + y)\n[PROOFSTEP]\nsimp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nhave H : c ^ n - c ^ m \u2260 0 := by simpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\n\u22a2 c ^ n - c ^ m \u2260 0\n[PROOFSTEP]\nsimpa only [sub_eq_zero, Ne.def] using (strictAnti_pow cpos cone).injective.ne hmn.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave : x \u2208 s := by\n convert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A\n rw [smul_smul, inv_mul_cancel H, one_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 x \u2208 s\n[PROOFSTEP]\nconvert s.smul_mem (c ^ n - c ^ m)\u207b\u00b9 A\n[GOAL]\ncase h.e'_4\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\n\u22a2 x = (c ^ n - c ^ m)\u207b\u00b9 \u2022 (c ^ n - c ^ m) \u2022 x\n[PROOFSTEP]\nrw [smul_smul, inv_mul_cancel H, one_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Submodule \u211d E\nhs : s \u2260 \u22a4\nx : E\nhx : \u00acx \u2208 s\nc : \u211d\ncpos : 0 < c\ncone : c < 1\nA\u271d : Metric.Bounded (range fun n => c ^ n \u2022 x)\nm n : \u2115\nhmn : m \u2260 n\ny : E\nhym : -(c ^ m \u2022 x) + y \u2208 s\nhyn : -(c ^ n \u2022 x) + y \u2208 s\nA : (c ^ n - c ^ m) \u2022 x \u2208 s\nH : c ^ n - c ^ m \u2260 0\nthis : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact hx this\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : s \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nrcases s.eq_bot_or_nonempty with (rfl | hne)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : \u22a5 \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc \u2191\u22a5 = 0\n[PROOFSTEP]\nrw [AffineSubspace.bot_coe, measure_empty]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : s \u2260 \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nrw [Ne.def, \u2190 AffineSubspace.direction_eq_top_iff_of_nonempty hne] at hs \n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nhne : Set.Nonempty \u2191s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nrcases hne with \u27e8x, hx : x \u2208 s\u27e9\n[GOAL]\ncase inr.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\ninst\u271d\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : AffineSubspace \u211d E\nhs : \u00acAffineSubspace.direction s = \u22a4\nx : E\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bc \u2191s = 0\n[PROOFSTEP]\nsimpa only [AffineSubspace.coe_direction_eq_vsub_set_right hx, vsub_eq_sub, sub_eq_add_neg, image_add_right, neg_neg,\n measure_preimage_add_right] using addHaar_submodule \u03bc s.direction hs\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave := addHaarMeasure_unique \u03bc (piIcc01 \u03b9)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Finite \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u03bc : Measure (\u03b9 \u2192 \u211d)\ninst\u271d : IsAddHaarMeasure \u03bc\nval\u271d : Fintype \u03b9\nthis : \u03bc = \u2191\u2191\u03bc \u2191(piIcc01 \u03b9) \u2022 addHaarMeasure (piIcc01 \u03b9)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nrw [this, addHaarMeasure_eq_volume_pi, Measure.map_smul, Real.map_linearMap_volume_pi_eq_smul_volume_pi hf, smul_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nlet \u03b9 := Fin (finrank \u211d E)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhaveI : FiniteDimensional \u211d (\u03b9 \u2192 \u211d) := by infer_instance\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\n\u22a2 FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d) := by simp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\n\u22a2 finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave e : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d := LinearEquiv.ofFinrankEq E (\u03b9 \u2192 \u211d) this\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 : \u2203 g, g = (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).comp (f.comp (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E)) := \u27e8_, rfl\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave gdet : LinearMap.det g = LinearMap.det f := by rw [hg]; exact LinearMap.det_conj f e\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 \u2191LinearMap.det g = \u2191LinearMap.det f\n[PROOFSTEP]\nrw [hg]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\n\u22a2 \u2191LinearMap.det (LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))) = \u2191LinearMap.det f\n[PROOFSTEP]\nexact LinearMap.det_conj f e\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nrw [\u2190 gdet] at hf \u22a2\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave fg : f = (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).comp (g.comp (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d)) :=\n by\n ext x\n simp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp, LinearEquiv.symm_apply_apply, hg]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\n\u22a2 f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nx : E\n\u22a2 \u2191f x = \u2191(LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)) x\n[PROOFSTEP]\nsimp only [LinearEquiv.coe_coe, Function.comp_apply, LinearMap.coe_comp, LinearEquiv.symm_apply_apply, hg]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nsimp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave Ce : Continuous e := (e : E \u2192\u2097[\u211d] \u03b9 \u2192 \u211d).continuous_of_finiteDimensional\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave Cg : Continuous g := LinearMap.continuous_of_finiteDimensional g\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave Cesymm : Continuous e.symm := (e.symm : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] E).continuous_of_finiteDimensional\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e) \u2218 \u2191g \u2218 \u2191e) \u03bc = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nrw [\u2190 map_map Cesymm.measurable (Cg.comp Ce).measurable, \u2190 map_map Cg.measurable Ce.measurable]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhaveI : IsAddHaarMeasure (map e \u03bc) := (e : E \u2243+ (\u03b9 \u2192 \u211d)).isAddHaarMeasure_map \u03bc Ce Cesymm\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave ecomp : e.symm \u2218 e = id := by ext x; simp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\n\u22a2 \u2191(LinearEquiv.symm e) \u2218 \u2191e = id\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\nx : E\n\u22a2 (\u2191(LinearEquiv.symm e) \u2218 \u2191e) x = id x\n[PROOFSTEP]\nsimp only [id.def, Function.comp_apply, LinearEquiv.symm_apply_apply]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\n\u03b9 : Type := Fin (finrank \u211d E)\nthis\u271d\u00b9 : FiniteDimensional \u211d (\u03b9 \u2192 \u211d)\nthis\u271d : finrank \u211d E = finrank \u211d (\u03b9 \u2192 \u211d)\ne : E \u2243\u2097[\u211d] \u03b9 \u2192 \u211d\ng : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det g \u2260 0\nhg : g = LinearMap.comp (\u2191e) (LinearMap.comp f \u2191(LinearEquiv.symm e))\ngdet : \u2191LinearMap.det g = \u2191LinearMap.det f\nfg : f = LinearMap.comp (\u2191(LinearEquiv.symm e)) (LinearMap.comp g \u2191e)\nCe : Continuous \u2191e\nCg : Continuous \u2191g\nCesymm : Continuous \u2191(LinearEquiv.symm e)\nthis : IsAddHaarMeasure (map (\u2191e) \u03bc)\necomp : \u2191(LinearEquiv.symm e) \u2218 \u2191e = id\n\u22a2 map (\u2191(LinearEquiv.symm e)) (map (\u2191g) (map (\u2191e) \u03bc)) = ENNReal.ofReal |(\u2191LinearMap.det g)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nrw [map_linearMap_addHaar_pi_eq_smul_addHaar hf (map e \u03bc), Measure.map_smul, map_map Cesymm.measurable Ce.measurable,\n ecomp, Measure.map_id]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\ns : Set E\n\u22a2 \u2191\u2191(map (\u2191f) \u03bc) s = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [map_linearMap_addHaar_eq_smul_addHaar \u03bc hf]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\nhf : \u2191LinearMap.det f \u2260 0\ns : Set E\n\u22a2 \u2191\u2191(ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 \u03bc) s = ENNReal.ofReal |(\u2191LinearMap.det f)\u207b\u00b9| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave A : LinearMap.det (f : E \u2192\u2097[\u211d] E) \u2260 0 := (LinearEquiv.isUnit_det' f).ne_zero\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' s) = ENNReal.ofReal |\u2191LinearMap.det \u2191(LinearEquiv.symm f)| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nconvert addHaar_preimage_linearMap \u03bc A s\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_1.h.e'_3\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2243\u2097[\u211d] E\ns : Set E\nA : \u2191LinearMap.det \u2191f \u2260 0\n\u22a2 \u2191LinearMap.det \u2191(LinearEquiv.symm f) = (\u2191LinearMap.det \u2191f)\u207b\u00b9\n[PROOFSTEP]\nsimp only [LinearEquiv.det_coe_symm]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrcases ne_or_eq (LinearMap.det f) 0 with (hf | hf)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nlet g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nchange \u03bc (g '' s) = _\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 \u2191\u2191\u03bc (\u2191g '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [ContinuousLinearEquiv.image_eq_preimage g s, addHaar_preimage_continuousLinearEquiv]\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f \u2260 0\ng : E \u2243L[\u211d] E := LinearEquiv.toContinuousLinearEquiv (LinearMap.equivOfDetNeZero f hf)\n\u22a2 ENNReal.ofReal |\u2191LinearMap.det \u2191\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm g))| * \u2191\u2191\u03bc s =\n ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f '' s) = ENNReal.ofReal |\u2191LinearMap.det f| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [hf, zero_mul, ENNReal.ofReal_zero, abs_zero]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0\n[PROOFSTEP]\nhave : \u03bc (LinearMap.range f) = 0 := addHaar_submodule \u03bc _ (LinearMap.range_lt_top_of_det_eq_zero hf).ne\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nf : E \u2192\u2097[\u211d] E\ns : Set E\nhf : \u2191LinearMap.det f = 0\nthis : \u2191\u2191\u03bc \u2191(LinearMap.range f) = 0\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191f a) '' s) = 0\n[PROOFSTEP]\nexact le_antisymm (le_trans (measure_mono (image_subset_range _ _)) this.le) (zero_le _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 map ((fun x x_1 => x \u2022 x_1) r) \u03bc = ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nlet f : E \u2192\u2097[\u211d] E := r \u2022 (1 : E \u2192\u2097[\u211d] E)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\n\u22a2 map ((fun x x_1 => x \u2022 x_1) r) \u03bc = ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nchange Measure.map f \u03bc = _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nhave hf : LinearMap.det f \u2260 0 :=\n by\n simp only [mul_one, LinearMap.det_smul, Ne.def, MonoidHom.map_one]\n intro h\n exact hr (pow_eq_zero h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\n\u22a2 \u2191LinearMap.det f \u2260 0\n[PROOFSTEP]\nsimp only [mul_one, LinearMap.det_smul, Ne.def, MonoidHom.map_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\n\u22a2 \u00acr ^ finrank \u211d E = 0\n[PROOFSTEP]\nintro h\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\nh : r ^ finrank \u211d E = 0\n\u22a2 False\n[PROOFSTEP]\nexact hr (pow_eq_zero h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\nf : E \u2192\u2097[\u211d] E := r \u2022 1\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 map (\u2191f) \u03bc = ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc\n[PROOFSTEP]\nsimp only [map_linearMap_addHaar_eq_smul_addHaar \u03bc hf, mul_one, LinearMap.det_smul, map_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191(map ((fun x x_1 => x \u2022 x_1) r) \u03bc) s = ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [map_addHaar_smul \u03bc hr, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\ns : Set E\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrcases ne_or_eq r 0 with (h | rfl)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\ns : Set E\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [\u2190 preimage_smul_inv\u2080 h, addHaar_preimage_smul \u03bc (inv_ne_zero h), inv_pow, inv_inv]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hs)\n[GOAL]\ncase inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u22a2 \u2191\u2191\u03bc (0 \u2022 \u2205) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc \u2205\n[PROOFSTEP]\nsimp only [measure_empty, mul_zero, smul_set_empty]\n[GOAL]\ncase inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (0 \u2022 s) = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [zero_smul_set hs, \u2190 singleton_zero]\n[GOAL]\ncase inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nby_cases h : finrank \u211d E = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhaveI : Subsingleton E := finrank_zero_iff.1 h\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : finrank \u211d E = 0\nthis : Subsingleton E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [h, one_mul, ENNReal.ofReal_one, abs_one, Subsingleton.eq_univ_of_nonempty hs, pow_zero,\n Subsingleton.eq_univ_of_nonempty (singleton_nonempty (0 : E))]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhaveI : Nontrivial E := nontrivial_of_finrank_pos (bot_lt_iff_ne_bot.2 h)\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : Set.Nonempty s\nh : \u00acfinrank \u211d E = 0\nthis : Nontrivial E\n\u22a2 \u2191\u2191\u03bc {0} = ENNReal.ofReal |0 ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [h, zero_mul, ENNReal.ofReal_zero, abs_zero, Ne.def, not_false_iff, zero_pow', measure_singleton]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : 0 \u2264 r\ns : Set E\n\u22a2 \u2191\u2191\u03bc (r \u2022 s) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [addHaar_smul, abs_pow, abs_of_nonneg hr]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\n\u22a2 NullMeasurableSet (r \u2022 s)\n[PROOFSTEP]\nobtain rfl | hs' := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhs : NullMeasurableSet \u2205\n\u22a2 NullMeasurableSet (r \u2022 \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (r \u2022 s)\n[PROOFSTEP]\nobtain rfl | hr := eq_or_ne r 0\n[GOAL]\ncase inr.inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nhs' : Set.Nonempty s\n\u22a2 NullMeasurableSet (0 \u2022 s)\n[PROOFSTEP]\nsimpa [zero_smul_set hs'] using nullMeasurableSet_singleton _\n[GOAL]\ncase inr.inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nhs : NullMeasurableSet s\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\n\u22a2 NullMeasurableSet (r \u2022 s)\n[PROOFSTEP]\nobtain \u27e8t, ht, hst\u27e9 := hs\n[GOAL]\ncase inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 NullMeasurableSet (r \u2022 s)\n[PROOFSTEP]\nrefine' \u27e8_, ht.const_smul_of_ne_zero hr, _\u27e9\n[GOAL]\ncase inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : s =\u1da0[ae \u03bc] t\n\u22a2 r \u2022 s =\u1da0[ae \u03bc] r \u2022 t\n[PROOFSTEP]\nrw [\u2190 measure_symmDiff_eq_zero_iff] at hst \u22a2\n[GOAL]\ncase inr.inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhs' : Set.Nonempty s\nhr : r \u2260 0\nt : Set E\nht : MeasurableSet t\nhst : \u2191\u2191\u03bc (s \u2206 t) = 0\n\u22a2 \u2191\u2191\u03bc ((r \u2022 s) \u2206 (r \u2022 t)) = 0\n[PROOFSTEP]\nrw [\u2190 smul_set_symmDiff\u2080 hr, addHaar_smul \u03bc, hst, mul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\ns : Set E\n\u22a2 \u2191\u2191\u03bc (\u2191(AffineMap.homothety x r) '' s) = \u2191\u2191\u03bc ((fun y => y + x) '' (r \u2022 (fun y => y + -x) '' s))\n[PROOFSTEP]\nsimp only [\u2190 image_smul, image_image, \u2190 sub_eq_add_neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\ns : Set E\n\u22a2 \u2191\u2191\u03bc ((fun a => \u2191(AffineMap.homothety x r) a) '' s) = \u2191\u2191\u03bc ((fun x_1 => r \u2022 (x_1 - x) + x) '' s)\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\ns : Set E\n\u22a2 \u2191\u2191\u03bc ((fun y => y + x) '' (r \u2022 (fun y => y + -x) '' s)) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [image_add_right, measure_preimage_add_right, addHaar_smul]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (ball x r) = \u2191\u2191\u03bc (ball 0 r)\n[PROOFSTEP]\nhave : ball (0 : E) r = (\u00b7 + \u00b7) x \u207b\u00b9' ball x r := by simp [preimage_add_ball]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r\n[PROOFSTEP]\nsimp [preimage_add_ball]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\nthis : ball 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' ball x r\n\u22a2 \u2191\u2191\u03bc (ball x r) = \u2191\u2191\u03bc (ball 0 r)\n[PROOFSTEP]\nrw [this, measure_preimage_add]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (closedBall 0 r)\n[PROOFSTEP]\nhave : closedBall (0 : E) r = (\u00b7 + \u00b7) x \u207b\u00b9' closedBall x r := by simp [preimage_add_closedBall]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\n\u22a2 closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r\n[PROOFSTEP]\nsimp [preimage_add_closedBall]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\u271d\ninst\u271d\u00b9\u2070 : MeasurableSpace E\u271d\ninst\u271d\u2079 : BorelSpace E\u271d\ninst\u271d\u2078 : FiniteDimensional \u211d E\u271d\n\u03bc\u271d : Measure E\u271d\ninst\u271d\u2077 : IsAddHaarMeasure \u03bc\u271d\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\ns : Set E\u271d\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nx : E\nr : \u211d\nthis : closedBall 0 r = (fun x x_1 => x + x_1) x \u207b\u00b9' closedBall x r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (closedBall 0 r)\n[PROOFSTEP]\nrw [this, measure_preimage_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n[PROOFSTEP]\nhave : ball (0 : E) (r * s) = r \u2022 ball (0 : E) s := by\n simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 ball 0 (r * s) = r \u2022 ball 0 s\n[PROOFSTEP]\nsimp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\nthis : ball 0 (r * s) = r \u2022 ball 0 s\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n[PROOFSTEP]\nsimp only [this, addHaar_smul, abs_of_nonneg hr.le, addHaar_ball_center, abs_pow]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : 0 < r\n\u22a2 \u2191\u2191\u03bc (ball x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrw [\u2190 addHaar_ball_mul_of_pos \u03bc x hr, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n[PROOFSTEP]\nrcases hr.eq_or_lt with (rfl | h)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\ns : \u211d\nhr : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bc (ball x (0 * s)) = ENNReal.ofReal (0 ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n[PROOFSTEP]\nsimp only [zero_pow (finrank_pos (K := \u211d) (V := E)), measure_empty, zero_mul, ENNReal.ofReal_zero, ball_zero]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n[PROOFSTEP]\nexact addHaar_ball_mul_of_pos \u03bc x h s\n[GOAL]\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (ball x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrw [\u2190 addHaar_ball_mul \u03bc x hr, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 \u2191\u2191\u03bc (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 s)\n[PROOFSTEP]\nhave : closedBall (0 : E) (r * s) = r \u2022 closedBall (0 : E) s := by\n simp [smul_closedBall' hr.ne' (0 : E), abs_of_nonneg hr.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\n\u22a2 closedBall 0 (r * s) = r \u2022 closedBall 0 s\n[PROOFSTEP]\nsimp [smul_closedBall' hr.ne' (0 : E), abs_of_nonneg hr.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 < r\ns : \u211d\nthis : closedBall 0 (r * s) = r \u2022 closedBall 0 s\n\u22a2 \u2191\u2191\u03bc (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 s)\n[PROOFSTEP]\nsimp only [this, addHaar_smul, abs_of_nonneg hr.le, addHaar_closedBall_center, abs_pow]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nhs : 0 \u2264 s\n\u22a2 \u2191\u2191\u03bc (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 s)\n[PROOFSTEP]\nhave : closedBall (0 : E) (r * s) = r \u2022 closedBall (0 : E) s := by simp [smul_closedBall r (0 : E) hs, abs_of_nonneg hr]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nhs : 0 \u2264 s\n\u22a2 closedBall 0 (r * s) = r \u2022 closedBall 0 s\n[PROOFSTEP]\nsimp [smul_closedBall r (0 : E) hs, abs_of_nonneg hr]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nhs : 0 \u2264 s\nthis : closedBall 0 (r * s) = r \u2022 closedBall 0 s\n\u22a2 \u2191\u2191\u03bc (closedBall x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 s)\n[PROOFSTEP]\nsimp only [this, addHaar_smul, abs_of_nonneg hr, addHaar_closedBall_center, abs_pow]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)\n[PROOFSTEP]\nrw [\u2190 addHaar_closedBall_mul \u03bc x hr zero_le_one, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (closedBall 0 1) = \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\napply le_antisymm _ (measure_mono ball_subset_closedBall)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nhave A :\n Tendsto (fun r : \u211d => ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (closedBall (0 : E) 1)) (\ud835\udcdd[<] 1)\n (\ud835\udcdd (ENNReal.ofReal ((1 : \u211d) ^ finrank \u211d E) * \u03bc (closedBall (0 : E) 1))) :=\n by\n refine' ENNReal.Tendsto.mul _ (by simp) tendsto_const_nhds (by simp)\n exact ENNReal.tendsto_ofReal ((tendsto_id'.2 nhdsWithin_le_nhds).pow _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1)\n (\ud835\udcdd (ENNReal.ofReal (1 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)))\n[PROOFSTEP]\nrefine' ENNReal.Tendsto.mul _ (by simp) tendsto_const_nhds (by simp)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 ENNReal.ofReal (1 ^ finrank \u211d E) \u2260 0 \u2228 \u2191\u2191\u03bc (closedBall 0 1) \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2191\u2191\u03bc (closedBall 0 1) \u2260 0 \u2228 ENNReal.ofReal (1 ^ finrank \u211d E) \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (ENNReal.ofReal (1 ^ finrank \u211d E)))\n[PROOFSTEP]\nexact ENNReal.tendsto_ofReal ((tendsto_id'.2 nhdsWithin_le_nhds).pow _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA :\n Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1)\n (\ud835\udcdd (ENNReal.ofReal (1 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nsimp only [one_pow, one_mul, ENNReal.ofReal_one] at A \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrefine' le_of_tendsto A _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2200\u1da0 (c : \u211d) in \ud835\udcdd[Iio 1] 1, ENNReal.ofReal (c ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrefine' mem_nhdsWithin_Iio_iff_exists_Ioo_subset.2 \u27e8(0 : \u211d), by simp, fun r hr => _\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 0 \u2208 Iio 1\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioo 0 1\n\u22a2 r \u2208 {x | (fun c => ENNReal.ofReal (c ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)) x}\n[PROOFSTEP]\ndsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioo 0 1\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrw [\u2190 addHaar_closedBall' \u03bc (0 : E) hr.1.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nA : Tendsto (fun r => ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1)) (\ud835\udcdd[Iio 1] 1) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioo 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall 0 r) \u2264 \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nexact measure_mono (closedBall_subset_ball hr.2)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nrw [addHaar_closedBall' \u03bc x hr, addHaar_closed_unit_ball_eq_addHaar_unit_ball]\n[GOAL]\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (ball x r)\n[PROOFSTEP]\nby_cases h : r < 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nh : r < 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (ball x r)\n[PROOFSTEP]\nrw [Metric.closedBall_eq_empty.mpr h, Metric.ball_eq_empty.mpr h.le]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nh : \u00acr < 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (ball x r)\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nh : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) = \u2191\u2191\u03bc (ball x r)\n[PROOFSTEP]\nrw [addHaar_closedBall \u03bc x h, addHaar_ball \u03bc x h]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\n[PROOFSTEP]\nrcases hr.lt_or_lt with (h | h)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\nh : r < 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\n[PROOFSTEP]\nsimp only [empty_diff, measure_empty, \u2190 closedBall_diff_ball, closedBall_eq_empty.2 h]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nx : E\nr : \u211d\nhr : r \u2260 0\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\n[PROOFSTEP]\nrw [\u2190 closedBall_diff_ball, measure_diff ball_subset_closedBall measurableSet_ball measure_ball_lt_top.ne,\n addHaar_ball_of_pos \u03bc _ h, addHaar_closedBall \u03bc _ h.le, tsub_self]\n[GOAL]\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\n[PROOFSTEP]\nrcases eq_or_ne r 0 with (rfl | h)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\n\u22a2 \u2191\u2191\u03bc (sphere x 0) = 0\n[PROOFSTEP]\nrw [sphere_zero, measure_singleton]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nh : r \u2260 0\n\u22a2 \u2191\u2191\u03bc (sphere x r) = 0\n[PROOFSTEP]\nexact addHaar_sphere_of_ne_zero \u03bc x h\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 s) / \u2191\u2191\u03bc ({y} + r \u2022 t) =\n ENNReal.ofReal (|r| ^ finrank \u211d E) * \u2191\u2191\u03bc s * (ENNReal.ofReal (|r| ^ finrank \u211d E) * \u2191\u2191\u03bc t)\u207b\u00b9\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, addHaar_smul, image_add_left, measure_preimage_add, abs_pow, singleton_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) * \u2191\u2191\u03bc s * (ENNReal.ofReal (|r| ^ finrank \u211d E) * \u2191\u2191\u03bc t)\u207b\u00b9 =\n ENNReal.ofReal (|r| ^ finrank \u211d E) * (ENNReal.ofReal (|r| ^ finrank \u211d E))\u207b\u00b9 * (\u2191\u2191\u03bc s * (\u2191\u2191\u03bc t)\u207b\u00b9)\n[PROOFSTEP]\nrw [ENNReal.mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) * \u2191\u2191\u03bc s * ((ENNReal.ofReal (|r| ^ finrank \u211d E))\u207b\u00b9 * (\u2191\u2191\u03bc t)\u207b\u00b9) =\n ENNReal.ofReal (|r| ^ finrank \u211d E) * (ENNReal.ofReal (|r| ^ finrank \u211d E))\u207b\u00b9 * (\u2191\u2191\u03bc s * (\u2191\u2191\u03bc t)\u207b\u00b9)\n[PROOFSTEP]\nring\n[GOAL]\ncase ha\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) \u2260 0 \u2228 \u2191\u2191\u03bc t \u2260 \u22a4\n[PROOFSTEP]\nsimp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne.def, true_or_iff]\n[GOAL]\ncase hb\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) \u2260 \u22a4 \u2228 \u2191\u2191\u03bc t \u2260 0\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_ne_top, true_or_iff, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) * (ENNReal.ofReal (|r| ^ finrank \u211d E))\u207b\u00b9 * (\u2191\u2191\u03bc s * (\u2191\u2191\u03bc t)\u207b\u00b9) = \u2191\u2191\u03bc s / \u2191\u2191\u03bc t\n[PROOFSTEP]\nrw [ENNReal.mul_inv_cancel, one_mul, div_eq_mul_inv]\n[GOAL]\ncase h0\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) \u2260 0\n[PROOFSTEP]\nsimp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne.def]\n[GOAL]\ncase ht\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d : Set E\nr : \u211d\nhr : r \u2260 0\nx y : E\ns t : Set E\n\u22a2 ENNReal.ofReal (|r| ^ finrank \u211d E) \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 IsUnifLocDoublingMeasure \u03bc\n[PROOFSTEP]\nrefine' \u27e8\u27e8(2 : \u211d\u22650) ^ finrank \u211d E, _\u27e9\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2200 (x : E), \u2191\u2191\u03bc (closedBall x (2 * \u03b5)) \u2264 \u2191(2 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall x \u03b5)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with r hr x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhr : r \u2208 Ioi 0\nx : E\n\u22a2 \u2191\u2191\u03bc (closedBall x (2 * r)) \u2264 \u2191(2 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nrw [addHaar_closedBall_mul \u03bc x zero_le_two (le_of_lt hr), addHaar_closedBall_center \u03bc x, ENNReal.ofReal,\n Real.toNNReal_pow zero_le_two]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns : Set E\nr : \u211d\nhr : r \u2208 Ioi 0\nx : E\n\u22a2 \u2191(Real.toNNReal 2 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 r) \u2264 \u2191(2 ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 r)\n[PROOFSTEP]\nsimp only [Real.toNNReal_ofNat, le_refl]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\n\u22a2 \u2191\u2191(Basis.addHaar b) (parallelepiped v) = ENNReal.ofReal |\u2191(Basis.det b) v|\n[PROOFSTEP]\nhave : FiniteDimensional \u211d G := FiniteDimensional.of_fintype_basis b\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 \u2191\u2191(Basis.addHaar b) (parallelepiped v) = ENNReal.ofReal |\u2191(Basis.det b) v|\n[PROOFSTEP]\nhave A : parallelepiped v = b.constr \u2115 v '' parallelepiped b :=\n by\n rw [image_parallelepiped]\n -- porting note: was `congr 1 with i` but Lean 4 `congr` applies `ext` first\n refine congr_arg _ <| funext fun i \u21a6 ?_\n exact (b.constr_basis \u2115 v i).symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 parallelepiped v = \u2191(\u2191(Basis.constr b \u2115) v) '' parallelepiped \u2191b\n[PROOFSTEP]\nrw [image_parallelepiped]\n -- porting note: was `congr 1 with i` but Lean 4 `congr` applies `ext` first\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\n\u22a2 parallelepiped v = parallelepiped (\u2191(\u2191(Basis.constr b \u2115) v) \u2218 \u2191b)\n[PROOFSTEP]\nrefine congr_arg _ <| funext fun i \u21a6 ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\ni : \u03b9\n\u22a2 v i = (\u2191(\u2191(Basis.constr b \u2115) v) \u2218 \u2191b) i\n[PROOFSTEP]\nexact (b.constr_basis \u2115 v i).symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : MeasurableSpace E\ninst\u271d\u00b9\u00b9 : BorelSpace E\ninst\u271d\u00b9\u2070 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2079 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : MeasurableSpace G\ninst\u271d : BorelSpace G\nb : Basis \u03b9 \u211d G\nv : \u03b9 \u2192 G\nthis : FiniteDimensional \u211d G\nA : parallelepiped v = \u2191(\u2191(Basis.constr b \u2115) v) '' parallelepiped \u2191b\n\u22a2 \u2191\u2191(Basis.addHaar b) (parallelepiped v) = ENNReal.ofReal |\u2191(Basis.det b) v|\n[PROOFSTEP]\nrw [A, addHaar_image_linearMap, b.addHaar_self, mul_one, \u2190 LinearMap.det_toMatrix b, \u2190\n Basis.toMatrix_eq_toMatrix_constr, Basis.det_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n\u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) = ENNReal.ofReal |\u2191\u03c9 v|\n[PROOFSTEP]\nconv_rhs => rw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n| ENNReal.ofReal |\u2191\u03c9 v|\n[PROOFSTEP]\nrw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n| ENNReal.ofReal |\u2191\u03c9 v|\n[PROOFSTEP]\nrw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n| ENNReal.ofReal |\u2191\u03c9 v|\n[PROOFSTEP]\nrw [\u03c9.eq_smul_basis_det (finBasisOfFinrankEq \u211d G _i.out)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\nv : Fin n \u2192 G\n\u22a2 \u2191\u2191(AlternatingMap.measure \u03c9) (parallelepiped v) =\n ENNReal.ofReal\n |\u2191(\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)) \u2022\n Basis.det (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))\n v|\n[PROOFSTEP]\nsimp only [addHaar_parallelepiped, AlternatingMap.measure, coe_nnreal_smul_apply, AlternatingMap.smul_apply,\n Algebra.id.smul_eq_mul, abs_mul, ENNReal.ofReal_mul (abs_nonneg _), Real.ennnorm_eq_ofReal_abs]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\n\u22a2 IsAddLeftInvariant (AlternatingMap.measure \u03c9)\n[PROOFSTEP]\nrw [AlternatingMap.measure]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\n\u22a2 IsAddLeftInvariant\n (\u2016\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n))\u2016\u208a \u2022\n Basis.addHaar (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\n\u22a2 IsLocallyFiniteMeasure (AlternatingMap.measure \u03c9)\n[PROOFSTEP]\nrw [AlternatingMap.measure]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : MeasurableSpace E\ninst\u271d\u00b9\u00b2 : BorelSpace E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b9\u2070 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ns : Set E\n\u03b9 : Type u_3\nG : Type u_4\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FiniteDimensional \u211d G\nn : \u2115\n_i : Fact (finrank \u211d G = n)\n\u03c9 : AlternatingMap \u211d G \u211d (Fin n)\n\u22a2 IsLocallyFiniteMeasure\n (\u2016\u2191\u03c9 \u2191(finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n))\u2016\u208a \u2022\n Basis.addHaar (finBasisOfFinrankEq \u211d G (_ : finrank \u211d G = n)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave A : Tendsto (fun r : \u211d => \u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n by\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h (eventually_of_forall fun b => zero_le _)\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _\n intro y hy\n have : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n apply smul_set_mono t_bound\n simpa [neg_add_eq_sub] using hy\n simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le, mem_closedBall_iff_norm, mul_one, sub_zero,\n smul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h (eventually_of_forall fun b => zero_le _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) \u2264 \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\napply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\n\u22a2 {x} + r \u2022 t \u2286 closedBall x r\n[PROOFSTEP]\nintro y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y \u2208 closedBall x r\n[PROOFSTEP]\nhave : y - x \u2208 r \u2022 closedBall (0 : E) 1 := by\n apply smul_set_mono t_bound\n simpa [neg_add_eq_sub] using hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y - x \u2208 r \u2022 closedBall 0 1\n[PROOFSTEP]\napply smul_set_mono t_bound\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\n\u22a2 y - x \u2208 r \u2022 t\n[PROOFSTEP]\nsimpa [neg_add_eq_sub] using hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nr : \u211d\nrpos : 0 < r\ny : E\nhy : y \u2208 {x} + r \u2022 t\nthis : y - x \u2208 r \u2022 closedBall 0 1\n\u22a2 y \u2208 closedBall x r\n[PROOFSTEP]\nsimpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le, mem_closedBall_iff_norm, mul_one, sub_zero,\n smul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave B :\n Tendsto (fun r : \u211d => \u03bc (closedBall x r) / \u03bc ({ x } + r \u2022 u)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (closedBall x 1) / \u03bc ({ x } + u))) :=\n by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n have : closedBall x r = { x } + r \u2022 closedBall (0 : E) 1 := by\n simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero, mul_one,\n singleton_add_closedBall, smul_zero]\n simp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne']\n simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n[PROOFSTEP]\napply tendsto_const_nhds.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x_1 => \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nhave : closedBall x r = { x } + r \u2022 closedBall (0 : E) 1 := by\n simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero, mul_one,\n singleton_add_closedBall, smul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 closedBall x r = {x} + r \u2022 closedBall 0 1\n[PROOFSTEP]\nsimp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero, mul_one,\n singleton_add_closedBall, smul_zero]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nsimp only [this, addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne']\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\nthis : closedBall x r = {x} + r \u2022 closedBall 0 1\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) = \u2191\u2191\u03bc (closedBall 0 1) / \u2191\u2191\u03bc u\n[PROOFSTEP]\nsimp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave C :\n Tendsto (fun r : \u211d => \u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({ x } + r \u2022 u)))\n (\ud835\udcdd[>] 0) (\ud835\udcdd (0 * (\u03bc (closedBall x 1) / \u03bc ({ x } + u)))) :=\n by\n apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)\n simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or, ENNReal.div_eq_top, h'u,\n false_or_iff, not_and, and_false_iff]\n intro aux\n exact (measure_closedBall_lt_top.ne aux).elim\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))))\n[PROOFSTEP]\napply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 0 \u2260 0 \u2228 \u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u) \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or, ENNReal.div_eq_top, h'u,\n false_or_iff, not_and, and_false_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\n\u22a2 \u2191\u2191\u03bc (closedBall x 1) = \u22a4 \u2192 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4\n[PROOFSTEP]\nintro aux\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\naux : \u2191\u2191\u03bc (closedBall x 1) = \u22a4\n\u22a2 \u00ac\u00ac\u2191\u2191\u03bc u = \u22a4\n[PROOFSTEP]\nexact (measure_closedBall_lt_top.ne aux).elim\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u))))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [zero_mul] at C \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\napply C.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u))) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi 0 \u2192\n \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc (closedBall x a) * (\u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc ({x} + a \u2022 u)) =\n \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\ncalc\n \u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc (closedBall x r) * (\u03bc (closedBall x r) / \u03bc ({ x } + r \u2022 u)) =\n \u03bc (closedBall x r) * (\u03bc (closedBall x r))\u207b\u00b9 * (\u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc ({ x } + r \u2022 u)) :=\n by simp only [div_eq_mul_inv]; ring\n _ = \u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc ({ x } + r \u2022 u) := by\n rw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne' measure_closedBall_lt_top.ne, one_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u))\n[PROOFSTEP]\nsimp only [div_eq_mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9) =\n \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) * (\u2191\u2191\u03bc ({x} + r \u2022 u))\u207b\u00b9)\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nt_bound : t \u2286 closedBall 0 1\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (closedBall x 1) / \u2191\u2191\u03bc ({x} + u)))\nC :\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc ({x} + r \u2022 u)))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) * (\u2191\u2191\u03bc (closedBall x r))\u207b\u00b9 * (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nrw [ENNReal.mul_inv_cancel (measure_closedBall_pos \u03bc x rpos).ne' measure_closedBall_lt_top.ne, one_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nset t' := R\u207b\u00b9 \u2022 t with ht'\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nset u' := R\u207b\u00b9 \u2022 u with hu'\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave A : Tendsto (fun r : \u211d => \u03bc (s \u2229 ({ x } + r \u2022 t')) / \u03bc ({ x } + r \u2022 u')) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n by\n apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 \u03bc s x h t' u'\n \u00b7\n simp only [h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff, ENNReal.ofReal_eq_zero,\n inv_eq_zero, inv_pow, Ne.def, or_self_iff, mul_eq_zero]\n \u00b7 refine (smul_set_mono t_bound).trans_eq ?_\n rw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le), inv_mul_cancel Rpos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 \u03bc s x h t' u'\n[GOAL]\ncase h'u\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 \u2191\u2191\u03bc u' \u2260 0\n[PROOFSTEP]\nsimp only [h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff, ENNReal.ofReal_eq_zero, inv_eq_zero,\n inv_pow, Ne.def, or_self_iff, mul_eq_zero]\n[GOAL]\ncase t_bound\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 t' \u2286 closedBall 0 1\n[PROOFSTEP]\nrefine (smul_set_mono t_bound).trans_eq ?_\n[GOAL]\ncase t_bound\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\n\u22a2 R\u207b\u00b9 \u2022 closedBall 0 R = closedBall 0 1\n[PROOFSTEP]\nrw [smul_closedBall _ _ Rpos.le, smul_zero, Real.norm_of_nonneg (inv_nonneg.2 Rpos.le), inv_mul_cancel Rpos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave B : Tendsto (fun r : \u211d => R * r) (\ud835\udcdd[>] 0) (\ud835\udcdd[>] (R * 0)) :=\n by\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n \u00b7 exact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n intro r rpos\n rw [mul_zero]\n exact mul_pos Rpos rpos\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 0))\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within\n[GOAL]\ncase h1\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (R * 0))\n[PROOFSTEP]\nexact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds\n[GOAL]\ncase h2\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi 0] 0, R * x \u2208 Ioi (R * 0)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 R * a \u2208 Ioi (R * 0)\n[PROOFSTEP]\nintro r rpos\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi (R * 0)\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : r \u2208 Ioi 0\n\u22a2 R * r \u2208 Ioi 0\n[PROOFSTEP]\nexact mul_pos Rpos rpos\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi (R * 0)] (R * 0))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [mul_zero] at B \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\napply (A.comp B).congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) \u2218 fun r => R * r) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r =>\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi 0 \u2192\n ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 R\u207b\u00b9 \u2022 u)) \u2218 fun r => R * r) a =\n \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 u)\n[PROOFSTEP]\nrintro r -\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 R\u207b\u00b9 \u2022 u)) \u2218 fun r => R * r) r =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nhave T : (R * r) \u2022 t' = r \u2022 t := by rw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\n\u22a2 (R * r) \u2022 t' = r \u2022 t\n[PROOFSTEP]\nrw [mul_comm, ht', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 R\u207b\u00b9 \u2022 u)) \u2218 fun r => R * r) r =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nhave U : (R * r) \u2022 u' = r \u2022 u := by rw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\n\u22a2 (R * r) \u2022 u' = r \u2022 u\n[PROOFSTEP]\nrw [mul_comm, hu', smul_smul, mul_assoc, mul_inv_cancel Rpos.ne', mul_one]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 ((fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 R\u207b\u00b9 \u2022 u)) \u2218 fun r => R * r) r =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt u : Set E\nh'u : \u2191\u2191\u03bc u \u2260 0\nR : \u211d\nRpos : 0 < R\nt_bound : t \u2286 closedBall 0 R\nt' : Set E := R\u207b\u00b9 \u2022 t\nht' : t' = R\u207b\u00b9 \u2022 t\nu' : Set E := R\u207b\u00b9 \u2022 u\nhu' : u' = R\u207b\u00b9 \u2022 u\nA : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 u')) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nB : Tendsto (fun r => R * r) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd[Ioi 0] 0)\nr : \u211d\nT : (R * r) \u2022 t' = r \u2022 t\nU : (R * r) \u2022 u' = r \u2022 u\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 t)) / \u2191\u2191\u03bc ({x} + (R * r) \u2022 R\u207b\u00b9 \u2022 u) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 u)\n[PROOFSTEP]\nrw [T, U]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' tendsto_order.2 \u27e8fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun \u03b5 (\u03b5pos : 0 < \u03b5) => _\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n[PROOFSTEP]\nrcases eq_or_ne (\u03bc t) 0 with (h't | h't)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n[PROOFSTEP]\napply eventually_of_forall fun r => ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n[PROOFSTEP]\nsuffices H : \u03bc (s \u2229 ({ x } + r \u2022 t)) = 0\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n[PROOFSTEP]\nrw [H]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\nH : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n\u22a2 0 / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n[PROOFSTEP]\nsimpa only [ENNReal.zero_div] using \u03b5pos\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0\n[PROOFSTEP]\napply le_antisymm _ (zero_le _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 0\n[PROOFSTEP]\ncalc\n \u03bc (s \u2229 ({ x } + r \u2022 t)) \u2264 \u03bc ({ x } + r \u2022 t) := measure_mono (inter_subset_right _ _)\n _ = 0 := by simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add, mul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t = 0\nr : \u211d\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) = 0\n[PROOFSTEP]\nsimp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add, mul_zero]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n[PROOFSTEP]\nobtain \u27e8n, npos, hn\u27e9 : \u2203 n : \u2115, 0 < n \u2227 \u03bc (t \\ closedBall 0 n) < \u03b5 / 2 * \u03bc t :=\n by\n have A : Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) :=\n by\n have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e := \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl, iUnion_closedBall_nat, compl_univ,\n inter_empty]\n simp only [this, measure_empty] at A \n have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't\n exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave A : Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) :=\n by\n have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e := \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), t \\ closedBall 0 \u2191n)))\n[PROOFSTEP]\nhave N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e := \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), t \\ closedBall 0 \u2191n)))\n[PROOFSTEP]\nrefine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nN : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 t \\ closedBall 0 \u2191n \u2264 t \\ closedBall 0 \u2191m\n[PROOFSTEP]\nexact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), t \\ closedBall 0 \u2191n)))\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl, iUnion_closedBall_nat, compl_univ,\n inter_empty]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), t \\ closedBall 0 \u2191n)))\n\u22a2 \u22c2 (n : \u2115), t \\ closedBall 0 \u2191n = \u2205\n[PROOFSTEP]\nsimp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl, iUnion_closedBall_nat, compl_univ,\n inter_empty]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 (n : \u2115), t \\ closedBall 0 \u2191n)))\nthis : \u22c2 (n : \u2115), t \\ closedBall 0 \u2191n = \u2205\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimp only [this, measure_empty] at A \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 (n : \u2115), t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nthis : \u22c2 (n : \u2115), t \\ closedBall 0 \u2191n = \u2205\nA : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0)\nI : 0 < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n[PROOFSTEP]\nhave L : Tendsto (fun r : \u211d => \u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n))) / \u03bc ({ x } + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 \u03bc s x h _ t h't n (Nat.cast_pos.2 npos) (inter_subset_right _ _)\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos \u03b5pos.ne'), self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 / 2 \u2192\n a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5\n[PROOFSTEP]\nrintro r hr (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n[PROOFSTEP]\nhave I : \u03bc (s \u2229 ({ x } + r \u2022 t)) \u2264 \u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({ x } + r \u2022 (t \\ closedBall 0 n)) :=\n calc\n \u03bc (s \u2229 ({ x } + r \u2022 t)) = \u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n)) \u222a s \u2229 ({ x } + r \u2022 (t \\ closedBall 0 n))) := by\n rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]\n _ \u2264 \u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc (s \u2229 ({ x } + r \u2022 (t \\ closedBall 0 n))) :=\n (measure_union_le _ _)\n _ \u2264 \u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({ x } + r \u2022 (t \\ closedBall 0 n)) :=\n add_le_add le_rfl (measure_mono (inter_subset_right _ _))\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 \u2191n)))\n[PROOFSTEP]\nrw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5\n[PROOFSTEP]\ncalc\n \u03bc (s \u2229 ({ x } + r \u2022 t)) / \u03bc ({ x } + r \u2022 t) \u2264\n (\u03bc (s \u2229 ({ x } + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({ x } + r \u2022 (t \\ closedBall 0 n))) / \u03bc ({ x } + r \u2022 t) :=\n mul_le_mul_right' I _\n _ < \u03b5 / 2 + \u03b5 / 2 := by\n rw [ENNReal.add_div]\n apply ENNReal.add_lt_add hr _\n rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]\n _ = \u03b5 := ENNReal.add_halves _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) <\n \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\nrw [ENNReal.add_div]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) +\n \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) <\n \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\napply ENNReal.add_lt_add hr _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nt : Set E\nht : MeasurableSet t\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nh't : \u2191\u2191\u03bc t \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t\nL : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nhr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\nrpos : 0 < r\nI : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2\n[PROOFSTEP]\nrwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave I : \u2200 u v, \u03bc u \u2260 0 \u2192 \u03bc u \u2260 \u221e \u2192 MeasurableSet v \u2192 \u03bc u / \u03bc u - \u03bc (v\u1d9c \u2229 u) / \u03bc u = \u03bc (v \u2229 u) / \u03bc u :=\n by\n intro u v uzero utop vmeas\n simp_rw [div_eq_mul_inv]\n rw [\u2190 ENNReal.sub_mul]; swap\n \u00b7 simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne.def, not_false_iff]\n congr 1\n apply ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono (inter_subset_right _ _)))\n rw [inter_comm _ u, inter_comm _ u]\n exact measure_inter_add_diff u vmeas\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n[PROOFSTEP]\nintro u v uzero utop vmeas\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u * (\u2191\u2191\u03bc u)\u207b\u00b9 - \u2191\u2191\u03bc (v\u1d9c \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 ENNReal.sub_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nswap\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 0 < \u2191\u2191\u03bc (v\u1d9c \u2229 u) \u2192 \u2191\u2191\u03bc (v\u1d9c \u2229 u) < \u2191\u2191\u03bc u \u2192 (\u2191\u2191\u03bc u)\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nsimp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 (\u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u)) * (\u2191\u2191\u03bc u)\u207b\u00b9 = \u2191\u2191\u03bc (v \u2229 u) * (\u2191\u2191\u03bc u)\u207b\u00b9\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc (v \u2229 u)\n[PROOFSTEP]\napply ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono (inter_subset_right _ _)))\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (v \u2229 u) + \u2191\u2191\u03bc (v\u1d9c \u2229 u) = \u2191\u2191\u03bc u\n[PROOFSTEP]\nrw [inter_comm _ u, inter_comm _ u]\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nu v : Set E\nuzero : \u2191\u2191\u03bc u \u2260 0\nutop : \u2191\u2191\u03bc u \u2260 \u22a4\nvmeas : MeasurableSet v\n\u22a2 \u2191\u2191\u03bc (u \u2229 v) + \u2191\u2191\u03bc (u \u2229 v\u1d9c) = \u2191\u2191\u03bc u\n[PROOFSTEP]\nexact measure_inter_add_diff u vmeas\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave L : Tendsto (fun r => \u03bc (s\u1d9c \u2229 closedBall x r) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n by\n have A : Tendsto (fun r => \u03bc (closedBall x r) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) :=\n by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n intro r hr\n rw [div_eq_mul_inv, ENNReal.mul_inv_cancel]\n \u00b7 exact (measure_closedBall_pos \u03bc _ hr).ne'\n \u00b7 exact measure_closedBall_lt_top.ne\n have B := ENNReal.Tendsto.sub A h (Or.inl ENNReal.one_ne_top)\n simp only [tsub_self] at B \n apply B.congr' _\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n convert I (closedBall x r) s\u1d9c (measure_closedBall_pos \u03bc _ rpos).ne' measure_closedBall_lt_top.ne hs.compl\n rw [compl_compl]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave A : Tendsto (fun r => \u03bc (closedBall x r) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) :=\n by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n intro r hr\n rw [div_eq_mul_inv, ENNReal.mul_inv_cancel]\n \u00b7 exact (measure_closedBall_pos \u03bc _ hr).ne'\n \u00b7 exact measure_closedBall_lt_top.ne\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\napply tendsto_const_nhds.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a)\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 1 = \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nrw [div_eq_mul_inv, ENNReal.mul_inv_cancel]\n[GOAL]\ncase h.h0\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 0\n[PROOFSTEP]\nexact (measure_closedBall_pos \u03bc _ hr).ne'\n[GOAL]\ncase h.ht\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (closedBall x r) \u2260 \u22a4\n[PROOFSTEP]\nexact measure_closedBall_lt_top.ne\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave B := ENNReal.Tendsto.sub A h (Or.inl ENNReal.one_ne_top)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (1 - 1))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [tsub_self] at B \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\napply B.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun a =>\n \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi 0 \u2192\n \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) =\n \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r) - \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) =\n \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nconvert I (closedBall x r) s\u1d9c (measure_closedBall_pos \u03bc _ rpos).ne' measure_closedBall_lt_top.ne hs.compl\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_5.h.e'_3.h.e'_3\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nA : Tendsto (fun r => \u2191\u2191\u03bc (closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nB :\n Tendsto (fun a => \u2191\u2191\u03bc (closedBall x a) / \u2191\u2191\u03bc (closedBall x a) - \u2191\u2191\u03bc (s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a))\n (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 s = s\u1d9c\u1d9c\n[PROOFSTEP]\nrw [compl_compl]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave L' : Tendsto (fun r : \u211d => \u03bc (s\u1d9c \u2229 ({ x } + r \u2022 t)) / \u03bc ({ x } + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n tendsto_addHaar_inter_smul_zero_of_density_zero \u03bc s\u1d9c x L t ht h''t\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave L'' : Tendsto (fun r : \u211d => \u03bc ({ x } + r \u2022 t) / \u03bc ({ x } + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) :=\n by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n rw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't h''t]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\napply tendsto_const_nhds.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 (fun x => 1) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 1 = \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nr : \u211d\nrpos : 0 < r\n\u22a2 1 = \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)\n[PROOFSTEP]\nrw [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne', ENNReal.div_self h't h''t]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave := ENNReal.Tendsto.sub L'' L' (Or.inl ENNReal.one_ne_top)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (1 - 0))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp only [tsub_zero] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\napply this.congr' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\n\u22a2 (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) =\u1da0[\ud835\udcdd[Ioi 0] 0]\n fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\n\u22a2 \u2200 (a : \u211d),\n a \u2208 Ioi 0 \u2192\n \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) =\n \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)\n[PROOFSTEP]\nrintro r (rpos : 0 < r)\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) =\n \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)\n[PROOFSTEP]\nrefine' I ({ x } + r \u2022 t) s _ _ hs\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 0\n[PROOFSTEP]\nsimp only [h't, abs_of_nonneg rpos.le, pow_pos rpos, addHaar_smul, image_add_left, ENNReal.ofReal_eq_zero, not_le,\n or_false_iff, Ne.def, measure_preimage_add, abs_pow, singleton_add, mul_eq_zero]\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nI :\n \u2200 (u v : Set E), \u2191\u2191\u03bc u \u2260 0 \u2192 \u2191\u2191\u03bc u \u2260 \u22a4 \u2192 MeasurableSet v \u2192 \u2191\u2191\u03bc u / \u2191\u2191\u03bc u - \u2191\u2191\u03bc (v\u1d9c \u2229 u) / \u2191\u2191\u03bc u = \u2191\u2191\u03bc (v \u2229 u) / \u2191\u2191\u03bc u\nL : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL' : Tendsto (fun r => \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\nL'' : Tendsto (fun r => \u2191\u2191\u03bc ({x} + r \u2022 t) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nthis :\n Tendsto (fun a => \u2191\u2191\u03bc ({x} + a \u2022 t) / \u2191\u2191\u03bc ({x} + a \u2022 t) - \u2191\u2191\u03bc (s\u1d9c \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t)) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd 1)\nr : \u211d\nrpos : 0 < r\n\u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) \u2260 \u22a4\n[PROOFSTEP]\nsimp [h''t, ENNReal.ofReal_ne_top, addHaar_smul, image_add_left, ENNReal.mul_eq_top, Ne.def, not_false_iff,\n measure_preimage_add, singleton_add, and_false_iff, false_and_iff, or_self_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave : Tendsto (fun r : \u211d => \u03bc (toMeasurable \u03bc s \u2229 ({ x } + r \u2022 t)) / \u03bc ({ x } + r \u2022 t)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1) :=\n by\n apply tendsto_addHaar_inter_smul_one_of_density_one_aux \u03bc _ (measurableSet_toMeasurable _ _) _ _ t ht h't h''t\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' h tendsto_const_nhds\n \u00b7 refine' eventually_of_forall fun r => mul_le_mul_right' _ _\n exact measure_mono (inter_subset_inter_left _ (subset_toMeasurable _ _))\n \u00b7 filter_upwards [self_mem_nhdsWithin]\n rintro r -\n apply ENNReal.div_le_of_le_mul\n rw [one_mul]\n exact measure_mono (inter_subset_right _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\napply tendsto_addHaar_inter_smul_one_of_density_one_aux \u03bc _ (measurableSet_toMeasurable _ _) _ _ t ht h't h''t\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\napply tendsto_of_tendsto_of_tendsto_of_le_of_le' h tendsto_const_nhds\n[GOAL]\ncase hgf\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0,\n \u2191\u2191\u03bc (s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b)\n[PROOFSTEP]\nrefine' eventually_of_forall fun r => mul_le_mul_right' _ _\n[GOAL]\ncase hgf\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r)\n[PROOFSTEP]\nexact measure_mono (inter_subset_inter_left _ (subset_toMeasurable _ _))\n[GOAL]\ncase hfh\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x b) / \u2191\u2191\u03bc (closedBall x b) \u2264 1\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x a) / \u2191\u2191\u03bc (closedBall x a) \u2264 1\n[PROOFSTEP]\nrintro r -\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r) \u2264 1\n[PROOFSTEP]\napply ENNReal.div_le_of_le_mul\n[GOAL]\ncase h.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 1 * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\ncase h.h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 closedBall x r) \u2264 \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nexact measure_mono (inter_subset_right _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine this.congr fun r => ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\napply measure_toMeasurable_inter_of_sigmaFinite\n[GOAL]\ncase e_a.hs\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ({x} + r \u2022 t)\n[PROOFSTEP]\nsimp only [image_add_left, singleton_add]\n[GOAL]\ncase e_a.hs\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nh''t : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : Tendsto (fun r => \u2191\u2191\u03bc (toMeasurable \u03bc s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nr : \u211d\n\u22a2 MeasurableSet ((fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 t))\n[PROOFSTEP]\napply (continuous_add_left (-x)).measurable (ht.const_smul\u2080 r)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\nobtain \u27e8t', t'_meas, t't, t'pos, t'top\u27e9 : \u2203 t', MeasurableSet t' \u2227 t' \u2286 t \u2227 0 < \u03bc t' \u2227 \u03bc t' < \u22a4 :=\n exists_subset_measure_lt_top ht h't.bot_lt\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1\n (tendsto_addHaar_inter_smul_one_of_density_one \u03bc s x h t' t'_meas t'pos.ne' t'top.ne)).1\n 0 zero_lt_one]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\n\u22a2 \u2200 (a : \u211d), 0 < \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t')) / \u2191\u2191\u03bc ({x} + a \u2022 t') \u2192 Set.Nonempty (s \u2229 ({x} + a \u2022 t))\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\nhave : \u03bc (s \u2229 ({ x } + r \u2022 t')) \u2260 0 := fun h' => by simp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr \n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nh' : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [ENNReal.not_lt_zero, ENNReal.zero_div, h'] at hr \n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\nhave : (s \u2229 ({ x } + r \u2022 t')).Nonempty := nonempty_of_measure_ne_zero this\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 Set.Nonempty (s \u2229 ({x} + r \u2022 t))\n[PROOFSTEP]\napply this.mono (inter_subset_inter Subset.rfl _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\ns\u271d s : Set E\nx : E\nh : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nt : Set E\nht : MeasurableSet t\nh't : \u2191\u2191\u03bc t \u2260 0\nt' : Set E\nt'_meas : MeasurableSet t'\nt't : t' \u2286 t\nt'pos : 0 < \u2191\u2191\u03bc t'\nt'top : \u2191\u2191\u03bc t' < \u22a4\nr : \u211d\nhr : 0 < \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) / \u2191\u2191\u03bc ({x} + r \u2022 t')\nthis\u271d : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t')) \u2260 0\nthis : Set.Nonempty (s \u2229 ({x} + r \u2022 t'))\n\u22a2 {x} + r \u2022 t' \u2286 {x} + r \u2022 t\n[PROOFSTEP]\nexact add_subset_add Subset.rfl (smul_set_mono t't)\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "llama_tokens": 117564, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.2827003840742062}} {"text": "[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type ?u.69019\nf : J \u2192 C\ninst\u271d : HasProduct f\nj j' : J\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type u_1\nf : J \u2192 C\ninst\u271d : HasProduct f\nj j' : J\nw : j = j'\n\u22a2 \u03c0 f j \u226b eqToHom (_ : f j = f j') = \u03c0 f j'\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type u_1\nf : J \u2192 C\ninst\u271d : HasProduct f\nj : J\n\u22a2 \u03c0 f j \u226b eqToHom (_ : f j = f j) = \u03c0 f j\n[PROOFSTEP]\nsimp\n -- The `simpNF` linter incorrectly identifies these as simp lemmas that could never apply.\n -- https://github.com/leanprover-community/mathlib4/issues/5049\n -- They are used by `simp` in `Sigma.whisker_equiv` below.\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type ?u.69987\nf : J \u2192 C\ninst\u271d : HasCoproduct f\nj j' : J\nw : j = j'\n\u22a2 f j = f j'\n[PROOFSTEP]\nsimp [w]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type u_1\nf : J \u2192 C\ninst\u271d : HasCoproduct f\nj j' : J\nw : j = j'\n\u22a2 eqToHom (_ : f j = f j') \u226b \u03b9 f j' = \u03b9 f j\n[PROOFSTEP]\ncases w\n[GOAL]\ncase refl\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type u_1\nf : J \u2192 C\ninst\u271d : HasCoproduct f\nj : J\n\u22a2 eqToHom (_ : f j = f j) \u226b \u03b9 f j = \u03b9 f j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf g : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasProduct f\ninst\u271d\u00b9 : HasProduct g\np : (b : \u03b2) \u2192 f b \u27f6 g b\ninst\u271d : \u2200 (i : \u03b2), Mono (p i)\n\u22a2 \u2200 (j : Discrete \u03b2), Mono (NatTrans.app (Discrete.natTrans fun X => p X.as) j)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf g : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasProduct f\ninst\u271d\u00b9 : HasProduct g\np : (b : \u03b2) \u2192 f b \u27f6 g b\ninst\u271d : \u2200 (i : \u03b2), Mono (p i)\n\u22a2 \u2200 (j : Discrete \u03b2), Mono (p j.as)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf g : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasCoproduct f\ninst\u271d\u00b9 : HasCoproduct g\np : (b : \u03b2) \u2192 f b \u27f6 g b\ninst\u271d : \u2200 (i : \u03b2), Epi (p i)\n\u22a2 \u2200 (j : Discrete \u03b2), Epi (NatTrans.app (Discrete.natTrans fun X => p X.as) j)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nf g : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasCoproduct f\ninst\u271d\u00b9 : HasCoproduct g\np : (b : \u03b2) \u2192 f b \u27f6 g b\ninst\u271d : \u2200 (i : \u03b2), Epi (p i)\n\u22a2 \u2200 (j : Discrete \u03b2), Epi (p j.as)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type ?u.78866\nK : Type ?u.78950\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct g\nk : K\n\u22a2 g (\u2191e (\u2191e.symm k)) = g k\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type ?u.94443\nK : Type ?u.94527\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (\u2191e j) \u2245 f j\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct g\nk : K\n\u22a2 g k = g (\u2191e (\u2191e.symm k))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nf : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun b => G.obj (f b)\nP : C\ng : (j : \u03b2) \u2192 P \u27f6 f j\n\u22a2 G.map (Pi.lift g) \u226b piComparison G f = Pi.lift fun j => G.map (g j)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nf : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun b => G.obj (f b)\nP : C\ng : (j : \u03b2) \u2192 P \u27f6 f j\nj : \u03b2\n\u22a2 (G.map (Pi.lift g) \u226b piComparison G f) \u226b Pi.\u03c0 (fun b => G.obj (f b)) j =\n (Pi.lift fun j => G.map (g j)) \u226b Pi.\u03c0 (fun b => G.obj (f b)) j\n[PROOFSTEP]\nsimp only [Discrete.functor_obj, Category.assoc, piComparison_comp_\u03c0, \u2190 G.map_comp, limit.lift_\u03c0, Fan.mk_pt,\n Fan.mk_\u03c0_app]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nf : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : \u03b2) \u2192 f j \u27f6 P\n\u22a2 sigmaComparison G f \u226b G.map (Sigma.desc g) = Sigma.desc fun j => G.map (g j)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nf : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun b => G.obj (f b)\nP : C\ng : (j : \u03b2) \u2192 f j \u27f6 P\nj : \u03b2\n\u22a2 Sigma.\u03b9 (fun b => G.obj (f b)) j \u226b sigmaComparison G f \u226b G.map (Sigma.desc g) =\n Sigma.\u03b9 (fun b => G.obj (f b)) j \u226b Sigma.desc fun j => G.map (g j)\n[PROOFSTEP]\nsimp only [Discrete.functor_obj, \u03b9_comp_sigmaComparison_assoc, \u2190 G.map_comp, colimit.\u03b9_desc, Cofan.mk_pt,\n Cofan.mk_\u03b9_app]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } = (Discrete.functor f).obj { as := j }\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 f default = f j\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 default = j\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cone (Discrete.functor f)\nj : Discrete \u03b2\n\u22a2 (fun s => NatTrans.app s.\u03c0 default) s \u226b\n NatTrans.app\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nhave h := Subsingleton.elim j default\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cone (Discrete.functor f)\nj : Discrete \u03b2\nh : j = default\n\u22a2 (fun s => NatTrans.app s.\u03c0 default) s \u226b\n NatTrans.app\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cone (Discrete.functor f)\n\u22a2 (fun s => NatTrans.app s.\u03c0 default) s \u226b\n NatTrans.app\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.\u03c0\n default =\n NatTrans.app s.\u03c0 default\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cone (Discrete.functor f)\nm :\n s.pt \u27f6\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.pt\nw :\n \u2200 (j : Discrete \u03b2),\n m \u226b\n NatTrans.app\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => NatTrans.app s.\u03c0 default) s\n[PROOFSTEP]\nspecialize w default\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cone (Discrete.functor f)\nm :\n s.pt \u27f6\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.pt\nw :\n m \u226b\n NatTrans.app\n { pt := f default,\n \u03c0 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j } =\n (Discrete.functor f).obj { as := j }) }.\u03c0\n default =\n NatTrans.app s.\u03c0 default\n\u22a2 m = (fun s => NatTrans.app s.\u03c0 default) s\n[PROOFSTEP]\nsimpa using w\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 (Discrete.functor f).obj { as := j } = ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 f j = f default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\nx\u271d : Discrete \u03b2\nj : \u03b2\n\u22a2 j = default\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\nj : Discrete \u03b2\n\u22a2 NatTrans.app\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.\u03b9\n j \u226b\n (fun s => NatTrans.app s.\u03b9 default) s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nhave h := Subsingleton.elim j default\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\nj : Discrete \u03b2\nh : j = default\n\u22a2 NatTrans.app\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.\u03b9\n j \u226b\n (fun s => NatTrans.app s.\u03b9 default) s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\n\u22a2 NatTrans.app\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.\u03b9\n default \u226b\n (fun s => NatTrans.app s.\u03b9 default) s =\n NatTrans.app s.\u03b9 default\n[PROOFSTEP]\napply Category.id_comp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\nm :\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.pt \u27f6\n s.pt\nw :\n \u2200 (j : Discrete \u03b2),\n NatTrans.app\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s => NatTrans.app s.\u03b9 default) s\n[PROOFSTEP]\nspecialize w default\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\nm :\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.pt \u27f6\n s.pt\nw :\n NatTrans.app\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.\u03b9\n default \u226b\n m =\n NatTrans.app s.\u03b9 default\n\u22a2 m = (fun s => NatTrans.app s.\u03b9 default) s\n[PROOFSTEP]\nerw [Category.id_comp] at w \n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Unique \u03b2\nf : \u03b2 \u2192 C\ns : Cocone (Discrete.functor f)\nm :\n { pt := f default,\n \u03b9 :=\n Discrete.natTrans fun x =>\n match x with\n | { as := j } =>\n eqToHom\n (_ :\n (Discrete.functor f).obj { as := j } =\n ((Functor.const (Discrete \u03b2)).obj (f default)).obj { as := j }) }.pt \u27f6\n s.pt\nw : m = NatTrans.app s.\u03b9 default\n\u22a2 m = (fun s => NatTrans.app s.\u03b9 default) s\n[PROOFSTEP]\nexact w\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 (reindex \u03b5 f).hom \u226b \u03c0 f (\u2191\u03b5 b) = \u03c0 (f \u2218 \u2191\u03b5) b\n[PROOFSTEP]\ndsimp [Pi.reindex]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 (HasLimit.isoOfEquivalence (Discrete.equivalence \u03b5)\n (Discrete.natIso fun x =>\n Iso.refl ((Discrete.functor f).obj ((Discrete.functor (Discrete.mk \u2218 \u2191\u03b5)).obj x)))).hom \u226b\n \u03c0 f (\u2191\u03b5 b) =\n \u03c0 (f \u2218 \u2191\u03b5) b\n[PROOFSTEP]\nsimp only [HasLimit.isoOfEquivalence_hom_\u03c0, Discrete.equivalence_inverse, Discrete.functor_obj, Function.comp_apply,\n Functor.id_obj, Discrete.equivalence_functor, Functor.comp_obj, Discrete.natIso_inv_app, Iso.refl_inv,\n Category.id_comp]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 limit.\u03c0 (Discrete.functor (f \u2218 \u2191\u03b5)) { as := \u2191\u03b5.symm (\u2191\u03b5 b) } \u226b\n (Discrete.functor f).map (NatTrans.app (Equivalence.counit (Discrete.equivalence \u03b5)) { as := \u2191\u03b5 b }) =\n \u03c0 (f \u2218 \u2191\u03b5) b\n[PROOFSTEP]\nexact limit.w (Discrete.functor (f \u2218 \u03b5)) (Discrete.eqToHom' (\u03b5.symm_apply_apply b))\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 (reindex \u03b5 f).inv \u226b \u03c0 (f \u2218 \u2191\u03b5) b = \u03c0 f (\u2191\u03b5 b)\n[PROOFSTEP]\nsimp [Iso.inv_comp_eq]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 \u03b9 (f \u2218 \u2191\u03b5) b \u226b (reindex \u03b5 f).hom = \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\ndsimp [Sigma.reindex]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 \u03b9 (f \u2218 \u2191\u03b5) b \u226b\n (HasColimit.isoOfEquivalence (Discrete.equivalence \u03b5)\n (Discrete.natIso fun x =>\n Iso.refl ((Discrete.functor f).obj ((Discrete.functor (Discrete.mk \u2218 \u2191\u03b5)).obj x)))).hom =\n \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\nsimp only [HasColimit.isoOfEquivalence_hom_\u03c0, Functor.id_obj, Discrete.functor_obj, Function.comp_apply,\n Discrete.equivalence_functor, Discrete.equivalence_inverse, Functor.comp_obj, Discrete.natIso_inv_app, Iso.refl_inv,\n Category.id_comp]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 (Discrete.functor (f \u2218 \u2191\u03b5)).map (NatTrans.app (Equivalence.unit (Discrete.equivalence \u03b5)) { as := b }) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) } =\n \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\nhave h := colimit.w (Discrete.functor f) (Discrete.eqToHom' (\u03b5.apply_symm_apply (\u03b5 b)))\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 (Discrete.functor (f \u2218 \u2191\u03b5)).map (NatTrans.app (Equivalence.unit (Discrete.equivalence \u03b5)) { as := b }) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) } =\n \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\nsimp only [Discrete.functor_obj] at h \n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 (Discrete.functor (f \u2218 \u2191\u03b5)).map (NatTrans.app (Equivalence.unit (Discrete.equivalence \u03b5)) { as := b }) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) } =\n \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\nerw [\u2190 h, eqToHom_map, eqToHom_map, eqToHom_trans_assoc]\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 eqToHom (_ : f (\u2191\u03b5 b) = f (\u2191\u03b5 b)) \u226b colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } = \u03b9 f (\u2191\u03b5 b)\ncase p\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 { as := b } = { as := \u2191\u03b5.symm (\u2191\u03b5 b) }\ncase p\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 { as := b } = { as := \u2191\u03b5.symm (\u2191\u03b5 b) }\ncase p\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 { as := b } = { as := \u2191\u03b5.symm (\u2191\u03b5 b) }\n[PROOFSTEP]\nall_goals {simp\n}\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 eqToHom (_ : f (\u2191\u03b5 b) = f (\u2191\u03b5 b)) \u226b colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } = \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\n{simp\n}\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 eqToHom (_ : f (\u2191\u03b5 b) = f (\u2191\u03b5 b)) \u226b colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } = \u03b9 f (\u2191\u03b5 b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase p\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 { as := b } = { as := \u2191\u03b5.symm (\u2191\u03b5 b) }\n[PROOFSTEP]\n{simp\n}\n[GOAL]\ncase p\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\nh :\n (Discrete.functor f).map (Discrete.eqToHom' (_ : \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) = \u2191\u03b5 b)) \u226b\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 b } =\n colimit.\u03b9 (Discrete.functor f) { as := \u2191\u03b5 (\u2191\u03b5.symm (\u2191\u03b5 b)) }\n\u22a2 { as := b } = { as := \u2191\u03b5.symm (\u2191\u03b5 b) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b3 : Type v\n\u03b5 : \u03b2 \u2243 \u03b3\nf : \u03b3 \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct (f \u2218 \u2191\u03b5)\nb : \u03b2\n\u22a2 \u03b9 f (\u2191\u03b5 b) \u226b (reindex \u03b5 f).inv = \u03b9 (f \u2218 \u2191\u03b5) b\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Products", "llama_tokens": 9589, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.2824528414032058}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nP Q : IsKernelPair f a b\n\u22a2 P = Q\n[PROOFSTEP]\ncases P\n[GOAL]\ncase mk\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nQ : IsKernelPair f a b\ntoCommSq\u271d : CommSq a b f f\nisLimit'\u271d : Nonempty (IsLimit (PullbackCone.mk a b (_ : a \u226b f = b \u226b f)))\n\u22a2 (_ : IsPullback a b f f) = Q\n[PROOFSTEP]\ncases Q\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ntoCommSq\u271d\u00b9 : CommSq a b f f\nisLimit'\u271d\u00b9 : Nonempty (IsLimit (PullbackCone.mk a b (_ : a \u226b f = b \u226b f)))\ntoCommSq\u271d : CommSq a b f f\nisLimit'\u271d : Nonempty (IsLimit (PullbackCone.mk a b (_ : a \u226b f = b \u226b f)))\n\u22a2 (_ : IsPullback a b f f) = (_ : IsPullback a b f f)\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nS : C\nk : IsKernelPair f a b\np q : S \u27f6 X\nw : p \u226b f = q \u226b f\n\u22a2 lift k p q w \u226b a = p \u2227 lift k p q w \u226b b = q\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nlet s' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) := PullbackCone.mk s.fst s.snd (s.condition_assoc _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\n\u22a2 { l //\n l \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s \u2227\n l \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s \u2227\n \u2200 {m : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt},\n m \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s \u2192\n m \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s \u2192 m = l }\n[PROOFSTEP]\nrefine'\n \u27e8big_k.isLimit.lift s', big_k.isLimit.fac _ WalkingCospan.left, big_k.isLimit.fac _ WalkingCospan.right, fun m\u2081 m\u2082 =>\n _\u27e9\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\nm\u271d : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt\nm\u2081 : m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s\nm\u2082 : m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s\n\u22a2 m\u271d = IsLimit.lift (IsPullback.isLimit big_k) s'\n[PROOFSTEP]\napply big_k.isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\nm\u271d : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt\nm\u2081 : m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s\nm\u2082 : m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s\n\u22a2 \u2200 (j : WalkingCospan),\n m\u271d \u226b NatTrans.app (IsPullback.cone big_k).\u03c0 j =\n IsLimit.lift (IsPullback.isLimit big_k) s' \u226b NatTrans.app (IsPullback.cone big_k).\u03c0 j\n[PROOFSTEP]\nrefine' (PullbackCone.mk a b _ : PullbackCone (f\u2081 \u226b f\u2082) _).equalizer_ext _ _\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\nm\u271d : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt\nm\u2081 : m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s\nm\u2082 : m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s\n\u22a2 a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082\n[PROOFSTEP]\napply reassoc_of% comm\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\nm\u271d : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt\nm\u2081 : m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s\nm\u2082 : m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s\n\u22a2 m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)) =\n IsLimit.lift (IsPullback.isLimit big_k) s' \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082))\n[PROOFSTEP]\napply m\u2081.trans (big_k.isLimit.fac s' WalkingCospan.left).symm\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ncomm : a \u226b f\u2081 = b \u226b f\u2081\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\ns : PullbackCone f\u2081 f\u2081\ns' : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n PullbackCone.mk (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 \u226b f\u2082 = PullbackCone.snd s \u226b f\u2081 \u226b f\u2082)\nm\u271d : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)).pt\nm\u2081 : m\u271d \u226b PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.fst s\nm\u2082 : m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 = b \u226b f\u2081)) = PullbackCone.snd s\n\u22a2 m\u271d \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)) =\n IsLimit.lift (IsPullback.isLimit big_k) s' \u226b PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082))\n[PROOFSTEP]\napply m\u2082.trans (big_k.isLimit.fac s' WalkingCospan.right).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nbig_k : IsKernelPair (f\u2081 \u226b f\u2082) a b\n\u22a2 a \u226b f\u2081 = b \u226b f\u2081\n[PROOFSTEP]\nrw [\u2190 cancel_mono f\u2082, assoc, assoc, big_k.w]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082\n[PROOFSTEP]\nrw [small_k.w_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 IsLimit (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082))\n[PROOFSTEP]\nrefine'\n PullbackCone.isLimitAux _ (fun s => small_k.lift s.fst s.snd (by rw [\u2190 cancel_mono f\u2082, assoc, s.condition, assoc]))\n (by simp) (by simp) _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\n\u22a2 PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081\n[PROOFSTEP]\nrw [\u2190 cancel_mono f\u2082, assoc, s.condition, assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)),\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.fst (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)) =\n PullbackCone.fst s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)),\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.snd (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)) =\n PullbackCone.snd s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)) (m : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt),\n (\u2200 (j : WalkingCospan),\n m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j) \u2192\n m =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s\n[PROOFSTEP]\nintro s m hm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s) (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s\n[PROOFSTEP]\napply small_k.isLimit.hom_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 \u2200 (j : WalkingCospan),\n m \u226b NatTrans.app (IsPullback.cone small_k).\u03c0 j =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n NatTrans.app (IsPullback.cone small_k).\u03c0 j\n[PROOFSTEP]\napply PullbackCone.equalizer_ext small_k.cone _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b PullbackCone.fst (IsPullback.cone small_k) =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.fst (IsPullback.cone small_k)\n[PROOFSTEP]\nexact (hm WalkingCospan.left).trans (by simp)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 NatTrans.app s.\u03c0 WalkingCospan.left =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.fst (IsPullback.cone small_k)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m \u226b PullbackCone.snd (IsPullback.cone small_k) =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.snd (IsPullback.cone small_k)\n[PROOFSTEP]\nexact (hm WalkingCospan.right).trans (by simp)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).pt\nhm :\n \u2200 (j : WalkingCospan), m \u226b NatTrans.app (PullbackCone.mk a b (_ : a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082)).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 NatTrans.app s.\u03c0 WalkingCospan.right =\n (fun s =>\n lift small_k (PullbackCone.fst s) (PullbackCone.snd s)\n (_ : PullbackCone.fst s \u226b f\u2081 = PullbackCone.snd s \u226b f\u2081))\n s \u226b\n PullbackCone.snd (IsPullback.cone small_k)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\n\u22a2 IsColimit (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f))\n[PROOFSTEP]\nlet t := k.isLimit.lift (PullbackCone.mk _ _ r.w)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\n\u22a2 IsColimit (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f))\n[PROOFSTEP]\nhave ht : t \u226b a = r.left := k.isLimit.fac _ WalkingCospan.left\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\n\u22a2 IsColimit (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f))\n[PROOFSTEP]\nhave kt : t \u226b b = r.right := k.isLimit.fac _ WalkingCospan.right\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\nkt : t \u226b b = RegularEpi.right\n\u22a2 IsColimit (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f))\n[PROOFSTEP]\nrefine'\n Cofork.IsColimit.mk _\n (fun s => Cofork.IsColimit.desc r.isColimit s.\u03c0 (by rw [\u2190 ht, assoc, s.condition, reassoc_of% kt])) (fun s => _)\n (fun s m w => _)\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\nkt : t \u226b b = RegularEpi.right\ns : Cofork a b\n\u22a2 RegularEpi.left \u226b Cofork.\u03c0 s = RegularEpi.right \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nrw [\u2190 ht, assoc, s.condition, reassoc_of% kt]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\nkt : t \u226b b = RegularEpi.right\ns : Cofork a b\n\u22a2 Cofork.\u03c0 (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f)) \u226b\n (fun s =>\n Cofork.IsColimit.desc RegularEpi.isColimit (Cofork.\u03c0 s)\n (_ : RegularEpi.left \u226b Cofork.\u03c0 s = RegularEpi.right \u226b Cofork.\u03c0 s))\n s =\n Cofork.\u03c0 s\n[PROOFSTEP]\napply Cofork.IsColimit.\u03c0_desc' r.isColimit\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\nkt : t \u226b b = RegularEpi.right\ns : Cofork a b\nm : (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f)).pt \u27f6 s.pt\nw : Cofork.\u03c0 (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f)) \u226b m = Cofork.\u03c0 s\n\u22a2 m =\n (fun s =>\n Cofork.IsColimit.desc RegularEpi.isColimit (Cofork.\u03c0 s)\n (_ : RegularEpi.left \u226b Cofork.\u03c0 s = RegularEpi.right \u226b Cofork.\u03c0 s))\n s\n[PROOFSTEP]\napply Cofork.IsColimit.hom_ext r.isColimit\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nk : IsKernelPair f a b\nr : RegularEpi f\nt : (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)).pt \u27f6\n (IsPullback.cone k).pt :=\n IsLimit.lift (IsPullback.isLimit k)\n (PullbackCone.mk RegularEpi.left RegularEpi.right (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f))\nht : t \u226b a = RegularEpi.left\nkt : t \u226b b = RegularEpi.right\ns : Cofork a b\nm : (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f)).pt \u27f6 s.pt\nw : Cofork.\u03c0 (Cofork.of\u03c0 f (_ : a \u226b f = b \u226b f)) \u226b m = Cofork.\u03c0 s\n\u22a2 Cofork.\u03c0 (Cofork.of\u03c0 f (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)) \u226b m =\n Cofork.\u03c0 (Cofork.of\u03c0 f (_ : RegularEpi.left \u226b f = RegularEpi.right \u226b f)) \u226b\n (fun s =>\n Cofork.IsColimit.desc RegularEpi.isColimit (Cofork.\u03c0 s)\n (_ : RegularEpi.left \u226b Cofork.\u03c0 s = RegularEpi.right \u226b Cofork.\u03c0 s))\n s\n[PROOFSTEP]\nexact w.trans (Cofork.IsColimit.\u03c0_desc' r.isColimit _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\n\u22a2 f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\n\u22a2 f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\n\u22a2 IsKernelPair pullback.fst\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n[PROOFSTEP]\nrefine'\n \u27e8\u27e8by rw [pullback.lift_fst, pullback.lift_fst]\u27e9,\n \u27e8PullbackCone.isLimitAux _\n (fun s => pullback.lift (s.fst \u226b pullback.fst) (h.lift (s.fst \u226b pullback.snd) (s.snd \u226b pullback.snd) _) _)\n (fun s => _) (fun s => _) (fun s m hm => _)\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\n\u22a2 pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b pullback.fst\n[PROOFSTEP]\nrw [pullback.lift_fst, pullback.lift_fst]\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g\n[PROOFSTEP]\nsimp_rw [Category.assoc, \u2190 pullback.condition, \u2190 Category.assoc, s.condition]\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g\n[PROOFSTEP]\nsimp only [assoc, lift_fst_assoc, pullback.condition]\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 (fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.fst\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)) =\n PullbackCone.fst s\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_3.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.fst\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst))) \u226b\n pullback.fst =\n PullbackCone.fst s \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.fst\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst))) \u226b\n pullback.snd =\n PullbackCone.fst s \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 (fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.snd\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)) =\n PullbackCone.snd s\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_4.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.snd\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst))) \u226b\n pullback.fst =\n PullbackCone.snd s \u226b pullback.fst\n[PROOFSTEP]\nsimp [s.condition]\n[GOAL]\ncase refine'_4.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\n\u22a2 ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n PullbackCone.snd\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst))) \u226b\n pullback.snd =\n PullbackCone.snd s \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_5\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\nm :\n s.pt \u27f6\n (PullbackCone.mk (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).pt\nhm :\n \u2200 (j : WalkingCospan),\n m \u226b\n NatTrans.app\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s\n[PROOFSTEP]\napply pullback.hom_ext\n[GOAL]\ncase refine'_5.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\nm :\n s.pt \u27f6\n (PullbackCone.mk (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).pt\nhm :\n \u2200 (j : WalkingCospan),\n m \u226b\n NatTrans.app\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m \u226b pullback.fst =\n (fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n pullback.fst\n[PROOFSTEP]\nsimpa using hm WalkingCospan.left =\u226b pullback.fst\n[GOAL]\ncase refine'_5.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\nm :\n s.pt \u27f6\n (PullbackCone.mk (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).pt\nhm :\n \u2200 (j : WalkingCospan),\n m \u226b\n NatTrans.app\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m \u226b pullback.snd =\n (fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n pullback.snd\n[PROOFSTEP]\napply PullbackCone.IsLimit.hom_ext h.isLimit\n[GOAL]\ncase refine'_5.h\u2081.h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\nm :\n s.pt \u27f6\n (PullbackCone.mk (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).pt\nhm :\n \u2200 (j : WalkingCospan),\n m \u226b\n NatTrans.app\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 (m \u226b pullback.snd) \u226b PullbackCone.fst (IsPullback.cone h) =\n ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n pullback.snd) \u226b\n PullbackCone.fst (IsPullback.cone h)\n[PROOFSTEP]\nsimpa using hm WalkingCospan.left =\u226b pullback.snd\n[GOAL]\ncase refine'_5.h\u2081.h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\na b : R \u27f6 X\u271d\nX Y Z A : C\ng : Y \u27f6 Z\na\u2081 a\u2082 : A \u27f6 Y\nh : IsKernelPair g a\u2081 a\u2082\nf : X \u27f6 Z\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback f (a\u2081 \u226b g)\ns : PullbackCone pullback.fst pullback.fst\nm :\n s.pt \u27f6\n (PullbackCone.mk (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).pt\nhm :\n \u2200 (j : WalkingCospan),\n m \u226b\n NatTrans.app\n (PullbackCone.mk\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g))\n (pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g))\n (_ :\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2081 \u226b g) \u226b\n pullback.fst =\n pullback.map f (a\u2081 \u226b g) f g (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (_ : f \u226b \ud835\udfd9 Z = \ud835\udfd9 X \u226b f) (_ : (a\u2081 \u226b g) \u226b \ud835\udfd9 Z = a\u2082 \u226b g) \u226b\n pullback.fst)).\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 (m \u226b pullback.snd) \u226b PullbackCone.snd (IsPullback.cone h) =\n ((fun s =>\n pullback.lift (PullbackCone.fst s \u226b pullback.fst)\n (lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g))\n (_ :\n (PullbackCone.fst s \u226b pullback.fst) \u226b f =\n lift h (PullbackCone.fst s \u226b pullback.snd) (PullbackCone.snd s \u226b pullback.snd)\n (_ : (PullbackCone.fst s \u226b pullback.snd) \u226b g = (PullbackCone.snd s \u226b pullback.snd) \u226b g) \u226b\n a\u2081 \u226b g))\n s \u226b\n pullback.snd) \u226b\n PullbackCone.snd (IsPullback.cone h)\n[PROOFSTEP]\nsimpa using hm WalkingCospan.right =\u226b pullback.snd\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\n\u22a2 Mono f\n[PROOFSTEP]\nobtain \u27e8l, h\u2081, h\u2082\u27e9 := Limits.PullbackCone.IsLimit.lift' h.isLimit (\ud835\udfd9 _) (\ud835\udfd9 _) (by simp [h.w])\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\n\u22a2 \ud835\udfd9 X \u226b f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp [h.w]\n[GOAL]\ncase mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l \u226b PullbackCone.fst (IsPullback.cone h) = \ud835\udfd9 X\nh\u2082 : l \u226b PullbackCone.snd (IsPullback.cone h) = \ud835\udfd9 X\n\u22a2 Mono f\n[PROOFSTEP]\nrw [IsPullback.cone_fst, \u2190 IsIso.eq_comp_inv, Category.id_comp] at h\u2081 \n[GOAL]\ncase mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l = inv a\nh\u2082 : l \u226b PullbackCone.snd (IsPullback.cone h) = \ud835\udfd9 X\n\u22a2 Mono f\n[PROOFSTEP]\nrw [h\u2081, IsIso.inv_comp_eq, Category.comp_id] at h\u2082 \n[GOAL]\ncase mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l = inv a\nh\u2082 : PullbackCone.snd (IsPullback.cone h) = a\n\u22a2 Mono f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.intro.right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l = inv a\nh\u2082 : PullbackCone.snd (IsPullback.cone h) = a\n\u22a2 \u2200 {Z : C} (g h : Z \u27f6 X), g \u226b f = h \u226b f \u2192 g = h\n[PROOFSTEP]\nintro Z g\u2081 g\u2082 e\n[GOAL]\ncase mk.intro.right_cancellation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z\u271d : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l = inv a\nh\u2082 : PullbackCone.snd (IsPullback.cone h) = a\nZ : C\ng\u2081 g\u2082 : Z \u27f6 X\ne : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nobtain \u27e8l', rfl, rfl\u27e9 := Limits.PullbackCone.IsLimit.lift' h.isLimit _ _ e\n[GOAL]\ncase mk.intro.right_cancellation.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z\u271d : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : IsIso a\nl : X \u27f6 (IsPullback.cone h).pt\nh\u2081 : l = inv a\nh\u2082 : PullbackCone.snd (IsPullback.cone h) = a\nZ : C\nl' : Z \u27f6 (IsPullback.cone h).pt\ne : (l' \u226b PullbackCone.fst (IsPullback.cone h)) \u226b f = (l' \u226b PullbackCone.snd (IsPullback.cone h)) \u226b f\n\u22a2 l' \u226b PullbackCone.fst (IsPullback.cone h) = l' \u226b PullbackCone.snd (IsPullback.cone h)\n[PROOFSTEP]\nrw [IsPullback.cone_fst, h\u2082]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : Mono f\n\u22a2 IsIso a\n[PROOFSTEP]\nrw [\u2190\n show _ = a from\n (Category.comp_id _).symm.trans\n ((IsKernelPair.id_of_mono f).isLimit.conePointUniqueUpToIso_inv_comp h.isLimit WalkingCospan.left)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nh : IsKernelPair f a b\ninst\u271d : Mono f\n\u22a2 IsIso\n (IsLimit.conePointUniqueUpToIso (IsPullback.isLimit (_ : IsKernelPair f (\ud835\udfd9 X) (\ud835\udfd9 X))) (IsPullback.isLimit h)).inv\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 IsKernelPair f a a\n[PROOFSTEP]\nchange IsPullback _ _ _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 IsPullback a a f f\n[PROOFSTEP]\nconvert (IsPullback.of_horiz_isIso \u27e8(rfl : a \u226b \ud835\udfd9 X = _)\u27e9).paste_vert (IsKernelPair.id_of_mono f)\n[GOAL]\ncase h.e'_8\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 a = a \u226b \ud835\udfd9 X\ncase h.e'_9\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nall_goals {simp\n}\n[GOAL]\ncase h.e'_8\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 a = a \u226b \ud835\udfd9 X\n[PROOFSTEP]\n{simp\n}\n[GOAL]\ncase h.e'_8\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 a = a \u226b \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_9\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\n{simp\n}\n[GOAL]\ncase h.e'_9\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\ninst\u271d\u00b9 : IsIso a\ninst\u271d : Mono f\n\u22a2 f = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.KernelPair", "llama_tokens": 24843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.40356685373537454, "lm_q1q2_score": 0.28219591172583075}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\ninst\u271d : Abelian D\nJ : GrothendieckTopology C\nj : \u2115\nF : Discrete (Fin j) \u2964 Sheaf J D\n\u22a2 HasLimit F\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Abelian", "llama_tokens": 103, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.4073334000459302, "lm_q1q2_score": 0.28213621400482797}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\n\u22a2 \u2200 \u2983X Y : Discrete WalkingPair\u2984 (f : X \u27f6 Y),\n ((Functor.const (Discrete WalkingPair)).obj (of R (\u2191M \u00d7 \u2191N))).map f \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N)) Y =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N)) X \u226b\n (pair M N).map f\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9\n[GOAL]\ncase mk.left.mk.left.up.up.refl\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\n\u22a2 ((Functor.const (Discrete WalkingPair)).obj (of R (\u2191M \u00d7 \u2191N))).map\n { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } } \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N))\n { as := WalkingPair.left } =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N))\n { as := WalkingPair.left } \u226b\n (pair M N).map { down := { down := (_ : { as := WalkingPair.left }.as = { as := WalkingPair.left }.as) } }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.right.mk.right.up.up.refl\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\n\u22a2 ((Functor.const (Discrete WalkingPair)).obj (of R (\u2191M \u00d7 \u2191N))).map\n { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } } \u226b\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N))\n { as := WalkingPair.right } =\n (fun j => Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N))\n { as := WalkingPair.right } \u226b\n (pair M N).map { down := { down := (_ : { as := WalkingPair.right }.as = { as := WalkingPair.right }.as) } }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\n\u22a2 \u2200 (s : Cone (pair M N)) (j : Discrete WalkingPair),\n (fun s =>\n LinearMap.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left }) (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nrintro s (\u27e8\u27e9 | \u27e8\u27e9)\n[GOAL]\ncase mk.left\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\ns : Cone (pair M N)\n\u22a2 (fun s => LinearMap.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left }) (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.\u03c0\n { as := WalkingPair.left } =\n NatTrans.app s.\u03c0 { as := WalkingPair.left }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.right\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\ns : Cone (pair M N)\n\u22a2 (fun s => LinearMap.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left }) (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s \u226b\n NatTrans.app\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.\u03c0\n { as := WalkingPair.right } =\n NatTrans.app s.\u03c0 { as := WalkingPair.right }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\ns : Cone (pair M N)\nm :\n s.pt \u27f6\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.pt\nw :\n \u2200 (j : Discrete WalkingPair),\n m \u226b\n NatTrans.app\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m =\n (fun s =>\n LinearMap.prod (NatTrans.app s.\u03c0 { as := WalkingPair.left }) (NatTrans.app s.\u03c0 { as := WalkingPair.right }))\n s\n[PROOFSTEP]\nsimp_rw [\u2190 w \u27e8WalkingPair.left\u27e9, \u2190 w \u27e8WalkingPair.right\u27e9]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM N : ModuleCat R\ns : Cone (pair M N)\nm :\n s.pt \u27f6\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.pt\nw :\n \u2200 (j : Discrete WalkingPair),\n m \u226b\n NatTrans.app\n { pt := of R (\u2191M \u00d7 \u2191N),\n \u03c0 :=\n NatTrans.mk fun j =>\n Discrete.casesOn j fun j => WalkingPair.casesOn j (LinearMap.fst R \u2191M \u2191N) (LinearMap.snd R \u2191M \u2191N) }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\n\u22a2 m = LinearMap.prod (m \u226b LinearMap.fst R \u2191M \u2191N) (m \u226b LinearMap.snd R \u2191M \u2191N)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Fan f\nx y : \u2191s.pt\n\u22a2 (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) (x + y) =\n (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) x + (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) y\n[PROOFSTEP]\nsimp only [Functor.const_obj_obj, map_add]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Fan f\nx y : \u2191s.pt\n\u22a2 (fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) x + \u2191(NatTrans.app s.\u03c0 { as := j }) y) =\n (fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) + fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) y\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Fan f\nr : R\nx : \u2191s.pt\n\u22a2 AddHom.toFun\n { toFun := fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) (x + y) =\n (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) x + (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) y) }\n (r \u2022 x) =\n \u2191(RingHom.id R) r \u2022\n AddHom.toFun\n { toFun := fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191s.pt),\n (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) (x + y) =\n (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) x + (fun x j => \u2191(NatTrans.app s.\u03c0 { as := j }) x) y) }\n x\n[PROOFSTEP]\nsimp only [Functor.const_obj_obj, map_smul]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Fan f\nr : R\nx : \u2191s.pt\n\u22a2 (fun j => r \u2022 \u2191(NatTrans.app s.\u03c0 { as := j }) x) = \u2191(RingHom.id R) r \u2022 fun j => \u2191(NatTrans.app s.\u03c0 { as := j }) x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b NatTrans.app { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.\u03c0 j =\n NatTrans.app s.\u03c0 j\n\u22a2 m = lift f s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b NatTrans.app { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.\u03c0 j =\n NatTrans.app s.\u03c0 j\nx : \u2191s.pt\n\u22a2 \u2191m x = \u2191(lift f s) x\n[PROOFSTEP]\nfunext j\n[GOAL]\ncase h.h\nR : Type u\ninst\u271d : Ring R\nJ : Type w\nf : J \u2192 ModuleCat R\ns : Cone (Discrete.functor f)\nm : s.pt \u27f6 { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b NatTrans.app { pt := of R ((j : J) \u2192 \u2191(f j)), \u03c0 := Discrete.natTrans fun j => LinearMap.proj j.as }.\u03c0 j =\n NatTrans.app s.\u03c0 j\nx : \u2191s.pt\nj : J\n\u22a2 \u2191m x j = \u2191(lift f s) x j\n[PROOFSTEP]\nexact congr_arg (fun g : s.pt \u27f6 f j => (g : s.pt \u2192 f j) x) (w \u27e8j\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Biproducts", "llama_tokens": 3486, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.28205689033908166}} {"text": "[GOAL]\n\u03b1 : Type v\ninst\u271d : Fintype \u03b1\n\u22a2 Small.{w, v} \u03b1\n[PROOFSTEP]\nrw [small_congr (Fintype.equivFin \u03b1)]\n[GOAL]\n\u03b1 : Type v\ninst\u271d : Fintype \u03b1\n\u22a2 Small.{w, 0} (Fin (Fintype.card \u03b1))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Small", "llama_tokens": 105, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.28198139768053615}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nE\u271d X\u271d : C\nf : 0 \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nx\u271d : Epi e\n\u22a2 0 \u226b e = f\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nP Q : C\ni : P \u2245 Q\nhP : Projective P\nE\u271d X\u271d : C\nf : Q \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\ne_epi : Epi e\nf' : P \u27f6 E\u271d\nhf' : f' \u226b e = i.hom \u226b f\n\u22a2 (i.inv \u226b f') \u226b e = f\n[PROOFSTEP]\nsimp [hf']\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryCoproduct P Q\ninst\u271d\u00b9 : Projective P\ninst\u271d : Projective Q\nE\u271d X\u271d : C\nf : P \u2a3f Q \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nepi : Epi e\n\u22a2 coprod.desc (factorThru (coprod.inl \u226b f) e) (factorThru (coprod.inr \u226b f) e) \u226b e = f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b2 : Type v\ng : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasCoproduct g\ninst\u271d : \u2200 (b : \u03b2), Projective (g b)\nE\u271d X\u271d : C\nf : \u2210 g \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nepi : Epi e\n\u22a2 (Sigma.desc fun b => factorThru (Sigma.\u03b9 g b \u226b f) e) \u226b e = f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Projective P\ninst\u271d : Projective Q\nE\u271d X\u271d : C\nf : P \u229e Q \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nepi : Epi e\n\u22a2 biprod.desc (factorThru (biprod.inl \u226b f) e) (factorThru (biprod.inr \u226b f) e) \u226b e = f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\n\u03b2 : Type v\ng : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBiproduct g\ninst\u271d : \u2200 (b : \u03b2), Projective (g b)\nE\u271d X\u271d : C\nf : \u2a01 g \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nepi : Epi e\n\u22a2 (biproduct.desc fun b => factorThru (biproduct.\u03b9 g b \u226b f) e) \u226b e = f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE\u271d X\u271d : D\nf : F.obj P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nrcases hP.factors (adj.unit.app P \u226b G.map f) (G.map g) with \u27e8f', hf'\u27e9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE\u271d X\u271d : D\nf : F.obj P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nf' : P \u27f6 G.obj E\u271d\nhf' : f' \u226b G.map g = NatTrans.app adj.unit P \u226b G.map f\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nuse F.map f' \u226b adj.counit.app _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE\u271d X\u271d : D\nf : F.obj P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nf' : P \u27f6 G.obj E\u271d\nhf' : f' \u226b G.map g = NatTrans.app adj.unit P \u226b G.map f\n\u22a2 (F.map f' \u226b NatTrans.app adj.counit E\u271d) \u226b g = f\n[PROOFSTEP]\nrw [Category.assoc, \u2190 Adjunction.counit_naturality, \u2190 Category.assoc, \u2190 F.map_comp, hf']\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE\u271d X\u271d : D\nf : F.obj P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nf' : P \u27f6 G.obj E\u271d\nhf' : f' \u226b G.map g = NatTrans.app adj.unit P \u226b G.map f\n\u22a2 F.map (NatTrans.app adj.unit P \u226b G.map f) \u226b NatTrans.app adj.counit X\u271d = f\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\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nP : C\nhP : Projective (F.obj P)\nE\u271d X\u271d : C\nf : P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nhaveI := Adjunction.leftAdjointPreservesColimits.{0, 0} adj\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nP : C\nhP : Projective (F.obj P)\nE\u271d X\u271d : C\nf : P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nrcases(@hP).1 (F.map f) (F.map g) with \u27e8f', hf'\u27e9\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nP : C\nhP : Projective (F.obj P)\nE\u271d X\u271d : C\nf : P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P \u27f6 F.obj E\u271d\nhf' : f' \u226b F.map g = F.map f\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nuse adj.unit.app _ \u226b G.map f' \u226b (inv <| adj.unit.app _)\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nP : C\nhP : Projective (F.obj P)\nE\u271d X\u271d : C\nf : P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P \u27f6 F.obj E\u271d\nhf' : f' \u226b F.map g = F.map f\n\u22a2 (NatTrans.app adj.unit P \u226b G.map f' \u226b inv (NatTrans.app adj.unit E\u271d)) \u226b g = f\n[PROOFSTEP]\nrefine' Faithful.map_injective (F := F) _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nP : C\nhP : Projective (F.obj P)\nE\u271d X\u271d : C\nf : P \u27f6 X\u271d\ng : E\u271d \u27f6 X\u271d\nx\u271d : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\nf' : F.obj P \u27f6 F.obj E\u271d\nhf' : f' \u226b F.map g = F.map f\n\u22a2 F.map ((NatTrans.app adj.unit P \u226b G.map f' \u226b inv (NatTrans.app adj.unit E\u271d)) \u226b g) = F.map f\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\n\u22a2 EnoughProjectives C \u2194 EnoughProjectives D\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\n\u22a2 EnoughProjectives C \u2192 EnoughProjectives D\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\n\u22a2 EnoughProjectives D \u2192 EnoughProjectives C\n[PROOFSTEP]\nall_goals intro H; constructor; intro X; constructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\n\u22a2 EnoughProjectives C \u2192 EnoughProjectives D\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives C\n\u22a2 EnoughProjectives D\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.presentation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives C\n\u22a2 \u2200 (X : D), Nonempty (ProjectivePresentation X)\n[PROOFSTEP]\nintro X\n[GOAL]\ncase mp.presentation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives C\nX : D\n\u22a2 Nonempty (ProjectivePresentation X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\n\u22a2 EnoughProjectives D \u2192 EnoughProjectives C\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives D\n\u22a2 EnoughProjectives C\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.presentation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives D\n\u22a2 \u2200 (X : C), Nonempty (ProjectivePresentation X)\n[PROOFSTEP]\nintro X\n[GOAL]\ncase mpr.presentation\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives D\nX : C\n\u22a2 Nonempty (ProjectivePresentation X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.presentation.val\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives C\nX : D\n\u22a2 ProjectivePresentation X\n[PROOFSTEP]\nexact F.symm.projectivePresentationOfMapProjectivePresentation _ (Nonempty.some (H.presentation (F.inverse.obj X)))\n[GOAL]\ncase mpr.presentation.val\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nF\u271d F : C \u224c D\nH : EnoughProjectives D\nX : C\n\u22a2 ProjectivePresentation X\n[PROOFSTEP]\nexact F.projectivePresentationOfMapProjectivePresentation X (Nonempty.some (H.presentation (F.functor.obj X)))\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n\u22a2 lift h f g hfg w \u226b f = h\n[PROOFSTEP]\nsimp only [Exact.lift]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n\u22a2 factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n f =\n h\n[PROOFSTEP]\nconv_lhs =>\n congr\n rfl\n rw [\u2190 imageSubobject_arrow_comp f]\n -- See the porting note on `Exact.epi`.\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n f\n[PROOFSTEP]\n congr\n rfl\n rw [\u2190 imageSubobject_arrow_comp f]\n -- See the porting note on `Exact.epi`.\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n f\n[PROOFSTEP]\n congr\n rfl\n rw [\u2190 imageSubobject_arrow_comp f]\n -- See the porting note on `Exact.epi`.\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n f\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f)\ncase a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n| f\n[PROOFSTEP]\nrw [\u2190 imageSubobject_arrow_comp f]\n -- See the porting note on `Exact.epi`.\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\n\u22a2 factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n factorThruImageSubobject f \u226b Subobject.arrow (imageSubobject f) =\n h\n[PROOFSTEP]\nhaveI := hfg.epi\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasEqualizers C\ninst\u271d\u00b9 : HasImages C\nP Q R S : C\ninst\u271d : Projective P\nh : P \u27f6 R\nf : Q \u27f6 R\ng : R \u27f6 S\nhfg : Exact f g\nw : h \u226b g = 0\nthis : Epi (imageToKernel f g (_ : f \u226b g = 0))\n\u22a2 factorThru (factorThru (factorThruKernelSubobject g h w) (imageToKernel f g (_ : f \u226b g = 0)))\n (factorThruImageSubobject f) \u226b\n factorThruImageSubobject f \u226b Subobject.arrow (imageSubobject f) =\n h\n[PROOFSTEP]\nrw [\u2190 Category.assoc, factorThru_comp, \u2190 imageToKernel_arrow f g, \u2190 Category.assoc,\n CategoryTheory.Projective.factorThru_comp, factorThruKernelSubobject_comp_arrow]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.Projective", "llama_tokens": 6204, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.28192409992955775}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.873, u_1} C\nX Y : C\nf : Y \u27f6 X\n\u22a2 \u2200 {Y_1 Z : C} {f_1 : Y_1 \u27f6 X},\n (fun Z => {g | \u2203 e, e \u226b f = g}) Y_1 f_1 \u2192 \u2200 (g : Z \u27f6 Y_1), (fun Z => {g | \u2203 e, e \u226b f = g}) Z (g \u226b f_1)\n[PROOFSTEP]\nrintro W Z g \u27e8e, rfl\u27e9 q\n[GOAL]\ncase intro\nC : Type u_1\ninst\u271d : Category.{?u.873, u_1} C\nX Y : C\nf : Y \u27f6 X\nW Z : C\ne : W \u27f6 Y\nq : Z \u27f6 W\n\u22a2 setOf (fun g => \u2203 e, e \u226b f = g) (q \u226b e \u226b f)\n[PROOFSTEP]\nrefine \u27e8q \u226b e, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.873, u_1} C\nX Y : C\nf : Y \u27f6 X\nW Z : C\ne : W \u27f6 Y\nq : Z \u27f6 W\n\u22a2 (q \u226b e) \u226b f = q \u226b e \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\n\u22a2 generate (Presieve.singleton f) = generateSingleton f\n[PROOFSTEP]\next Z g\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\n\u22a2 (generate (Presieve.singleton f)).arrows g \u2194 (generateSingleton f).arrows g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\n\u22a2 (generate (Presieve.singleton f)).arrows g \u2192 (generateSingleton f).arrows g\n[PROOFSTEP]\nrintro \u27e8W, i, p, \u27e8\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y\u271d : C\nf : Y\u271d \u27f6 X\nZ Y : C\ni : Z \u27f6 Y\u271d\n\u22a2 (generateSingleton f).arrows (i \u226b f)\n[PROOFSTEP]\nexact \u27e8i, rfl\u27e9\n[GOAL]\ncase h.mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\n\u22a2 (generateSingleton f).arrows g \u2192 (generate (Presieve.singleton f)).arrows g\n[PROOFSTEP]\nrintro \u27e8g, h\u27e9\n[GOAL]\ncase h.mpr.intro\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng\u271d : Z \u27f6 X\ng : Z \u27f6 Y\nh : g \u226b f = g\u271d\n\u22a2 (generate (Presieve.singleton f)).arrows g\u271d\n[PROOFSTEP]\nexact \u27e8Y, g, f, \u27e8\u27e9, h\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\n\u22a2 Epi f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\n\u22a2 \u2200 {Z : C} (g h : X \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\n[PROOFSTEP]\nintro W m\u2081 m\u2082 h\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\nW : C\nm\u2081 m\u2082 : X \u27f6 W\nh : f \u226b m\u2081 = f \u226b m\u2082\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2082 = EffectiveEpi.desc f (f \u226b m\u2082) (fun {Z} g\u2081 g\u2082 h => by simp only [\u2190 Category.assoc, h]) :=\n EffectiveEpi.uniq _ _ _ _ rfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\nW : C\nm\u2081 m\u2082 : X \u27f6 W\nh\u271d : f \u226b m\u2081 = f \u226b m\u2082\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 g\u2081 \u226b f \u226b m\u2082 = g\u2082 \u226b f \u226b m\u2082\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, h]\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\nW : C\nm\u2081 m\u2082 : X \u27f6 W\nh : f \u226b m\u2081 = f \u226b m\u2082\nthis : m\u2082 = EffectiveEpi.desc f (f \u226b m\u2082) (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b f \u226b m\u2082 = g\u2082 \u226b f \u226b m\u2082)\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.7498, u_1} C\nX Y : C\nf : Y \u27f6 X\ninst\u271d : EffectiveEpi f\nW : C\nm\u2081 m\u2082 : X \u27f6 W\nh : f \u226b m\u2081 = f \u226b m\u2082\nthis : m\u2082 = EffectiveEpi.desc f (f \u226b m\u2082) (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b f \u226b m\u2082 = g\u2082 \u226b f \u226b m\u2082)\n\u22a2 m\u2081 = EffectiveEpi.desc f (f \u226b m\u2082) (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b f \u226b m\u2082 = g\u2082 \u226b f \u226b m\u2082)\n[PROOFSTEP]\nexact EffectiveEpi.uniq _ _ _ _ h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk f).left) \u226b f = (Over.mk f).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\n\u22a2 \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nintro Z g\u2081 g\u2082 h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nlet Y' : D := \u27e8Over.mk f, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk f).left) \u226b f = (Over.mk f).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nlet Z' : D := \u27e8Over.mk (g\u2081 \u226b f), g\u2081, rfl\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) }\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nlet g\u2081' : Z' \u27f6 Y' := Over.homMk g\u2081\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) }\ng\u2081' : Z' \u27f6 Y' := Over.homMk g\u2081\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nlet g\u2082' : Z' \u27f6 Y' := Over.homMk g\u2082 (by simp [h])\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) }\ng\u2081' : Z' \u27f6 Y' := Over.homMk g\u2081\n\u22a2 g\u2082 \u226b Y'.obj.hom = Z'.obj.hom\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) }\ng\u2081' : Z' \u27f6 Y' := Over.homMk g\u2081\ng\u2082' : Z' \u27f6 Y' := Over.homMk g\u2082\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nchange F.map g\u2081' \u226b _ = F.map g\u2082' \u226b _\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nZ : C\ng\u2081 g\u2082 : Z \u27f6 Y\nh : g\u2081 \u226b f = g\u2082 \u226b f\nY' : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) }\ng\u2081' : Z' \u27f6 Y' := Over.homMk g\u2081\ng\u2082' : Z' \u27f6 Y' := Over.homMk g\u2082\n\u22a2 F.map g\u2081' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n F.map g\u2082' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n[PROOFSTEP]\nsimp only [S.w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\n\u22a2 \u2200 (s : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows))\n (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b\n (fun S =>\n EffectiveEpiStruct.desc Hf\n (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y) (h : g\u2081 \u226b f = g\u2082 \u226b f),\n let Y' := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) };\n let Z' := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) };\n let g\u2081' := Over.homMk g\u2081;\n let g\u2082' := Over.homMk g\u2082;\n F.map g\u2081' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n F.map g\u2082' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }))\n s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nrintro S \u27e8T, g, hT\u27e9\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\n\u22a2 NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9\n { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) } \u226b\n (fun S =>\n EffectiveEpiStruct.desc Hf\n (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y) (h : g\u2081 \u226b f = g\u2082 \u226b f),\n let Y' := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) };\n let Z' := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) };\n let g\u2081' := Over.homMk g\u2081;\n let g\u2082' := Over.homMk g\u2082;\n F.map g\u2081' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n F.map g\u2082' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }))\n S =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\n\u22a2 T.hom \u226b\n EffectiveEpiStruct.desc Hf (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) }) =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nnth_rewrite 1 [\u2190 hT, Category.assoc, Hf.fac]\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nlet y : D := \u27e8Over.mk f, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk f).left) \u226b f = (Over.mk f).hom\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\ny : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nlet x : D := \u27e8Over.mk T.hom, g, hT\u27e9\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\ny : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nx : D := { obj := Over.mk T.hom, property := (_ : \u2203 e, e \u226b f = (Over.mk T.hom).hom) }\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nlet g' : x \u27f6 y := Over.homMk g\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\ny : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nx : D := { obj := Over.mk T.hom, property := (_ : \u2203 e, e \u226b f = (Over.mk T.hom).hom) }\ng' : x \u27f6 y := Over.homMk g\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nchange F.map g' \u226b _ = _\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\ny : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nx : D := { obj := Over.mk T.hom, property := (_ : \u2203 e, e \u226b f = (Over.mk T.hom).hom) }\ng' : x \u27f6 y := Over.homMk g\n\u22a2 F.map g' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nrw [S.w]\n[GOAL]\ncase mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nT : Over X\ng : (\ud835\udfed C).obj T.left \u27f6 Y\nhT : g \u226b f = T.hom\ny : D := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\nx : D := { obj := Over.mk T.hom, property := (_ : \u2203 e, e \u226b f = (Over.mk T.hom).hom) }\ng' : x \u27f6 y := Over.homMk g\n\u22a2 NatTrans.app S.\u03b9 x = NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 e, e \u226b f = T.hom) }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\n\u22a2 \u2200 (s : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows))\n (m : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 s.pt),\n (\u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192\n m =\n (fun S =>\n EffectiveEpiStruct.desc Hf\n (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y) (h : g\u2081 \u226b f = g\u2082 \u226b f),\n let Y' := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) };\n let Z' := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) };\n let g\u2081' := Over.homMk g\u2081;\n let g\u2082' := Over.homMk g\u2082;\n F.map g\u2081' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n F.map g\u2082' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }))\n s\n[PROOFSTEP]\nintro S m hm\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nm : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m =\n (fun S =>\n EffectiveEpiStruct.desc Hf\n (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y) (h : g\u2081 \u226b f = g\u2082 \u226b f),\n let Y' := { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) };\n let Z' := { obj := Over.mk (g\u2081 \u226b f), property := (_ : \u2203 e, e \u226b f = (Over.mk (g\u2081 \u226b f)).hom) };\n let g\u2081' := Over.homMk g\u2081;\n let g\u2082' := Over.homMk g\u2082;\n F.map g\u2081' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } =\n F.map g\u2082' \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }))\n S\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nm : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m =\n EffectiveEpiStruct.desc Hf (NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) })\n (_ :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = f) })\n[PROOFSTEP]\ngeneralize_proofs h1 h2\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nm : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nh1 : \u2203 e, e \u226b f = f\nh2 :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 }\n\u22a2 m = EffectiveEpiStruct.desc Hf (NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 }) h2\n[PROOFSTEP]\napply Hf.uniq _ h2\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nm : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nh1 : \u2203 e, e \u226b f = f\nh2 :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 }\n\u22a2 f \u226b m = NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 }\n[PROOFSTEP]\nexact hm \u27e8Over.mk f, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.8526, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : EffectiveEpiStruct f\nD : Type (max u_1 ?u.8526) := FullSubcategory fun T => (Sieve.generateSingleton f).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateSingleton f).arrows\nS : Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows)\nm : (Presieve.cocone (Sieve.generateSingleton f).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\nh1 : \u2203 e, e \u226b f = f\nh2 :\n \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y),\n g\u2081 \u226b f = g\u2082 \u226b f \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk f, property := h1 }\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk f).left) \u226b f = (Over.mk f).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 \u2200 \u2983X_1 Y_1 : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom\u2984 (f_1 : X_1 \u27f6 Y_1),\n (Presieve.diagram (Sieve.generateSingleton f).arrows).map f_1 \u226b\n (fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e)\n Y_1 =\n (fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e)\n X_1 \u226b\n ((Functor.const (FullSubcategory fun f_2 => (Sieve.generateSingleton f).arrows f_2.hom)).obj W).map f_1\n[PROOFSTEP]\nrintro \u27e8A, hA\u27e9 \u27e8B, hB\u27e9 (q : A \u27f6 B)\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nA : Over X\nhA : (Sieve.generateSingleton f).arrows A.hom\nB : Over X\nhB : (Sieve.generateSingleton f).arrows B.hom\nq : A \u27f6 B\n\u22a2 (Presieve.diagram (Sieve.generateSingleton f).arrows).map q \u226b\n (fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e)\n { obj := B, property := hB } =\n (fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e)\n { obj := A, property := hA } \u226b\n ((Functor.const (FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom)).obj W).map q\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nA : Over X\nhA : (Sieve.generateSingleton f).arrows A.hom\nB : Over X\nhB : (Sieve.generateSingleton f).arrows B.hom\nq : A \u27f6 B\n\u22a2 q.left \u226b Exists.choose hB \u226b e = (Exists.choose hA \u226b e) \u226b \ud835\udfd9 W\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, Category.comp_id]\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nA : Over X\nhA : (Sieve.generateSingleton f).arrows A.hom\nB : Over X\nhB : (Sieve.generateSingleton f).arrows B.hom\nq : A \u27f6 B\n\u22a2 (q.left \u226b Exists.choose hB) \u226b e = Exists.choose hA \u226b e\n[PROOFSTEP]\napply h\n[GOAL]\ncase mk.mk.a\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nA : Over X\nhA : (Sieve.generateSingleton f).arrows A.hom\nB : Over X\nhB : (Sieve.generateSingleton f).arrows B.hom\nq : A \u27f6 B\n\u22a2 (q.left \u226b Exists.choose hB) \u226b f = Exists.choose hA \u226b f\n[PROOFSTEP]\nrw [Category.assoc, hB.choose_spec, hA.choose_spec, Over.w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\n\u22a2 \u2200 {W : C} (e : Y \u27f6 W) (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e),\n f \u226b\n (fun {W} e h => IsColimit.desc Hf (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e))) e\n (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) =\n e\n[PROOFSTEP]\nintro W e h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 f \u226b\n (fun {W} e h => IsColimit.desc Hf (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e))) e\n (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) =\n e\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 f \u226b\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e } =\n e\n[PROOFSTEP]\nhave := Hf.fac (aux e h) \u27e8Over.mk f, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk f).left) \u226b f = (Over.mk f).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nthis :\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9\n { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) } \u226b\n IsColimit.desc Hf (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)) =\n NatTrans.app (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)).\u03b9\n { obj := Over.mk f, property := (_ : \u2203 e, e \u226b f = (Over.mk f).hom) }\n\u22a2 f \u226b\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e } =\n e\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nthis :\n f \u226b\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e } =\n Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b e\n\u22a2 f \u226b\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e } =\n e\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nthis :\n f \u226b\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e } =\n Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b e\n\u22a2 Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b e = e\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b e = e\n[PROOFSTEP]\nnth_rewrite 2 [\u2190 Category.id_comp e]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b e = \ud835\udfd9 Y \u226b e\n[PROOFSTEP]\napply h\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\n\u22a2 Exists.choose (_ : \u2203 e, e \u226b f = f) \u226b f = \ud835\udfd9 Y \u226b f\n[PROOFSTEP]\ngeneralize_proofs hh\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nhh : \u2203 e, e \u226b f = f\n\u22a2 Exists.choose hh \u226b f = \ud835\udfd9 Y \u226b f\n[PROOFSTEP]\nrw [hh.choose_spec, Category.id_comp]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\n\u22a2 \u2200 {W : C} (e : Y \u27f6 W) (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) (m : X \u27f6 W),\n f \u226b m = e \u2192\n m =\n (fun {W} e h => IsColimit.desc Hf (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e))) e\n (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)\n[PROOFSTEP]\nintro W e h m hm\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\n\u22a2 m =\n (fun {W} e h => IsColimit.desc Hf (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e))) e\n (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\n\u22a2 m =\n IsColimit.desc Hf\n { pt := W, \u03b9 := NatTrans.mk fun x => Exists.choose (_ : (Sieve.generateSingleton f).arrows x.obj.hom) \u226b e }\n[PROOFSTEP]\napply Hf.uniq (aux e h)\n[GOAL]\ncase x\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\n\u22a2 \u2200 (j : FullSubcategory fun f_1 => (Sieve.generateSingleton f).arrows f_1.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9 j \u226b m =\n NatTrans.app (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)).\u03b9 j\n[PROOFSTEP]\nrintro \u27e8A, g, hA\u27e9\n[GOAL]\ncase x.mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\nA : Over X\ng : (\ud835\udfed C).obj A.left \u27f6 Y\nhA : g \u226b f = A.hom\n\u22a2 NatTrans.app (Presieve.cocone (Sieve.generateSingleton f).arrows).\u03b9\n { obj := A, property := (_ : \u2203 e, e \u226b f = A.hom) } \u226b\n m =\n NatTrans.app (aux e (_ : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)).\u03b9\n { obj := A, property := (_ : \u2203 e, e \u226b f = A.hom) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase x.mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\nA : Over X\ng : (\ud835\udfed C).obj A.left \u27f6 Y\nhA : g \u226b f = A.hom\n\u22a2 A.hom \u226b m = Exists.choose (_ : \u2203 e, e \u226b f = A.hom) \u226b e\n[PROOFSTEP]\nnth_rewrite 1 [\u2190 hA, Category.assoc, hm]\n[GOAL]\ncase x.mk.intro\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\nA : Over X\ng : (\ud835\udfed C).obj A.left \u27f6 Y\nhA : g \u226b f = A.hom\n\u22a2 g \u226b e = Exists.choose (_ : \u2203 e, e \u226b f = A.hom) \u226b e\n[PROOFSTEP]\napply h\n[GOAL]\ncase x.mk.intro.a\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\nA : Over X\ng : (\ud835\udfed C).obj A.left \u27f6 Y\nhA : g \u226b f = A.hom\n\u22a2 g \u226b f = Exists.choose (_ : \u2203 e, e \u226b f = A.hom) \u226b f\n[PROOFSTEP]\ngeneralize_proofs hh\n[GOAL]\ncase x.mk.intro.a\nC : Type u_1\ninst\u271d : Category.{?u.20742, u_1} C\nX Y : C\nf : Y \u27f6 X\nHf : IsColimit (Presieve.cocone (Sieve.generateSingleton f).arrows)\naux : {W : C} \u2192\n (e : Y \u27f6 W) \u2192\n (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192\n Cocone (Presieve.diagram (Sieve.generateSingleton f).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } => Exists.choose hT \u226b e }\nW : C\ne : Y \u27f6 W\nh : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e\nm : X \u27f6 W\nhm : f \u226b m = e\nA : Over X\ng : (\ud835\udfed C).obj A.left \u27f6 Y\nhA : g \u226b f = A.hom\nhh : \u2203 e, e \u226b f = A.hom\n\u22a2 g \u226b f = Exists.choose hh \u226b f\n[PROOFSTEP]\nrwa [hh.choose_spec]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.singleton f) \u2194 EffectiveEpi f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.singleton f) \u2192 EffectiveEpi f\n[PROOFSTEP]\nintro (h : Nonempty _)\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (IsColimit (Presieve.cocone (generate (Presieve.singleton f)).arrows))\n\u22a2 EffectiveEpi f\n[PROOFSTEP]\nrw [Sieve.generateSingleton_eq] at h \n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (IsColimit (Presieve.cocone (generateSingleton f).arrows))\n\u22a2 EffectiveEpi f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.effectiveEpi\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (IsColimit (Presieve.cocone (generateSingleton f).arrows))\n\u22a2 Nonempty (EffectiveEpiStruct f)\n[PROOFSTEP]\napply Nonempty.map (effectiveEpiStructOfIsColimit _) h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\n\u22a2 EffectiveEpi f \u2192 Presieve.EffectiveEpimorphic (Presieve.singleton f)\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (EffectiveEpiStruct f)\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.singleton f)\n[PROOFSTEP]\nshow Nonempty _\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (EffectiveEpiStruct f)\n\u22a2 Nonempty (IsColimit (Presieve.cocone (generate (Presieve.singleton f)).arrows))\n[PROOFSTEP]\nrw [Sieve.generateSingleton_eq]\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf : Y \u27f6 X\nh : Nonempty (EffectiveEpiStruct f)\n\u22a2 Nonempty (IsColimit (Presieve.cocone (generateSingleton f).arrows))\n[PROOFSTEP]\napply Nonempty.map (isColimitOfEffectiveEpiStruct _) h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.36898, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 B},\n (fun Y => {f | \u2203 a g, g \u226b \u03c0 a = f}) Y f \u2192 \u2200 (g : Z \u27f6 Y), (fun Y => {f | \u2203 a g, g \u226b \u03c0 a = f}) Z (g \u226b f)\n[PROOFSTEP]\nrintro Y\u2081 Y\u2082 g\u2081 \u27e8a, q, rfl\u27e9 e\n[GOAL]\ncase intro.intro\nC : Type u_1\ninst\u271d : Category.{?u.36898, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY\u2081 Y\u2082 : C\na : \u03b1\nq : Y\u2081 \u27f6 X a\ne : Y\u2082 \u27f6 Y\u2081\n\u22a2 setOf (fun f => \u2203 a g, g \u226b \u03c0 a = f) (e \u226b q \u226b \u03c0 a)\n[PROOFSTEP]\nrefine \u27e8a, e \u226b q, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.36898, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY\u2081 Y\u2082 : C\na : \u03b1\nq : Y\u2081 \u27f6 X a\ne : Y\u2082 \u27f6 Y\u2081\n\u22a2 (e \u226b q) \u226b \u03c0 a = e \u226b q \u226b \u03c0 a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\n\u22a2 generate (Presieve.ofArrows X \u03c0) = generateFamily X \u03c0\n[PROOFSTEP]\next Y g\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY : C\ng : Y \u27f6 B\n\u22a2 (generate (Presieve.ofArrows X \u03c0)).arrows g \u2194 (generateFamily X \u03c0).arrows g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY : C\ng : Y \u27f6 B\n\u22a2 (generate (Presieve.ofArrows X \u03c0)).arrows g \u2192 (generateFamily X \u03c0).arrows g\n[PROOFSTEP]\nrintro \u27e8W, g, f, \u27e8a\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY\u271d Y : C\na : \u03b1\ng : Y\u271d \u27f6 X a\n\u22a2 (generateFamily X \u03c0).arrows (g \u226b \u03c0 a)\n[PROOFSTEP]\nexact \u27e8a, g, rfl\u27e9\n[GOAL]\ncase h.mpr\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY : C\ng : Y \u27f6 B\n\u22a2 (generateFamily X \u03c0).arrows g \u2192 (generate (Presieve.ofArrows X \u03c0)).arrows g\n[PROOFSTEP]\nrintro \u27e8a, g, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nY : C\na : \u03b1\ng : Y \u27f6 X a\n\u22a2 (generate (Presieve.ofArrows X \u03c0)).arrows (g \u226b \u03c0 a)\n[PROOFSTEP]\nrefine \u27e8_, g, \u03c0 a, \u27e8a\u27e9, rfl\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.43458, u_1} C\n\u03b1 : Unit \u2192 C\n\u22a2 \u2200 {W : C} (e : (a : Unit) \u2192 \u03b1 a \u27f6 W)\n (h :\n \u2200 {Z : C} (a\u2081 a\u2082 : Unit) (g\u2081 : Z \u27f6 \u03b1 a\u2081) (g\u2082 : Z \u27f6 \u03b1 a\u2082), g\u2081 \u226b \ud835\udfd9 (\u03b1 ()) = g\u2082 \u226b \ud835\udfd9 (\u03b1 ()) \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n (a : Unit),\n \ud835\udfd9 (\u03b1 ()) \u226b\n (fun {W} e x => e ()) e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : Unit) (g\u2081 : Z \u27f6 \u03b1 a\u2081) (g\u2082 : Z \u27f6 \u03b1 a\u2082),\n g\u2081 \u226b \ud835\udfd9 (\u03b1 ()) = g\u2082 \u226b \ud835\udfd9 (\u03b1 ()) \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) =\n e a\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.43458, u_1} C\n\u03b1 : Unit \u2192 C\n\u22a2 \u2200 {W : C} (e : (a : Unit) \u2192 \u03b1 a \u27f6 W)\n (h :\n \u2200 {Z : C} (a\u2081 a\u2082 : Unit) (g\u2081 : Z \u27f6 \u03b1 a\u2081) (g\u2082 : Z \u27f6 \u03b1 a\u2082), g\u2081 \u226b \ud835\udfd9 (\u03b1 ()) = g\u2082 \u226b \ud835\udfd9 (\u03b1 ()) \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n (m : \u03b1 () \u27f6 W),\n (\u2200 (a : Unit), \ud835\udfd9 (\u03b1 ()) \u226b m = e a) \u2192\n m =\n (fun {W} e x => e ()) e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : Unit) (g\u2081 : Z \u27f6 \u03b1 a\u2081) (g\u2082 : Z \u27f6 \u03b1 a\u2082),\n g\u2081 \u226b \ud835\udfd9 (\u03b1 ()) = g\u2082 \u226b \ud835\udfd9 (\u03b1 ()) \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.51706, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 \u03c0 a \u226b\n EffectiveEpiFamily.desc X \u03c0 e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) =\n e a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.52695, u_1} C\nB W Q : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\nq : W \u27f6 Q\n\u22a2 \u03c0 a \u226b\n EffectiveEpiFamily.desc X \u03c0 e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u226b\n q =\n e a \u226b q\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nhave : m\u2082 = EffectiveEpiFamily.desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082) (fun a\u2081 a\u2082 g\u2081 g\u2082 h => by simp only [\u2190 Category.assoc, h]) :=\n by apply EffectiveEpiFamily.uniq; intro; rfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh\u271d : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\nZ\u271d : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z\u271d \u27f6 X a\u2081\ng\u2082 : Z\u271d \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\n\u22a2 g\u2081 \u226b (fun a => \u03c0 a \u226b m\u2082) a\u2081 = g\u2082 \u226b (fun a => \u03c0 a \u226b m\u2082) a\u2082\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, h]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\n\u22a2 m\u2082 =\n desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082)\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b \u03c0 a\u2081 \u226b m\u2082 = g\u2082 \u226b \u03c0 a\u2082 \u226b m\u2082)\n[PROOFSTEP]\napply EffectiveEpiFamily.uniq\n[GOAL]\ncase hm\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\n\u22a2 \u2200 (a : \u03b1), \u03c0 a \u226b m\u2082 = \u03c0 a \u226b m\u2082\n[PROOFSTEP]\nintro\n[GOAL]\ncase hm\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\na\u271d : \u03b1\n\u22a2 \u03c0 a\u271d \u226b m\u2082 = \u03c0 a\u271d \u226b m\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\nthis :\n m\u2082 =\n desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082)\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b \u03c0 a\u2081 \u226b m\u2082 = g\u2082 \u226b \u03c0 a\u2082 \u226b m\u2082)\n\u22a2 m\u2081 = m\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_3, u_1} C\nB W : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d : EffectiveEpiFamily X \u03c0\nm\u2081 m\u2082 : B \u27f6 W\nh : \u2200 (a : \u03b1), \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082\nthis :\n m\u2082 =\n desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082)\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b \u03c0 a\u2081 \u226b m\u2082 = g\u2082 \u226b \u03c0 a\u2082 \u226b m\u2082)\n\u22a2 m\u2081 =\n desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082)\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b \u03c0 a\u2081 \u226b m\u2082 = g\u2082 \u226b \u03c0 a\u2082 \u226b m\u2082)\n[PROOFSTEP]\nexact EffectiveEpiFamily.uniq _ _ _ _ _ h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\n\u22a2 Epi (Sigma.desc \u03c0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\n\u22a2 \u2200 {Z : C} (g h : B \u27f6 Z), Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h \u2192 g = h\n[PROOFSTEP]\nintro Z g h H\n[GOAL]\ncase left_cancellation\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\nZ : C\ng h : B \u27f6 Z\nH : Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h\n\u22a2 g = h\n[PROOFSTEP]\napply EffectiveEpiFamily.hom_ext X \u03c0\n[GOAL]\ncase left_cancellation.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\nZ : C\ng h : B \u27f6 Z\nH : Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h\n\u22a2 \u2200 (a : \u03b1), \u03c0 a \u226b g = \u03c0 a \u226b h\n[PROOFSTEP]\nintro a\n[GOAL]\ncase left_cancellation.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\nZ : C\ng h : B \u27f6 Z\nH : Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h\na : \u03b1\n\u22a2 \u03c0 a \u226b g = \u03c0 a \u226b h\n[PROOFSTEP]\nsuffices (Sigma.\u03b9 _ a \u226b Sigma.desc \u03c0) \u226b g = (Sigma.\u03b9 _ a \u226b Sigma.desc \u03c0) \u226b h by simpa only [colimit.\u03b9_desc] using this\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\nZ : C\ng h : B \u27f6 Z\nH : Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h\na : \u03b1\nthis : (Sigma.\u03b9 (fun b => X b) a \u226b Sigma.desc \u03c0) \u226b g = (Sigma.\u03b9 (fun b => X b) a \u226b Sigma.desc \u03c0) \u226b h\n\u22a2 \u03c0 a \u226b g = \u03c0 a \u226b h\n[PROOFSTEP]\nsimpa only [colimit.\u03b9_desc] using this\n[GOAL]\ncase left_cancellation.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.55691, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\ninst\u271d\u00b9 : EffectiveEpiFamily X \u03c0\ninst\u271d : HasCoproduct X\nZ : C\ng h : B \u27f6 Z\nH : Sigma.desc \u03c0 \u226b g = Sigma.desc \u03c0 \u226b h\na : \u03b1\n\u22a2 (Sigma.\u03b9 (fun b => X b) a \u226b Sigma.desc \u03c0) \u226b g = (Sigma.\u03b9 (fun b => X b) a \u226b Sigma.desc \u03c0) \u226b h\n[PROOFSTEP]\nsimp only [Category.assoc, H]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\na : \u03b1\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a)).left) \u226b \u03c0 a = (Over.mk (\u03c0 a)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\n\u22a2 \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082\n[PROOFSTEP]\nintro Z a\u2081 a\u2082 g\u2081 g\u2082 h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\n\u22a2 g\u2081 \u226b\n (fun a => NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a => NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nlet A\u2081 : D := \u27e8Over.mk (\u03c0 a\u2081), a\u2081, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a\u2081)).left) \u226b \u03c0 a\u2081 = (Over.mk (\u03c0 a\u2081)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nlet A\u2082 : D := \u27e8Over.mk (\u03c0 a\u2082), a\u2082, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a\u2082)).left) \u226b \u03c0 a\u2082 = (Over.mk (\u03c0 a\u2082)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\nA\u2082 : D := { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2082)).hom) }\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nlet Z' : D := \u27e8Over.mk (g\u2081 \u226b \u03c0 a\u2081), a\u2081, g\u2081, rfl\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\nA\u2082 : D := { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2082)).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b \u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (g\u2081 \u226b \u03c0 a\u2081)).hom) }\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nlet i\u2081 : Z' \u27f6 A\u2081 := Over.homMk g\u2081\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\nA\u2082 : D := { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2082)).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b \u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (g\u2081 \u226b \u03c0 a\u2081)).hom) }\ni\u2081 : Z' \u27f6 A\u2081 := Over.homMk g\u2081\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nlet i\u2082 : Z' \u27f6 A\u2082 := Over.homMk g\u2082\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\nA\u2082 : D := { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2082)).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b \u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (g\u2081 \u226b \u03c0 a\u2081)).hom) }\ni\u2081 : Z' \u27f6 A\u2081 := Over.homMk g\u2081\ni\u2082 : Z' \u27f6 A\u2082 := Over.homMk g\u2082\n\u22a2 g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nchange F.map i\u2081 \u226b _ = F.map i\u2082 \u226b _\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nZ : C\na\u2081 a\u2082 : \u03b1\ng\u2081 : Z \u27f6 X a\u2081\ng\u2082 : Z \u27f6 X a\u2082\nh : g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082\nA\u2081 : D := { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2081)).hom) }\nA\u2082 : D := { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (\u03c0 a\u2082)).hom) }\nZ' : D := { obj := Over.mk (g\u2081 \u226b \u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk (g\u2081 \u226b \u03c0 a\u2081)).hom) }\ni\u2081 : Z' \u27f6 A\u2081 := Over.homMk g\u2081\ni\u2082 : Z' \u27f6 A\u2082 := Over.homMk g\u2082\n\u22a2 F.map i\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n F.map i\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }\n[PROOFSTEP]\nsimp only [S.w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\n\u22a2 \u2200 (s : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows))\n (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b\n (fun S =>\n EffectiveEpiFamilyStruct.desc H\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082))\n s =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\nintro S \u27e8T, a, (g : T.left \u27f6 X a), hT\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\n\u22a2 NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9\n { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) } \u226b\n (fun S =>\n EffectiveEpiFamilyStruct.desc H\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082))\n S =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\n\u22a2 T.hom \u226b\n EffectiveEpiFamilyStruct.desc H\n (fun a => NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) }) =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nnth_rewrite 1 [\u2190 hT, Category.assoc, H.fac]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nlet A : D := \u27e8Over.mk (\u03c0 a), a, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a)).left) \u226b \u03c0 a = (Over.mk (\u03c0 a)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\nA : D := { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nlet B : D := \u27e8Over.mk T.hom, a, g, hT\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\u271d\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\nA : D := { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\nB : D := { obj := Over.mk T.hom, property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk T.hom).hom) }\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nlet i : B \u27f6 A := Over.homMk g\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\u271d\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\nA : D := { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\nB : D := { obj := Over.mk T.hom, property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk T.hom).hom) }\ni : B \u27f6 A := Over.homMk g\n\u22a2 g \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nchange F.map i \u226b _ = _\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\u271d\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\nA : D := { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\nB : D := { obj := Over.mk T.hom, property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk T.hom).hom) }\ni : B \u27f6 A := Over.homMk g\n\u22a2 F.map i \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) } =\n NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nrw [S.w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nT : Over B\u271d\na : \u03b1\ng : T.left \u27f6 X a\nhT : g \u226b \u03c0 a = T.hom\nA : D := { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\nB : D := { obj := Over.mk T.hom, property := (_ : \u2203 a g, g \u226b \u03c0 a = (Over.mk T.hom).hom) }\ni : B \u27f6 A := Over.homMk g\n\u22a2 NatTrans.app S.\u03b9 B = NatTrans.app S.\u03b9 { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\n\u22a2 \u2200 (s : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows))\n (m : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 s.pt),\n (\u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j) \u2192\n m =\n (fun S =>\n EffectiveEpiFamilyStruct.desc H\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082))\n s\n[PROOFSTEP]\nintro S m hm\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nm : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m =\n (fun S =>\n EffectiveEpiFamilyStruct.desc H\n (fun a =>\n NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2081 =\n g\u2082 \u226b\n (fun a =>\n NatTrans.app S.\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) })\n a\u2082))\n S\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nm : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 m =\n EffectiveEpiFamilyStruct.desc H\n (fun a => NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) })\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192\n g\u2081 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2081), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2081) } =\n g\u2082 \u226b NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a\u2082), property := (_ : \u2203 a g, g \u226b \u03c0 a = \u03c0 a\u2082) })\n[PROOFSTEP]\napply H.uniq\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nm : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\n\u22a2 \u2200 (a : \u03b1), \u03c0 a \u226b m = NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) }\n[PROOFSTEP]\nintro a\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nm : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\na : \u03b1\n\u22a2 \u03c0 a \u226b m = NatTrans.app S.\u03b9 { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a) }\n[PROOFSTEP]\nexact hm \u27e8Over.mk (\u03c0 a), a, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.59810, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : EffectiveEpiFamilyStruct X \u03c0\nD : Type (max u_1 ?u.59810) := FullSubcategory fun T => (Sieve.generateFamily X \u03c0).arrows T.hom\nF : D \u2964 C := Presieve.diagram (Sieve.generateFamily X \u03c0).arrows\nS : Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows)\nm : (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).pt \u27f6 S.pt\nhm :\n \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m = NatTrans.app S.\u03b9 j\na : \u03b1\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a)).left) \u226b \u03c0 a = (Over.mk (\u03c0 a)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\n\u22a2 \u2200 \u2983X_1 Y : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom\u2984 (f : X_1 \u27f6 Y),\n (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows).map f \u226b\n (fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT))\n Y =\n (fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT))\n X_1 \u226b\n ((Functor.const (FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom)).obj W).map f\n[PROOFSTEP]\nintro \u27e8A, a, (g\u2081 : A.left \u27f6 _), ha\u27e9 \u27e8B, b, (g\u2082 : B.left \u27f6 _), hb\u27e9 (q : A \u27f6 B)\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\n\u22a2 (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows).map q \u226b\n (fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT))\n { obj := B, property := (_ : \u2203 a g, g \u226b \u03c0 a = B.hom) } =\n (fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT))\n { obj := A, property := (_ : \u2203 a g, g \u226b \u03c0 a = A.hom) } \u226b\n ((Functor.const (FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom)).obj W).map q\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\n\u22a2 q.left \u226b\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) = B.hom) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) =\n (Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom)) = A.hom) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom))) \u226b\n \ud835\udfd9 W\n[PROOFSTEP]\nrw [Category.comp_id, \u2190 Category.assoc]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\n\u22a2 (q.left \u226b Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) = B.hom)) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) =\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom)) = A.hom) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom))\n[PROOFSTEP]\napply h\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\n\u22a2 (q.left \u226b Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) = B.hom)) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) =\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom)) = A.hom) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom))\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\n\u22a2 q.left \u226b\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) = B.hom) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = B.hom)) =\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom)) = A.hom) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = A.hom))\n[PROOFSTEP]\ngeneralize_proofs h1 h2 h3 h4\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB\u271d : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\u271d\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nA : Over B\u271d\na : \u03b1\ng\u2081 : A.left \u27f6 X a\nha : g\u2081 \u226b \u03c0 a = A.hom\nB : Over B\u271d\nb : \u03b1\ng\u2082 : B.left \u27f6 X b\nhb : g\u2082 \u226b \u03c0 b = B.hom\nq : A \u27f6 B\nh1 : \u2203 a g, g \u226b \u03c0 a = B.hom\nh2 : \u2203 g, g \u226b \u03c0 (Exists.choose h1) = B.hom\nh3 : \u2203 a g, g \u226b \u03c0 a = A.hom\nh4 : \u2203 g, g \u226b \u03c0 (Exists.choose h3) = A.hom\n\u22a2 q.left \u226b Exists.choose h2 \u226b \u03c0 (Exists.choose h1) = Exists.choose h4 \u226b \u03c0 (Exists.choose h3)\n[PROOFSTEP]\nrw [h2.choose_spec, h4.choose_spec, Over.w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\n\u22a2 \u2200 {W : C} (e : (a : \u03b1) \u2192 X a \u27f6 W)\n (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) (a : \u03b1),\n \u03c0 a \u226b\n (fun {W} e h =>\n IsColimit.desc H\n (aux e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)))\n e (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) =\n e a\n[PROOFSTEP]\nintro W e h a\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 \u03c0 a \u226b\n (fun {W} e h =>\n IsColimit.desc H\n (aux e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)))\n e (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) =\n e a\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 \u03c0 a \u226b\n IsColimit.desc H\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n Exists.choose\n (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) = x.obj.hom) \u226b\n e (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) } =\n e a\n[PROOFSTEP]\nhave := H.fac (aux e h) \u27e8Over.mk (\u03c0 a), a, \ud835\udfd9 _, by simp\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj (Over.mk (\u03c0 a)).left) \u226b \u03c0 a = (Over.mk (\u03c0 a)).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\nthis :\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) } \u226b\n IsColimit.desc H\n (aux e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)) =\n NatTrans.app\n (aux e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)).\u03b9\n { obj := Over.mk (\u03c0 a), property := (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = (Over.mk (\u03c0 a)).hom) }\n\u22a2 \u03c0 a \u226b\n IsColimit.desc H\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n Exists.choose\n (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) = x.obj.hom) \u226b\n e (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) } =\n e a\n[PROOFSTEP]\ndsimp at this \n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\nthis :\n \u03c0 a \u226b\n IsColimit.desc H\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n Exists.choose\n (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) = x.obj.hom) \u226b\n e (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) } =\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n e (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a))\n\u22a2 \u03c0 a \u226b\n IsColimit.desc H\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n Exists.choose\n (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) = x.obj.hom) \u226b\n e (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) } =\n e a\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\nthis :\n \u03c0 a \u226b\n IsColimit.desc H\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n Exists.choose\n (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) = x.obj.hom) \u226b\n e (Exists.choose (_ : (Sieve.generateFamily X \u03c0).arrows x.obj.hom)) } =\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n e (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a))\n\u22a2 Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n e (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) =\n e a\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n e (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) =\n e a\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Category.id_comp (e a)]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n| e a\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (e a)]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n| e a\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (e a)]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n| e a\n[PROOFSTEP]\nrw [\u2190 Category.id_comp (e a)]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n e (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) =\n \ud835\udfd9 (X a) \u226b e a\n[PROOFSTEP]\napply h\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\n\u22a2 Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) = \u03c0 a) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a)) =\n \ud835\udfd9 (X a) \u226b \u03c0 a\n[PROOFSTEP]\ngeneralize_proofs h1 h2\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\na : \u03b1\nh1 : \u2203 a_1 g, g \u226b \u03c0 a_1 = \u03c0 a\nh2 : \u2203 g, g \u226b \u03c0 (Exists.choose h1) = \u03c0 a\n\u22a2 Exists.choose h2 \u226b \u03c0 (Exists.choose h1) = \ud835\udfd9 (X a) \u226b \u03c0 a\n[PROOFSTEP]\nrw [h2.choose_spec, Category.id_comp]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\n\u22a2 \u2200 {W : C} (e : (a : \u03b1) \u2192 X a \u27f6 W)\n (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n (m : B \u27f6 W),\n (\u2200 (a : \u03b1), \u03c0 a \u226b m = e a) \u2192\n m =\n (fun {W} e h =>\n IsColimit.desc H\n (aux e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)))\n e (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n[PROOFSTEP]\nintro W e h m hm\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\n\u22a2 m =\n (fun {W} e h =>\n IsColimit.desc H\n (aux e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)))\n e (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)\n[PROOFSTEP]\napply H.uniq (aux e h)\n[GOAL]\ncase x\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\n\u22a2 \u2200 (j : FullSubcategory fun f => (Sieve.generateFamily X \u03c0).arrows f.hom),\n NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9 j \u226b m =\n NatTrans.app\n (aux e\n (_ :\n \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)).\u03b9\n j\n[PROOFSTEP]\nrintro \u27e8T, a, (g : T.left \u27f6 _), ha\u27e9\n[GOAL]\ncase x.mk.intro.intro\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nha : g \u226b \u03c0 a = T.hom\n\u22a2 NatTrans.app (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows).\u03b9\n { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) } \u226b\n m =\n NatTrans.app\n (aux e\n (_ : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082)).\u03b9\n { obj := T, property := (_ : \u2203 a g, g \u226b \u03c0 a = T.hom) }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase x.mk.intro.intro\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nha : g \u226b \u03c0 a = T.hom\n\u22a2 T.hom \u226b m =\n Exists.choose (_ : \u2203 g_1, g_1 \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom)) = T.hom) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom))\n[PROOFSTEP]\nnth_rewrite 1 [\u2190 ha, Category.assoc, hm]\n[GOAL]\ncase x.mk.intro.intro\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nha : g \u226b \u03c0 a = T.hom\n\u22a2 g \u226b e a =\n Exists.choose (_ : \u2203 g_1, g_1 \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom)) = T.hom) \u226b\n e (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom))\n[PROOFSTEP]\napply h\n[GOAL]\ncase x.mk.intro.intro.a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nha : g \u226b \u03c0 a = T.hom\n\u22a2 g \u226b \u03c0 a =\n Exists.choose (_ : \u2203 g_1, g_1 \u226b \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom)) = T.hom) \u226b\n \u03c0 (Exists.choose (_ : \u2203 a g, g \u226b \u03c0 a = T.hom))\n[PROOFSTEP]\ngeneralize_proofs h1 h2\n[GOAL]\ncase x.mk.intro.intro.a\nC : Type u_1\ninst\u271d : Category.{?u.74817, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nH : IsColimit (Presieve.cocone (Sieve.generateFamily X \u03c0).arrows)\naux : {W : C} \u2192\n (e : (a : \u03b1) \u2192 X a \u27f6 W) \u2192\n (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082) \u2192\n Cocone (Presieve.diagram (Sieve.generateFamily X \u03c0).arrows) :=\n fun {W} e h =>\n { pt := W,\n \u03b9 :=\n NatTrans.mk fun x =>\n match x with\n | { obj := T, property := hT } =>\n Exists.choose (_ : \u2203 g, g \u226b \u03c0 (Exists.choose hT) = T.hom) \u226b e (Exists.choose hT) }\nW : C\ne : (a : \u03b1) \u2192 X a \u27f6 W\nh : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082), g\u2081 \u226b \u03c0 a\u2081 = g\u2082 \u226b \u03c0 a\u2082 \u2192 g\u2081 \u226b e a\u2081 = g\u2082 \u226b e a\u2082\nm : B \u27f6 W\nhm : \u2200 (a : \u03b1), \u03c0 a \u226b m = e a\nT : Over B\na : \u03b1\ng : T.left \u27f6 X a\nha : g \u226b \u03c0 a = T.hom\nh1 : \u2203 a g, g \u226b \u03c0 a = T.hom\nh2 : \u2203 g, g \u226b \u03c0 (Exists.choose h1) = T.hom\n\u22a2 g \u226b \u03c0 a = Exists.choose h2 \u226b \u03c0 (Exists.choose h1)\n[PROOFSTEP]\nrwa [h2.choose_spec]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.ofArrows X \u03c0) \u2194 EffectiveEpiFamily X \u03c0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.ofArrows X \u03c0) \u2192 EffectiveEpiFamily X \u03c0\n[PROOFSTEP]\nintro (h : Nonempty _)\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X \u03c0)).arrows))\n\u22a2 EffectiveEpiFamily X \u03c0\n[PROOFSTEP]\nrw [Sieve.generateFamily_eq] at h \n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X \u03c0).arrows))\n\u22a2 EffectiveEpiFamily X \u03c0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.effectiveEpiFamily\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (IsColimit (Presieve.cocone (generateFamily X \u03c0).arrows))\n\u22a2 Nonempty (EffectiveEpiFamilyStruct X \u03c0)\n[PROOFSTEP]\napply Nonempty.map (effectiveEpiFamilyStructOfIsColimit _ _) h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\n\u22a2 EffectiveEpiFamily X \u03c0 \u2192 Presieve.EffectiveEpimorphic (Presieve.ofArrows X \u03c0)\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (EffectiveEpiFamilyStruct X \u03c0)\n\u22a2 Presieve.EffectiveEpimorphic (Presieve.ofArrows X \u03c0)\n[PROOFSTEP]\nshow Nonempty _\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (EffectiveEpiFamilyStruct X \u03c0)\n\u22a2 Nonempty (IsColimit (Presieve.cocone (generate (Presieve.ofArrows X \u03c0)).arrows))\n[PROOFSTEP]\nrw [Sieve.generateFamily_eq]\n[GOAL]\ncase mpr.mk\nC : Type u_1\ninst\u271d : Category.{u_3, u_1} C\nB : C\n\u03b1 : Type u_2\nX : \u03b1 \u2192 C\n\u03c0 : (a : \u03b1) \u2192 X a \u27f6 B\nh : Nonempty (EffectiveEpiFamilyStruct X \u03c0)\n\u22a2 Nonempty (IsColimit (Presieve.cocone (generateFamily X \u03c0).arrows))\n[PROOFSTEP]\napply Nonempty.map (isColimitOfEffectiveEpiFamilyStruct _ _) h\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.EffectiveEpimorphic", "llama_tokens": 55249, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370308082623216, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.28135931178618534}} {"text": "[GOAL]\nB : Type u\u2081\ninst\u271d\u2075 : Quiver B\ninst\u271d\u2074 : (a b : B) \u2192 Quiver (a \u27f6 b)\nC : Type u\u2082\ninst\u271d\u00b3 : Quiver C\ninst\u271d\u00b2 : (a b : C) \u2192 Quiver (a \u27f6 b)\nD : Type u\u2083\ninst\u271d\u00b9 : Quiver D\ninst\u271d : (a b : D) \u2192 Quiver (a \u27f6 b)\nF\u271d F : PrelaxFunctor B C\nG : PrelaxFunctor C D\nsrc\u271d : B \u2964q D := \u2191F \u22d9q \u2191G\na\u271d b\u271d : B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 g\u271d\n\u22a2 { obj := src\u271d.obj, map := fun {X Y} => src\u271d.map }.map f\u271d \u27f6 { obj := src\u271d.obj, map := fun {X Y} => src\u271d.map }.map g\u271d\n[PROOFSTEP]\nexact G.map\u2082 (F.map\u2082 \u03b7)\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na : B\n\u22a2 (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map (\ud835\udfd9 a) \u27f6\n \ud835\udfd9 ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).obj a)\n[PROOFSTEP]\nexact (G.mapFunctor _ _).map (F.mapId a) \u226b G.mapId (F.obj a)\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\n\u22a2 (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map (f \u226b g) \u27f6\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map g\n[PROOFSTEP]\nexact (G.mapFunctor _ _).map (F.mapComp f g) \u226b G.mapComp (F.map f) (F.map g)\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d f'\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 f'\u271d\ng : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b7 \u25b7 g) \u226b\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n f'\u271d g =\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n f\u271d g \u226b\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7 \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d f'\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 f'\u271d\ng : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b7 \u25b7 g)) \u226b\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f'\u271d g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f'\u271d) ((\u2191F.toPrelaxFunctor).map g) =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f\u271d g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f\u271d) ((\u2191F.toPrelaxFunctor).map g)) \u226b\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7) \u25b7\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map g)\n[PROOFSTEP]\nrw [\u2190 map\u2082_comp_assoc, mapComp_naturality_left, map\u2082_comp_assoc, mapComp_naturality_left, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\n\u03b7 : a\u271d \u27f6 b\u271d\n\u22a2 \u2200 {g g' : b\u271d \u27f6 c\u271d} (\u03b7_1 : g \u27f6 g'),\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b7 \u25c1 \u03b7_1) \u226b\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n \u03b7 g' =\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n \u03b7 g \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map \u03b7 \u25c1\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7_1\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\n\u03b7 : a\u271d \u27f6 b\u271d\n\u22a2 \u2200 {g g' : b\u271d \u27f6 c\u271d} (\u03b7_1 : g \u27f6 g'),\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b7 \u25c1 \u03b7_1)) \u226b\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F \u03b7 g') \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map \u03b7) ((\u2191F.toPrelaxFunctor).map g') =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F \u03b7 g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map \u03b7) ((\u2191F.toPrelaxFunctor).map g)) \u226b\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map \u03b7) \u25c1\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7_1)\n[PROOFSTEP]\nintros\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\n\u03b7 : a\u271d \u27f6 b\u271d\ng\u271d g'\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : g\u271d \u27f6 g'\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b7 \u25c1 \u03b7\u271d)) \u226b\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F \u03b7 g'\u271d) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map \u03b7) ((\u2191F.toPrelaxFunctor).map g'\u271d) =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F \u03b7 g\u271d) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map \u03b7) ((\u2191F.toPrelaxFunctor).map g\u271d)) \u226b\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map \u03b7) \u25c1\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7\u271d)\n[PROOFSTEP]\nrw [\u2190 map\u2082_comp_assoc, mapComp_naturality_right, map\u2082_comp_assoc, mapComp_naturality_right, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b1_ f g h).hom \u226b\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n f (g \u226b h) \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n g h =\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n (f \u226b g) h \u226b\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n f g \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h \u226b\n (\u03b1_ ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map g)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h)).hom\n[PROOFSTEP]\ndsimp\n -- porting note: if you use the `map\u2082_associator_aux` hack in the definition of\n -- `map\u2082_associator` then the `simp only` call below does not seem to apply `map\u2082_associator`\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b1_ f g h).hom) \u226b\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f (g \u226b h)) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map (g \u226b h))) \u226b\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f) \u25c1\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F g h) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)) =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F (f \u226b g) h) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map (f \u226b g)) ((\u2191F.toPrelaxFunctor).map h)) \u226b\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g)) \u25b7\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map h) \u226b\n (\u03b1_ ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f))\n ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map g))\n ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map h))).hom\n[PROOFSTEP]\nsimp only [map\u2082_associator, \u2190 map\u2082_comp_assoc, \u2190 mapComp_naturality_right_assoc, whiskerLeft_comp, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor\n (mapComp F (f \u226b g) h \u226b\n mapComp F f g \u25b7 (\u2191F.toPrelaxFunctor).map h \u226b\n (\u03b1_ ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)).hom) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g \u226b (\u2191F.toPrelaxFunctor).map h) \u226b\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f) \u25c1\n mapComp G ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h) =\n PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F (f \u226b g) h) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map (f \u226b g)) ((\u2191F.toPrelaxFunctor).map h) \u226b\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g)) \u25b7\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map h) \u226b\n (\u03b1_ ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f))\n ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map g))\n ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map h))).hom\n[PROOFSTEP]\nsimp only [map\u2082_associator, map\u2082_comp, mapComp_naturality_left_assoc, comp_whiskerRight, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d : B\nf : a\u271d \u27f6 b\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03bb_ f).hom =\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n (\ud835\udfd9 a\u271d) f \u226b\n (fun a =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj a)).map (mapId F a) \u226b\n mapId G ((\u2191F.toPrelaxFunctor).obj a))\n a\u271d \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u226b\n (\u03bb_ ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d : B\nf : a\u271d \u27f6 b\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03bb_ f).hom) =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F (\ud835\udfd9 a\u271d) f) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map (\ud835\udfd9 a\u271d)) ((\u2191F.toPrelaxFunctor).map f)) \u226b\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapId F a\u271d) \u226b mapId G ((\u2191F.toPrelaxFunctor).obj a\u271d)) \u25b7\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f) \u226b\n (\u03bb_ ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f))).hom\n[PROOFSTEP]\nsimp only [map\u2082_leftUnitor, map\u2082_comp, mapComp_naturality_left_assoc, comp_whiskerRight, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d : B\nf : a\u271d \u27f6 b\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03c1_ f).hom =\n (fun {a b c} f g =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj c)).map (mapComp F f g) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g))\n f (\ud835\udfd9 b\u271d) \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n (fun a =>\n (mapFunctor G ((\u2191F.toPrelaxFunctor).obj a) ((\u2191F.toPrelaxFunctor).obj a)).map (mapId F a) \u226b\n mapId G ((\u2191F.toPrelaxFunctor).obj a))\n b\u271d \u226b\n (\u03c1_ ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d F : OplaxFunctor B C\nG : OplaxFunctor C D\nsrc\u271d : PrelaxFunctor B D := PrelaxFunctor.comp F.toPrelaxFunctor G.toPrelaxFunctor\na\u271d b\u271d : B\nf : a\u271d \u27f6 b\u271d\n\u22a2 PrelaxFunctor.map\u2082 G.toPrelaxFunctor (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03c1_ f).hom) =\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapComp F f (\ud835\udfd9 b\u271d)) \u226b\n mapComp G ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map (\ud835\udfd9 b\u271d))) \u226b\n (\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f) \u25c1\n (PrelaxFunctor.map\u2082 G.toPrelaxFunctor (mapId F b\u271d) \u226b mapId G ((\u2191F.toPrelaxFunctor).obj b\u271d)) \u226b\n (\u03c1_ ((\u2191G.toPrelaxFunctor).map ((\u2191F.toPrelaxFunctor).map f))).hom\n[PROOFSTEP]\nsimp only [map\u2082_rightUnitor, map\u2082_comp, mapComp_naturality_right_assoc, whiskerLeft_comp, assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf : a\u271d \u27f6 b\u271d\ng h : b\u271d \u27f6 c\u271d\n\u03b7 : g \u27f6 h\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (f \u25c1 \u03b7) =\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') f g).hom \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7 \u226b\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') f h).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf : a\u271d \u27f6 b\u271d\ng h : b\u271d \u27f6 c\u271d\n\u03b7 : g \u27f6 h\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (f \u25c1 \u03b7) =\n (OplaxFunctor.PseudoCore.mapCompIso F' f g).hom \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7 \u226b\n (OplaxFunctor.PseudoCore.mapCompIso F' f h).inv\n[PROOFSTEP]\nrw [F'.mapCompIso_hom f g, \u2190 F.mapComp_naturality_right_assoc, \u2190 F'.mapCompIso_hom f h, hom_inv_id, comp_id]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 g\u271d\nh : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b7 \u25b7 h) =\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') f\u271d h).hom \u226b\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7 \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h \u226b\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') g\u271d h).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 g\u271d\nh : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b7 \u25b7 h) =\n (OplaxFunctor.PseudoCore.mapCompIso F' f\u271d h).hom \u226b\n PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7 \u25b7 (\u2191F.toPrelaxFunctor).map h \u226b\n (OplaxFunctor.PseudoCore.mapCompIso F' g\u271d h).inv\n[PROOFSTEP]\nrw [F'.mapCompIso_hom _ h, \u2190 F.mapComp_naturality_left_assoc, \u2190 F'.mapCompIso_hom _ h, hom_inv_id, comp_id]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b1_ f g h).hom =\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') (f \u226b g) h).hom \u226b\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') f g).hom \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h \u226b\n (\u03b1_ ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map g)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h)).hom \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') g h).inv \u226b\n ((fun {a b c} => OplaxFunctor.PseudoCore.mapCompIso F') f (g \u226b h)).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u00b2 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b9 : Bicategory C\nD : Type u\u2083\ninst\u271d : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\nF' : OplaxFunctor.PseudoCore F\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b1_ f g h).hom =\n (OplaxFunctor.PseudoCore.mapCompIso F' (f \u226b g) h).hom \u226b\n (OplaxFunctor.PseudoCore.mapCompIso F' f g).hom \u25b7 (\u2191F.toPrelaxFunctor).map h \u226b\n (\u03b1_ ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)).hom \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 (OplaxFunctor.PseudoCore.mapCompIso F' g h).inv \u226b\n (OplaxFunctor.PseudoCore.mapCompIso F' f (g \u226b h)).inv\n[PROOFSTEP]\nrw [F'.mapCompIso_hom (f \u226b g) h, F'.mapCompIso_hom f g, \u2190 F.map\u2082_associator_assoc, \u2190 F'.mapCompIso_hom f (g \u226b h), \u2190\n F'.mapCompIso_hom g h, hom_inv_whiskerLeft_assoc, hom_inv_id, comp_id]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf : a\u271d \u27f6 b\u271d\ng h : b\u271d \u27f6 c\u271d\n\u03b7 : g \u27f6 h\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (f \u25c1 \u03b7) =\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) f g).hom \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7 \u226b\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) f h).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf : a\u271d \u27f6 b\u271d\ng h : b\u271d \u27f6 c\u271d\n\u03b7 : g \u27f6 h\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (f \u25c1 \u03b7) =\n OplaxFunctor.mapComp F f g \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7 \u226b inv (OplaxFunctor.mapComp F f h)\n[PROOFSTEP]\nrw [\u2190 assoc, IsIso.eq_comp_inv, F.mapComp_naturality_right]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 g\u271d\nh : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b7 \u25b7 h) =\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) f\u271d h).hom \u226b\n PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } \u03b7 \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h \u226b\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) g\u271d h).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d : B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : f\u271d \u27f6 g\u271d\nh : b\u271d \u27f6 c\u271d\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b7 \u25b7 h) =\n OplaxFunctor.mapComp F f\u271d h \u226b\n PrelaxFunctor.map\u2082 F.toPrelaxFunctor \u03b7 \u25b7 (\u2191F.toPrelaxFunctor).map h \u226b inv (OplaxFunctor.mapComp F g\u271d h)\n[PROOFSTEP]\nrw [\u2190 assoc, IsIso.eq_comp_inv, F.mapComp_naturality_left]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 { toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d } (\u03b1_ f g h).hom =\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) (f \u226b g) h).hom \u226b\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) f g).hom \u25b7\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h \u226b\n (\u03b1_ ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map g)\n ((\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map h)).hom \u226b\n (\u2191{ toPrefunctor := \u2191src\u271d, map\u2082 := fun {a b} {f g} => PrelaxFunctor.map\u2082 src\u271d }).map f \u25c1\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) g h).inv \u226b\n ((fun {a b c} f g => asIso (OplaxFunctor.mapComp F f g)) f (g \u226b h)).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b1_ f g h).hom =\n OplaxFunctor.mapComp F (f \u226b g) h \u226b\n OplaxFunctor.mapComp F f g \u25b7 (\u2191F.toPrelaxFunctor).map h \u226b\n (\u03b1_ ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)).hom \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 inv (OplaxFunctor.mapComp F g h) \u226b inv (OplaxFunctor.mapComp F f (g \u226b h))\n[PROOFSTEP]\nsimp only [\u2190 assoc]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b1_ f g h).hom =\n (((OplaxFunctor.mapComp F (f \u226b g) h \u226b OplaxFunctor.mapComp F f g \u25b7 (\u2191F.toPrelaxFunctor).map h) \u226b\n (\u03b1_ ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)).hom) \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 inv (OplaxFunctor.mapComp F g h)) \u226b\n inv (OplaxFunctor.mapComp F f (g \u226b h))\n[PROOFSTEP]\nrw [IsIso.eq_comp_inv, \u2190 inv_whiskerLeft, IsIso.eq_comp_inv]\n[GOAL]\nB : Type u\u2081\ninst\u271d\u2074 : Bicategory B\nC : Type u\u2082\ninst\u271d\u00b3 : Bicategory C\nD : Type u\u2083\ninst\u271d\u00b2 : Bicategory D\nF\u271d : Pseudofunctor B C\nF : OplaxFunctor B C\ninst\u271d\u00b9 : \u2200 (a : B), IsIso (OplaxFunctor.mapId F a)\ninst\u271d : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c), IsIso (OplaxFunctor.mapComp F f g)\nsrc\u271d : PrelaxFunctor B C := F.toPrelaxFunctor\na\u271d b\u271d c\u271d d\u271d : B\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 (PrelaxFunctor.map\u2082 F.toPrelaxFunctor (\u03b1_ f g h).hom \u226b OplaxFunctor.mapComp F f (g \u226b h)) \u226b\n (\u2191F.toPrelaxFunctor).map f \u25c1 OplaxFunctor.mapComp F g h =\n (OplaxFunctor.mapComp F (f \u226b g) h \u226b OplaxFunctor.mapComp F f g \u25b7 (\u2191F.toPrelaxFunctor).map h) \u226b\n (\u03b1_ ((\u2191F.toPrelaxFunctor).map f) ((\u2191F.toPrelaxFunctor).map g) ((\u2191F.toPrelaxFunctor).map h)).hom\n[PROOFSTEP]\nsimp only [assoc, F.map\u2082_associator]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.Functor", "llama_tokens": 13398, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.28134441761855017}} {"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf g : CentroidHom \u03b1\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\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\ng : CentroidHom \u03b1\ntoAddMonoidHom\u271d : \u03b1 \u2192+ \u03b1\nmap_mul_left'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = a * ZeroHom.toFun (\u2191toAddMonoidHom\u271d) b\nmap_mul_right'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = ZeroHom.toFun (\u2191toAddMonoidHom\u271d) a * b\nh :\n (fun f => f.toFun)\n { toAddMonoidHom := toAddMonoidHom\u271d, map_mul_left' := map_mul_left'\u271d, map_mul_right' := map_mul_right'\u271d } =\n (fun f => f.toFun) g\n\u22a2 { toAddMonoidHom := toAddMonoidHom\u271d, map_mul_left' := map_mul_left'\u271d, map_mul_right' := map_mul_right'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\ntoAddMonoidHom\u271d\u00b9 : \u03b1 \u2192+ \u03b1\nmap_mul_left'\u271d\u00b9 : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) (a * b) = a * ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) b\nmap_mul_right'\u271d\u00b9 : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) (a * b) = ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) a * b\ntoAddMonoidHom\u271d : \u03b1 \u2192+ \u03b1\nmap_mul_left'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = a * ZeroHom.toFun (\u2191toAddMonoidHom\u271d) b\nmap_mul_right'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = ZeroHom.toFun (\u2191toAddMonoidHom\u271d) a * b\nh :\n (fun f => f.toFun)\n { toAddMonoidHom := toAddMonoidHom\u271d\u00b9, map_mul_left' := map_mul_left'\u271d\u00b9, map_mul_right' := map_mul_right'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toAddMonoidHom := toAddMonoidHom\u271d, map_mul_left' := map_mul_left'\u271d, map_mul_right' := map_mul_right'\u271d }\n\u22a2 { toAddMonoidHom := toAddMonoidHom\u271d\u00b9, map_mul_left' := map_mul_left'\u271d\u00b9, map_mul_right' := map_mul_right'\u271d\u00b9 } =\n { toAddMonoidHom := toAddMonoidHom\u271d, map_mul_left' := map_mul_left'\u271d, map_mul_right' := map_mul_right'\u271d }\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase mk.mk.e_toAddMonoidHom.h\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\ntoAddMonoidHom\u271d\u00b9 : \u03b1 \u2192+ \u03b1\nmap_mul_left'\u271d\u00b9 : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) (a * b) = a * ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) b\nmap_mul_right'\u271d\u00b9 : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) (a * b) = ZeroHom.toFun (\u2191toAddMonoidHom\u271d\u00b9) a * b\ntoAddMonoidHom\u271d : \u03b1 \u2192+ \u03b1\nmap_mul_left'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = a * ZeroHom.toFun (\u2191toAddMonoidHom\u271d) b\nmap_mul_right'\u271d : \u2200 (a b : \u03b1), ZeroHom.toFun (\u2191toAddMonoidHom\u271d) (a * b) = ZeroHom.toFun (\u2191toAddMonoidHom\u271d) a * b\nh :\n (fun f => f.toFun)\n { toAddMonoidHom := toAddMonoidHom\u271d\u00b9, map_mul_left' := map_mul_left'\u271d\u00b9, map_mul_right' := map_mul_right'\u271d\u00b9 } =\n (fun f => f.toFun)\n { toAddMonoidHom := toAddMonoidHom\u271d, map_mul_left' := map_mul_left'\u271d, map_mul_right' := map_mul_right'\u271d }\nx : \u03b1\n\u22a2 \u2191toAddMonoidHom\u271d\u00b9 x = \u2191toAddMonoidHom\u271d x\n[PROOFSTEP]\nexact congrFun h x\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nf' : \u03b1 \u2192 \u03b1\nh : f' = \u2191f\nsrc\u271d : \u03b1 \u2192+ \u03b1 := AddMonoidHom.copy f.toAddMonoidHom f' h\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nsimp_rw [h, map_mul_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nf' : \u03b1 \u2192 \u03b1\nh : f' = \u2191f\nsrc\u271d : \u03b1 \u2192+ \u03b1 := AddMonoidHom.copy f.toAddMonoidHom f' h\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (\u2191src\u271d) 0 = 0) },\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nsimp_rw [h, map_mul_right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\ng f\u2081 f\u2082 : CentroidHom \u03b1\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\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f + \u2191g\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nshow f (a * b) + g (a * b) = a * (f b + g b)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f + \u2191g\na b : \u03b1\n\u22a2 \u2191f (a * b) + \u2191g (a * b) = a * (\u2191f b + \u2191g b)\n[PROOFSTEP]\nsimp [map_mul_left, mul_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f + \u2191g\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nshow f (a * b) + g (a * b) = (f a + g a) * b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f + \u2191g\na b : \u03b1\n\u22a2 \u2191f (a * b) + \u2191g (a * b) = (\u2191f a + \u2191g a) * b\n[PROOFSTEP]\nsimp [map_mul_right, add_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nn : \u2115\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nchange n \u2022 f (a * b) = a * n \u2022 f b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nn : \u2115\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 n \u2022 \u2191f (a * b) = a * n \u2022 \u2191f b\n[PROOFSTEP]\nrw [map_mul_left f, \u2190 mul_smul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nn : \u2115\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nchange n \u2022 f (a * b) = n \u2022 f a * b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nn : \u2115\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 n \u2022 \u2191f (a * b) = n \u2022 \u2191f a * b\n[PROOFSTEP]\nrw [map_mul_right f, \u2190 smul_mul_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn : \u2115\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ n\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\na b : \u03b1\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.zero\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 \u2191(toEnd f ^ Nat.succ n) (a * b) = a * \u2191(toEnd f ^ Nat.succ n) b\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 \u2191(toEnd f * toEnd f ^ n) (a * b) = a * \u2191(toEnd f * toEnd f ^ n) b\n[PROOFSTEP]\nexact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn : \u2115\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ n\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\na b : \u03b1\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.zero\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n a *\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n a *\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 \u2191(toEnd f ^ Nat.succ n) (a * b) = \u2191(toEnd f ^ Nat.succ n) a * b\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nf : CentroidHom \u03b1\nn\u271d : \u2115\nsrc\u271d\u00b9 : AddMonoid.End \u03b1 := toEnd f ^ n\u271d\na b : \u03b1\nn : \u2115\nih :\n let src := toEnd f ^ n;\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) })\n a *\n b\nsrc\u271d : AddMonoid.End \u03b1 := toEnd f ^ Nat.succ n\n\u22a2 \u2191(toEnd f * toEnd f ^ n) (a * b) = \u2191(toEnd f * toEnd f ^ n) a * b\n[PROOFSTEP]\nexact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nT S : CentroidHom \u03b1\na b : \u03b1\n\u22a2 (\u2191T \u2218 \u2191S) (a * b) = (\u2191S \u2218 \u2191T) (a * b)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nT S : CentroidHom \u03b1\na b : \u03b1\n\u22a2 \u2191T (\u2191S (a * b)) = \u2191S (\u2191T (a * b))\n[PROOFSTEP]\nrw [map_mul_right, map_mul_left, \u2190 map_mul_right, \u2190 map_mul_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := -\u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nchange -f (a * b) = a * (-f b)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := -\u2191f\na b : \u03b1\n\u22a2 -\u2191f (a * b) = a * -\u2191f b\n[PROOFSTEP]\nsimp [map_mul_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := -\u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nchange -f (a * b) = (-f a) * b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := -\u2191f\na b : \u03b1\n\u22a2 -\u2191f (a * b) = -\u2191f a * b\n[PROOFSTEP]\nsimp [map_mul_right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f - \u2191g\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nchange (FunLike.coe f - FunLike.coe g) (a * b) = a * (FunLike.coe f - FunLike.coe g) b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f - \u2191g\na b : \u03b1\n\u22a2 (\u2191f - \u2191g) (a * b) = a * (\u2191f - \u2191g) b\n[PROOFSTEP]\nsimp [map_mul_left, mul_sub]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f - \u2191g\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nchange (FunLike.coe f - FunLike.coe g) (a * b) = ((FunLike.coe f - FunLike.coe g) a) * b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nf g : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := \u2191f - \u2191g\na b : \u03b1\n\u22a2 (\u2191f - \u2191g) (a * b) = (\u2191f - \u2191g) a * b\n[PROOFSTEP]\nsimp [map_mul_right, sub_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nn : \u2124\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n a *\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n b\n[PROOFSTEP]\nchange n \u2022 f (a * b) = a * n \u2022 f b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nn : \u2124\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 n \u2022 \u2191f (a * b) = a * n \u2022 \u2191f b\n[PROOFSTEP]\nrw [map_mul_left f, \u2190 mul_smul_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nn : \u2124\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n (a * b) =\n ZeroHom.toFun\n (\u2191{ toZeroHom := \u2191src\u271d,\n map_add' :=\n (_ : \u2200 (x y : \u03b1), ZeroHom.toFun (\u2191src\u271d) (x + y) = ZeroHom.toFun (\u2191src\u271d) x + ZeroHom.toFun (\u2191src\u271d) y) })\n a *\n b\n[PROOFSTEP]\nchange n \u2022 f (a * b) = n \u2022 f a * b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalNonAssocRing \u03b1\nn : \u2124\nf : CentroidHom \u03b1\nsrc\u271d : \u03b1 \u2192+ \u03b1 := SMul.smul n \u2191f\na b : \u03b1\n\u22a2 n \u2022 \u2191f (a * b) = n \u2022 \u2191f a * b\n[PROOFSTEP]\nrw [map_mul_right f, \u2190 smul_mul_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\nh : \u2200 (a b : \u03b1), (\u2200 (r : \u03b1), a * r * b = 0) \u2192 a = 0 \u2228 b = 0\nsrc\u271d : Ring (CentroidHom \u03b1) := instRing\nf g : CentroidHom \u03b1\n\u22a2 f * g = g * f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\nh : \u2200 (a b : \u03b1), (\u2200 (r : \u03b1), a * r * b = 0) \u2192 a = 0 \u2228 b = 0\nsrc\u271d : Ring (CentroidHom \u03b1) := instRing\nf g : CentroidHom \u03b1\na\u271d : \u03b1\n\u22a2 \u2191(f * g) a\u271d = \u2191(g * f) a\u271d\n[PROOFSTEP]\nrefine' sub_eq_zero.1 ((or_self_iff _).1 <| (h _ _) fun r \u21a6 _)\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\nh : \u2200 (a b : \u03b1), (\u2200 (r : \u03b1), a * r * b = 0) \u2192 a = 0 \u2228 b = 0\nsrc\u271d : Ring (CentroidHom \u03b1) := instRing\nf g : CentroidHom \u03b1\na\u271d r : \u03b1\n\u22a2 (\u2191(f * g) a\u271d - \u2191(g * f) a\u271d) * r * (\u2191(f * g) a\u271d - \u2191(g * f) a\u271d) = 0\n[PROOFSTEP]\nrw [mul_assoc, sub_mul, sub_eq_zero, \u2190 map_mul_right, \u2190 map_mul_right, coe_mul, coe_mul, comp_mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Centroid", "llama_tokens": 10593, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.3960681662740417, "lm_q1q2_score": 0.2808199081714713}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 a :: l \u2208 subchain s \u2194 a \u2208 s \u2227 l \u2208 subchain s \u2227 \u2200 (b : \u03b1), b \u2208 head? l \u2192 a < b\n[PROOFSTEP]\nsimp only [subchain, mem_setOf_eq, forall_mem_cons, chain'_cons', and_left_comm, and_comm, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 [a] \u2208 subchain s \u2194 a \u2208 s\n[PROOFSTEP]\nsimp [cons_mem_subchain_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\ncases' (le_top : s.chainHeight \u2264 \u22a4).eq_or_lt with ha ha\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : chainHeight s = \u22a4\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nrw [chainHeight_eq_iSup_subtype] at ha \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : chainHeight s < \u22a4\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nrw [chainHeight_eq_iSup_subtype] at ha \n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : \u2a06 (l : \u2191(subchain s)), \u2191(length \u2191l) = \u22a4\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nobtain \u27e8_, \u27e8\u27e8l, h\u2081, h\u2082\u27e9, rfl\u27e9, h\u2083\u27e9 := not_bddAbove_iff'.mp ((WithTop.iSup_coe_eq_top _).mp ha) n\n[GOAL]\ncase inl.intro.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : \u2a06 (l : \u2191(subchain s)), \u2191(length \u2191l) = \u22a4\nl : List \u03b1\nh\u2081 : Chain' (fun x x_1 => x < x_1) l\nh\u2082 : \u2200 (i : \u03b1), i \u2208 l \u2192 i \u2208 s\nh\u2083 :\n \u00ac(fun x => \u2191(length \u2191x)) { val := l, property := (_ : Chain' (fun x x_1 => x < x_1) l \u2227 \u2200 (i : \u03b1), i \u2208 l \u2192 i \u2208 s) } \u2264\n n\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nexact\n \u27e8l.take n, \u27e8h\u2081.take _, fun x h \u21a6 h\u2082 _ <| take_subset _ _ h\u27e9,\n (l.length_take n).trans <| min_eq_left <| le_of_not_ge h\u2083\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : \u2a06 (l : \u2191(subchain s)), \u2191(length \u2191l) < \u22a4\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nrw [ENat.iSup_coe_lt_top] at ha \n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : BddAbove (range fun l => length \u2191l)\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nobtain \u27e8\u27e8l, h\u2081, h\u2082\u27e9, e : l.length = _\u27e9 := Nat.sSup_mem (Set.range_nonempty _) ha\n[GOAL]\ncase inr.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : BddAbove (range fun l => length \u2191l)\nl : List \u03b1\nh\u2081 : Chain' (fun x x_1 => x < x_1) l\nh\u2082 : \u2200 (i : \u03b1), i \u2208 l \u2192 i \u2208 s\ne : length l = sSup (range fun l => length \u2191l)\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nrefine' \u27e8l.take n, \u27e8h\u2081.take _, fun x h \u21a6 h\u2082 _ <| take_subset _ _ h\u27e9, (l.length_take n).trans <| min_eq_left <| _\u27e9\n[GOAL]\ncase inr.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nn : \u2115\nhn : \u2191n \u2264 chainHeight s\nha : BddAbove (range fun l => length \u2191l)\nl : List \u03b1\nh\u2081 : Chain' (fun x x_1 => x < x_1) l\nh\u2082 : \u2200 (i : \u03b1), i \u2208 l \u2192 i \u2208 s\ne : length l = sSup (range fun l => length \u2191l)\n\u22a2 n \u2264 length l\n[PROOFSTEP]\nrwa [e, \u2190 Nat.cast_le (\u03b1 := \u2115\u221e), sSup_range, ENat.coe_iSup ha, \u2190 chainHeight_eq_iSup_subtype]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\n\u22a2 TFAE [\u2191n \u2264 chainHeight s, \u2203 l, l \u2208 subchain s \u2227 length l = n, \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nexact s.exists_chain_of_le_chainHeight\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\n\u22a2 TFAE [\u2191n \u2264 chainHeight s, \u2203 l, l \u2208 subchain s \u2227 length l = n, \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l]\n[PROOFSTEP]\ntfae_have 2 \u2192 3\n[GOAL]\ncase tfae_2_to_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\n\u22a2 (\u2203 l, l \u2208 subchain s \u2227 length l = n) \u2192 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\n[PROOFSTEP]\nrintro \u27e8l, hls, he\u27e9\n[GOAL]\ncase tfae_2_to_3.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\nl : List \u03b1\nhls : l \u2208 subchain s\nhe : length l = n\n\u22a2 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\n[PROOFSTEP]\nexact \u27e8l, hls, he.ge\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\ntfae_2_to_3 : (\u2203 l, l \u2208 subchain s \u2227 length l = n) \u2192 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\n\u22a2 TFAE [\u2191n \u2264 chainHeight s, \u2203 l, l \u2208 subchain s \u2227 length l = n, \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l]\n[PROOFSTEP]\ntfae_have 3 \u2192 1\n[GOAL]\ncase tfae_3_to_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\ntfae_2_to_3 : (\u2203 l, l \u2208 subchain s \u2227 length l = n) \u2192 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\n\u22a2 (\u2203 l, l \u2208 subchain s \u2227 n \u2264 length l) \u2192 \u2191n \u2264 chainHeight s\n[PROOFSTEP]\nrintro \u27e8l, hs, hn\u27e9\n[GOAL]\ncase tfae_3_to_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\ntfae_2_to_3 : (\u2203 l, l \u2208 subchain s \u2227 length l = n) \u2192 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\nl : List \u03b1\nhs : l \u2208 subchain s\nhn : n \u2264 length l\n\u22a2 \u2191n \u2264 chainHeight s\n[PROOFSTEP]\nexact le_iSup\u2082_of_le l hs (WithTop.coe_le_coe.2 hn)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn : \u2115\ntfae_1_to_2 : \u2191n \u2264 chainHeight s \u2192 \u2203 l, l \u2208 subchain s \u2227 length l = n\ntfae_2_to_3 : (\u2203 l, l \u2208 subchain s \u2227 length l = n) \u2192 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l\ntfae_3_to_1 : (\u2203 l, l \u2208 subchain s \u2227 n \u2264 length l) \u2192 \u2191n \u2264 chainHeight s\n\u22a2 TFAE [\u2191n \u2264 chainHeight s, \u2203 l, l \u2208 subchain s \u2227 length l = n, \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 chainHeight s = \u22a4 \u2194 \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\n[PROOFSTEP]\nrefine' \u27e8fun h n \u21a6 le_chainHeight_iff.1 (le_top.trans_eq h.symm), fun h \u21a6 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\n\u22a2 chainHeight s = \u22a4\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nh : chainHeight s \u2260 \u22a4\n\u22a2 \u2203 n, \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 length l \u2260 n\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := WithTop.ne_top_iff_exists.1 h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nh : chainHeight s \u2260 \u22a4\nn : \u2115\nhn : \u2191n = chainHeight s\n\u22a2 \u2203 n, \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 length l \u2260 n\n[PROOFSTEP]\nexact\n \u27e8n + 1, fun l hs \u21a6\n (Nat.lt_succ_iff.2 <| Nat.cast_le.1 <| (length_le_chainHeight_of_mem_subchain hs).trans_eq hn.symm).ne\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 1 \u2264 chainHeight s \u2194 Set.Nonempty s\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, Set.le_chainHeight_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 (\u2203 l, l \u2208 subchain s \u2227 length l = 1) \u2194 Set.Nonempty s\n[PROOFSTEP]\nsimp only [length_eq_one, @and_comm (_ \u2208 _), @eq_comm _ _ [_], exists_exists_eq_and, singleton_mem_subchain_iff,\n Set.Nonempty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 chainHeight s = 0 \u2194 s = \u2205\n[PROOFSTEP]\nrw [\u2190 not_iff_not, \u2190 Ne.def, \u2190 ENat.one_le_iff_ne_zero, one_le_chainHeight_iff, nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\nn m : \u2115\n\u22a2 \u2191n \u2264 chainHeight s + \u2191m \u2194 \u2203 l, l \u2208 subchain s \u2227 n \u2264 length l + m\n[PROOFSTEP]\nsimp_rw [\u2190 tsub_le_iff_right, \u2190 ENat.coe_sub, (le_chainHeight_TFAE s (n - m)).out 0 2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m \u2194\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\n[PROOFSTEP]\nrefine'\n \u27e8fun e l h \u21a6 le_chainHeight_add_nat_iff.1 ((add_le_add_right (length_le_chainHeight_of_mem_subchain h) _).trans e),\n fun H \u21a6 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nby_cases s.chainHeight = \u22a4\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nby_cases s.chainHeight = \u22a4\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : chainHeight s = \u22a4\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nsuffices t.chainHeight = \u22a4 by\n rw [this, top_add]\n exact le_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : chainHeight s = \u22a4\nthis : chainHeight t = \u22a4\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nrw [this, top_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : chainHeight s = \u22a4\nthis : chainHeight t = \u22a4\n\u22a2 chainHeight s + \u2191n \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : chainHeight s = \u22a4\n\u22a2 chainHeight t = \u22a4\n[PROOFSTEP]\nrw [chainHeight_eq_top_iff] at h \u22a2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\n\u22a2 \u2200 (n : \u2115), \u2203 l, l \u2208 subchain t \u2227 length l = n\n[PROOFSTEP]\nintro k\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\nk : \u2115\n\u22a2 \u2203 l, l \u2208 subchain t \u2227 length l = k\n[PROOFSTEP]\nhave := (le_chainHeight_TFAE t k).out 1 2\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\nk : \u2115\nthis : (\u2203 l, l \u2208 subchain t \u2227 length l = k) \u2194 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\n\u22a2 \u2203 l, l \u2208 subchain t \u2227 length l = k\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\nk : \u2115\nthis : (\u2203 l, l \u2208 subchain t \u2227 length l = k) \u2194 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\n\u22a2 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\n[PROOFSTEP]\nobtain \u27e8l, hs, hl\u27e9 := h (k + m)\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\nk : \u2115\nthis : (\u2203 l, l \u2208 subchain t \u2227 length l = k) \u2194 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\nl : List \u03b1\nhs : l \u2208 subchain s\nhl : length l = k + m\n\u22a2 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\n[PROOFSTEP]\nobtain \u27e8l', ht, hl'\u27e9 := H l hs\n[GOAL]\ncase pos.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u2200 (n : \u2115), \u2203 l, l \u2208 subchain s \u2227 length l = n\nk : \u2115\nthis : (\u2203 l, l \u2208 subchain t \u2227 length l = k) \u2194 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\nl : List \u03b1\nhs : l \u2208 subchain s\nhl : length l = k + m\nl' : List \u03b2\nht : l' \u2208 subchain t\nhl' : length l + n \u2264 length l' + m\n\u22a2 \u2203 l, l \u2208 subchain t \u2227 k \u2264 length l\n[PROOFSTEP]\nexact \u27e8l', ht, (add_le_add_iff_right m).1 <| _root_.trans (hl.symm.trans_le le_self_add) hl'\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u00acchainHeight s = \u22a4\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := WithTop.ne_top_iff_exists.1 h\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u00acchainHeight s = \u22a4\nk : \u2115\nhk : \u2191k = chainHeight s\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nobtain \u27e8l, hs, hl\u27e9 := le_chainHeight_iff.1 hk.le\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u00acchainHeight s = \u22a4\nk : \u2115\nhk : \u2191k = chainHeight s\nl : List \u03b1\nhs : l \u2208 subchain s\nhl : length l = k\n\u22a2 chainHeight s + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nrw [\u2190 hk, \u2190 hl]\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\nn m : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l + n \u2264 length l' + m\nh : \u00acchainHeight s = \u22a4\nk : \u2115\nhk : \u2191k = chainHeight s\nl : List \u03b1\nhs : l \u2208 subchain s\nhl : length l = k\n\u22a2 \u2191(length l) + \u2191n \u2264 chainHeight t + \u2191m\n[PROOFSTEP]\nexact le_chainHeight_add_nat_iff.2 (H l hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 TFAE\n [chainHeight s \u2264 chainHeight t, \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l = length l',\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l']\n[PROOFSTEP]\ntfae_have 1 \u2194 3\n[GOAL]\ncase tfae_1_iff_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 chainHeight s \u2264 chainHeight t \u2194 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n[PROOFSTEP]\nconvert \u2190 chainHeight_add_le_chainHeight_add s t 0 0\n[GOAL]\ncase h.e'_1.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 chainHeight s + \u21910 = chainHeight s\n[PROOFSTEP]\napply add_zero\n[GOAL]\ncase h.e'_1.h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 chainHeight t + \u21910 = chainHeight t\n[PROOFSTEP]\napply add_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\ntfae_1_iff_3 :\n chainHeight s \u2264 chainHeight t \u2194 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n\u22a2 TFAE\n [chainHeight s \u2264 chainHeight t, \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l = length l',\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l']\n[PROOFSTEP]\ntfae_have 2 \u2194 3\n[GOAL]\ncase tfae_2_iff_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\ntfae_1_iff_3 :\n chainHeight s \u2264 chainHeight t \u2194 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n\u22a2 (\u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l = length l') \u2194\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n[PROOFSTEP]\nrefine' forall\u2082_congr fun l hl \u21a6 _\n[GOAL]\ncase tfae_2_iff_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\ntfae_1_iff_3 :\n chainHeight s \u2264 chainHeight t \u2194 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 (\u2203 l', l' \u2208 subchain t \u2227 length l = length l') \u2194 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n[PROOFSTEP]\nsimp_rw [\u2190 (le_chainHeight_TFAE t l.length).out 1 2, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t\u271d : Set \u03b1\nl : List \u03b1\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\ntfae_1_iff_3 :\n chainHeight s \u2264 chainHeight t \u2194 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\ntfae_2_iff_3 :\n (\u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l = length l') \u2194\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l'\n\u22a2 TFAE\n [chainHeight s \u2264 chainHeight t, \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l = length l',\n \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain t \u2227 length l \u2264 length l']\n[PROOFSTEP]\ntfae_finish\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 chainHeight (f '' s) = chainHeight s\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 chainHeight (f '' s) \u2264 chainHeight s\n[PROOFSTEP]\nrw [chainHeight_le_chainHeight_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 chainHeight s \u2264 chainHeight (f '' s)\n[PROOFSTEP]\nrw [chainHeight_le_chainHeight_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l = length l'\n[PROOFSTEP]\nsuffices \u2200 l \u2208 (f '' s).subchain, \u2203 l' \u2208 s.subchain, map f l' = l\n by\n intro l hl\n obtain \u27e8l', h\u2081, rfl\u27e9 := this l hl\n exact \u27e8l', h\u2081, length_map _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nthis : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l\n\u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l = length l'\n[PROOFSTEP]\nintro l hl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nthis : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l\nl : List \u03b2\nhl : l \u2208 subchain (f '' s)\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 length l = length l'\n[PROOFSTEP]\nobtain \u27e8l', h\u2081, rfl\u27e9 := this l hl\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nthis : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l\nl' : List \u03b1\nh\u2081 : l' \u2208 subchain s\nhl : map f l' \u2208 subchain (f '' s)\n\u22a2 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 length (map f l') = length l'_1\n[PROOFSTEP]\nexact \u27e8l', h\u2081, length_map _ _\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l\n[PROOFSTEP]\nintro l\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b2\n\u22a2 l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l\n[PROOFSTEP]\ninduction' l with x xs hx\n[GOAL]\ncase a.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 [] \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = []\n[PROOFSTEP]\nexact fun _ \u21a6 \u27e8nil, \u27e8trivial, fun x h \u21a6 (not_mem_nil x h).elim\u27e9, rfl\u27e9\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b2\nxs : List \u03b2\nhx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs\n\u22a2 x :: xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs\n[PROOFSTEP]\nintro h\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b2\nxs : List \u03b2\nhx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs\nh : x :: xs \u2208 subchain (f '' s)\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs\n[PROOFSTEP]\nrw [cons_mem_subchain_iff] at h \n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b2\nxs : List \u03b2\nhx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs\nh : x \u2208 f '' s \u2227 xs \u2208 subchain (f '' s) \u2227 \u2200 (b : \u03b2), b \u2208 head? xs \u2192 x < b\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs\n[PROOFSTEP]\nobtain \u27e8\u27e8x, hx', rfl\u27e9, h\u2081, h\u2082\u27e9 := h\n[GOAL]\ncase a.cons.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nxs : List \u03b2\nhx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs\nx : \u03b1\nhx' : x \u2208 s\nh\u2081 : xs \u2208 subchain (f '' s)\nh\u2082 : \u2200 (b : \u03b2), b \u2208 head? xs \u2192 f x < b\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = f x :: xs\n[PROOFSTEP]\nobtain \u27e8l', h\u2083, rfl\u27e9 := hx h\u2081\n[GOAL]\ncase a.cons.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b1\nhx' : x \u2208 s\nl' : List \u03b1\nh\u2083 : l' \u2208 subchain s\nhx : map f l' \u2208 subchain (f '' s) \u2192 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = map f l'\nh\u2081 : map f l' \u2208 subchain (f '' s)\nh\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f l') \u2192 f x < b\n\u22a2 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = f x :: map f l'\n[PROOFSTEP]\nrefine' \u27e8x :: l', Set.cons_mem_subchain_iff.mpr \u27e8hx', h\u2083, _\u27e9, rfl\u27e9\n[GOAL]\ncase a.cons.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b1\nhx' : x \u2208 s\nl' : List \u03b1\nh\u2083 : l' \u2208 subchain s\nhx : map f l' \u2208 subchain (f '' s) \u2192 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = map f l'\nh\u2081 : map f l' \u2208 subchain (f '' s)\nh\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f l') \u2192 f x < b\n\u22a2 \u2200 (b : \u03b1), b \u2208 head? l' \u2192 x < b\n[PROOFSTEP]\ncases l'\n[GOAL]\ncase a.cons.intro.intro.intro.intro.intro.intro.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b1\nhx' : x \u2208 s\nh\u2083 : [] \u2208 subchain s\nhx : map f [] \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = map f []\nh\u2081 : map f [] \u2208 subchain (f '' s)\nh\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f []) \u2192 f x < b\n\u22a2 \u2200 (b : \u03b1), b \u2208 head? [] \u2192 x < b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.cons.intro.intro.intro.intro.intro.intro.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nx : \u03b1\nhx' : x \u2208 s\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh\u2083 : head\u271d :: tail\u271d \u2208 subchain s\nhx : map f (head\u271d :: tail\u271d) \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = map f (head\u271d :: tail\u271d)\nh\u2081 : map f (head\u271d :: tail\u271d) \u2208 subchain (f '' s)\nh\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f (head\u271d :: tail\u271d)) \u2192 f x < b\n\u22a2 \u2200 (b : \u03b1), b \u2208 head? (head\u271d :: tail\u271d) \u2192 x < b\n[PROOFSTEP]\nsimpa [\u2190 hf] using h\u2082\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\n\u22a2 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain (f '' s) \u2227 length l = length l'\n[PROOFSTEP]\nintro l hl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 \u2203 l', l' \u2208 subchain (f '' s) \u2227 length l = length l'\n[PROOFSTEP]\nrefine' \u27e8l.map f, \u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 Chain' (fun x x_1 => x < x_1) (map f l)\n[PROOFSTEP]\nsimp_rw [chain'_map, \u2190 hf]\n[GOAL]\ncase a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 Chain' (fun a b => a < b) l\n[PROOFSTEP]\nexact hl.1\n[GOAL]\ncase a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 \u2200 (i : \u03b2), i \u2208 map f l \u2192 i \u2208 f '' s\n[PROOFSTEP]\nintro _ e\n[GOAL]\ncase a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\ni\u271d : \u03b2\ne : i\u271d \u2208 map f l\n\u22a2 i\u271d \u2208 f '' s\n[PROOFSTEP]\nobtain \u27e8a, ha, rfl\u27e9 := mem_map.mp e\n[GOAL]\ncase a.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\na : \u03b1\nha : a \u2208 l\ne : f a \u2208 map f l\n\u22a2 f a \u2208 f '' s\n[PROOFSTEP]\nexact Set.mem_image_of_mem _ (hl.2 _ ha)\n[GOAL]\ncase a.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns\u271d t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y\ns : Set \u03b1\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 length l = length (map f l)\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 chainHeight (\u2191ofDual \u207b\u00b9' s) = chainHeight s\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 chainHeight (\u2191ofDual \u207b\u00b9' s) \u2264 chainHeight s\n[PROOFSTEP]\nrw [chainHeight_le_chainHeight_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 \u2200 (l : List \u03b1\u1d52\u1d48), l \u2208 subchain (\u2191ofDual \u207b\u00b9' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l = length l'\n[PROOFSTEP]\nrintro l \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\u1d52\u1d48\nh\u2081 : Chain' (fun x x_1 => x < x_1) l\nh\u2082 : \u2200 (i : \u03b1\u1d52\u1d48), i \u2208 l \u2192 i \u2208 \u2191ofDual \u207b\u00b9' s\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 length l = length l'\n[PROOFSTEP]\nexact \u27e8l.reverse, \u27e8chain'_reverse.mpr h\u2081, fun i h \u21a6 h\u2082 i (mem_reverse.mp h)\u27e9, (length_reverse _).symm\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 chainHeight s \u2264 chainHeight (\u2191ofDual \u207b\u00b9' s)\n[PROOFSTEP]\nrw [chainHeight_le_chainHeight_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain (\u2191ofDual \u207b\u00b9' s) \u2227 length l = length l'\n[PROOFSTEP]\nrintro l \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\ns t : Set \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nh\u2081 : Chain' (fun x x_1 => x < x_1) l\nh\u2082 : \u2200 (i : \u03b1), i \u2208 l \u2192 i \u2208 s\n\u22a2 \u2203 l', l' \u2208 subchain (\u2191ofDual \u207b\u00b9' s) \u2227 length l = length l'\n[PROOFSTEP]\nexact \u27e8l.reverse, \u27e8chain'_reverse.mpr h\u2081, fun i h \u21a6 h\u2082 i (mem_reverse.mp h)\u27e9, (length_reverse _).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight s = \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight s \u2264 \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i)\n[PROOFSTEP]\nrefine' iSup\u2082_le _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 \u2200 (i : List \u03b1), i \u2208 subchain s \u2192 \u2191(length i) \u2264 \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i)\n[PROOFSTEP]\nrintro (_ | \u27e8x, xs\u27e9) h\n[GOAL]\ncase a.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nh : [] \u2208 subchain s\n\u22a2 \u2191(length []) \u2264 \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\n\u22a2 \u2191(length (x :: xs)) \u2264 \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i)\n[PROOFSTEP]\napply le_trans _ (le_iSup\u2082 x (cons_mem_subchain_iff.mp h).1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\n\u22a2 \u2191(length (x :: xs)) \u2264 chainHeight (s \u2229 Ici x)\n[PROOFSTEP]\napply length_le_chainHeight_of_mem_subchain\n[GOAL]\ncase hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\n\u22a2 x :: xs \u2208 subchain (s \u2229 Ici x)\n[PROOFSTEP]\nrefine' \u27e8h.1, fun i hi \u21a6 \u27e8h.2 i hi, _\u27e9\u27e9\n[GOAL]\ncase hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\ni : \u03b1\nhi : i \u2208 x :: xs\n\u22a2 i \u2208 Ici x\n[PROOFSTEP]\ncases hi\n[GOAL]\ncase hl.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\n\u22a2 x \u2208 Ici x\n[PROOFSTEP]\nexact left_mem_Ici\n[GOAL]\ncase hl.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\ni : \u03b1\na\u271d : Mem i xs\n\u22a2 i \u2208 Ici x\n[PROOFSTEP]\nrename_i hi\n[GOAL]\ncase hl.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\ni : \u03b1\nhi : Mem i xs\n\u22a2 i \u2208 Ici x\n[PROOFSTEP]\ncases' chain'_iff_pairwise.mp h.1 with _ _ h'\n[GOAL]\ncase hl.tail.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nx : \u03b1\nxs : List \u03b1\nh : x :: xs \u2208 subchain s\ni : \u03b1\nhi : Mem i xs\na\u271d : List.Pairwise (fun x x_1 => x < x_1) xs\nh' : \u2200 (a' : \u03b1), a' \u2208 xs \u2192 x < a'\n\u22a2 i \u2208 Ici x\n[PROOFSTEP]\nexact (h' _ hi).le\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Ici i) \u2264 chainHeight s\n[PROOFSTEP]\nexact iSup\u2082_le fun i _ \u21a6 chainHeight_mono <| Set.inter_subset_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight s = \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (s \u2229 Iic i)\n[PROOFSTEP]\nsimp_rw [\u2190 chainHeight_dual (_ \u2229 _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight s = \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (\u2191ofDual \u207b\u00b9' (s \u2229 Iic i))\n[PROOFSTEP]\nrw [\u2190 chainHeight_dual, chainHeight_eq_iSup_Ici]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 \u2a06 (i : \u03b1\u1d52\u1d48) (_ : i \u2208 \u2191ofDual \u207b\u00b9' s), chainHeight (\u2191ofDual \u207b\u00b9' s \u2229 Ici i) =\n \u2a06 (i : \u03b1) (_ : i \u2208 s), chainHeight (\u2191ofDual \u207b\u00b9' (s \u2229 Iic i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 chainHeight (insert a s) = chainHeight s + 1\n[PROOFSTEP]\nrw [\u2190 add_zero (insert a s).chainHeight]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 chainHeight (insert a s) + 0 = chainHeight s + 1\n[PROOFSTEP]\nchange (insert a s).chainHeight + (0 : \u2115) = s.chainHeight + (1 : \u2115)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 chainHeight (insert a s) + \u21910 = chainHeight s + \u21911\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 chainHeight (insert a s) + \u21910 \u2264 chainHeight s + \u21911\n[PROOFSTEP]\nrw [chainHeight_add_le_chainHeight_add]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 chainHeight s + \u21911 \u2264 chainHeight (insert a s) + \u21910\n[PROOFSTEP]\nrw [chainHeight_add_le_chainHeight_add]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 \u2200 (l : List \u03b1), l \u2208 subchain (insert a s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l + 0 \u2264 length l' + 1\n[PROOFSTEP]\nrintro (_ | \u27e8y, ys\u27e9) h\n[GOAL]\ncase a.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nh : [] \u2208 subchain (insert a s)\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 length [] + 0 \u2264 length l' + 1\n[PROOFSTEP]\nexact \u27e8[], nil_mem_subchain _, zero_le _\u27e9\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\ny : \u03b1\nys : List \u03b1\nh : y :: ys \u2208 subchain (insert a s)\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 length (y :: ys) + 0 \u2264 length l' + 1\n[PROOFSTEP]\nhave h' := cons_mem_subchain_iff.mp h\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\ny : \u03b1\nys : List \u03b1\nh : y :: ys \u2208 subchain (insert a s)\nh' : y \u2208 insert a s \u2227 ys \u2208 subchain (insert a s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\n\u22a2 \u2203 l', l' \u2208 subchain s \u2227 length (y :: ys) + 0 \u2264 length l' + 1\n[PROOFSTEP]\nrefine' \u27e8ys, \u27e8h'.2.1.1, fun i hi \u21a6 _\u27e9, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\ny : \u03b1\nys : List \u03b1\nh : y :: ys \u2208 subchain (insert a s)\nh' : y \u2208 insert a s \u2227 ys \u2208 subchain (insert a s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\n\u22a2 length (y :: ys) + 0 \u2264 length ys + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\ny : \u03b1\nys : List \u03b1\nh : y :: ys \u2208 subchain (insert a s)\nh' : y \u2208 insert a s \u2227 ys \u2208 subchain (insert a s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\ni : \u03b1\nhi : i \u2208 ys\n\u22a2 i \u2208 s\n[PROOFSTEP]\napply (h'.2.1.2 i hi).resolve_left\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\ny : \u03b1\nys : List \u03b1\nh : y :: ys \u2208 subchain (insert a s)\nh' : y \u2208 insert a s \u2227 ys \u2208 subchain (insert a s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\ni : \u03b1\nhi : i \u2208 ys\n\u22a2 \u00aci = a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase a.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\ny : \u03b1\nys : List \u03b1\ni : \u03b1\nhi : i \u2208 ys\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 i < b\nh : y :: ys \u2208 subchain (insert i s)\nh' : y \u2208 insert i s \u2227 ys \u2208 subchain (insert i s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\n\u22a2 False\n[PROOFSTEP]\ncases' chain'_iff_pairwise.mp h.1 with _ _ hy\n[GOAL]\ncase a.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\ny : \u03b1\nys : List \u03b1\ni : \u03b1\nhi : i \u2208 ys\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 i < b\nh : y :: ys \u2208 subchain (insert i s)\nh' : y \u2208 insert i s \u2227 ys \u2208 subchain (insert i s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\na\u271d : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y < a'\n\u22a2 False\n[PROOFSTEP]\ncases' h'.1 with h' h'\n[GOAL]\ncase a.cons.cons.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\ny : \u03b1\nys : List \u03b1\ni : \u03b1\nhi : i \u2208 ys\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 i < b\nh : y :: ys \u2208 subchain (insert i s)\nh'\u271d : y \u2208 insert i s \u2227 ys \u2208 subchain (insert i s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\na\u271d : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y < a'\nh' : y = i\n\u22a2 False\ncase a.cons.cons.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\ny : \u03b1\nys : List \u03b1\ni : \u03b1\nhi : i \u2208 ys\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 i < b\nh : y :: ys \u2208 subchain (insert i s)\nh'\u271d : y \u2208 insert i s \u2227 ys \u2208 subchain (insert i s) \u2227 \u2200 (b : \u03b1), b \u2208 head? ys \u2192 y < b\na\u271d : List.Pairwise (fun x x_1 => x < x_1) ys\nhy : \u2200 (a' : \u03b1), a' \u2208 ys \u2192 y < a'\nh' : y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexacts [(hy _ hi).ne h', not_le_of_gt (hy _ hi) (hx _ h').le]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\n\u22a2 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain (insert a s) \u2227 length l + 1 \u2264 length l' + 0\n[PROOFSTEP]\nintro l hl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 \u2203 l', l' \u2208 subchain (insert a s) \u2227 length l + 1 \u2264 length l' + 0\n[PROOFSTEP]\nrefine' \u27e8a :: l, \u27e8_, _\u27e9, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 length l + 1 \u2264 length (a :: l) + 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 Chain' (fun x x_1 => x < x_1) (a :: l)\n[PROOFSTEP]\nrw [chain'_cons']\n[GOAL]\ncase a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 (\u2200 (y : \u03b1), y \u2208 head? l \u2192 a < y) \u2227 Chain' (fun x x_1 => x < x_1) l\n[PROOFSTEP]\nexact \u27e8fun y hy \u21a6 hx _ (hl.2 _ (mem_of_mem_head? hy)), hl.1\u27e9\n[GOAL]\ncase a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 \u2200 (i : \u03b1), i \u2208 a :: l \u2192 i \u2208 insert a s\n[PROOFSTEP]\nrintro x (_ | _)\n[GOAL]\ncase a.refine'_2.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 a \u2208 insert a s\ncase a.refine'_2.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nhx : \u2200 (b : \u03b1), b \u2208 s \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nx : \u03b1\na\u271d : Mem x l\n\u22a2 x \u2208 insert a s\n[PROOFSTEP]\nexacts [Or.inl (Set.mem_singleton a), Or.inr (hl.2 x \u2039_\u203a)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2208 s \u2192 b < a\n\u22a2 chainHeight (insert a s) = chainHeight s + 1\n[PROOFSTEP]\nrw [\u2190 chainHeight_dual, \u2190 chainHeight_dual s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\na : \u03b1\nha : \u2200 (b : \u03b1), b \u2208 s \u2192 b < a\n\u22a2 chainHeight (\u2191ofDual \u207b\u00b9' insert a s) = chainHeight (\u2191ofDual \u207b\u00b9' s) + 1\n[PROOFSTEP]\nexact chainHeight_insert_of_forall_gt _ ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight (s \u222a t) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nclassical\nrefine' iSup\u2082_le fun l hl \u21a6 _\nlet l\u2081 := l.filter (\u00b7 \u2208 s)\nlet l\u2082 := l.filter (\u00b7 \u2208 t)\nhave hl\u2081 : \u2191l\u2081.length \u2264 s.chainHeight :=\n by\n apply Set.length_le_chainHeight_of_mem_subchain\n exact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\nhave hl\u2082 : \u2191l\u2082.length \u2264 t.chainHeight :=\n by\n apply Set.length_le_chainHeight_of_mem_subchain\n exact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\nrefine' le_trans _ (add_le_add hl\u2081 hl\u2082)\nsimp_rw [\u2190 Nat.cast_add, \u2190 Multiset.coe_card, \u2190 Multiset.card_add, \u2190 Multiset.coe_filter]\nrw [Multiset.filter_add_filter, Multiset.filter_eq_self.mpr, Multiset.card_add, Nat.cast_add]\nexacts [le_add_right rfl.le, hl.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\n\u22a2 chainHeight (s \u222a t) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nrefine' iSup\u2082_le fun l hl \u21a6 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\n\u22a2 \u2191(length l) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nlet l\u2081 := l.filter (\u00b7 \u2208 s)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\n\u22a2 \u2191(length l) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nlet l\u2082 := l.filter (\u00b7 \u2208 t)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\n\u22a2 \u2191(length l) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nhave hl\u2081 : \u2191l\u2081.length \u2264 s.chainHeight :=\n by\n apply Set.length_le_chainHeight_of_mem_subchain\n exact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\n\u22a2 \u2191(length l\u2081) \u2264 chainHeight s\n[PROOFSTEP]\napply Set.length_le_chainHeight_of_mem_subchain\n[GOAL]\ncase hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\n\u22a2 l\u2081 \u2208 subchain s\n[PROOFSTEP]\nexact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\ni : \u03b1\nh : i \u2208 l\u2081\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsimpa using (of_mem_filter h : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\n\u22a2 \u2191(length l) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nhave hl\u2082 : \u2191l\u2082.length \u2264 t.chainHeight :=\n by\n apply Set.length_le_chainHeight_of_mem_subchain\n exact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\n\u22a2 \u2191(length l\u2082) \u2264 chainHeight t\n[PROOFSTEP]\napply Set.length_le_chainHeight_of_mem_subchain\n[GOAL]\ncase hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\n\u22a2 l\u2082 \u2208 subchain t\n[PROOFSTEP]\nexact \u27e8hl.1.sublist (filter_sublist _), fun i h \u21a6 by simpa using (of_mem_filter h : _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\ni : \u03b1\nh : i \u2208 l\u2082\n\u22a2 i \u2208 t\n[PROOFSTEP]\nsimpa using (of_mem_filter h : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\nhl\u2082 : \u2191(length l\u2082) \u2264 chainHeight t\n\u22a2 \u2191(length l) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nrefine' le_trans _ (add_le_add hl\u2081 hl\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\nhl\u2082 : \u2191(length l\u2082) \u2264 chainHeight t\n\u22a2 \u2191(length l) \u2264 \u2191(length l\u2081) + \u2191(length l\u2082)\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.cast_add, \u2190 Multiset.coe_card, \u2190 Multiset.card_add, \u2190 Multiset.coe_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\nhl\u2082 : \u2191(length l\u2082) \u2264 chainHeight t\n\u22a2 \u2191(\u2191Multiset.card \u2191l) \u2264 \u2191(\u2191Multiset.card (Multiset.filter (fun b => b \u2208 s) \u2191l + Multiset.filter (fun b => b \u2208 t) \u2191l))\n[PROOFSTEP]\nrw [Multiset.filter_add_filter, Multiset.filter_eq_self.mpr, Multiset.card_add, Nat.cast_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\nhl\u2082 : \u2191(length l\u2082) \u2264 chainHeight t\n\u22a2 \u2191(\u2191Multiset.card \u2191l) \u2264 \u2191(\u2191Multiset.card \u2191l) + \u2191(\u2191Multiset.card (Multiset.filter (fun a => a \u2208 s \u2227 a \u2208 t) \u2191l))\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\ninst\u271d : Preorder \u03b1\nl : List \u03b1\nhl : l \u2208 subchain (s \u222a t)\nl\u2081 : List \u03b1 := filter (fun x => decide (x \u2208 s)) l\nl\u2082 : List \u03b1 := filter (fun x => decide (x \u2208 t)) l\nhl\u2081 : \u2191(length l\u2081) \u2264 chainHeight s\nhl\u2082 : \u2191(length l\u2082) \u2264 chainHeight t\n\u22a2 \u2200 (a : \u03b1), a \u2208 \u2191l \u2192 a \u2208 s \u2228 a \u2208 t\n[PROOFSTEP]\nexacts [le_add_right rfl.le, hl.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\n\u22a2 chainHeight (s \u222a t) = chainHeight s + chainHeight t\n[PROOFSTEP]\ncases h : t.chainHeight\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nh : chainHeight t = none\n\u22a2 chainHeight (s \u222a t) = chainHeight s + none\n[PROOFSTEP]\nrw [WithTop.none_eq_top, add_top, eq_top_iff, \u2190 WithTop.none_eq_top, \u2190 h]\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nh : chainHeight t = none\n\u22a2 chainHeight t \u2264 chainHeight (s \u222a t)\n[PROOFSTEP]\nexact Set.chainHeight_mono (Set.subset_union_right _ _)\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\n\u22a2 chainHeight (s \u222a t) = chainHeight s + some val\u271d\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase some.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\n\u22a2 chainHeight (s \u222a t) \u2264 chainHeight s + some val\u271d\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase some.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\n\u22a2 chainHeight (s \u222a t) \u2264 chainHeight s + chainHeight t\n[PROOFSTEP]\nexact chainHeight_union_le\n[GOAL]\ncase some.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\n\u22a2 chainHeight s + some val\u271d \u2264 chainHeight (s \u222a t)\n[PROOFSTEP]\nrw [WithTop.some_eq_coe, \u2190 add_zero (s \u222a t).chainHeight, \u2190 WithTop.coe_zero, ENat.some_eq_coe,\n chainHeight_add_le_chainHeight_add]\n[GOAL]\ncase some.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\n\u22a2 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain (s \u222a t) \u2227 length l + val\u271d \u2264 length l' + 0\n[PROOFSTEP]\nintro l hl\n[GOAL]\ncase some.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nval\u271d : \u2115\nh : chainHeight t = some val\u271d\nl : List \u03b1\nhl : l \u2208 subchain s\n\u22a2 \u2203 l', l' \u2208 subchain (s \u222a t) \u2227 length l + val\u271d \u2264 length l' + 0\n[PROOFSTEP]\nobtain \u27e8l', hl', rfl\u27e9 := exists_chain_of_le_chainHeight t h.symm.le\n[GOAL]\ncase some.a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\n\u22a2 \u2203 l'_1, l'_1 \u2208 subchain (s \u222a t) \u2227 length l + length l' \u2264 length l'_1 + 0\n[PROOFSTEP]\nrefine' \u27e8l ++ l', \u27e8Chain'.append hl.1 hl'.1 fun x hx y hy \u21a6 _, fun i hi \u21a6 _\u27e9, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\n\u22a2 length l + length l' \u2264 length (l ++ l') + 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.a.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\nx : \u03b1\nhx : x \u2208 getLast? l\ny : \u03b1\nhy : y \u2208 head? l'\n\u22a2 x < y\n[PROOFSTEP]\nexact H x (hl.2 _ <| mem_of_mem_getLast? hx) y (hl'.2 _ <| mem_of_mem_head? hy)\n[GOAL]\ncase some.a.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\ni : \u03b1\nhi : i \u2208 l ++ l'\n\u22a2 i \u2208 s \u222a t\n[PROOFSTEP]\nrw [mem_append] at hi \n[GOAL]\ncase some.a.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\ni : \u03b1\nhi : i \u2208 l \u2228 i \u2208 l'\n\u22a2 i \u2208 s \u222a t\n[PROOFSTEP]\ncases' hi with hi hi\n[GOAL]\ncase some.a.intro.intro.refine'_2.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\ni : \u03b1\nhi : i \u2208 l\n\u22a2 i \u2208 s \u222a t\ncase some.a.intro.intro.refine'_2.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Preorder \u03b1\ns t : Set \u03b1\nH : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a < b\nl : List \u03b1\nhl : l \u2208 subchain s\nl' : List \u03b1\nhl' : l' \u2208 subchain t\nh : chainHeight t = some (length l')\ni : \u03b1\nhi : i \u2208 l'\n\u22a2 i \u2208 s \u222a t\n[PROOFSTEP]\nexacts [Or.inl (hl.2 _ hi), Or.inr (hl'.2 _ hi)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\n\u22a2 WellFoundedGT \u2191s\n[PROOFSTEP]\nhaveI : IsTrans { x // x \u2208 s } (\u2191\u00b7 < \u2191\u00b7) := inferInstance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\n\u22a2 WellFoundedGT \u2191s\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := WithTop.ne_top_iff_exists.1 hs\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\n\u22a2 WellFoundedGT \u2191s\n[PROOFSTEP]\nrefine' \u27e8RelEmbedding.wellFounded_iff_no_descending_seq.2 \u27e8fun f \u21a6 _\u27e9\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\n\u22a2 False\n[PROOFSTEP]\nrefine' n.lt_succ_self.not_le (WithTop.coe_le_coe.1 <| hn.symm \u25b8 _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\n\u22a2 \u2191(Nat.succ n) \u2264 chainHeight s\n[PROOFSTEP]\nrefine'\n le_iSup\u2082_of_le _\n \u27e8chain'_map_of_chain' ((\u2191) : { x // x \u2208 s } \u2192 \u03b1) (fun _ _ \u21a6 id)\n (chain'_iff_pairwise.2 <| pairwise_ofFn.2 fun i j \u21a6 f.map_rel_iff.2),\n fun i h \u21a6 _\u27e9\n _\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\n\u22a2 \u2115\n[PROOFSTEP]\nexact n.succ\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\ni : \u03b1\nh : i \u2208 map Subtype.val (ofFn fun i => \u2191f \u2191i)\n\u22a2 i \u2208 s\n[PROOFSTEP]\nobtain \u27e8a, -, rfl\u27e9 := mem_map.1 h\n[GOAL]\ncase intro.refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\na : { x // x \u2208 s }\nh : \u2191a \u2208 map Subtype.val (ofFn fun i => \u2191f \u2191i)\n\u22a2 \u2191a \u2208 s\n[PROOFSTEP]\nexact a.prop\n[GOAL]\ncase intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\n\u22a2 \u2191(Nat.succ n) \u2264 \u2191(length (map Subtype.val (ofFn fun i => \u2191f \u2191i)))\n[PROOFSTEP]\nrw [length_map, length_ofFn]\n[GOAL]\ncase intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\nthis : IsTrans { x // x \u2208 s } fun x x_1 => x < x_1\nn : \u2115\nhn : \u2191n = chainHeight s\nf : (fun x x_1 => x > x_1) \u21aar fun x x_1 => x > x_1\n\u22a2 \u2191(Nat.succ n) \u2264 \u2191(Nat.succ n)\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t : Set \u03b1\ninst\u271d : Preorder \u03b1\ns : Set \u03b1\nhs : chainHeight s \u2260 \u22a4\n\u22a2 chainHeight (\u2191ofDual \u207b\u00b9' s) \u2260 \u22a4\n[PROOFSTEP]\nrwa [chainHeight_dual]\n", "meta": {"mathlib_filename": "Mathlib.Order.Height", "llama_tokens": 27991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.2804747141350864}} {"text": "[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\n\u22a2 ContinuousOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082') (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nhave h\u2081 := (compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082)).continuous\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\n\u22a2 ContinuousOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082') (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nhave h\u2082 := (ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3)).continuous\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\n\u22a2 ContinuousOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082') (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nhave h\u2083 := continuousOn_coordChange \ud835\udd5c\u2081 e\u2081' e\u2081\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\n\u22a2 ContinuousOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082') (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nhave h\u2084 := continuousOn_coordChange \ud835\udd5c\u2082 e\u2082 e\u2082'\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\n\u22a2 ContinuousOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082') (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nrefine' ((h\u2081.comp_continuousOn (h\u2084.mono _)).clm_comp (h\u2082.comp_continuousOn (h\u2083.mono _))).congr _\n[GOAL]\ncase refine'_1\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\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 refine'_2\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\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 refine'_3\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\n\u22a2 EqOn (continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082')\n (fun x =>\n comp ((\u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082)) \u2218 fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) x)\n ((\u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3)) \u2218 fun b =>\n \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b))\n x))\n (e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet))\n[PROOFSTEP]\nintro b _\n[GOAL]\ncase refine'_3\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\nb : B\na\u271d : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\n\u22a2 continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082' b =\n (fun x =>\n comp ((\u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082)) \u2218 fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) x)\n ((\u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3)) \u2218 fun b =>\n \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b))\n x))\n b\n[PROOFSTEP]\next L v\n[GOAL]\ncase refine'_3.h.h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\nb : B\na\u271d : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 \u2191(\u2191(continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082' b) L) v =\n \u2191(\u2191((fun x =>\n comp ((\u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082)) \u2218 fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) x)\n ((\u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3)) \u2218 fun b =>\n \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b))\n x))\n b)\n L)\n v\n[PROOFSTEP]\ndsimp [continuousLinearMapCoordChange]\n[GOAL]\ncase refine'_3.h.h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b3 : MemTrivializationAtlas e\u2081\ninst\u271d\u00b2 : MemTrivializationAtlas e\u2081'\ninst\u271d\u00b9 : MemTrivializationAtlas e\u2082\ninst\u271d : MemTrivializationAtlas e\u2082'\nh\u2081 : Continuous \u2191(compSL F\u2081 F\u2082 F\u2082 \u03c3 (RingHom.id \ud835\udd5c\u2082))\nh\u2082 : Continuous \u2191(ContinuousLinearMap.flip (compSL F\u2081 F\u2081 F\u2082 (RingHom.id \ud835\udd5c\u2081) \u03c3))\nh\u2083 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b)) (e\u2081'.baseSet \u2229 e\u2081.baseSet)\nh\u2084 : ContinuousOn (fun b => \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)) (e\u2082.baseSet \u2229 e\u2082'.baseSet)\nb : B\na\u271d : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b)\n (\u2191L (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b))) v)) =\n \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b) (\u2191L (\u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b) v))\n[PROOFSTEP]\nrw [ContinuousLinearEquiv.symm_symm]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\n\u22a2 (fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst) (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n ((fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj) (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n { proj := x, snd := L }) =\n { proj := x, snd := L }\n[PROOFSTEP]\nsimp only [TotalSpace.mk_inj]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\n\u22a2 comp\n (Trivialization.symmL \ud835\udd5c\u2082 e\u2082\n ((fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj)\n (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n { proj := x, snd := L }).fst)\n (comp\n (comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 { proj := x, snd := L }.proj)\n (comp L (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 { proj := x, snd := L }.proj)))\n (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081\n ((fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj)\n (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n { proj := x, snd := L }).fst)) =\n L\n[PROOFSTEP]\next (v : E\u2081 x)\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\nv : E\u2081 x\n\u22a2 \u2191(comp\n (Trivialization.symmL \ud835\udd5c\u2082 e\u2082\n ((fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj)\n (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n { proj := x, snd := L }).fst)\n (comp\n (comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 { proj := x, snd := L }.proj)\n (comp L (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 { proj := x, snd := L }.proj)))\n (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081\n ((fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj)\n (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n { proj := x, snd := L }).fst)))\n v =\n \u2191L v\n[PROOFSTEP]\ndsimp only [comp_apply]\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\nv : E\u2081 x\n\u22a2 \u2191(Trivialization.symmL \ud835\udd5c\u2082 e\u2082 x)\n (\u2191(Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x)\n (\u2191L (\u2191(Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x) (\u2191(Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 x) v)))) =\n \u2191L v\n[PROOFSTEP]\nrw [Trivialization.symmL_continuousLinearMapAt, Trivialization.symmL_continuousLinearMapAt]\n[GOAL]\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\nv : E\u2081 x\n\u22a2 x \u2208 e\u2081.baseSet\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : TotalSpace (F\u2081 \u2192SL[\u03c3] F\u2082) (Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082)\nx : B\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\nx\u271d : { proj := x, snd := L } \u2208 TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)\nh\u2081 : { proj := x, snd := L }.proj \u2208 e\u2081.baseSet\nh\u2082 : { proj := x, snd := L }.proj \u2208 e\u2082.baseSet\nv : E\u2081 x\n\u22a2 x \u2208 e\u2082.baseSet\n[PROOFSTEP]\nexacts [h\u2081, h\u2082]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\n\u22a2 (fun p =>\n (p.proj,\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 p.proj) (comp p.snd (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 p.proj))))\n ((fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst) (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n (x, f)) =\n (x, f)\n[PROOFSTEP]\nsimp only [Prod.mk_inj_left]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\n\u22a2 comp\n (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082\n ((fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst)\n (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n (x, f)).proj)\n (comp\n (comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 (x, f).fst) (comp f (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 (x, f).fst)))\n (Trivialization.symmL \ud835\udd5c\u2081 e\u2081\n ((fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst)\n (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n (x, f)).proj)) =\n f\n[PROOFSTEP]\next v\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\nv : F\u2081\n\u22a2 \u2191(comp\n (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082\n ((fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst)\n (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n (x, f)).proj)\n (comp\n (comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 (x, f).fst)\n (comp f (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 (x, f).fst)))\n (Trivialization.symmL \ud835\udd5c\u2081 e\u2081\n ((fun p =>\n { proj := p.fst,\n snd :=\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 p.fst)\n (comp p.snd (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 p.fst)) })\n (x, f)).proj)))\n v =\n \u2191f v\n[PROOFSTEP]\ndsimp only [comp_apply]\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\nv : F\u2081\n\u22a2 \u2191(Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x)\n (\u2191(Trivialization.symmL \ud835\udd5c\u2082 e\u2082 x)\n (\u2191f (\u2191(Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 x) (\u2191(Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x) v)))) =\n \u2191f v\n[PROOFSTEP]\nrw [Trivialization.continuousLinearMapAt_symmL, Trivialization.continuousLinearMapAt_symmL]\n[GOAL]\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\nv : F\u2081\n\u22a2 x \u2208 e\u2081.baseSet\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nx\u271d\u00b9 : B \u00d7 (F\u2081 \u2192SL[\u03c3] F\u2082)\nx : B\nf : F\u2081 \u2192SL[\u03c3] F\u2082\nx\u271d : (x, f) \u2208 (e\u2081.baseSet \u2229 e\u2082.baseSet) \u00d7\u02e2 univ\nh\u2081 : (x, f).fst \u2208 e\u2081.baseSet\nh\u2082 : (x, f).fst \u2208 e\u2082.baseSet\nright\u271d : (x, f).snd \u2208 univ\nv : F\u2081\n\u22a2 x \u2208 e\u2082.baseSet\n[PROOFSTEP]\nexacts [h\u2081, h\u2082]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u2074 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\ninst\u271d\u00b9 : \u2200 (x : B), ContinuousAdd (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nx : B\nx\u271d : x \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\nL L' : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\n\u22a2 comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x) (comp (L + L') (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x)) =\n (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L }).snd +\n (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L' }).snd\n[PROOFSTEP]\nsimp_rw [add_comp, comp_add]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u2074 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\ninst\u271d\u00b9 : \u2200 (x : B), ContinuousAdd (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nx : B\nx\u271d : x \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\nL L' : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\n\u22a2 comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x) (comp L (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x)) +\n comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x) (comp L' (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x)) =\n (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L }).snd +\n (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L' }).snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u2074 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\ninst\u271d\u00b9 : \u2200 (x : B), ContinuousAdd (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nx : B\nx\u271d : x \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\nc : \ud835\udd5c\u2082\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\n\u22a2 comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x) (comp (c \u2022 L) (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x)) =\n c \u2022 (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L }).snd\n[PROOFSTEP]\nsimp_rw [smul_comp, comp_smul\u209b\u2097, RingHom.id_apply]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2078 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2077 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2076 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u00b9\u00b9 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u00b9\u2070 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2079 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2078 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2076 : FiberBundle F\u2082 E\u2082\ninst\u271d\u2075 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u2074 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\ninst\u271d\u00b9 : \u2200 (x : B), ContinuousAdd (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nx : B\nx\u271d : x \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\nc : \ud835\udd5c\u2082\nL : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 x\n\u22a2 c \u2022 comp (Trivialization.continuousLinearMapAt \ud835\udd5c\u2082 e\u2082 x) (comp L (Trivialization.symmL \ud835\udd5c\u2081 e\u2081 x)) =\n c \u2022 (\u2191(continuousLinearMap \u03c3 e\u2081 e\u2082) { proj := x, snd := L }).snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet\nL : F\u2081 \u2192SL[\u03c3] F\u2082\n\u22a2 Pretrivialization.symm (continuousLinearMap \u03c3 e\u2081 e\u2082) b L =\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 b) (comp L (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 b))\n[PROOFSTEP]\nrw [symm_apply]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet\nL : F\u2081 \u2192SL[\u03c3] F\u2082\n\u22a2 cast\n (_ :\n Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 (\u2191(LocalEquiv.symm (continuousLinearMap \u03c3 e\u2081 e\u2082).toLocalEquiv) (b, L)).proj =\n Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 b)\n (\u2191(LocalEquiv.symm (continuousLinearMap \u03c3 e\u2081 e\u2082).toLocalEquiv) (b, L)).snd =\n comp (Trivialization.symmL \ud835\udd5c\u2082 e\u2082 b) (comp L (Trivialization.continuousLinearMapAt \ud835\udd5c\u2081 e\u2081 b))\ncase hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet\nL : F\u2081 \u2192SL[\u03c3] F\u2082\n\u22a2 b \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet\nL : F\u2081 \u2192SL[\u03c3] F\u2082\n\u22a2 b \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet\n[PROOFSTEP]\nexact hb\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\n\u22a2 \u2191(continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082' b) L =\n (\u2191(continuousLinearMap \u03c3 e\u2081' e\u2082')\n { proj := b, snd := Pretrivialization.symm (continuousLinearMap \u03c3 e\u2081 e\u2082) b L }).snd\n[PROOFSTEP]\next v\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 \u2191(\u2191(continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082' b) L) v =\n \u2191(\u2191(continuousLinearMap \u03c3 e\u2081' e\u2082')\n { proj := b, snd := Pretrivialization.symm (continuousLinearMap \u03c3 e\u2081 e\u2082) b L }).snd\n v\n[PROOFSTEP]\nsimp_rw [continuousLinearMapCoordChange, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.arrowCongrSL_apply,\n continuousLinearMap_apply, continuousLinearMap_symm_apply' \u03c3 e\u2081 e\u2082 hb.1, comp_apply, ContinuousLinearEquiv.coe_coe,\n ContinuousLinearEquiv.symm_symm, Trivialization.continuousLinearMapAt_apply, Trivialization.symmL_apply]\n[GOAL]\ncase h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 \u2191(Trivialization.coordChangeL \ud835\udd5c\u2082 e\u2082 e\u2082' b) (\u2191L (\u2191(Trivialization.coordChangeL \ud835\udd5c\u2081 e\u2081' e\u2081 b) v)) =\n \u2191(Trivialization.linearMapAt \ud835\udd5c\u2082 e\u2082'\n { proj := b, snd := Pretrivialization.symm (continuousLinearMap \u03c3 e\u2081 e\u2082) b L }.proj)\n (Trivialization.symm e\u2082 b (\u2191L (\u2191(Trivialization.linearMapAt \ud835\udd5c\u2081 e\u2081 b) (Trivialization.symm e\u2081' b v))))\n[PROOFSTEP]\nrw [e\u2082.coordChangeL_apply e\u2082', e\u2081'.coordChangeL_apply e\u2081, e\u2081.coe_linearMapAt_of_mem hb.1.1,\n e\u2082'.coe_linearMapAt_of_mem hb.2.2]\n[GOAL]\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 b \u2208 e\u2081'.baseSet \u2229 e\u2081.baseSet\ncase h.hb\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\nita : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d\u2074 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b3 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081\ninst\u271d\u00b2 : Trivialization.IsLinear \ud835\udd5c\u2081 e\u2081'\ninst\u271d\u00b9 : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082\ninst\u271d : Trivialization.IsLinear \ud835\udd5c\u2082 e\u2082'\nb : B\nhb : b \u2208 e\u2081.baseSet \u2229 e\u2082.baseSet \u2229 (e\u2081'.baseSet \u2229 e\u2082'.baseSet)\nL : F\u2081 \u2192SL[\u03c3] F\u2082\nv : F\u2081\n\u22a2 b \u2208 e\u2082.baseSet \u2229 e\u2082'.baseSet\n[PROOFSTEP]\nexacts [\u27e8hb.2.1, hb.1.1\u27e9, \u27e8hb.1.2, hb.2.2\u27e9]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\n\u22a2 \u2200 (e : Pretrivialization (F\u2081 \u2192SL[\u03c3] F\u2082) TotalSpace.proj),\n e \u2208 {e | \u2203 e\u2081 e\u2082 x x_1, e = continuousLinearMap \u03c3 e\u2081 e\u2082} \u2192 Pretrivialization.IsLinear \ud835\udd5c\u2082 e\n[PROOFSTEP]\nrintro _ \u27e8e\u2081, he\u2081, e\u2082, he\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081\u271d e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082\u271d e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\nhe\u2081 : Trivialization F\u2082 TotalSpace.proj\ne\u2082 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas he\u2081\n\u22a2 Pretrivialization.IsLinear \ud835\udd5c\u2082 (continuousLinearMap \u03c3 e\u2081 he\u2081)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\n\u22a2 \u2200 (e : Pretrivialization (F\u2081 \u2192SL[\u03c3] F\u2082) TotalSpace.proj),\n e \u2208 {e | \u2203 e\u2081 e\u2082 x x_1, e = continuousLinearMap \u03c3 e\u2081 e\u2082} \u2192\n \u2200 (e' : Pretrivialization (F\u2081 \u2192SL[\u03c3] F\u2082) TotalSpace.proj),\n e' \u2208 {e | \u2203 e\u2081 e\u2082 x x_1, e = continuousLinearMap \u03c3 e\u2081 e\u2082} \u2192\n \u2203 f,\n ContinuousOn f (e.baseSet \u2229 e'.baseSet) \u2227\n \u2200 (b : B),\n b \u2208 e.baseSet \u2229 e'.baseSet \u2192\n \u2200 (v : F\u2081 \u2192SL[\u03c3] F\u2082), \u2191(f b) v = (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).snd\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 intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081\u271d e\u2081'\u271d : Trivialization F\u2081 TotalSpace.proj\ne\u2082\u271d e\u2082'\u271d : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\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 \u2203 f,\n ContinuousOn f ((continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet \u2229 (continuousLinearMap \u03c3 e\u2081' e\u2082').baseSet) \u2227\n \u2200 (b : B),\n b \u2208 (continuousLinearMap \u03c3 e\u2081 e\u2082).baseSet \u2229 (continuousLinearMap \u03c3 e\u2081' e\u2082').baseSet \u2192\n \u2200 (v : F\u2081 \u2192SL[\u03c3] F\u2082),\n \u2191(f b) v =\n (\u2191(continuousLinearMap \u03c3 e\u2081' e\u2082')\n { proj := b, snd := Pretrivialization.symm (continuousLinearMap \u03c3 e\u2081 e\u2082) b v }).snd\n[PROOFSTEP]\nexact\n \u27e8continuousLinearMapCoordChange \u03c3 e\u2081 e\u2081' e\u2082 e\u2082', continuousOn_continuousLinearMapCoordChange,\n continuousLinearMapCoordChange_apply \u03c3 e\u2081 e\u2081' e\u2082 e\u2082'\u27e9\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\n\u22a2 \u2200 (b : B),\n Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nintro b\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\n\u22a2 Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nlet L\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 := (trivializationAt F\u2081 E\u2081 b).continuousLinearEquivAt \ud835\udd5c\u2081 b (mem_baseSet_trivializationAt _ _ _)\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\n\u22a2 Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nlet L\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 := (trivializationAt F\u2082 E\u2082 b).continuousLinearEquivAt \ud835\udd5c\u2082 b (mem_baseSet_trivializationAt _ _ _)\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u22a2 Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nlet \u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := L\u2081.arrowCongrSL L\u2082\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\n\u22a2 Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nhave : Inducing fun x => (b, \u03c6 x) := inducing_const_prod.mpr \u03c6.toHomeomorph.inducing\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\nthis : Inducing fun x => (b, \u2191\u03c6 x)\n\u22a2 Inducing\n (\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5.h.h.e'_4\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\nthis : Inducing fun x => (b, \u2191\u03c6 x)\nx\u271d : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 b\n\u22a2 ((\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n x\u271d).snd =\n \u2191\u03c6 x\u271d\n[PROOFSTEP]\next f\n[GOAL]\ncase h.e'_5.h.h.e'_4.h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\nthis : Inducing fun x => (b, \u2191\u03c6 x)\nx\u271d : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 b\nf : F\u2081\n\u22a2 \u2191((\u2191((fun x => continuousLinearMap \u03c3 (trivializationAt F\u2081 E\u2081 x) (trivializationAt F\u2082 E\u2082 x)) b) \u2218 TotalSpace.mk b)\n x\u271d).snd\n f =\n \u2191(\u2191\u03c6 x\u271d) f\n[PROOFSTEP]\ndsimp [Pretrivialization.continuousLinearMap_apply]\n[GOAL]\ncase h.e'_5.h.h.e'_4.h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\nthis : Inducing fun x => (b, \u2191\u03c6 x)\nx\u271d : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 b\nf : F\u2081\n\u22a2 \u2191(Trivialization.linearMapAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b)\n (\u2191x\u271d (Trivialization.symm (trivializationAt F\u2081 E\u2081 b) b f)) =\n (\u2191(trivializationAt F\u2082 E\u2082 b) { proj := b, snd := \u2191x\u271d (Trivialization.symm (trivializationAt F\u2081 E\u2081 b) b f) }).snd\n[PROOFSTEP]\nrw [Trivialization.linearMapAt_def_of_mem _ (mem_baseSet_trivializationAt _ _ _)]\n[GOAL]\ncase h.e'_5.h.h.e'_4.h\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nb : B\nL\u2081 : E\u2081 b \u2243L[\ud835\udd5c\u2081] F\u2081 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F\u2081 E\u2081 b) b (_ : b \u2208 (trivializationAt F\u2081 E\u2081 b).baseSet)\nL\u2082 : E\u2082 b \u2243L[\ud835\udd5c\u2082] F\u2082 :=\n Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet)\n\u03c6 : (E\u2081 b \u2192SL[\u03c3] E\u2082 b) \u2243L[\ud835\udd5c\u2082] F\u2081 \u2192SL[\u03c3] F\u2082 := ContinuousLinearEquiv.arrowCongrSL L\u2081 L\u2082\nthis : Inducing fun x => (b, \u2191\u03c6 x)\nx\u271d : Bundle.ContinuousLinearMap \u03c3 E\u2081 E\u2082 b\nf : F\u2081\n\u22a2 \u2191\u2191(Trivialization.linearEquivAt \ud835\udd5c\u2082 (trivializationAt F\u2082 E\u2082 b) b (_ : b \u2208 (trivializationAt F\u2082 E\u2082 b).baseSet))\n (\u2191x\u271d (Trivialization.symm (trivializationAt F\u2081 E\u2081 b) b f)) =\n (\u2191(trivializationAt F\u2082 E\u2082 b) { proj := b, snd := \u2191x\u271d (Trivialization.symm (trivializationAt F\u2081 E\u2081 b) b f) }).snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\n\u22a2 MemTrivializationAtlas e\u2081\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2079 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ni\u03c3 : RingHomIsometric \u03c3\nB : Type u_3\nF\u2081 : Type u_4\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2081\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c\u2081 F\u2081\nE\u2081 : B \u2192 Type u_5\ninst\u271d\u00b9\u2076 : (x : B) \u2192 AddCommGroup (E\u2081 x)\ninst\u271d\u00b9\u2075 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E\u2081 x)\ninst\u271d\u00b9\u2074 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\u2082\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082 F\u2082\nE\u2082 : B \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 AddCommGroup (E\u2082 x)\ninst\u271d\u00b9\u2070 : (x : B) \u2192 Module \ud835\udd5c\u2082 (E\u2082 x)\ninst\u271d\u2079 : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ninst\u271d\u2078 : TopologicalSpace B\ne\u2081 e\u2081' : Trivialization F\u2081 TotalSpace.proj\ne\u2082 e\u2082' : Trivialization F\u2082 TotalSpace.proj\ninst\u271d\u2077 : (x : B) \u2192 TopologicalSpace (E\u2081 x)\ninst\u271d\u2076 : FiberBundle F\u2081 E\u2081\ninst\u271d\u2075 : VectorBundle \ud835\udd5c\u2081 F\u2081 E\u2081\ninst\u271d\u2074 : (x : B) \u2192 TopologicalSpace (E\u2082 x)\ninst\u271d\u00b3 : FiberBundle F\u2082 E\u2082\ninst\u271d\u00b2 : VectorBundle \ud835\udd5c\u2082 F\u2082 E\u2082\ninst\u271d\u00b9 : \u2200 (x : B), TopologicalAddGroup (E\u2082 x)\ninst\u271d : \u2200 (x : B), ContinuousSMul \ud835\udd5c\u2082 (E\u2082 x)\nhe\u2081 : MemTrivializationAtlas e\u2081\nhe\u2082 : MemTrivializationAtlas e\u2082\n\u22a2 MemTrivializationAtlas e\u2082\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.VectorBundle.Hom", "llama_tokens": 38884, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.36296921241058616, "lm_q1q2_score": 0.2801554265148809}} {"text": "[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : Nonempty X\ninst\u271d\u00b9 : MeasurableSpace X\ns : Set X\ninst\u271d : HasCountableSeparatingOn X MeasurableSet s\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 s\nhgm : NullMeasurable g\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\n\u22a2 \u2203 c, g =\u1d50[\u03bc] const \u03b1 c\n[PROOFSTEP]\nrefine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : Nonempty X\ninst\u271d\u00b9 : MeasurableSpace X\ns : Set X\ninst\u271d : HasCountableSeparatingOn X MeasurableSet s\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 s\nhgm : NullMeasurable g\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\n\u22a2 \u2200 (U : Set X), MeasurableSet U \u2192 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 U) \u2228 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acg x \u2208 U\n[PROOFSTEP]\nrefine fun U hU \u21a6 h.ae_mem_or_ae_nmem\u2080 (s := g \u207b\u00b9' U) (hgm hU) ?_b\n[GOAL]\ncase _b\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : Nonempty X\ninst\u271d\u00b9 : MeasurableSpace X\ns : Set X\ninst\u271d : HasCountableSeparatingOn X MeasurableSet s\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 s\nhgm : NullMeasurable g\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\nU : Set X\nhU : MeasurableSet U\n\u22a2 f \u207b\u00b9' (g \u207b\u00b9' U) =\u1d50[\u03bc] g \u207b\u00b9' U\n[PROOFSTEP]\nrefine (hg_eq.mono fun x hx \u21a6 ?_).set_eq\n[GOAL]\ncase _b\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : Nonempty X\ninst\u271d\u00b9 : MeasurableSpace X\ns : Set X\ninst\u271d : HasCountableSeparatingOn X MeasurableSet s\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhs : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 s\nhgm : NullMeasurable g\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\nU : Set X\nhU : MeasurableSet U\nx : \u03b1\nhx : (g \u2218 f) x = g x\n\u22a2 x \u2208 f \u207b\u00b9' (g \u207b\u00b9' U) \u2194 x \u2208 g \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 preimage_comp, mem_preimage, mem_preimage, hx]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : Nonempty X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : HasCountableSeparatingOn X MeasurableSet univ\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : PreErgodic f\nhgm : Measurable g\nhg_eq : g \u2218 f = g\nU : Set X\nhU : MeasurableSet U\n\u22a2 f \u207b\u00b9' (g \u207b\u00b9' U) = g \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 preimage_comp, hg_eq]\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MetrizableSpace X\ninst\u271d : Nonempty X\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhgm : AEStronglyMeasurable g \u03bc\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\n\u22a2 \u2203 c, g =\u1d50[\u03bc] const \u03b1 c\n[PROOFSTEP]\nborelize X\n[GOAL]\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MetrizableSpace X\ninst\u271d : Nonempty X\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhgm : AEStronglyMeasurable g \u03bc\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\nthis\u271d\u00b9 : MeasurableSpace X := borel X\nthis\u271d : BorelSpace X\n\u22a2 \u2203 c, g =\u1d50[\u03bc] const \u03b1 c\n[PROOFSTEP]\nrcases hgm.isSeparable_ae_range with \u27e8t, ht, hgt\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MetrizableSpace X\ninst\u271d : Nonempty X\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhgm : AEStronglyMeasurable g \u03bc\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\nthis\u271d\u00b9 : MeasurableSpace X := borel X\nthis\u271d : BorelSpace X\nt : Set X\nht : IsSeparable t\nhgt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 t\n\u22a2 \u2203 c, g =\u1d50[\u03bc] const \u03b1 c\n[PROOFSTEP]\nhaveI := ht.secondCountableTopology\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : MetrizableSpace X\ninst\u271d : Nonempty X\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 X\nh : QuasiErgodic f\nhgm : AEStronglyMeasurable g \u03bc\nhg_eq : g \u2218 f =\u1d50[\u03bc] g\nthis\u271d\u00b9 : MeasurableSpace X := borel X\nthis\u271d : BorelSpace X\nt : Set X\nht : IsSeparable t\nhgt : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x \u2208 t\nthis : SecondCountableTopology \u2191t\n\u22a2 \u2203 c, g =\u1d50[\u03bc] const \u03b1 c\n[PROOFSTEP]\nexact h.ae_eq_const_of_ae_eq_comp_of_ae_range\u2080 hgt hgm.aemeasurable.nullMeasurable hg_eq\n", "meta": {"mathlib_filename": "Mathlib.Dynamics.Ergodic.Function", "llama_tokens": 1981, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.28014291746307257}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\n\u22a2 \u2191(expand R p) f = sum f fun e a => \u2191C a * (X ^ p) ^ e\n[PROOFSTEP]\nsimp [expand, eval\u2082]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nr : R\n\u22a2 \u2191(expand R p) (\u2191(monomial q) r) = \u2191(monomial (q * p)) r\n[PROOFSTEP]\nsimp_rw [\u2190 smul_X_eq_monomial, AlgHom.map_smul, AlgHom.map_pow, expand_X, mul_comm, pow_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nr : R\n\u22a2 \u2191(expand R p) (\u2191(expand R q) (\u2191C r)) = \u2191(expand R (p * q)) (\u2191C r)\n[PROOFSTEP]\nsimp_rw [expand_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf\u271d f g : R[X]\nihf : \u2191(expand R p) (\u2191(expand R q) f) = \u2191(expand R (p * q)) f\nihg : \u2191(expand R p) (\u2191(expand R q) g) = \u2191(expand R (p * q)) g\n\u22a2 \u2191(expand R p) (\u2191(expand R q) (f + g)) = \u2191(expand R (p * q)) (f + g)\n[PROOFSTEP]\nsimp_rw [AlgHom.map_add, ihf, ihg]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nn : \u2115\nr : R\nx\u271d : \u2191(expand R p) (\u2191(expand R q) (\u2191C r * X ^ n)) = \u2191(expand R (p * q)) (\u2191C r * X ^ n)\n\u22a2 \u2191(expand R p) (\u2191(expand R q) (\u2191C r * X ^ (n + 1))) = \u2191(expand R (p * q)) (\u2191C r * X ^ (n + 1))\n[PROOFSTEP]\nsimp_rw [AlgHom.map_mul, expand_C, AlgHom.map_pow, expand_X, AlgHom.map_pow, expand_X, pow_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\n\u22a2 \u2191(expand R 0) f = \u2191C (eval 1 f)\n[PROOFSTEP]\nsimp [expand]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nr : R\n\u22a2 \u2191(expand R 1) (\u2191C r) = \u2191C r\n[PROOFSTEP]\nrw [expand_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf\u271d f g : R[X]\nihf : \u2191(expand R 1) f = f\nihg : \u2191(expand R 1) g = g\n\u22a2 \u2191(expand R 1) (f + g) = f + g\n[PROOFSTEP]\nrw [AlgHom.map_add, ihf, ihg]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nn : \u2115\nr : R\nx\u271d : \u2191(expand R 1) (\u2191C r * X ^ n) = \u2191C r * X ^ n\n\u22a2 \u2191(expand R 1) (\u2191C r * X ^ (n + 1)) = \u2191C r * X ^ (n + 1)\n[PROOFSTEP]\nrw [AlgHom.map_mul, expand_C, AlgHom.map_pow, expand_X, pow_one]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\n\u22a2 \u2191(expand R (p ^ Nat.zero)) f = (\u2191(expand R p))^[Nat.zero] f\n[PROOFSTEP]\nrw [pow_zero, expand_one, Function.iterate_zero, id]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nn : \u2115\nih : \u2191(expand R (p ^ n)) f = (\u2191(expand R p))^[n] f\n\u22a2 \u2191(expand R (p ^ Nat.succ n)) f = (\u2191(expand R p))^[Nat.succ n] f\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', pow_succ, expand_mul, ih]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\n\u22a2 \u2191derivative (\u2191(expand R p) f) = \u2191(expand R p) (\u2191derivative f) * (\u2191p * X ^ (p - 1))\n[PROOFSTEP]\nrw [coe_expand, derivative_eval\u2082_C, derivative_pow, C_eq_nat_cast, derivative_X, mul_one]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191(expand R p) f) n = if p \u2223 n then coeff f (n / p) else 0\n[PROOFSTEP]\nsimp only [expand_eq_sum]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\n\u22a2 coeff (sum f fun e a => \u2191C a * (X ^ p) ^ e) n = if p \u2223 n then coeff f (n / p) else 0\n[PROOFSTEP]\nsimp_rw [coeff_sum, \u2190 pow_mul, C_mul_X_pow_eq_monomial, coeff_monomial, sum]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\n\u22a2 (\u2211 x in support f, if p * x = n then coeff f x else 0) = if p \u2223 n then coeff f (n / p) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\n\u22a2 (\u2211 x in support f, if p * x = n then coeff f x else 0) = coeff f (n / p)\n[PROOFSTEP]\nrw [Finset.sum_eq_single (n / p), Nat.mul_div_cancel' h, if_pos rfl]\n[GOAL]\ncase pos.h\u2080\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\n\u22a2 \u2200 (b : \u2115), b \u2208 support f \u2192 b \u2260 n / p \u2192 (if p * b = n then coeff f b else 0) = 0\n[PROOFSTEP]\nintro b _ hb2\n[GOAL]\ncase pos.h\u2080\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nb : \u2115\na\u271d : b \u2208 support f\nhb2 : b \u2260 n / p\n\u22a2 (if p * b = n then coeff f b else 0) = 0\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase pos.h\u2080.hnc\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nb : \u2115\na\u271d : b \u2208 support f\nhb2 : b \u2260 n / p\n\u22a2 \u00acp * b = n\n[PROOFSTEP]\nintro hb3\n[GOAL]\ncase pos.h\u2080.hnc\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nb : \u2115\na\u271d : b \u2208 support f\nhb2 : b \u2260 n / p\nhb3 : p * b = n\n\u22a2 False\n[PROOFSTEP]\napply hb2\n[GOAL]\ncase pos.h\u2080.hnc\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nb : \u2115\na\u271d : b \u2208 support f\nhb2 : b \u2260 n / p\nhb3 : p * b = n\n\u22a2 b = n / p\n[PROOFSTEP]\nrw [\u2190 hb3, Nat.mul_div_cancel_left b hp]\n[GOAL]\ncase pos.h\u2081\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\n\u22a2 \u00acn / p \u2208 support f \u2192 (if p * (n / p) = n then coeff f (n / p) else 0) = 0\n[PROOFSTEP]\nintro hn\n[GOAL]\ncase pos.h\u2081\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nhn : \u00acn / p \u2208 support f\n\u22a2 (if p * (n / p) = n then coeff f (n / p) else 0) = 0\n[PROOFSTEP]\nrw [not_mem_support_iff.1 hn]\n[GOAL]\ncase pos.h\u2081\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nhn : \u00acn / p \u2208 support f\n\u22a2 (if p * (n / p) = n then 0 else 0) = 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nhn : \u00acn / p \u2208 support f\nh\u271d : p * (n / p) = n\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : p \u2223 n\nhn : \u00acn / p \u2208 support f\nh\u271d : \u00acp * (n / p) = n\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : \u00acp \u2223 n\n\u22a2 (\u2211 x in support f, if p * x = n then coeff f x else 0) = 0\n[PROOFSTEP]\nrw [Finset.sum_eq_zero]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : \u00acp \u2223 n\n\u22a2 \u2200 (x : \u2115), x \u2208 support f \u2192 (if p * x = n then coeff f x else 0) = 0\n[PROOFSTEP]\nintro k _\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : \u00acp \u2223 n\nk : \u2115\na\u271d : k \u2208 support f\n\u22a2 (if p * k = n then coeff f k else 0) = 0\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase neg.hnc\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\nh : \u00acp \u2223 n\nk : \u2115\na\u271d : k \u2208 support f\n\u22a2 \u00acp * k = n\n[PROOFSTEP]\nexact fun hkn => h \u27e8k, hkn.symm\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191(expand R p) f) (n * p) = coeff f n\n[PROOFSTEP]\nrw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nn : \u2115\n\u22a2 coeff (\u2191(expand R p) f) (p * n) = coeff f n\n[PROOFSTEP]\nrw [mul_comm, coeff_expand_mul hp]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q n : \u2115\nhn : 0 < n\ng g' : R[X]\nH : \u2191(expand R n) g = \u2191(expand R n) g'\nk : \u2115\n\u22a2 coeff g k = coeff g' k\n[PROOFSTEP]\nrw [\u2190 coeff_expand_mul hn, H, coeff_expand_mul hn]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : 0 < p\nf : R[X]\nr : R\n\u22a2 \u2191(expand R p) f = \u2191C r \u2194 f = \u2191C r\n[PROOFSTEP]\nrw [\u2190 expand_C, expand_inj hp, expand_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\ncases' p.eq_zero_or_pos with hp hp\n[GOAL]\ncase inl\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p = 0\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\nrw [hp, coe_expand, pow_zero, mul_zero, \u2190 C_1, eval\u2082_hom, natDegree_C]\n[GOAL]\ncase inr\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\nby_cases hf : f = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : f = 0\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\nrw [hf, AlgHom.map_zero, natDegree_zero, zero_mul]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\nhave hf1 : expand R p f \u2260 0 := mt (expand_eq_zero hp).1 hf\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 natDegree (\u2191(expand R p) f) = natDegree f * p\n[PROOFSTEP]\nrw [\u2190 WithBot.coe_eq_coe]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 \u2191(natDegree (\u2191(expand R p) f)) = \u2191(natDegree f * p)\n[PROOFSTEP]\nconvert (degree_eq_natDegree hf1).symm\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 \u2191(natDegree f * p) = degree (\u2191(expand R p) f)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 degree (\u2191(expand R p) f) = \u2191(natDegree f * p)\n[PROOFSTEP]\nrefine' le_antisymm ((degree_le_iff_coeff_zero _ _).2 fun n hn => _) _\n[GOAL]\ncase h.e'_3.refine'_1\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhn : \u2191(natDegree f * p) < \u2191n\n\u22a2 coeff (\u2191(expand R p) f) n = 0\n[PROOFSTEP]\nrw [coeff_expand hp]\n[GOAL]\ncase h.e'_3.refine'_1\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhn : \u2191(natDegree f * p) < \u2191n\n\u22a2 (if p \u2223 n then coeff f (n / p) else 0) = 0\n[PROOFSTEP]\nsplit_ifs with hpn\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhn : \u2191(natDegree f * p) < \u2191n\nhpn : p \u2223 n\n\u22a2 coeff f (n / p) = 0\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhn : \u2191(natDegree f * p) < \u2191n\nhpn : p \u2223 n\n\u22a2 natDegree f < n / p\n[PROOFSTEP]\ncontrapose! hn\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhpn : p \u2223 n\nhn : n / p \u2264 natDegree f\n\u22a2 \u2191n \u2264 \u2191(natDegree f * p)\n[PROOFSTEP]\nerw [WithBot.coe_le_coe, \u2190 Nat.div_mul_cancel hpn]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhpn : p \u2223 n\nhn : n / p \u2264 natDegree f\n\u22a2 \u2191(n / p * p) \u2264 natDegree f * p\n[PROOFSTEP]\nexact Nat.mul_le_mul_right p hn\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\nn : \u2115\nhn : \u2191(natDegree f * p) < \u2191n\nhpn : \u00acp \u2223 n\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.refine'_2\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 \u2191(natDegree f * p) \u2264 degree (\u2191(expand R p) f)\n[PROOFSTEP]\nrefine' le_degree_of_ne_zero _\n[GOAL]\ncase h.e'_3.refine'_2\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 coeff (\u2191(expand R p) f) (Mul.mul (natDegree f) p) \u2260 0\n[PROOFSTEP]\nerw [coeff_expand_mul hp, \u2190 leadingCoeff]\n[GOAL]\ncase h.e'_3.refine'_2\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : p > 0\nhf : \u00acf = 0\nhf1 : \u2191(expand R p) f \u2260 0\n\u22a2 leadingCoeff f \u2260 0\n[PROOFSTEP]\nexact mt leadingCoeff_eq_zero.1 hf\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : 0 < p\nh : Monic f\n\u22a2 Monic (\u2191(Polynomial.expand R p) f)\n[PROOFSTEP]\nrw [Monic.def, Polynomial.leadingCoeff, natDegree_expand, coeff_expand hp]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nf : R[X]\nhp : 0 < p\nh : Monic f\n\u22a2 (if p \u2223 natDegree f * p then coeff f (natDegree f * p / p) else 0) = 1\n[PROOFSTEP]\nsimp [hp, h]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\n\u22a2 map f (\u2191(expand R p) q) = \u2191(expand S p) (map f q)\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : p = 0\n\u22a2 map f (\u2191(expand R p) q) = \u2191(expand S p) (map f q)\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : \u00acp = 0\n\u22a2 map f (\u2191(expand R p) q) = \u2191(expand S p) (map f q)\n[PROOFSTEP]\next\n[GOAL]\ncase neg.a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : \u00acp = 0\nn\u271d : \u2115\n\u22a2 coeff (map f (\u2191(expand R p) q)) n\u271d = coeff (\u2191(expand S p) (map f q)) n\u271d\n[PROOFSTEP]\nrw [coeff_map, coeff_expand (Nat.pos_of_ne_zero hp), coeff_expand (Nat.pos_of_ne_zero hp)]\n[GOAL]\ncase neg.a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : \u00acp = 0\nn\u271d : \u2115\n\u22a2 \u2191f (if p \u2223 n\u271d then coeff q (n\u271d / p) else 0) = if p \u2223 n\u271d then coeff (map f q) (n\u271d / p) else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : \u00acp = 0\nn\u271d : \u2115\nh\u271d : p \u2223 n\u271d\n\u22a2 \u2191f (if p \u2223 n\u271d then coeff q (n\u271d / p) else 0) = coeff (map f q) (n\u271d / p)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d p : \u2115\nf : R \u2192+* S\nq : R[X]\nhp : \u00acp = 0\nn\u271d : \u2115\nh\u271d : \u00acp \u2223 n\u271d\n\u22a2 \u2191f (if p \u2223 n\u271d then coeff q (n\u271d / p) else 0) = 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nP : R[X]\nr : R\n\u22a2 eval r (\u2191(expand R p) P) = eval (r ^ p) P\n[PROOFSTEP]\nrefine' Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => _) fun n a _ => by simp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nP : R[X]\nr a : R\n\u22a2 eval r (\u2191(expand R p) (\u2191C a)) = eval (r ^ p) (\u2191C a)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nP : R[X]\nr : R\nn : \u2115\na : R\nx\u271d : eval r (\u2191(expand R p) (\u2191C a * X ^ n)) = eval (r ^ p) (\u2191C a * X ^ n)\n\u22a2 eval r (\u2191(expand R p) (\u2191C a * X ^ (n + 1))) = eval (r ^ p) (\u2191C a * X ^ (n + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nP : R[X]\nr : R\nf g : R[X]\nhf : eval r (\u2191(expand R p) f) = eval (r ^ p) f\nhg : eval r (\u2191(expand R p) g) = eval (r ^ p) g\n\u22a2 eval r (\u2191(expand R p) (f + g)) = eval (r ^ p) (f + g)\n[PROOFSTEP]\nrw [AlgHom.map_add, eval_add, eval_add, hf, hg]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np\u271d q : \u2115\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : \u2115\nP : R[X]\nr : A\n\u22a2 \u2191(aeval r) (\u2191(expand R p) P) = \u2191(aeval (r ^ p)) P\n[PROOFSTEP]\nrefine' Polynomial.induction_on P (fun a => by simp) (fun f g hf hg => _) fun n a _ => by simp\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np\u271d q : \u2115\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : \u2115\nP : R[X]\nr : A\na : R\n\u22a2 \u2191(aeval r) (\u2191(expand R p) (\u2191C a)) = \u2191(aeval (r ^ p)) (\u2191C a)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np\u271d q : \u2115\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : \u2115\nP : R[X]\nr : A\nn : \u2115\na : R\nx\u271d : \u2191(aeval r) (\u2191(expand R p) (\u2191C a * X ^ n)) = \u2191(aeval (r ^ p)) (\u2191C a * X ^ n)\n\u22a2 \u2191(aeval r) (\u2191(expand R p) (\u2191C a * X ^ (n + 1))) = \u2191(aeval (r ^ p)) (\u2191C a * X ^ (n + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np\u271d q : \u2115\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : \u2115\nP : R[X]\nr : A\nf g : R[X]\nhf : \u2191(aeval r) (\u2191(expand R p) f) = \u2191(aeval (r ^ p)) f\nhg : \u2191(aeval r) (\u2191(expand R p) g) = \u2191(aeval (r ^ p)) g\n\u22a2 \u2191(aeval r) (\u2191(expand R p) (f + g)) = \u2191(aeval (r ^ p)) (f + g)\n[PROOFSTEP]\nrw [AlgHom.map_add, aeval_add, aeval_add, hf, hg]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\n\u22a2 coeff (contract p f) n = coeff f (n * p)\n[PROOFSTEP]\nsimp only [contract, coeff_monomial, sum_ite_eq', finset_sum_coeff, mem_range, not_lt, ite_eq_left_iff]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\n\u22a2 natDegree f + 1 \u2264 n \u2192 0 = coeff f (n * p)\n[PROOFSTEP]\nintro hn\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\nhn : natDegree f + 1 \u2264 n\n\u22a2 0 = coeff f (n * p)\n[PROOFSTEP]\napply (coeff_eq_zero_of_natDegree_lt _).symm\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\nhn : natDegree f + 1 \u2264 n\n\u22a2 natDegree f < n * p\n[PROOFSTEP]\ncalc\n f.natDegree < f.natDegree + 1 := Nat.lt_succ_self _\n _ \u2264 n * 1 := by simpa only [mul_one] using hn\n _ \u2264 n * p := mul_le_mul_of_nonneg_left (show 1 \u2264 p from hp.bot_lt) (zero_le n)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q p : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\nhn : natDegree f + 1 \u2264 n\n\u22a2 natDegree f + 1 \u2264 n * 1\n[PROOFSTEP]\nsimpa only [mul_one] using hn\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nhp : p \u2260 0\n\u22a2 contract p (\u2191(expand R p) f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nhp : p \u2260 0\nn\u271d : \u2115\n\u22a2 coeff (contract p (\u2191(expand R p) f)) n\u271d = coeff f n\u271d\n[PROOFSTEP]\nsimp [coeff_contract hp, coeff_expand hp.bot_lt, Nat.mul_div_cancel _ hp.bot_lt]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\n\u22a2 \u2191(expand R p) (contract p f) = f\n[PROOFSTEP]\next n\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\n\u22a2 coeff (\u2191(expand R p) (contract p f)) n = coeff f n\n[PROOFSTEP]\nrw [coeff_expand hp.bot_lt, coeff_contract hp]\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\n\u22a2 (if p \u2223 n then coeff f (n / p * p) else 0) = coeff f n\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : p \u2223 n\n\u22a2 coeff f (n / p * p) = coeff f n\n[PROOFSTEP]\nrw [Nat.div_mul_cancel h]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 n\n\u22a2 0 = coeff f n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase neg.zero\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nh : \u00acp \u2223 Nat.zero\n\u22a2 0 = coeff f Nat.zero\n[PROOFSTEP]\nexact absurd (dvd_zero p) h\n[GOAL]\ncase neg.succ\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nhave := coeff_derivative f n\n[GOAL]\ncase neg.succ\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nthis : coeff (\u2191derivative f) n = coeff f (n + 1) * (\u2191n + 1)\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nrw [hf, coeff_zero, zero_eq_mul] at this \n[GOAL]\ncase neg.succ\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nthis : coeff f (n + 1) = 0 \u2228 \u2191n + 1 = 0\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\ncases' this with h'\n[GOAL]\ncase neg.succ.inl\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nh' : coeff f (n + 1) = 0\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nrw [h']\n[GOAL]\ncase neg.succ.inr\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nh\u271d : \u2191n + 1 = 0\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nrename_i _ _ _ _ h'\n[GOAL]\ncase neg.succ.inr\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nh' : \u2191n + 1 = 0\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_succ, CharP.cast_eq_zero_iff R p] at h' \n[GOAL]\ncase neg.succ.inr\nR : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type v\ninst\u271d\u00b2 : CommSemiring S\np q : \u2115\ninst\u271d\u00b9 : CharP R p\ninst\u271d : NoZeroDivisors R\nf : R[X]\nhf : \u2191derivative f = 0\nhp : p \u2260 0\nn : \u2115\nh : \u00acp \u2223 Nat.succ n\nh' : p \u2223 Nat.succ n\n\u22a2 0 = coeff f (Nat.succ n)\n[PROOFSTEP]\nexact absurd h' h\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\n\u22a2 map (frobenius R p) (\u2191(expand R p) f) = f ^ p\n[PROOFSTEP]\nrefine' f.induction_on' (fun a b ha hb => _) fun n a => _\n[GOAL]\ncase refine'_1\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf a b : R[X]\nha : map (frobenius R p) (\u2191(expand R p) a) = a ^ p\nhb : map (frobenius R p) (\u2191(expand R p) b) = b ^ p\n\u22a2 map (frobenius R p) (\u2191(expand R p) (a + b)) = (a + b) ^ p\n[PROOFSTEP]\nrw [AlgHom.map_add, Polynomial.map_add, ha, hb, add_pow_char]\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn : \u2115\na : R\n\u22a2 map (frobenius R p) (\u2191(expand R p) (\u2191(monomial n) a)) = \u2191(monomial n) a ^ p\n[PROOFSTEP]\nrw [expand_monomial, map_monomial, \u2190 C_mul_X_pow_eq_monomial, \u2190 C_mul_X_pow_eq_monomial, mul_pow, \u2190 C.map_pow,\n frobenius_def]\n[GOAL]\ncase refine'_2\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn : \u2115\na : R\n\u22a2 \u2191C (a ^ p) * X ^ (n * p) = \u2191C (a ^ p) * (X ^ n) ^ p\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn : \u2115\n\u22a2 map (frobenius R p ^ n) (\u2191(expand R (p ^ n)) f) = f ^ p ^ n\n[PROOFSTEP]\ninduction' n with _ n_ih\n[GOAL]\ncase zero\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\n\u22a2 map (frobenius R p ^ Nat.zero) (\u2191(expand R (p ^ Nat.zero)) f) = f ^ p ^ Nat.zero\n[PROOFSTEP]\nsimp [RingHom.one_def]\n[GOAL]\ncase succ\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn\u271d : \u2115\nn_ih : map (frobenius R p ^ n\u271d) (\u2191(expand R (p ^ n\u271d)) f) = f ^ p ^ n\u271d\n\u22a2 map (frobenius R p ^ Nat.succ n\u271d) (\u2191(expand R (p ^ Nat.succ n\u271d)) f) = f ^ p ^ Nat.succ n\u271d\n[PROOFSTEP]\nsymm\n[GOAL]\ncase succ\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\np q : \u2115\ninst\u271d : CharP R p\nhp : Fact (Nat.Prime p)\nf : R[X]\nn\u271d : \u2115\nn_ih : map (frobenius R p ^ n\u271d) (\u2191(expand R (p ^ n\u271d)) f) = f ^ p ^ n\u271d\n\u22a2 f ^ p ^ Nat.succ n\u271d = map (frobenius R p ^ Nat.succ n\u271d) (\u2191(expand R (p ^ Nat.succ n\u271d)) f)\n[PROOFSTEP]\nrw [pow_succ', pow_mul, \u2190 n_ih, \u2190 expand_char, pow_succ, RingHom.mul_def, \u2190 map_map, mul_comm, expand_mul, \u2190 map_expand]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\n\u22a2 IsLocalRingHom \u2191(expand R p)\n[PROOFSTEP]\nrefine' \u27e8fun f hf1 => _\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (\u2191\u2191(expand R p) f)\n\u22a2 IsUnit f\n[PROOFSTEP]\nnorm_cast at hf1 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (\u2191(expand R p) f)\n\u22a2 IsUnit f\n[PROOFSTEP]\nhave hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit hf1)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (\u2191(expand R p) f)\nhf2 : \u2191(expand R p) f = \u2191C (coeff (\u2191(expand R p) f) 0)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [coeff_expand hp, if_pos (dvd_zero _), p.zero_div] at hf2 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (\u2191(expand R p) f)\nhf2 : \u2191(expand R p) f = \u2191C (coeff f 0)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [hf2, isUnit_C] at hf1 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : \u2191(expand R p) f = \u2191C (coeff f 0)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [expand_eq_C hp] at hf2 \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : 0 < p\nf : R[X]\nhf1 : IsUnit (coeff f 0)\nhf2 : f = \u2191C (coeff f 0)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrwa [hf2, isUnit_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : p \u2260 0\nf : R[X]\nn : \u2115\nhf : Irreducible (\u2191(expand R (p ^ Nat.zero)) f)\n\u22a2 Irreducible f\n[PROOFSTEP]\nrwa [pow_zero, expand_one] at hf \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : p \u2260 0\nf : R[X]\nn\u271d n : \u2115\nih : Irreducible (\u2191(expand R (p ^ n)) f) \u2192 Irreducible f\nhf : Irreducible (\u2191(expand R (p ^ Nat.succ n)) f)\n\u22a2 Irreducible (\u2191(expand R p) (\u2191(expand R (p ^ n)) f))\n[PROOFSTEP]\nrw [pow_succ] at hf \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : \u2115\nhp : p \u2260 0\nf : R[X]\nn\u271d n : \u2115\nih : Irreducible (\u2191(expand R (p ^ n)) f) \u2192 Irreducible f\nhf : Irreducible (\u2191(expand R (p * p ^ n)) f)\n\u22a2 Irreducible (\u2191(expand R p) (\u2191(expand R (p ^ n)) f))\n[PROOFSTEP]\nrwa [expand_expand]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Expand", "llama_tokens": 14970, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.28011547995408753}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x\n[PROOFSTEP]\nlet p := fun i : \u2115 => (Fix.approx f i x).Dom\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x\n[PROOFSTEP]\nhave : p (Nat.find h') := Nat.find_spec h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x\n[PROOFSTEP]\ngeneralize hk : Nat.find h' = k\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nreplace hk : Nat.find h' = k + (@Upto.zero p).val := hk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nrw [hk] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nrevert hk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 Part.fix f x = Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\ndsimp [Part.fix]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n (assert (\u2203 i, (Fix.approx f i x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x) =\n Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nrw [assert_pos h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\n\u22a2 p (k + \u2191Upto.zero) \u2192\n Nat.find h' = k + \u2191Upto.zero \u2192\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\ngeneralize Upto.zero = z\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n\u22a2 p (k + \u2191z) \u2192\n Nat.find h' = k + \u2191z \u2192\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nintro _this hk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n Fix.approx f (Nat.succ k) x\n[PROOFSTEP]\nsuffices : \u2200 x', WellFounded.fix (Part.fix.proof_1 f x h') (fixAux f) z x' = Fix.approx f (succ k) x'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\nthis :\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ k) x'\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n Fix.approx f (Nat.succ k) x\ncase this\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ k) x'\n[PROOFSTEP]\nexact this _\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ k) x'\n[PROOFSTEP]\ninduction k generalizing z with\n| zero =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext x : 1\n rw [assert_neg]\n rfl\n rw [Nat.zero_add] at _this \n simpa only [not_not, Coe]\n| succ n n_ih =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext : 1\n have hh : \u00ac(Fix.approx f z.val x).Dom := by\n apply Nat.find_min h'\n rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n apply Nat.lt_of_succ_le\n apply Nat.le_add_left\n rw [succ_add_eq_succ_add] at _this hk \n rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ k) x'\n[PROOFSTEP]\ninduction k generalizing z with\n| zero =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext x : 1\n rw [assert_neg]\n rfl\n rw [Nat.zero_add] at _this \n simpa only [not_not, Coe]\n| succ n n_ih =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext : 1\n have hh : \u00ac(Fix.approx f z.val x).Dom := by\n apply Nat.find_min h'\n rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n apply Nat.lt_of_succ_le\n apply Nat.le_add_left\n rw [succ_add_eq_succ_add] at _this hk \n rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n[GOAL]\ncase this.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ Nat.zero) x'\n[PROOFSTEP]\n\n| zero =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext x : 1\n rw [assert_neg]\n rfl\n rw [Nat.zero_add] at _this \n simpa only [not_not, Coe]\n[GOAL]\ncase this.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ Nat.zero) x'\n[PROOFSTEP]\nintro x'\n[GOAL]\ncase this.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ Nat.zero) x'\n[PROOFSTEP]\nrw [Fix.approx, WellFounded.fix_eq, fixAux]\n[GOAL]\ncase this.zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 f\n (fun x_1 =>\n assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n x_1)\n x' =\n f (Fix.approx f Nat.zero) x'\n[PROOFSTEP]\ncongr\n[GOAL]\ncase this.zero.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n x_1) =\n Fix.approx f Nat.zero\n[PROOFSTEP]\next x : 1\n[GOAL]\ncase this.zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x\u271d).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x => (Fix.approx f x x\u271d).Dom)) (fixAux f) (Upto.succ z h) x) =\n Fix.approx f Nat.zero x\n[PROOFSTEP]\nrw [assert_neg]\n[GOAL]\ncase this.zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 none = Fix.approx f Nat.zero x\ncase this.zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase this.zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom\n[PROOFSTEP]\nrw [Nat.zero_add] at _this \n[GOAL]\ncase this.zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p \u2191z\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom\n[PROOFSTEP]\nsimpa only [not_not, Coe]\n[GOAL]\ncase this.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ (Nat.succ n)) x'\n[PROOFSTEP]\n\n| succ n n_ih =>\n intro x'\n rw [Fix.approx, WellFounded.fix_eq, fixAux]\n congr\n ext : 1\n have hh : \u00ac(Fix.approx f z.val x).Dom := by\n apply Nat.find_min h'\n rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n apply Nat.lt_of_succ_le\n apply Nat.le_add_left\n rw [succ_add_eq_succ_add] at _this hk \n rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n[GOAL]\ncase this.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ (Nat.succ n)) x'\n[PROOFSTEP]\nintro x'\n[GOAL]\ncase this.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ (Nat.succ n)) x'\n[PROOFSTEP]\nrw [Fix.approx, WellFounded.fix_eq, fixAux]\n[GOAL]\ncase this.succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 f\n (fun x_1 =>\n assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n x_1)\n x' =\n f (Fix.approx f (n + 1)) x'\n[PROOFSTEP]\ncongr\n[GOAL]\ncase this.succ.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n x_1) =\n Fix.approx f (n + 1)\n[PROOFSTEP]\next : 1\n[GOAL]\ncase this.succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n Fix.approx f (n + 1) x\u271d\n[PROOFSTEP]\nhave hh : \u00ac(Fix.approx f z.val x).Dom := by\n apply Nat.find_min h'\n rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n apply Nat.lt_of_succ_le\n apply Nat.le_add_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u00ac(Fix.approx f (\u2191z) x).Dom\n[PROOFSTEP]\napply Nat.find_min h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < Nat.find h'\n[PROOFSTEP]\nrw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < n + Nat.succ \u2191z\n[PROOFSTEP]\napply Nat.lt_of_succ_le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 Nat.succ \u2191z \u2264 n + Nat.succ \u2191z\n[PROOFSTEP]\napply Nat.le_add_left\n[GOAL]\ncase this.succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n Fix.approx f (n + 1) x\u271d\n[PROOFSTEP]\nrw [succ_add_eq_succ_add] at _this hk \n[GOAL]\ncase this.succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n \u2200 (z : Upto p),\n p (n + \u2191z) \u2192\n Nat.find h' = n + \u2191z \u2192\n \u2200 (x' : \u03b1),\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (n + Nat.succ \u2191z)\nhk : Nat.find h' = n + Nat.succ \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n Fix.approx f (n + 1) x\u271d\n[PROOFSTEP]\nrw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u00ac\u2203 i, (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = none\n[PROOFSTEP]\ndsimp [Part.fix]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u00ac\u2203 i, (Fix.approx f i x).Dom\n\u22a2 (assert (\u2203 i, (Fix.approx f i x).Dom) fun h =>\n WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x) =\n none\n[PROOFSTEP]\nrw [assert_neg h']\n", "meta": {"mathlib_filename": "Mathlib.Control.Fix", "llama_tokens": 10440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.3738758227716966, "lm_q1q2_score": 0.280065598131026}} {"text": "[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\n\u22a2 1 = M.mk F { fst := j, snd := 1 }\n[PROOFSTEP]\napply M.mk_eq\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\n\u22a2 \u2203 k f g, \u2191(F.map f) { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd = \u2191(F.map g) { fst := j, snd := 1 }.snd\n[PROOFSTEP]\nrefine' \u27e8max' _ j, IsFiltered.leftToMax _ j, IsFiltered.rightToMax _ j, _\u27e9\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\n\u22a2 \u2191(F.map (IsFiltered.leftToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.snd =\n \u2191(F.map (IsFiltered.rightToMax { fst := Nonempty.some (_ : Nonempty J), snd := 1 }.fst j))\n { fst := j, snd := 1 }.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx x' y : (j : J) \u00d7 \u2191(F.obj j)\nhxx' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) x x'\n\u22a2 colimitMulAux F x y = colimitMulAux F x' y\n[PROOFSTEP]\ncases' x with j\u2081 x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx' y : (j : J) \u00d7 \u2191(F.obj j)\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nhxx' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := x } x'\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } y = colimitMulAux F x' y\n[PROOFSTEP]\ncases' y with j\u2082 y\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx' : (j : J) \u00d7 \u2191(F.obj j)\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nhxx' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := x } x'\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } { fst := j\u2082, snd := y } = colimitMulAux F x' { fst := j\u2082, snd := y }\n[PROOFSTEP]\ncases' x' with j\u2083 x'\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nhxx' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := x } { fst := j\u2083, snd := x' }\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } { fst := j\u2082, snd := y } =\n colimitMulAux F { fst := j\u2083, snd := x' } { fst := j\u2082, snd := y }\n[PROOFSTEP]\nobtain \u27e8l, f, g, hfg\u27e9 := hxx'\n[GOAL]\ncase mk.mk.mk.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : (F \u22d9 forget MonCat).map f { fst := j\u2081, snd := x }.snd = (F \u22d9 forget MonCat).map g { fst := j\u2083, snd := x' }.snd\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } { fst := j\u2082, snd := y } =\n colimitMulAux F { fst := j\u2083, snd := x' } { fst := j\u2082, snd := y }\n[PROOFSTEP]\nsimp at hfg \n[GOAL]\ncase mk.mk.mk.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } { fst := j\u2082, snd := y } =\n colimitMulAux F { fst := j\u2083, snd := x' } { fst := j\u2082, snd := y }\n[PROOFSTEP]\nobtain \u27e8s, \u03b1, \u03b2, \u03b3, h\u2081, h\u2082, h\u2083\u27e9 :=\n IsFiltered.tulip (IsFiltered.leftToMax j\u2081 j\u2082) (IsFiltered.rightToMax j\u2081 j\u2082) (IsFiltered.rightToMax j\u2083 j\u2082)\n (IsFiltered.leftToMax j\u2083 j\u2082) f g\n[GOAL]\ncase mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 colimitMulAux F { fst := j\u2081, snd := x } { fst := j\u2082, snd := y } =\n colimitMulAux F { fst := j\u2083, snd := x' } { fst := j\u2082, snd := y }\n[PROOFSTEP]\napply M.mk_eq\n[GOAL]\ncase mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2203 k f g,\n \u2191(F.map f)\n { fst := IsFiltered.max { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd =\n \u2191(F.map g)\n { fst := IsFiltered.max { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2083, snd := x' }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd\n[PROOFSTEP]\nuse s, \u03b1, \u03b3\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1)\n { fst := IsFiltered.max { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd =\n \u2191(F.map \u03b3)\n { fst := IsFiltered.max { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2083, snd := x' }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2083, snd := x' }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082)) x * \u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082)) y) =\n \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.leftToMax j\u2083 j\u2082)) x' * \u2191(F.map (IsFiltered.rightToMax j\u2083 j\u2082)) y)\n[PROOFSTEP]\nsimp_rw [MonoidHom.map_mul]\n -- Porting note : Lean cannot seem to use lemmas from concrete categories directly\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082)) x) * \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082)) y) =\n \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.leftToMax j\u2083 j\u2082)) x') * \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.rightToMax j\u2083 j\u2082)) y)\n[PROOFSTEP]\nchange (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _ = (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082) \u226b F.map \u03b1) x * \u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082) \u226b F.map \u03b1) y =\n \u2191(F.map (IsFiltered.leftToMax j\u2083 j\u2082) \u226b F.map \u03b3) x' * \u2191(F.map (IsFiltered.rightToMax j\u2083 j\u2082) \u226b F.map \u03b3) y\n[PROOFSTEP]\nsimp_rw [\u2190 F.map_comp, h\u2081, h\u2082, h\u2083, F.map_comp]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map f \u226b F.map \u03b2) x * \u2191(F.map (IsFiltered.rightToMax j\u2083 j\u2082) \u226b F.map \u03b3) y =\n \u2191(F.map g \u226b F.map \u03b2) x' * \u2191(F.map (IsFiltered.rightToMax j\u2083 j\u2082) \u226b F.map \u03b3) y\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map f \u226b F.map \u03b2) x = \u2191(F.map g \u226b F.map \u03b2) x'\n[PROOFSTEP]\nchange F.map _ (F.map _ _) = F.map _ (F.map _ _)\n[GOAL]\ncase h.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\nj\u2083 : J\nx' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := x }.fst \u27f6 l\ng : { fst := j\u2083, snd := x' }.fst \u27f6 l\nhfg : \u2191(F.map f) x = \u2191(F.map g) x'\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2083 j\u2082 \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = IsFiltered.rightToMax j\u2083 j\u2082 \u226b \u03b3\nh\u2083 : IsFiltered.leftToMax j\u2083 j\u2082 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b2) (\u2191(F.map f) x) = \u2191(F.map \u03b2) (\u2191(F.map g) x')\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y y' : (j : J) \u00d7 \u2191(F.obj j)\nhyy' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) y y'\n\u22a2 colimitMulAux F x y = colimitMulAux F x y'\n[PROOFSTEP]\ncases' y with j\u2081 y\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y' : (j : J) \u00d7 \u2191(F.obj j)\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nhyy' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := y } y'\n\u22a2 colimitMulAux F x { fst := j\u2081, snd := y } = colimitMulAux F x y'\n[PROOFSTEP]\ncases' x with j\u2082 x\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\ny' : (j : J) \u00d7 \u2191(F.obj j)\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nhyy' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := y } y'\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\n\u22a2 colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2081, snd := y } = colimitMulAux F { fst := j\u2082, snd := x } y'\n[PROOFSTEP]\ncases' y' with j\u2083 y'\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nhyy' : Types.FilteredColimit.Rel (F \u22d9 forget MonCat) { fst := j\u2081, snd := y } { fst := j\u2083, snd := y' }\n\u22a2 colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2081, snd := y } =\n colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2083, snd := y' }\n[PROOFSTEP]\nobtain \u27e8l, f, g, hfg\u27e9 := hyy'\n[GOAL]\ncase mk.mk.mk.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : (F \u22d9 forget MonCat).map f { fst := j\u2081, snd := y }.snd = (F \u22d9 forget MonCat).map g { fst := j\u2083, snd := y' }.snd\n\u22a2 colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2081, snd := y } =\n colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2083, snd := y' }\n[PROOFSTEP]\nsimp at hfg \n[GOAL]\ncase mk.mk.mk.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\n\u22a2 colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2081, snd := y } =\n colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2083, snd := y' }\n[PROOFSTEP]\nobtain \u27e8s, \u03b1, \u03b2, \u03b3, h\u2081, h\u2082, h\u2083\u27e9 :=\n IsFiltered.tulip (IsFiltered.rightToMax j\u2082 j\u2081) (IsFiltered.leftToMax j\u2082 j\u2081) (IsFiltered.leftToMax j\u2082 j\u2083)\n (IsFiltered.rightToMax j\u2082 j\u2083) f g\n[GOAL]\ncase mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2081, snd := y } =\n colimitMulAux F { fst := j\u2082, snd := x } { fst := j\u2083, snd := y' }\n[PROOFSTEP]\napply M.mk_eq\n[GOAL]\ncase mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2203 k f g,\n \u2191(F.map f)\n { fst := IsFiltered.max { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst))\n { fst := j\u2082, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst))\n { fst := j\u2081, snd := y }.snd }.snd =\n \u2191(F.map g)\n { fst := IsFiltered.max { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst))\n { fst := j\u2082, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst))\n { fst := j\u2083, snd := y' }.snd }.snd\n[PROOFSTEP]\nuse s, \u03b1, \u03b3\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1)\n { fst := IsFiltered.max { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst))\n { fst := j\u2082, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2082, snd := x }.fst { fst := j\u2081, snd := y }.fst))\n { fst := j\u2081, snd := y }.snd }.snd =\n \u2191(F.map \u03b3)\n { fst := IsFiltered.max { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst))\n { fst := j\u2082, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2082, snd := x }.fst { fst := j\u2083, snd := y' }.fst))\n { fst := j\u2083, snd := y' }.snd }.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2081)) x * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2081)) y) =\n \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) x * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) y')\n[PROOFSTEP]\nsimp_rw [MonoidHom.map_mul]\n -- Porting note : Lean cannot seem to use lemmas from concrete categories directly\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2081)) x) * \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2081)) y) =\n \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) x) * \u2191(F.map \u03b3) (\u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) y')\n[PROOFSTEP]\nchange (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _ = (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2081) \u226b F.map \u03b1) x * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2081) \u226b F.map \u03b1) y =\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083) \u226b F.map \u03b3) x * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083) \u226b F.map \u03b3) y'\n[PROOFSTEP]\nsimp_rw [\u2190 F.map_comp, h\u2081, h\u2082, h\u2083, F.map_comp]\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083) \u226b F.map \u03b3) x * \u2191(F.map f \u226b F.map \u03b2) y =\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083) \u226b F.map \u03b3) x * \u2191(F.map g \u226b F.map \u03b2) y'\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map f \u226b F.map \u03b2) y = \u2191(F.map g \u226b F.map \u03b2) y'\n[PROOFSTEP]\nchange F.map _ (F.map _ _) = F.map _ (F.map _ _)\n[GOAL]\ncase h.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj\u2081 : J\ny : \u2191(F.obj j\u2081)\nj\u2082 : J\nx : \u2191(F.obj j\u2082)\nj\u2083 : J\ny' : \u2191(F.obj j\u2083)\nl : J\nf : { fst := j\u2081, snd := y }.fst \u27f6 l\ng : { fst := j\u2083, snd := y' }.fst \u27f6 l\nhfg : \u2191(F.map f) y = \u2191(F.map g) y'\ns : J\n\u03b1 : IsFiltered.max j\u2082 j\u2081 \u27f6 s\n\u03b2 : l \u27f6 s\n\u03b3 : IsFiltered.max j\u2082 j\u2083 \u27f6 s\nh\u2081 : IsFiltered.rightToMax j\u2082 j\u2081 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.leftToMax j\u2082 j\u2081 \u226b \u03b1 = IsFiltered.leftToMax j\u2082 j\u2083 \u226b \u03b3\nh\u2083 : IsFiltered.rightToMax j\u2082 j\u2083 \u226b \u03b3 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b2) (\u2191(F.map f) y) = \u2191(F.map \u03b2) (\u2191(F.map g) y')\n[PROOFSTEP]\nrw [hfg]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y : M F\n\u22a2 M F\n[PROOFSTEP]\nrefine' Quot.lift\u2082 (colimitMulAux F) _ _ x y\n[GOAL]\ncase refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y : M F\n\u22a2 \u2200 (a b\u2081 b\u2082 : (j : J) \u00d7 \u2191(F.obj j)),\n Types.Quot.Rel (F \u22d9 forget MonCat) b\u2081 b\u2082 \u2192 colimitMulAux F a b\u2081 = colimitMulAux F a b\u2082\n[PROOFSTEP]\nintro x y y' h\n[GOAL]\ncase refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx y y' : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) y y'\n\u22a2 colimitMulAux F x y = colimitMulAux F x y'\n[PROOFSTEP]\napply colimitMulAux_eq_of_rel_right\n[GOAL]\ncase refine'_1.hyy'\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx y y' : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) y y'\n\u22a2 Types.FilteredColimit.Rel (F \u22d9 forget MonCat) y y'\n[PROOFSTEP]\napply Types.FilteredColimit.rel_of_quot_rel\n[GOAL]\ncase refine'_1.hyy'.a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx y y' : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) y y'\n\u22a2 Types.Quot.Rel (F \u22d9 forget MonCat) y y'\n[PROOFSTEP]\nexact h\n[GOAL]\ncase refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y : M F\n\u22a2 \u2200 (a\u2081 a\u2082 b : (j : J) \u00d7 \u2191(F.obj j)),\n Types.Quot.Rel (F \u22d9 forget MonCat) a\u2081 a\u2082 \u2192 colimitMulAux F a\u2081 b = colimitMulAux F a\u2082 b\n[PROOFSTEP]\nintro x x' y h\n[GOAL]\ncase refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx x' y : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) x x'\n\u22a2 colimitMulAux F x y = colimitMulAux F x' y\n[PROOFSTEP]\napply colimitMulAux_eq_of_rel_left\n[GOAL]\ncase refine'_2.hxx'\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx x' y : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) x x'\n\u22a2 Types.FilteredColimit.Rel (F \u22d9 forget MonCat) x x'\n[PROOFSTEP]\napply Types.FilteredColimit.rel_of_quot_rel\n[GOAL]\ncase refine'_2.hxx'.a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx\u271d y\u271d : M F\nx x' y : (j : J) \u00d7 \u2191(F.obj j)\nh : Types.Quot.Rel (F \u22d9 forget MonCat) x x'\n\u22a2 Types.Quot.Rel (F \u22d9 forget MonCat) x x'\n[PROOFSTEP]\nexact h\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nx y : (j : J) \u00d7 \u2191(F.obj j)\nk : J\nf : x.fst \u27f6 k\ng : y.fst \u27f6 k\n\u22a2 M.mk F x * M.mk F y = M.mk F { fst := k, snd := \u2191(F.map f) x.snd * \u2191(F.map g) y.snd }\n[PROOFSTEP]\ncases' x with j\u2081 x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\ny : (j : J) \u00d7 \u2191(F.obj j)\nk : J\ng : y.fst \u27f6 k\nj\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\n\u22a2 M.mk F { fst := j\u2081, snd := x } * M.mk F y =\n M.mk F { fst := k, snd := \u2191(F.map f) { fst := j\u2081, snd := x }.snd * \u2191(F.map g) y.snd }\n[PROOFSTEP]\ncases' y with j\u2082 y\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\n\u22a2 M.mk F { fst := j\u2081, snd := x } * M.mk F { fst := j\u2082, snd := y } =\n M.mk F { fst := k, snd := \u2191(F.map f) { fst := j\u2081, snd := x }.snd * \u2191(F.map g) { fst := j\u2082, snd := y }.snd }\n[PROOFSTEP]\nobtain \u27e8s, \u03b1, \u03b2, h\u2081, h\u2082\u27e9 := IsFiltered.bowtie (IsFiltered.leftToMax j\u2081 j\u2082) f (IsFiltered.rightToMax j\u2081 j\u2082) g\n[GOAL]\ncase mk.mk.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 M.mk F { fst := j\u2081, snd := x } * M.mk F { fst := j\u2082, snd := y } =\n M.mk F { fst := k, snd := \u2191(F.map f) { fst := j\u2081, snd := x }.snd * \u2191(F.map g) { fst := j\u2082, snd := y }.snd }\n[PROOFSTEP]\napply M.mk_eq\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 \u2203 k_1 f_1 g_1,\n \u2191(F.map f_1)\n { fst := IsFiltered.max { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd =\n \u2191(F.map g_1)\n { fst := k, snd := \u2191(F.map f) { fst := j\u2081, snd := x }.snd * \u2191(F.map g) { fst := j\u2082, snd := y }.snd }.snd\n[PROOFSTEP]\nuse s, \u03b1, \u03b2\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1)\n { fst := IsFiltered.max { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst { fst := j\u2082, snd := y }.fst))\n { fst := j\u2082, snd := y }.snd }.snd =\n \u2191(F.map \u03b2) { fst := k, snd := \u2191(F.map f) { fst := j\u2081, snd := x }.snd * \u2191(F.map g) { fst := j\u2082, snd := y }.snd }.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082)) x * \u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082)) y) =\n \u2191(F.map \u03b2) (\u2191(F.map f) x * \u2191(F.map g) y)\n[PROOFSTEP]\nsimp_rw [MonoidHom.map_mul]\n -- Porting note : Lean cannot seem to use lemmas from concrete categories directly\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082)) x) * \u2191(F.map \u03b1) (\u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082)) y) =\n \u2191(F.map \u03b2) (\u2191(F.map f) x) * \u2191(F.map \u03b2) (\u2191(F.map g) y)\n[PROOFSTEP]\nchange (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _ = (F.map _ \u226b F.map _) _ * (F.map _ \u226b F.map _) _\n[GOAL]\ncase h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nk j\u2081 : J\nx : \u2191(F.obj j\u2081)\nf : { fst := j\u2081, snd := x }.fst \u27f6 k\nj\u2082 : J\ny : \u2191(F.obj j\u2082)\ng : { fst := j\u2082, snd := y }.fst \u27f6 k\ns : J\n\u03b1 : IsFiltered.max j\u2081 j\u2082 \u27f6 s\n\u03b2 : k \u27f6 s\nh\u2081 : IsFiltered.leftToMax j\u2081 j\u2082 \u226b \u03b1 = f \u226b \u03b2\nh\u2082 : IsFiltered.rightToMax j\u2081 j\u2082 \u226b \u03b1 = g \u226b \u03b2\n\u22a2 \u2191(F.map (IsFiltered.leftToMax j\u2081 j\u2082) \u226b F.map \u03b1) x * \u2191(F.map (IsFiltered.rightToMax j\u2081 j\u2082) \u226b F.map \u03b1) y =\n \u2191(F.map f \u226b F.map \u03b2) x * \u2191(F.map g \u226b F.map \u03b2) y\n[PROOFSTEP]\nsimp_rw [\u2190 F.map_comp, h\u2081, h\u2082]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx : M F\n\u22a2 1 * x = x\n[PROOFSTEP]\nrefine Quot.inductionOn x ?_\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx : M F\n\u22a2 \u2200 (a : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n 1 * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a = Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a\n[PROOFSTEP]\nintro x\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nx : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\n\u22a2 1 * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x = Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nj : J\nx : (F \u22d9 forget MonCat).obj j\n\u22a2 1 * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x } =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x }\n[PROOFSTEP]\nrw [colimit_one_eq F j, colimit_mul_mk_eq F \u27e8j, 1\u27e9 \u27e8j, x\u27e9 j (\ud835\udfd9 j) (\ud835\udfd9 j), MonoidHom.map_one, one_mul, F.map_id]\n -- Porting note : `id_apply` does not work here, but the two sides are def-eq\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nj : J\nx : (F \u22d9 forget MonCat).obj j\n\u22a2 M.mk F { fst := j, snd := \u2191(\ud835\udfd9 (F.obj { fst := j, snd := x }.fst)) { fst := j, snd := x }.snd } =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x }\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx : M F\n\u22a2 x * 1 = x\n[PROOFSTEP]\nrefine Quot.inductionOn x ?_\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx : M F\n\u22a2 \u2200 (a : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a * 1 = Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a\n[PROOFSTEP]\nintro x\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nx : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x * 1 = Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x\n[PROOFSTEP]\ncases' x with j x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nj : J\nx : (F \u22d9 forget MonCat).obj j\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x } * 1 =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x }\n[PROOFSTEP]\nrw [colimit_one_eq F j, colimit_mul_mk_eq F \u27e8j, x\u27e9 \u27e8j, 1\u27e9 j (\ud835\udfd9 j) (\ud835\udfd9 j), MonoidHom.map_one, mul_one, F.map_id]\n -- Porting note : `id_apply` does not work here, but the two sides are def-eq\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d\u00b9 : One (M F) := colimitOne F\nsrc\u271d : Mul (M F) := colimitMul F\nx\u271d : M F\nj : J\nx : (F \u22d9 forget MonCat).obj j\n\u22a2 M.mk F { fst := j, snd := \u2191(\ud835\udfd9 (F.obj { fst := j, snd := x }.fst)) { fst := j, snd := x }.snd } =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := x }\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nx y z : M F\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\nrefine Quot.induction_on\u2083 x y z ?_\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nx y z : M F\n\u22a2 \u2200 (a b c : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) c =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) c)\n[PROOFSTEP]\nclear x y z\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\n\u22a2 \u2200 (a b c : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) c =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) c)\n[PROOFSTEP]\nintro x y z\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nx y z : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z)\n[PROOFSTEP]\ncases' x with j\u2081 x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\ny z : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z)\n[PROOFSTEP]\ncases' y with j\u2082 y\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nz : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2082, snd := y } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2082, snd := y } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) z)\n[PROOFSTEP]\ncases' z with j\u2083 z\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2082, snd := y } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2083, snd := z } =\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2081, snd := x } *\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2082, snd := y } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j\u2083, snd := z })\n[PROOFSTEP]\nchange M.mk F _ * M.mk F _ * M.mk F _ = M.mk F _ * M.mk F _\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 M.mk F { fst := j\u2081, snd := x } * M.mk F { fst := j\u2082, snd := y } * M.mk F { fst := j\u2083, snd := z } =\n M.mk F { fst := j\u2081, snd := x } *\n M.mk F\n { fst := IsFiltered.max { fst := j\u2082, snd := y }.fst { fst := j\u2083, snd := z }.fst,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax { fst := j\u2082, snd := y }.fst { fst := j\u2083, snd := z }.fst))\n { fst := j\u2082, snd := y }.snd *\n \u2191(F.map (IsFiltered.rightToMax { fst := j\u2082, snd := y }.fst { fst := j\u2083, snd := z }.fst))\n { fst := j\u2083, snd := z }.snd }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 M.mk F { fst := j\u2081, snd := x } * M.mk F { fst := j\u2082, snd := y } * M.mk F { fst := j\u2083, snd := z } =\n M.mk F { fst := j\u2081, snd := x } *\n M.mk F\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }\n[PROOFSTEP]\nrw [colimit_mul_mk_eq F \u27e8j\u2081, x\u27e9 \u27e8j\u2082, y\u27e9 (IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083))\n (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)) (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax _ _),\n colimit_mul_mk_eq F \u27e8(IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083)), _\u27e9 \u27e8j\u2083, z\u27e9 (IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083))\n (\ud835\udfd9 _) (IsFiltered.rightToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax _ _),\n colimit_mul_mk_eq.{v, u} F \u27e8j\u2081, x\u27e9 \u27e8IsFiltered.max j\u2082 j\u2083, _\u27e9 _ (IsFiltered.leftToMax _ _) (IsFiltered.rightToMax _ _)]\n[GOAL]\ncase mk.mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 M.mk F\n { fst := IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083),\n snd :=\n \u2191(F.map\n (\ud835\udfd9\n { fst := IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083),\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2082, snd := y }.snd }.fst))\n { fst := IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083),\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2082, snd := y }.snd }.snd *\n \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2083, snd := z }.snd } =\n M.mk F\n {\n fst :=\n IsFiltered.max { fst := j\u2081, snd := x }.fst\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.fst,\n snd :=\n \u2191(F.map\n (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map\n (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.fst))\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.snd }\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase mk.mk.mk.e_a.e_snd\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 \u2191(F.map\n (\ud835\udfd9\n { fst := IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083),\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2082, snd := y }.snd }.fst))\n { fst := IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083),\n snd :=\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) { fst := j\u2081, snd := x }.snd *\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2082, snd := y }.snd }.snd *\n \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n { fst := j\u2083, snd := z }.snd =\n \u2191(F.map\n (IsFiltered.leftToMax { fst := j\u2081, snd := x }.fst\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.fst))\n { fst := j\u2081, snd := x }.snd *\n \u2191(F.map\n (IsFiltered.rightToMax { fst := j\u2081, snd := x }.fst\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.fst))\n { fst := IsFiltered.max j\u2082 j\u2083,\n snd := \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z }.snd\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.mk.mk.e_a.e_snd\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 \u2191(F.map (\ud835\udfd9 (IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083))))\n (\u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) x *\n \u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) y) *\n \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083 \u226b IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) z =\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) x *\n \u2191(F.map (IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083)))\n (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y * \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z)\n[PROOFSTEP]\nrw [F.map_id, show \u2200 x, (\ud835\udfd9 (F.obj (IsFiltered.max j\u2081 (IsFiltered.max j\u2082 j\u2083)))) x = x from fun _ => rfl, mul_assoc,\n MonoidHom.map_mul, F.map_comp, F.map_comp]\n[GOAL]\ncase mk.mk.mk.e_a.e_snd\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nsrc\u271d : MulOneClass (M F) := colimitMulOneClass F\nj\u2081 : J\nx : (F \u22d9 forget MonCat).obj j\u2081\nj\u2082 : J\ny : (F \u22d9 forget MonCat).obj j\u2082\nj\u2083 : J\nz : (F \u22d9 forget MonCat).obj j\u2083\n\u22a2 \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) x *\n (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083) \u226b F.map (IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) y *\n \u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083) \u226b F.map (IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) z) =\n \u2191(F.map (IsFiltered.leftToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) x *\n (\u2191(F.map (IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) (\u2191(F.map (IsFiltered.leftToMax j\u2082 j\u2083)) y) *\n \u2191(F.map (IsFiltered.rightToMax j\u2081 (IsFiltered.max j\u2082 j\u2083))) (\u2191(F.map (IsFiltered.rightToMax j\u2082 j\u2083)) z))\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\nx y : \u2191(F.obj j)\n\u22a2 OneHom.toFun\n { toFun := NatTrans.app (Types.colimitCocone (F \u22d9 forget MonCat)).\u03b9 j,\n map_one' := (_ : M.mk F { fst := j, snd := 1 } = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := NatTrans.app (Types.colimitCocone (F \u22d9 forget MonCat)).\u03b9 j,\n map_one' := (_ : M.mk F { fst := j, snd := 1 } = 1) }\n x *\n OneHom.toFun\n { toFun := NatTrans.app (Types.colimitCocone (F \u22d9 forget MonCat)).\u03b9 j,\n map_one' := (_ : M.mk F { fst := j, snd := 1 } = 1) }\n y\n[PROOFSTEP]\nconvert (colimit_mul_mk_eq.{v, u} F \u27e8j, x\u27e9 \u27e8j, y\u27e9 j (\ud835\udfd9 j) (\ud835\udfd9 j)).symm\n[GOAL]\ncase h.e'_2.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\nx y : \u2191(F.obj j)\ne_1\u271d : \u2191(colimit F) = M F\n\u22a2 OneHom.toFun\n { toFun := NatTrans.app (Types.colimitCocone (F \u22d9 forget MonCat)).\u03b9 j,\n map_one' := (_ : M.mk F { fst := j, snd := 1 } = 1) }\n (x * y) =\n M.mk F { fst := j, snd := \u2191(F.map (\ud835\udfd9 j)) { fst := j, snd := x }.snd * \u2191(F.map (\ud835\udfd9 j)) { fst := j, snd := y }.snd }\n[PROOFSTEP]\nrw [F.map_id]\n[GOAL]\ncase h.e'_2.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nj : J\nx y : \u2191(F.obj j)\ne_1\u271d : \u2191(colimit F) = M F\n\u22a2 OneHom.toFun\n { toFun := NatTrans.app (Types.colimitCocone (F \u22d9 forget MonCat)).\u03b9 j,\n map_one' := (_ : M.mk F { fst := j, snd := 1 } = 1) }\n (x * y) =\n M.mk F\n { fst := j,\n snd :=\n \u2191(\ud835\udfd9 (F.obj { fst := j, snd := x }.fst)) { fst := j, snd := x }.snd *\n \u2191(\ud835\udfd9 (F.obj { fst := j, snd := x }.fst)) { fst := j, snd := y }.snd }\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\n\u22a2 IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1\n[PROOFSTEP]\nrw [colimit_one_eq F IsFiltered.Nonempty.some]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\n\u22a2 IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t)\n (M.mk F { fst := Nonempty.some (_ : Nonempty J), snd := 1 }) =\n 1\n[PROOFSTEP]\nexact MonoidHom.map_one _\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\nx y : \u2191(colimit F)\n\u22a2 OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ : IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n x *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n y\n[PROOFSTEP]\nrefine Quot.induction_on\u2082 x y ?_\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\nx y : \u2191(colimit F)\n\u22a2 \u2200 (a b : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b)\n[PROOFSTEP]\nclear x y\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\n\u22a2 \u2200 (a b : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j),\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) a) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) b)\n[PROOFSTEP]\nintro x y\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\nx y : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\n\u22a2 OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ : IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x * Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) x) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y)\n[PROOFSTEP]\ncases' x with i x\n[GOAL]\ncase mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ny : (j : J) \u00d7 (F \u22d9 forget MonCat).obj j\ni : J\nx : (F \u22d9 forget MonCat).obj i\n\u22a2 OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ : IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := i, snd := x } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := i, snd := x }) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) y)\n[PROOFSTEP]\ncases' y with j y\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ : IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := i, snd := x } *\n Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := y }) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := i, snd := x }) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := y })\n[PROOFSTEP]\nrw [colimit_mul_mk_eq F \u27e8i, x\u27e9 \u27e8j, y\u27e9 (max' i j) (IsFiltered.leftToMax i j) (IsFiltered.rightToMax i j)]\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ : IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (M.mk F\n { fst := IsFiltered.max i j,\n snd :=\n \u2191(F.map (IsFiltered.leftToMax i j)) { fst := i, snd := x }.snd *\n \u2191(F.map (IsFiltered.rightToMax i j)) { fst := j, snd := y }.snd }) =\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := i, snd := x }) *\n OneHom.toFun\n { toFun := IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t),\n map_one' :=\n (_ :\n IsColimit.desc (Types.colimitCoconeIsColimit (F \u22d9 forget MonCat)) ((forget MonCat).mapCocone t) 1 = 1) }\n (Quot.mk (Types.Quot.Rel (F \u22d9 forget MonCat)) { fst := j, snd := y })\n[PROOFSTEP]\ndsimp [Types.colimitCoconeIsColimit]\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 \u2191(NatTrans.app t.\u03b9 (IsFiltered.max i j))\n (\u2191(F.map (IsFiltered.leftToMax i j)) x * \u2191(F.map (IsFiltered.rightToMax i j)) y) =\n \u2191(NatTrans.app t.\u03b9 i) x * \u2191(NatTrans.app t.\u03b9 j) y\n[PROOFSTEP]\nrw [MonoidHom.map_mul]\n -- Porting note : `rw` can't see through coercion is actually forgetful functor,\n -- so can't rewrite `t.w_apply`\n[GOAL]\ncase mk.mk\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 \u2191(NatTrans.app t.\u03b9 (IsFiltered.max i j)) (\u2191(F.map (IsFiltered.leftToMax i j)) x) *\n \u2191(NatTrans.app t.\u03b9 (IsFiltered.max i j)) (\u2191(F.map (IsFiltered.rightToMax i j)) y) =\n \u2191(NatTrans.app t.\u03b9 i) x * \u2191(NatTrans.app t.\u03b9 j) y\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mk.mk.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 \u2191(NatTrans.app t.\u03b9 (IsFiltered.max i j)) (\u2191(F.map (IsFiltered.leftToMax i j)) x) = \u2191(NatTrans.app t.\u03b9 i) x\n[PROOFSTEP]\nexact t.w_apply _ _\n[GOAL]\ncase mk.mk.e_a\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 MonCat\ninst\u271d : IsFiltered J\nt : Cocone F\ni : J\nx : (F \u22d9 forget MonCat).obj i\nj : J\ny : (F \u22d9 forget MonCat).obj j\n\u22a2 \u2191(NatTrans.app t.\u03b9 (IsFiltered.max i j)) (\u2191(F.map (IsFiltered.rightToMax i j)) y) = \u2191(NatTrans.app t.\u03b9 j) y\n[PROOFSTEP]\nexact t.w_apply _ _\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : \u2191(M F)\n\u22a2 x * y = y * x\n[PROOFSTEP]\nrefine Quot.induction_on\u2082 x y ?_\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : \u2191(M F)\n\u22a2 \u2200 (a b : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j),\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) a *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) b =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) b *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) a\n[PROOFSTEP]\nclear x y\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\n\u22a2 \u2200 (a b : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j),\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) a *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) b =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) b *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) a\n[PROOFSTEP]\nintro x y\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\n\u22a2 Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x\n[PROOFSTEP]\nlet k := max' x.1 y.1\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\nk : J := IsFiltered.max x.fst y.fst\n\u22a2 Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x\n[PROOFSTEP]\nlet f := IsFiltered.leftToMax x.1 y.1\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\nk : J := IsFiltered.max x.fst y.fst\nf : x.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.leftToMax x.fst y.fst\n\u22a2 Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x\n[PROOFSTEP]\nlet g := IsFiltered.rightToMax x.1 y.1\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\nk : J := IsFiltered.max x.fst y.fst\nf : x.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.leftToMax x.fst y.fst\ng : y.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.rightToMax x.fst y.fst\n\u22a2 Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y =\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) y *\n Quot.mk (Types.Quot.Rel ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat)) x\n[PROOFSTEP]\nrw [colimit_mul_mk_eq.{v, u} (F \u22d9 forget\u2082 CommMonCat MonCat) x y k f g,\n colimit_mul_mk_eq.{v, u} (F \u22d9 forget\u2082 CommMonCat MonCat) y x k g f]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\nk : J := IsFiltered.max x.fst y.fst\nf : x.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.leftToMax x.fst y.fst\ng : y.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.rightToMax x.fst y.fst\n\u22a2 MonCat.FilteredColimits.M.mk (F \u22d9 forget\u2082 CommMonCat MonCat)\n { fst := k,\n snd := \u2191((F \u22d9 forget\u2082 CommMonCat MonCat).map f) x.snd * \u2191((F \u22d9 forget\u2082 CommMonCat MonCat).map g) y.snd } =\n MonCat.FilteredColimits.M.mk (F \u22d9 forget\u2082 CommMonCat MonCat)\n { fst := k,\n snd := \u2191((F \u22d9 forget\u2082 CommMonCat MonCat).map g) y.snd * \u2191((F \u22d9 forget\u2082 CommMonCat MonCat).map f) x.snd }\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF : J \u2964 CommMonCat\nsrc\u271d : MonCat := M F\nx y : (j : J) \u00d7 ((F \u22d9 forget\u2082 CommMonCat MonCat) \u22d9 forget MonCat).obj j\nk : J := IsFiltered.max x.fst y.fst\nf : x.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.leftToMax x.fst y.fst\ng : y.fst \u27f6 IsFiltered.max x.fst y.fst := IsFiltered.rightToMax x.fst y.fst\n\u22a2 MonCat.FilteredColimits.M.mk (F \u22d9 forget\u2082 CommMonCat MonCat)\n { fst := IsFiltered.max x.fst y.fst,\n snd :=\n \u2191((forget\u2082 CommMonCat MonCat).map (F.map (IsFiltered.leftToMax x.fst y.fst))) x.snd *\n \u2191((forget\u2082 CommMonCat MonCat).map (F.map (IsFiltered.rightToMax x.fst y.fst))) y.snd } =\n MonCat.FilteredColimits.M.mk (F \u22d9 forget\u2082 CommMonCat MonCat)\n { fst := IsFiltered.max x.fst y.fst,\n snd :=\n \u2191((forget\u2082 CommMonCat MonCat).map (F.map (IsFiltered.rightToMax x.fst y.fst))) y.snd *\n \u2191((forget\u2082 CommMonCat MonCat).map (F.map (IsFiltered.leftToMax x.fst y.fst))) x.snd }\n[PROOFSTEP]\nrw [mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.MonCat.FilteredColimits", "llama_tokens": 31107, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.27977499079579654}} {"text": "[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\n\u22a2 IsProper IF.I\n[PROOFSTEP]\ncases' IF.F.nonempty with w h\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\nw : P\nh : w \u2208 \u2191IF.F\n\u22a2 IsProper IF.I\n[PROOFSTEP]\napply isProper_of_not_mem (_ : w \u2209 IF.I)\n[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\nw : P\nh : w \u2208 \u2191IF.F\n\u22a2 \u00acw \u2208 IF.I\n[PROOFSTEP]\nrwa [\u2190 IF.compl_I_eq_F] at h \n[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\nsrc\u271d : IsProper IF.I := I_isProper IF\n\u22a2 IsPFilter (\u2191IF.I)\u1d9c\n[PROOFSTEP]\nrw [IF.compl_I_eq_F]\n[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : PrimePair P\nsrc\u271d : IsProper IF.I := I_isProper IF\n\u22a2 IsPFilter \u2191IF.F\n[PROOFSTEP]\nexact IF.F.isPFilter\n[GOAL]\nP : Type u_1\ninst\u271d : SemilatticeInf P\nx\u271d y\u271d : P\nI : Ideal P\nhI : IsPrime I\nx y : P\n\u22a2 x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nP : Type u_1\ninst\u271d : SemilatticeInf P\nx\u271d y\u271d : P\nI : Ideal P\nhI : IsPrime I\nx y : P\n\u22a2 \u00acx \u2208 I \u2227 \u00acy \u2208 I \u2192 \u00acx \u2293 y \u2208 I\n[PROOFSTEP]\nlet F := hI.compl_filter.toPFilter\n[GOAL]\nP : Type u_1\ninst\u271d : SemilatticeInf P\nx\u271d y\u271d : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (\u2191I)\u1d9c)\n\u22a2 \u00acx \u2208 I \u2227 \u00acy \u2208 I \u2192 \u00acx \u2293 y \u2208 I\n[PROOFSTEP]\nshow x \u2208 F \u2227 y \u2208 F \u2192 x \u2293 y \u2208 F\n[GOAL]\nP : Type u_1\ninst\u271d : SemilatticeInf P\nx\u271d y\u271d : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (\u2191I)\u1d9c)\n\u22a2 x \u2208 F \u2227 y \u2208 F \u2192 x \u2293 y \u2208 F\n[PROOFSTEP]\nexact fun h => inf_mem h.1 h.2\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 IsPrime I\n[PROOFSTEP]\nrw [IsPrime_iff]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 IsProper I \u2227 IsPFilter (\u2191I)\u1d9c\n[PROOFSTEP]\nuse\u2039_\u203a\n[GOAL]\ncase right\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 IsPFilter (\u2191I)\u1d9c\n[PROOFSTEP]\nrefine .of_def ?_ ?_ ?_\n[GOAL]\ncase right.refine_1\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 Set.Nonempty (\u2191I)\u1d9c\n[PROOFSTEP]\nexact Set.nonempty_compl.2 (I.IsProper_iff.1 \u2039_\u203a)\n[GOAL]\ncase right.refine_2\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 DirectedOn (fun x x_1 => x \u2265 x_1) (\u2191I)\u1d9c\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\ncase right.refine_2\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx\u271d y\u271d : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\nx : P\nhx : x \u2208 (\u2191I)\u1d9c\ny : P\nhy : y \u2208 (\u2191I)\u1d9c\n\u22a2 \u2203 z, z \u2208 (\u2191I)\u1d9c \u2227 (fun x x_1 => x \u2265 x_1) x z \u2227 (fun x x_1 => x \u2265 x_1) y z\n[PROOFSTEP]\nexact \u27e8x \u2293 y, fun h => (hI h).elim hx hy, inf_le_left, inf_le_right\u27e9\n[GOAL]\ncase right.refine_3\nP : Type u_1\ninst\u271d\u00b9 : SemilatticeInf P\nx y : P\nI : Ideal P\ninst\u271d : IsProper I\nhI : \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n\u22a2 \u2200 {x y : P}, x \u2264 y \u2192 x \u2208 (\u2191I)\u1d9c \u2192 y \u2208 (\u2191I)\u1d9c\n[PROOFSTEP]\nexact @mem_compl_of_ge _ _ _\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\n\u22a2 IsPrime I\n[PROOFSTEP]\nrw [isPrime_iff_mem_or_mem]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\n\u22a2 \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\nintro x y\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\n\u22a2 x \u2293 y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 I\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\n\u22a2 \u00acx \u2208 I \u2227 \u00acy \u2208 I \u2192 \u00acx \u2293 y \u2208 I\n[PROOFSTEP]\nrintro \u27e8hx, hynI\u27e9 hxy\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\n\u22a2 False\n[PROOFSTEP]\napply hynI\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\n\u22a2 y \u2208 I\n[PROOFSTEP]\nlet J := I \u2294 principal x\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave hJuniv : (J : Set P) = Set.univ := IsMaximal.maximal_proper (lt_sup_principal_of_not_mem \u2039_\u203a)\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave hyJ : y \u2208 \u2191J := Set.eq_univ_iff_forall.mp hJuniv y\n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\nhyJ : y \u2208 \u2191J\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrw [coe_sup_eq] at hyJ \n[GOAL]\ncase intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\nhyJ : y \u2208 {x_1 | \u2203 i, i \u2208 I \u2227 \u2203 j, j \u2208 principal x \u2227 x_1 = i \u2294 j}\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrcases hyJ with \u27e8a, ha, b, hb, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\na : P\nha : a \u2208 I\nb : P\nhb : b \u2208 principal x\nhy : y = a \u2294 b\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase intro.intro.intro.intro.intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\na : P\nha : a \u2208 I\nb : P\nhb : b \u2208 principal x\nhy : y = a \u2294 b\n\u22a2 a \u2294 b \u2208 I\n[PROOFSTEP]\nrefine' sup_mem ha (I.lower (le_inf hb _) hxy)\n[GOAL]\ncase intro.intro.intro.intro.intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\na : P\nha : a \u2208 I\nb : P\nhb : b \u2208 principal x\nhy : y = a \u2294 b\n\u22a2 b \u2264 y\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase intro.intro.intro.intro.intro\nP : Type u_1\ninst\u271d\u00b9 : DistribLattice P\nI : Ideal P\ninst\u271d : IsMaximal I\nx y : P\nhx : \u00acx \u2208 I\nhynI : \u00acy \u2208 I\nhxy : x \u2293 y \u2208 I\nJ : Ideal P := I \u2294 principal x\nhJuniv : \u2191J = Set.univ\na : P\nha : a \u2208 I\nb : P\nhb : b \u2208 principal x\nhy : y = a \u2294 b\n\u22a2 b \u2264 a \u2294 b\n[PROOFSTEP]\nexact le_sup_right\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsPrime I\n\u22a2 x \u2208 I \u2228 x\u1d9c \u2208 I\n[PROOFSTEP]\napply hI.mem_or_mem\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsPrime I\n\u22a2 x \u2293 x\u1d9c \u2208 I\n[PROOFSTEP]\nrw [inf_compl_eq_bot]\n[GOAL]\nP : Type u_1\ninst\u271d : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : IsPrime I\n\u22a2 \u22a5 \u2208 I\n[PROOFSTEP]\nexact I.bot_mem\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx : P\nI : Ideal P\ninst\u271d : IsProper I\nh : \u2200 {x : P}, x \u2208 I \u2228 x\u1d9c \u2208 I\n\u22a2 IsPrime I\n[PROOFSTEP]\nsimp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx : P\nI : Ideal P\ninst\u271d : IsProper I\nh : \u2200 {x : P}, x \u2208 I \u2228 x\u1d9c \u2208 I\n\u22a2 \u2200 {x y : P}, x \u2293 y \u2208 I \u2192 \u00acx \u2208 I \u2192 y \u2208 I\n[PROOFSTEP]\nintro x y hxy hxI\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsProper I\nh : \u2200 {x : P}, x \u2208 I \u2228 x\u1d9c \u2208 I\nx y : P\nhxy : x \u2293 y \u2208 I\nhxI : \u00acx \u2208 I\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave hxcI : x\u1d9c \u2208 I := h.resolve_left hxI\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsProper I\nh : \u2200 {x : P}, x \u2208 I \u2228 x\u1d9c \u2208 I\nx y : P\nhxy : x \u2293 y \u2208 I\nhxI : \u00acx \u2208 I\nhxcI : x\u1d9c \u2208 I\n\u22a2 y \u2208 I\n[PROOFSTEP]\nhave ass : x \u2293 y \u2294 y \u2293 x\u1d9c \u2208 I := sup_mem hxy (I.lower inf_le_right hxcI)\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsProper I\nh : \u2200 {x : P}, x \u2208 I \u2228 x\u1d9c \u2208 I\nx y : P\nhxy : x \u2293 y \u2208 I\nhxI : \u00acx \u2208 I\nhxcI : x\u1d9c \u2208 I\nass : x \u2293 y \u2294 y \u2293 x\u1d9c \u2208 I\n\u22a2 y \u2208 I\n[PROOFSTEP]\nrwa [inf_comm, sup_inf_inf_compl] at ass \n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx : P\nI : Ideal P\ninst\u271d : IsPrime I\n\u22a2 IsMaximal I\n[PROOFSTEP]\nsimp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx : P\nI : Ideal P\ninst\u271d : IsPrime I\n\u22a2 \u2200 \u2983J : Ideal P\u2984, I < J \u2192 \u2200 (x : P), x \u2208 \u2191J\n[PROOFSTEP]\nintro J hIJ x\n[GOAL]\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsPrime I\nJ : Ideal P\nhIJ : I < J\nx : P\n\u22a2 x \u2208 \u2191J\n[PROOFSTEP]\nrcases Set.exists_of_ssubset hIJ with \u27e8y, hyJ, hyI\u27e9\n[GOAL]\ncase intro.intro\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsPrime I\nJ : Ideal P\nhIJ : I < J\nx y : P\nhyJ : y \u2208 \u2191J\nhyI : \u00acy \u2208 \u2191I\n\u22a2 x \u2208 \u2191J\n[PROOFSTEP]\nsuffices ass : x \u2293 y \u2294 x \u2293 y\u1d9c \u2208 J\n[GOAL]\ncase intro.intro\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsPrime I\nJ : Ideal P\nhIJ : I < J\nx y : P\nhyJ : y \u2208 \u2191J\nhyI : \u00acy \u2208 \u2191I\nass : x \u2293 y \u2294 x \u2293 y\u1d9c \u2208 J\n\u22a2 x \u2208 \u2191J\n[PROOFSTEP]\nrwa [sup_inf_inf_compl] at ass \n[GOAL]\ncase ass\nP : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra P\nx\u271d : P\nI : Ideal P\ninst\u271d : IsPrime I\nJ : Ideal P\nhIJ : I < J\nx y : P\nhyJ : y \u2208 \u2191J\nhyI : \u00acy \u2208 \u2191I\n\u22a2 x \u2293 y \u2294 x \u2293 y\u1d9c \u2208 J\n[PROOFSTEP]\nexact sup_mem (J.lower inf_le_right hyJ) (hIJ.le <| I.lower inf_le_right <| IsPrime.mem_compl_of_not_mem \u2039_\u203a hyI)\n[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : Ideal.PrimePair P\n\u22a2 IsIdeal (\u2191IF.F)\u1d9c\n[PROOFSTEP]\nrw [IF.compl_F_eq_I]\n[GOAL]\nP : Type u_1\ninst\u271d : Preorder P\nIF : Ideal.PrimePair P\n\u22a2 IsIdeal \u2191IF.I\n[PROOFSTEP]\nexact IF.I.isIdeal\n", "meta": {"mathlib_filename": "Mathlib.Order.PrimeIdeal", "llama_tokens": 5405, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.27960226837805585}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain rfl | m_pos := m.eq_zero_or_pos\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nn a b : \u2115\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = 0 \u2228 card x = 0 + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 0) \u2227\n card (filter (fun i => card i = 0 + 1) Q.parts) = b\n[PROOFSTEP]\nrefine' \u27e8\u22a5, by simp, _, by simpa using hs.symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nn a b : \u2115\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n\u22a2 \u2200 (x : Finset \u03b1), x \u2208 \u22a5.parts \u2192 card x = 0 \u2228 card x = 0 + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nn a b : \u2115\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n\u22a2 card (filter (fun i => card i = 0 + 1) \u22a5.parts) = b\n[PROOFSTEP]\nsimpa using hs.symm\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nn a b : \u2115\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n\u22a2 \u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) \u22a5.parts) id) \u2264 0\n[PROOFSTEP]\nsimp only [le_zero_iff, card_eq_zero, mem_biUnion, exists_prop, mem_filter, id.def, and_assoc,\n sdiff_eq_empty_iff_subset, subset_iff]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nn a b : \u2115\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n\u22a2 \u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 \u2200 \u2983x_1 : \u03b1\u2984, x_1 \u2208 x \u2192 \u2203 a, a \u2208 \u22a5.parts \u2227 (\u2200 \u2983x_2 : \u03b1\u2984, x_2 \u2208 a \u2192 x_2 \u2208 x) \u2227 x_1 \u2208 a\n[PROOFSTEP]\nexact fun x hx a ha =>\n \u27e8{ a }, mem_map_of_mem _ (P.le hx ha), singleton_subset_iff.2 ha, mem_singleton_self _\u27e9\n -- Prove the case `m > 0` by strong induction on `s`\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nm_pos : m > 0\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\ninduction' s using Finset.strongInduction with s ih generalizing a b\n[GOAL]\ncase inr.H\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nby_cases hab : a = 0 \u2227 b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : a = 0 \u2227 b = 0\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nsimp only [hab.1, hab.2, add_zero, zero_mul, eq_comm, card_eq_zero, Finset.bot_eq_empty] at hs \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhab : a = 0 \u2227 b = 0\nhs : s = \u2205\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nsubst hs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\nhs : a\u271d * m + b\u271d * (m + 1) = card s\nm_pos : m > 0\na b : \u2115\nhab : a = 0 \u2227 b = 0\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 \u2205 \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition \u2205\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nhave : P = Finpartition.empty _ := Unique.eq_default (\u03b1 := Finpartition \u22a5) P\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\nhs : a\u271d * m + b\u271d * (m + 1) = card s\nm_pos : m > 0\na b : \u2115\nhab : a = 0 \u2227 b = 0\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 \u2205 \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition \u2205\nthis : P = Finpartition.empty (Finset \u03b1)\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nexact \u27e8Finpartition.empty _, by simp, by simp [this], by simp [hab.2]\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\nhs : a\u271d * m + b\u271d * (m + 1) = card s\nm_pos : m > 0\na b : \u2115\nhab : a = 0 \u2227 b = 0\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 \u2205 \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition \u2205\nthis : P = Finpartition.empty (Finset \u03b1)\n\u22a2 \u2200 (x : Finset \u03b1), x \u2208 (Finpartition.empty (Finset \u03b1)).parts \u2192 card x = m \u2228 card x = m + 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\nhs : a\u271d * m + b\u271d * (m + 1) = card s\nm_pos : m > 0\na b : \u2115\nhab : a = 0 \u2227 b = 0\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 \u2205 \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition \u2205\nthis : P = Finpartition.empty (Finset \u03b1)\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) (Finpartition.empty (Finset \u03b1)).parts) id) \u2264 m\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\nhs : a\u271d * m + b\u271d * (m + 1) = card s\nm_pos : m > 0\na b : \u2115\nhab : a = 0 \u2227 b = 0\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 \u2205 \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition \u2205\nthis : P = Finpartition.empty (Finset \u03b1)\n\u22a2 card (filter (fun i => card i = m + 1) (Finpartition.empty (Finset \u03b1)).parts) = b\n[PROOFSTEP]\nsimp [hab.2]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nsimp_rw [not_and_or, \u2190 Ne.def, \u2190 pos_iff_ne_zero] at hab \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nset n := if 0 < a then m else m + 1 with hn\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8hn\u2080, hn\u2081, hn\u2082, hn\u2083\u27e9 :\n 0 < n \u2227\n n \u2264 m + 1 \u2227 n \u2264 a * m + b * (m + 1) \u2227 ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = s.card - n :=\n by\n rw [hn, \u2190 hs]\n split_ifs with h <;> rw [tsub_mul, one_mul]\n \u00b7 refine' \u27e8m_pos, le_succ _, le_add_right (le_mul_of_pos_left \u20390 < a\u203a), _\u27e9\n rw [tsub_add_eq_add_tsub (le_mul_of_pos_left h)]\n \u00b7 refine' \u27e8succ_pos', le_rfl, le_add_left (le_mul_of_pos_left <| hab.resolve_left \u2039\u00ac0 < a\u203a), _\u27e9\n rw [\u2190 add_tsub_assoc_of_le (le_mul_of_pos_left <| hab.resolve_left \u2039\u00ac0 < a\u203a)]\n /- We will call the inductive hypothesis on a partition of `s \\ t` for a carefully chosen `t \u2286 s`.\n To decide which, however, we must distinguish the case where all parts of `P` have size `m` (in\n which case we take `t` to be an arbitrary subset of `s` of size `n`) from the case where at\n least one part `u` of `P` has size `m + 1` (in which case we take `t` to be an arbitrary subset\n of `u` of size `n`). The rest of each branch is just tedious calculations to satisfy the\n induction hypothesis. -/\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n\u22a2 0 < n \u2227\n n \u2264 m + 1 \u2227\n n \u2264 a * m + b * (m + 1) \u2227 (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\n[PROOFSTEP]\nrw [hn, \u2190 hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n\u22a2 (0 < if 0 < a then m else m + 1) \u2227\n (if 0 < a then m else m + 1) \u2264 m + 1 \u2227\n (if 0 < a then m else m + 1) \u2264 a * m + b * (m + 1) \u2227\n (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) =\n a * m + b * (m + 1) - if 0 < a then m else m + 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n\u22a2 0 < m \u2227 m \u2264 m + 1 \u2227 m \u2264 a * m + b * (m + 1) \u2227 (a - 1) * m + b * (m + 1) = a * m + b * (m + 1) - m\n[PROOFSTEP]\nrw [tsub_mul, one_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : \u00ac0 < a\n\u22a2 0 < m + 1 \u2227 m + 1 \u2264 m + 1 \u2227 m + 1 \u2264 a * m + b * (m + 1) \u2227 a * m + (b - 1) * (m + 1) = a * m + b * (m + 1) - (m + 1)\n[PROOFSTEP]\nrw [tsub_mul, one_mul]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n\u22a2 0 < m \u2227 m \u2264 m + 1 \u2227 m \u2264 a * m + b * (m + 1) \u2227 a * m - m + b * (m + 1) = a * m + b * (m + 1) - m\n[PROOFSTEP]\nrefine' \u27e8m_pos, le_succ _, le_add_right (le_mul_of_pos_left \u20390 < a\u203a), _\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n\u22a2 a * m - m + b * (m + 1) = a * m + b * (m + 1) - m\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub (le_mul_of_pos_left h)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : \u00ac0 < a\n\u22a2 0 < m + 1 \u2227\n m + 1 \u2264 m + 1 \u2227 m + 1 \u2264 a * m + b * (m + 1) \u2227 a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)\n[PROOFSTEP]\nrefine' \u27e8succ_pos', le_rfl, le_add_left (le_mul_of_pos_left <| hab.resolve_left \u2039\u00ac0 < a\u203a), _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : \u00ac0 < a\n\u22a2 a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)\n[PROOFSTEP]\nrw [\u2190 add_tsub_assoc_of_le (le_mul_of_pos_left <| hab.resolve_left \u2039\u00ac0 < a\u203a)]\n /- We will call the inductive hypothesis on a partition of `s \\ t` for a carefully chosen `t \u2286 s`.\n To decide which, however, we must distinguish the case where all parts of `P` have size `m` (in\n which case we take `t` to be an arbitrary subset of `s` of size `n`) from the case where at\n least one part `u` of `P` has size `m + 1` (in which case we take `t` to be an arbitrary subset\n of `u` of size `n`). The rest of each branch is just tedious calculations to satisfy the\n induction hypothesis. -/\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nby_cases h : \u2200 u \u2208 P.parts, card u < m + 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8t, hts, htn\u27e9 := exists_smaller_set s n (hn\u2082.trans_eq hs)\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nhave ht : t.Nonempty := by rwa [\u2190 card_pos, htn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\n\u22a2 Finset.Nonempty t\n[PROOFSTEP]\nrwa [\u2190 card_pos, htn]\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nhave hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = (s \\ t).card := by\n rw [card_sdiff \u2039t \u2286 s\u203a, htn, hn\u2083]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\n\u22a2 (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n[PROOFSTEP]\nrw [card_sdiff \u2039t \u2286 s\u203a, htn, hn\u2083]\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8R, hR\u2081, _, hR\u2083\u27e9 :=\n @ih (s \\ t) (sdiff_ssubset hts \u2039t.Nonempty\u203a) (if 0 < a then a - 1 else a) (if 0 < a then b else b - 1) (P.avoid t)\n hcard\n[GOAL]\ncase pos.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nrefine' \u27e8R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel hts), _, _, _\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts \u2192 card x = m \u2228 card x = m + 1\n[PROOFSTEP]\nsimp only [extend_parts, mem_insert, forall_eq_or_imp, and_iff_left hR\u2081, htn, hn]\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 (if 0 < a then m else m + 1) = m \u2228 (if 0 < a then m else m + 1) = m + 1\n[PROOFSTEP]\nexact ite_eq_or_eq _ _ _\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 P.parts \u2192\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y \u2286 x) (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts) id) \u2264\n m\n[PROOFSTEP]\nexact fun x hx => (card_le_of_subset <| sdiff_subset _ _).trans (lt_succ_iff.1 <| h _ hx)\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (filter (fun i => card i = m + 1) (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts) = b\n[PROOFSTEP]\nsimp_rw [extend_parts, filter_insert, htn, m.succ_ne_self.symm.ite_eq_right_iff]\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card\n (if \u00ac0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b\n[PROOFSTEP]\nsplit_ifs with ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : 0 < a\n\u22a2 card (filter (fun i => card i = m + 1) R.parts) = b\n[PROOFSTEP]\nrw [hR\u2083, if_pos ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : \u00ac0 < a\n\u22a2 card (insert t (filter (fun i => card i = m + 1) R.parts)) = b\n[PROOFSTEP]\nrw [card_insert_of_not_mem, hR\u2083, if_neg ha, tsub_add_cancel_of_le]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : \u00ac0 < a\n\u22a2 1 \u2264 b\n[PROOFSTEP]\nexact hab.resolve_left ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : \u00ac0 < a\n\u22a2 \u00act \u2208 filter (fun i => card i = m + 1) R.parts\n[PROOFSTEP]\nintro H\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\nt : Finset \u03b1\nhts : t \u2286 s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nleft\u271d : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : \u00ac0 < a\nH : t \u2208 filter (fun i => card i = m + 1) R.parts\n\u22a2 False\n[PROOFSTEP]\nexact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u00ac\u2200 (u : Finset \u03b1), u \u2208 P.parts \u2192 card u < m + 1\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : \u2203 u, u \u2208 P.parts \u2227 m + 1 \u2264 card u\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8u, hu\u2081, hu\u2082\u27e9 := h\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8t, htu, htn\u27e9 := exists_smaller_set _ _ (hn\u2081.trans hu\u2082)\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nhave ht : t.Nonempty := by rwa [\u2190 card_pos, htn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\n\u22a2 Finset.Nonempty t\n[PROOFSTEP]\nrwa [\u2190 card_pos, htn]\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nhave hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = (s \\ t).card := by\n rw [card_sdiff (htu.trans <| P.le hu\u2081), htn, hn\u2083]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\n\u22a2 (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n[PROOFSTEP]\nrw [card_sdiff (htu.trans <| P.le hu\u2081), htn, hn\u2083]\n[GOAL]\ncase neg.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nobtain \u27e8R, hR\u2081, hR\u2082, hR\u2083\u27e9 :=\n @ih (s \\ t) (sdiff_ssubset (htu.trans <| P.le hu\u2081) ht) (if 0 < a then a - 1 else a) (if 0 < a then b else b - 1)\n (P.avoid t) hcard\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\n[PROOFSTEP]\nrefine' \u27e8R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel <| htu.trans <| P.le hu\u2081), _, _, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts \u2192 card x = m \u2228 card x = m + 1\n[PROOFSTEP]\nsimp only [mem_insert, forall_eq_or_imp, extend_parts, and_iff_left hR\u2081, htn, hn]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 (if 0 < a then m else m + 1) = m \u2228 (if 0 < a then m else m + 1) = m + 1\n[PROOFSTEP]\nexact ite_eq_or_eq _ _ _\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 P.parts \u2192\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y \u2286 x) (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts) id) \u2264\n m\n[PROOFSTEP]\nconv in _ \u2208 _ => rw [\u2190 insert_erase hu\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\n| x \u2208 P.parts\n[PROOFSTEP]\nrw [\u2190 insert_erase hu\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\n| x \u2208 P.parts\n[PROOFSTEP]\nrw [\u2190 insert_erase hu\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\n| x \u2208 P.parts\n[PROOFSTEP]\nrw [\u2190 insert_erase hu\u2081]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 \u2200 (x : Finset \u03b1),\n x \u2208 insert u (erase P.parts u) \u2192\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y \u2286 x) (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts) id) \u2264\n m\n[PROOFSTEP]\nsimp only [and_imp, mem_insert, forall_eq_or_imp, Ne.def, extend_parts]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (u \\ Finset.biUnion (filter (fun y => y \u2286 u) (insert t R.parts)) id) \u2264 m \u2227\n \u2200 (a : Finset \u03b1),\n a \u2208 erase P.parts u \u2192 card (a \\ Finset.biUnion (filter (fun y => y \u2286 a) (insert t R.parts)) id) \u2264 m\n[PROOFSTEP]\nrefine' \u27e8_, fun x hx => (card_le_of_subset _).trans <| hR\u2082 x _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (u \\ Finset.biUnion (filter (fun y => y \u2286 u) (insert t R.parts)) id) \u2264 m\n[PROOFSTEP]\nsimp only [filter_insert, if_pos htu, biUnion_insert, mem_erase, id.def]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (u \\ (t \u222a Finset.biUnion (filter (fun y => y \u2286 u) R.parts) id)) \u2264 m\n[PROOFSTEP]\nobtain rfl | hut := eq_or_ne u t\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nhtu : u \u2286 u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P u).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (u \\ (u \u222a Finset.biUnion (filter (fun y => y \u2286 u) R.parts) id)) \u2264 m\n[PROOFSTEP]\nrw [sdiff_eq_empty_iff_subset.2 (subset_union_left _ _)]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nhtu : u \u2286 u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P u).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card \u2205 \u2264 m\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u \u2260 t\n\u22a2 card (u \\ (t \u222a Finset.biUnion (filter (fun y => y \u2286 u) R.parts) id)) \u2264 m\n[PROOFSTEP]\nrefine'\n (card_le_of_subset fun i => _).trans\n (hR\u2082 (u \\ t) <| P.mem_avoid.2 \u27e8u, hu\u2081, fun i => hut <| i.antisymm htu, rfl\u27e9)\n -- Porting note: `not_and` required because `\u2203 x \u2208 s, p x` is defined differently\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u \u2260 t\ni : \u03b1\n\u22a2 i \u2208 u \\ (t \u222a Finset.biUnion (filter (fun y => y \u2286 u) R.parts) id) \u2192\n i \u2208 (u \\ t) \\ Finset.biUnion (filter (fun y => y \u2286 u \\ t) R.parts) id\n[PROOFSTEP]\nsimp only [not_exists, not_and, mem_biUnion, and_imp, mem_union, mem_filter, mem_sdiff, id.def, not_or]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u \u2260 t\ni : \u03b1\n\u22a2 i \u2208 u \u2192\n \u00aci \u2208 t \u2192\n (\u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 x \u2286 u \u2192 \u00aci \u2208 x) \u2192\n (i \u2208 u \u2227 \u00aci \u2208 t) \u2227 \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 x \u2286 u \\ t \u2192 \u00aci \u2208 x\n[PROOFSTEP]\nexact fun hi\u2081 hi\u2082 hi\u2083 => \u27e8\u27e8hi\u2081, hi\u2082\u27e9, fun x hx hx' => hi\u2083 _ hx <| hx'.trans <| sdiff_subset _ _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\nhx : x \u2208 erase P.parts u\n\u22a2 x \\ Finset.biUnion (filter (fun y => y \u2286 x) (insert t R.parts)) id \u2286\n x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id\n[PROOFSTEP]\napply sdiff_subset_sdiff Subset.rfl (biUnion_subset_biUnion_of_subset_left _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\nhx : x \u2208 erase P.parts u\n\u22a2 filter (fun y => y \u2286 x) R.parts \u2286 filter (fun y => y \u2286 x) (insert t R.parts)\n[PROOFSTEP]\nexact filter_subset_filter _ (subset_insert _ _)\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\nhx : x \u2208 erase P.parts u\n\u22a2 x \u2208 (avoid P t).parts\n[PROOFSTEP]\nsimp only [avoid, ofErase, mem_erase, mem_image, bot_eq_empty]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset \u03b1\nhx : x \u2208 erase P.parts u\n\u22a2 x \u2260 \u2205 \u2227 \u2203 a, a \u2208 P.parts \u2227 a \\ t = x\n[PROOFSTEP]\nexact\n \u27e8(nonempty_of_mem_parts _ <| mem_of_mem_erase hx).ne_empty, _, mem_of_mem_erase hx,\n (disjoint_of_subset_right htu <| P.disjoint (mem_of_mem_erase hx) hu\u2081 <| ne_of_mem_erase hx).sdiff_eq_left\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card (filter (fun i => card i = m + 1) (extend R (_ : t \u2260 \u2205) (_ : Disjoint (s \\ t) t) (_ : s \\ t \u2294 t = s)).parts) = b\n[PROOFSTEP]\nsimp only [extend_parts, filter_insert, htn, hn, m.succ_ne_self.symm.ite_eq_right_iff]\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n\u22a2 card\n (if \u00ac0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : 0 < a\n\u22a2 card (filter (fun i => card i = m + 1) R.parts) = b\n[PROOFSTEP]\nrw [hR\u2083, if_pos h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : \u00ac0 < a\n\u22a2 card (insert t (filter (fun i => card i = m + 1) R.parts)) = b\n[PROOFSTEP]\nrw [card_insert_of_not_mem, hR\u2083, if_neg h, Nat.sub_add_cancel (hab.resolve_left h)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : \u00ac0 < a\n\u22a2 \u00act \u2208 filter (fun i => card i = m + 1) R.parts\n[PROOFSTEP]\nintro H\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Finset \u03b1\nm n\u271d a\u271d b\u271d : \u2115\nP\u271d : Finpartition s\u271d\nhs\u271d : a\u271d * m + b\u271d * (m + 1) = card s\u271d\nm_pos : m > 0\ns : Finset \u03b1\nih :\n \u2200 (t : Finset \u03b1),\n t \u2282 s \u2192\n \u2200 {a b : \u2115} {P : Finpartition t},\n a * m + b * (m + 1) = card t \u2192\n \u2203 Q,\n (\u2200 (x : Finset \u03b1), x \u2208 Q.parts \u2192 card x = m \u2228 card x = m + 1) \u2227\n (\u2200 (x : Finset \u03b1), x \u2208 P.parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) Q.parts) id) \u2264 m) \u2227\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : \u2115\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a \u2228 0 < b\nn : \u2115 := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn\u2080 : 0 < n\nhn\u2081 : n \u2264 m + 1\nhn\u2082 : n \u2264 a * m + b * (m + 1)\nhn\u2083 : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset \u03b1\nhu\u2081 : u \u2208 P.parts\nhu\u2082 : m + 1 \u2264 card u\nt : Finset \u03b1\nhtu : t \u2286 u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR\u2081 : \u2200 (x : Finset \u03b1), x \u2208 R.parts \u2192 card x = m \u2228 card x = m + 1\nhR\u2082 : \u2200 (x : Finset \u03b1), x \u2208 (avoid P t).parts \u2192 card (x \\ Finset.biUnion (filter (fun y => y \u2286 x) R.parts) id) \u2264 m\nhR\u2083 : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : \u00ac0 < a\nH : t \u2208 filter (fun i => card i = m + 1) R.parts\n\u22a2 False\n[PROOFSTEP]\nexact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 card (filter (fun u => card u = m) (equitabilise h).parts) = a\n[PROOFSTEP]\nrefine' (mul_eq_mul_right_iff.1 <| (add_left_inj (b * (m + 1))).1 _).resolve_right hm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) = a * m + b * (m + 1)\n[PROOFSTEP]\nrw [h, \u2190 (P.equitabilise h).sum_card_parts]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i\n[PROOFSTEP]\nhave hunion :\n (P.equitabilise h).parts =\n ((P.equitabilise h).parts.filter fun u => u.card = m) \u222a (P.equitabilise h).parts.filter fun u => u.card = m + 1 :=\n by\n rw [\u2190 filter_or, filter_true_of_mem]\n exact fun x => card_eq_of_mem_parts_equitabilise\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts\n[PROOFSTEP]\nrw [\u2190 filter_or, filter_true_of_mem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 \u2200 (x : Finset \u03b1), x \u2208 (equitabilise h).parts \u2192 card x = m \u2228 card x = m + 1\n[PROOFSTEP]\nexact fun x => card_eq_of_mem_parts_equitabilise\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts\n\u22a2 card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i\n[PROOFSTEP]\nnth_rw 2 [hunion]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts\n\u22a2 card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum\n (filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts)\n fun i => card i\n[PROOFSTEP]\nrw [sum_union, sum_const_nat fun x hx => (mem_filter.1 hx).2, sum_const_nat fun x hx => (mem_filter.1 hx).2,\n P.card_filter_equitabilise_big]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts\n\u22a2 Disjoint (filter (fun u => card u = m) (equitabilise h).parts)\n (filter (fun u => card u = m + 1) (equitabilise h).parts)\n[PROOFSTEP]\nrefine' disjoint_filter_filter' _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts \u222a filter (fun u => card u = m + 1) (equitabilise h).parts\n\u22a2 Disjoint (fun u => card u = m) fun u => card u = m + 1\n[PROOFSTEP]\nintro x ha hb i h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh\u271d : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h\u271d).parts =\n filter (fun u => card u = m) (equitabilise h\u271d).parts \u222a filter (fun u => card u = m + 1) (equitabilise h\u271d).parts\nx : Finset \u03b1 \u2192 Prop\nha : x \u2264 fun u => card u = m\nhb : x \u2264 fun u => card u = m + 1\ni : Finset \u03b1\nh : x i\n\u22a2 \u22a5 i\n[PROOFSTEP]\napply succ_ne_self m _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh\u271d : a * m + b * (m + 1) = card s\nhm : m \u2260 0\nhunion :\n (equitabilise h\u271d).parts =\n filter (fun u => card u = m) (equitabilise h\u271d).parts \u222a filter (fun u => card u = m + 1) (equitabilise h\u271d).parts\nx : Finset \u03b1 \u2192 Prop\nha : x \u2264 fun u => card u = m\nhb : x \u2264 fun u => card u = m + 1\ni : Finset \u03b1\nh : x i\n\u22a2 succ m = m\n[PROOFSTEP]\nexact (hb i h).symm.trans (ha i h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 card (equitabilise h).parts = a + b\n[PROOFSTEP]\nrw [\u2190 filter_true_of_mem fun x => card_eq_of_mem_parts_equitabilise, filter_or, card_union_eq,\n P.card_filter_equitabilise_small _ hm, P.card_filter_equitabilise_big]\n -- Porting note: was `infer_instance`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m \u2260 0\n\u22a2 Disjoint (filter (fun x => card x = m) (equitabilise h).parts)\n (filter (fun x => card x = m + 1) (equitabilise h).parts)\n[PROOFSTEP]\nexact disjoint_filter.2 fun x _ h\u2080 h\u2081 => Nat.succ_ne_self m <| h\u2081.symm.trans h\u2080\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhn : n \u2260 0\nhs : n \u2264 card s\n\u22a2 \u2203 P, IsEquipartition P \u2227 card P.parts = n\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero] at hn \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhn : 0 < n\nhs : n \u2264 card s\n\u22a2 \u2203 P, IsEquipartition P \u2227 card P.parts = n\n[PROOFSTEP]\nhave : (n - s.card % n) * (s.card / n) + s.card % n * (s.card / n + 1) = s.card := by\n rw [tsub_mul, mul_add, \u2190 add_assoc, tsub_add_cancel_of_le (Nat.mul_le_mul_right _ (mod_lt _ hn).le), mul_one,\n add_comm, mod_add_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhn : 0 < n\nhs : n \u2264 card s\n\u22a2 (n - card s % n) * (card s / n) + card s % n * (card s / n + 1) = card s\n[PROOFSTEP]\nrw [tsub_mul, mul_add, \u2190 add_assoc, tsub_add_cancel_of_le (Nat.mul_le_mul_right _ (mod_lt _ hn).le), mul_one, add_comm,\n mod_add_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhn : 0 < n\nhs : n \u2264 card s\nthis : (n - card s % n) * (card s / n) + card s % n * (card s / n + 1) = card s\n\u22a2 \u2203 P, IsEquipartition P \u2227 card P.parts = n\n[PROOFSTEP]\nrefine' \u27e8(indiscrete (card_pos.1 <| hn.trans_le hs).ne_empty).equitabilise this, equitabilise_isEquipartition, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nm n a b : \u2115\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhn : 0 < n\nhs : n \u2264 card s\nthis : (n - card s % n) * (card s / n) + card s % n * (card s / n + 1) = card s\n\u22a2 card (equitabilise this).parts = n\n[PROOFSTEP]\nrw [card_parts_equitabilise _ _ (Nat.div_pos hs hn).ne', tsub_add_cancel_of_le (mod_lt _ hn).le]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise", "llama_tokens": 51970, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.27902851134008505}} {"text": "[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\ni : I\nx\u271d\u00b3 x\u271d\u00b2 : C i\nx\u271d\u00b9 x\u271d : x\u271d\u00b3 \u27f6 x\u271d\u00b2\nh : (incl i).map x\u271d\u00b9 = (incl i).map x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\ninjection h\n[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF\u271d : (i : I) \u2192 C i \u2964 D\nF G : (i : I) \u00d7 C i \u2964 D\nh : (i : I) \u2192 incl i \u22d9 F \u27f6 incl i \u22d9 G\n\u22a2 \u2200 \u2983X Y : (i : I) \u00d7 C i\u2984 (f : X \u27f6 Y),\n F.map f \u226b\n (fun x =>\n match x with\n | { fst := j, snd := X } => NatTrans.app (h j) X)\n Y =\n (fun x =>\n match x with\n | { fst := j, snd := X } => NatTrans.app (h j) X)\n X \u226b\n G.map f\n[PROOFSTEP]\nrintro \u27e8j, X\u27e9 \u27e8_, _\u27e9 \u27e8f\u27e9\n[GOAL]\ncase mk.mk.mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF\u271d : (i : I) \u2192 C i \u2964 D\nF G : (i : I) \u00d7 C i \u2964 D\nh : (i : I) \u2192 incl i \u22d9 F \u27f6 incl i \u22d9 G\nj : I\nX Y\u271d : C j\nf : X \u27f6 Y\u271d\n\u22a2 F.map (SigmaHom.mk f) \u226b\n (fun x =>\n match x with\n | { fst := j, snd := X } => NatTrans.app (h j) X)\n { fst := j, snd := Y\u271d } =\n (fun x =>\n match x with\n | { fst := j, snd := X } => NatTrans.app (h j) X)\n { fst := j, snd := X } \u226b\n G.map (SigmaHom.mk f)\n[PROOFSTEP]\napply (h j).naturality\n[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\n\u22a2 \u2200 (X : (i : I) \u00d7 C i),\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (\ud835\udfd9 X) =\n \ud835\udfd9 ({ obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.obj X)\n[PROOFSTEP]\nrintro \u27e8i, X\u27e9\n[GOAL]\ncase mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\ni : I\nX : C i\n\u22a2 { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (\ud835\udfd9 { fst := i, snd := X }) =\n \ud835\udfd9 ({ obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.obj { fst := i, snd := X })\n[PROOFSTEP]\napply (F i).map_id\n[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\n\u22a2 \u2200 {X Y Z : (i : I) \u00d7 C i} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (f \u226b g) =\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map f \u226b\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map g\n[PROOFSTEP]\nrintro \u27e8i, X\u27e9 \u27e8_, Y\u27e9 \u27e8_, Z\u27e9 \u27e8f\u27e9 \u27e8g\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\ni : I\nX Y\u271d\u00b9 : C i\nf : X \u27f6 Y\u271d\u00b9\nY\u271d : C i\ng : Y\u271d\u00b9 \u27f6 Y\u271d\n\u22a2 { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f \u226b SigmaHom.mk g) =\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk f) \u226b\n { obj := fun X => (F X.fst).obj X.snd, map := fun {X Y} g => descMap F X Y g }.map (SigmaHom.mk g)\n[PROOFSTEP]\napply (F i).map_comp\n[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\nq : (i : I) \u00d7 C i \u2964 D\nh : (i : I) \u2192 incl i \u22d9 q \u2245 F i\n\u22a2 \u2200 {X Y : (i : I) \u00d7 C i} (f : X \u27f6 Y),\n q.map f \u226b\n ((fun x =>\n match x with\n | { fst := i, snd := X } => (h i).app X)\n Y).hom =\n ((fun x =>\n match x with\n | { fst := i, snd := X } => (h i).app X)\n X).hom \u226b\n (desc F).map f\n[PROOFSTEP]\nrintro \u27e8i, X\u27e9 \u27e8_, _\u27e9 \u27e8f\u27e9\n[GOAL]\ncase mk.mk.mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : (i : I) \u2192 C i \u2964 D\nq : (i : I) \u00d7 C i \u2964 D\nh : (i : I) \u2192 incl i \u22d9 q \u2245 F i\ni : I\nX Y\u271d : C i\nf : X \u27f6 Y\u271d\n\u22a2 q.map (SigmaHom.mk f) \u226b\n ((fun x =>\n match x with\n | { fst := i, snd := X } => (h i).app X)\n { fst := i, snd := Y\u271d }).hom =\n ((fun x =>\n match x with\n | { fst := i, snd := X } => (h i).app X)\n { fst := i, snd := X }).hom \u226b\n (desc F).map (SigmaHom.mk f)\n[PROOFSTEP]\napply (h i).hom.naturality f\n[GOAL]\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : I \u2192 Type u\u2081\ninst\u271d : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i)\nF G : (i : I) \u2192 C i \u2964 D i\n\u03b1 : (i : I) \u2192 F i \u27f6 G i\n\u22a2 \u2200 \u2983X Y : (i : I) \u00d7 C i\u2984 (f : X \u27f6 Y),\n (Functor.sigma F).map f \u226b (fun f => SigmaHom.mk (NatTrans.app (\u03b1 f.fst) f.snd)) Y =\n (fun f => SigmaHom.mk (NatTrans.app (\u03b1 f.fst) f.snd)) X \u226b (Functor.sigma G).map f\n[PROOFSTEP]\nrintro \u27e8i, X\u27e9 \u27e8_, _\u27e9 \u27e8f\u27e9\n[GOAL]\ncase mk.mk.mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : I \u2192 Type u\u2081\ninst\u271d : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i)\nF G : (i : I) \u2192 C i \u2964 D i\n\u03b1 : (i : I) \u2192 F i \u27f6 G i\ni : I\nX Y\u271d : C i\nf : X \u27f6 Y\u271d\n\u22a2 (Functor.sigma F).map (SigmaHom.mk f) \u226b\n (fun f => SigmaHom.mk (NatTrans.app (\u03b1 f.fst) f.snd)) { fst := i, snd := Y\u271d } =\n (fun f => SigmaHom.mk (NatTrans.app (\u03b1 f.fst) f.snd)) { fst := i, snd := X } \u226b (Functor.sigma G).map (SigmaHom.mk f)\n[PROOFSTEP]\nchange SigmaHom.mk _ = SigmaHom.mk _\n[GOAL]\ncase mk.mk.mk\nI : Type w\u2081\nC : I \u2192 Type u\u2081\ninst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i)\nD : I \u2192 Type u\u2081\ninst\u271d : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i)\nF G : (i : I) \u2192 C i \u2964 D i\n\u03b1 : (i : I) \u2192 F i \u27f6 G i\ni : I\nX Y\u271d : C i\nf : X \u27f6 Y\u271d\n\u22a2 SigmaHom.mk\n ((F { fst := i, snd := X }.fst).map f \u226b\n NatTrans.app (\u03b1 { fst := i, snd := Y\u271d }.fst) { fst := i, snd := Y\u271d }.snd) =\n SigmaHom.mk\n (NatTrans.app (\u03b1 { fst := i, snd := X }.fst) { fst := i, snd := X }.snd \u226b (G { fst := i, snd := X }.fst).map f)\n[PROOFSTEP]\nrw [(\u03b1 i).naturality]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sigma.Basic", "llama_tokens": 3095, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.2789070454488447}} {"text": "[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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\nh : \u2200 (B : D) (A : C), (A \u27f9 i.obj B) \u2208 Functor.essImage i\nB\u271d : C\nhB : B\u271d \u2208 Functor.essImage i\nA : C\n\u22a2 (A \u27f9 B\u271d) \u2208 Functor.essImage i\n[PROOFSTEP]\nrcases hB with \u27e8B', \u27e8iB'\u27e9\u27e9\n[GOAL]\ncase intro.intro\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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\nh : \u2200 (B : D) (A : C), (A \u27f9 i.obj B) \u2208 Functor.essImage i\nB\u271d A : C\nB' : D\niB' : i.obj B' \u2245 B\u271d\n\u22a2 (A \u27f9 B\u271d) \u2208 Functor.essImage i\n[PROOFSTEP]\nexact Functor.essImage.ofIso ((exp A).mapIso iB') (h B' 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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\n\u22a2 ExponentialIdeal (subterminalInclusion C)\n[PROOFSTEP]\napply ExponentialIdeal.mk'\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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\n\u22a2 \u2200 (B : Subterminals C) (A : C), (A \u27f9 (subterminalInclusion C).obj B) \u2208 Functor.essImage (subterminalInclusion C)\n[PROOFSTEP]\nintro B A\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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\nB : Subterminals C\nA : C\n\u22a2 (A \u27f9 (subterminalInclusion C).obj B) \u2208 Functor.essImage (subterminalInclusion C)\n[PROOFSTEP]\nrefine' \u27e8\u27e8A \u27f9 B.1, fun Z g h => _\u27e9, \u27e8Iso.refl _\u27e9\u27e9\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\ni : D \u2964 C\ninst\u271d\u00b9 : HasFiniteProducts C\ninst\u271d : CartesianClosed C\nB : Subterminals C\nA Z : C\ng h : Z \u27f6 A \u27f9 B.obj\n\u22a2 g = h\n[PROOFSTEP]\nexact uncurry_injective (B.2 (CartesianClosed.uncurry g) (CartesianClosed.uncurry h))\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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\n\u22a2 i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i \u2245 i \u22d9 exp A\n[PROOFSTEP]\nsymm\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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\n\u22a2 i \u22d9 exp A \u2245 i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i\n[PROOFSTEP]\napply NatIso.ofComponents _ _\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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\n\u22a2 (X : D) \u2192 (i \u22d9 exp A).obj X \u2245 (i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i).obj 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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\nX : D\n\u22a2 (i \u22d9 exp A).obj X \u2245 (i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i).obj X\n[PROOFSTEP]\nhaveI := Functor.essImage.unit_isIso (ExponentialIdeal.exp_closed (i.obj_mem_essImage X) A)\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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\nX : D\nthis : IsIso (NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u27f9 i.obj X))\n\u22a2 (i \u22d9 exp A).obj X \u2245 (i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i).obj X\n[PROOFSTEP]\napply asIso ((Adjunction.ofRightAdjoint i).unit.app (A \u27f9 i.obj 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\ni : D \u2964 C\ninst\u271d\u00b3 : HasFiniteProducts C\ninst\u271d\u00b2 : CartesianClosed C\nA : C\ninst\u271d\u00b9 : Reflective i\ninst\u271d : ExponentialIdeal i\n\u22a2 \u2200 {X Y : D} (f : X \u27f6 Y),\n (i \u22d9 exp A).map f \u226b (asIso (NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u27f9 i.obj Y))).hom =\n (asIso (NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u27f9 i.obj X))).hom \u226b\n (i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i).map f\n[PROOFSTEP]\nsimp [asIso]\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\ni : D \u2964 C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : CartesianClosed C\ninst\u271d : Reflective i\nh : (A : C) \u2192 i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i \u2245 i \u22d9 exp A\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\napply ExponentialIdeal.mk'\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : CartesianClosed C\ninst\u271d : Reflective i\nh : (A : C) \u2192 i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i \u2245 i \u22d9 exp A\n\u22a2 \u2200 (B : D) (A : C), (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nintro B A\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : CartesianClosed C\ninst\u271d : Reflective i\nh : (A : C) \u2192 i \u22d9 exp A \u22d9 leftAdjoint i \u22d9 i \u2245 i \u22d9 exp A\nB : D\nA : C\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nexact \u27e8_, \u27e8(h A).app B\u27e9\u27e9\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\nlet ir := Adjunction.ofRightAdjoint i\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\nlet L : C \u2964 D := leftAdjoint i\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\nlet \u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\nlet \u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\n\u22a2 ExponentialIdeal i\n[PROOFSTEP]\napply ExponentialIdeal.mk'\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\n\u22a2 \u2200 (B : D) (A : C), (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nintro B A\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nlet q : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B\n[GOAL]\ncase q\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\n\u22a2 i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B := ?q\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\napply CartesianClosed.curry (ir.homEquiv _ _ _)\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\n\u22a2 (leftAdjoint i).obj (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) \u27f6 B\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B) ?m.26486)\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\napply _ \u226b (ir.homEquiv _ _).symm ((exp.ev A).app (i.obj B))\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\n\u22a2 (leftAdjoint i).obj (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) \u27f6\n (leftAdjoint i).obj ((exp A \u22d9 prod.functor.obj A).obj (i.obj B))\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n (?m.26547 \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nrefine' prodComparison L A _ \u226b Limits.prod.map (\ud835\udfd9 _) (\u03b5.app _) \u226b inv (prodComparison _ _ _)\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nhave : \u03b7.app (A \u27f9 i.obj B) \u226b q = \ud835\udfd9 (A \u27f9 i.obj B) := by\n dsimp\n rw [\u2190 curry_natural_left, curry_eq_iff, uncurry_id_eq_ev, \u2190 ir.homEquiv_naturality_left, ir.homEquiv_apply_eq, assoc,\n assoc, prodComparison_natural_assoc, L.map_id, \u2190 prod.map_id_comp_assoc, ir.left_triangle_components,\n prod.map_id_id, id_comp]\n apply IsIso.hom_inv_id_assoc\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\n\u22a2 NatTrans.app \u03b7 (A \u27f9 i.obj B) \u226b q = \ud835\udfd9 (A \u27f9 i.obj B)\n[PROOFSTEP]\ndsimp\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\n\u22a2 NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u27f9 i.obj B) \u226b\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f i.obj ((leftAdjoint i).obj (A \u27f9 i.obj B))) B)\n ((prodComparison (leftAdjoint i) A (i.obj ((leftAdjoint i).obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 ((leftAdjoint i).obj A))\n (NatTrans.app (Adjunction.ofRightAdjoint i).counit ((leftAdjoint i).obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f A \u27f9 i.obj B) B).symm\n (NatTrans.app (exp.ev A) (i.obj B)))) =\n \ud835\udfd9 (A \u27f9 i.obj B)\n[PROOFSTEP]\nrw [\u2190 curry_natural_left, curry_eq_iff, uncurry_id_eq_ev, \u2190 ir.homEquiv_naturality_left, ir.homEquiv_apply_eq, assoc,\n assoc, prodComparison_natural_assoc, L.map_id, \u2190 prod.map_id_comp_assoc, ir.left_triangle_components, prod.map_id_id,\n id_comp]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\n\u22a2 prodComparison (leftAdjoint i) A (A \u27f9 i.obj B) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B)) \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f A \u27f9 i.obj B) B).symm\n (NatTrans.app (exp.ev A) (i.obj B)) =\n \u2191(Adjunction.homEquiv ir (A \u2a2f A \u27f9 i.obj B) B).symm (NatTrans.app (exp.ev A) (i.obj B))\n[PROOFSTEP]\napply IsIso.hom_inv_id_assoc\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\nthis : NatTrans.app \u03b7 (A \u27f9 i.obj B) \u226b q = \ud835\udfd9 (A \u27f9 i.obj B)\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\nhaveI : IsSplitMono (\u03b7.app (A \u27f9 i.obj B)) := IsSplitMono.mk' \u27e8_, this\u27e9\n[GOAL]\ncase h\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i)\nir : leftAdjoint i \u22a3 i := Adjunction.ofRightAdjoint i\nL : C \u2964 D := leftAdjoint i\n\u03b7 : \ud835\udfed C \u27f6 L \u22d9 i := ir.unit\n\u03b5 : i \u22d9 L \u27f6 \ud835\udfed D := ir.counit\nB : D\nA : C\nq : i.obj (L.obj (A \u27f9 i.obj B)) \u27f6 A \u27f9 i.obj B :=\n CartesianClosed.curry\n (\u2191(Adjunction.homEquiv ir (A \u2a2f i.obj (L.obj (A \u27f9 i.obj B))) B)\n ((prodComparison L A (i.obj (L.obj (A \u27f9 i.obj B))) \u226b\n prod.map (\ud835\udfd9 (L.obj A)) (NatTrans.app \u03b5 (L.obj (A \u27f9 i.obj B))) \u226b\n inv (prodComparison (leftAdjoint i) A (A \u27f9 i.obj B))) \u226b\n \u2191(Adjunction.homEquiv ir ((exp A \u22d9 prod.functor.obj A).obj (i.obj B)) B).symm\n (NatTrans.app (exp.ev A) (i.obj B))))\nthis\u271d : NatTrans.app \u03b7 (A \u27f9 i.obj B) \u226b q = \ud835\udfd9 (A \u27f9 i.obj B)\nthis : IsSplitMono (NatTrans.app \u03b7 (A \u27f9 i.obj B))\n\u22a2 (A \u27f9 i.obj B) \u2208 Functor.essImage i\n[PROOFSTEP]\napply mem_essImage_of_unit_isSplitMono\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB : D\n\u22a2 MonoidalCategory.tensorLeft B \u22a3 i \u22d9 exp (i.obj B) \u22d9 leftAdjoint i\n[PROOFSTEP]\napply Adjunction.restrictFullyFaithful i i (exp.adjunction (i.obj B))\n[GOAL]\ncase comm1\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB : D\n\u22a2 i \u22d9 prod.functor.obj (i.obj B) \u2245 MonoidalCategory.tensorLeft B \u22d9 i\n[PROOFSTEP]\nsymm\n[GOAL]\ncase comm1\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB : D\n\u22a2 MonoidalCategory.tensorLeft B \u22d9 i \u2245 i \u22d9 prod.functor.obj (i.obj B)\n[PROOFSTEP]\nrefine' NatIso.ofComponents (fun X => _) (fun f => _)\n[GOAL]\ncase comm1.refine'_1\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB X : D\n\u22a2 (MonoidalCategory.tensorLeft B \u22d9 i).obj X \u2245 (i \u22d9 prod.functor.obj (i.obj B)).obj X\n[PROOFSTEP]\nhaveI := Adjunction.rightAdjointPreservesLimits.{0, 0} (Adjunction.ofRightAdjoint i)\n[GOAL]\ncase comm1.refine'_1\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB X : D\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 (MonoidalCategory.tensorLeft B \u22d9 i).obj X \u2245 (i \u22d9 prod.functor.obj (i.obj B)).obj X\n[PROOFSTEP]\napply asIso (prodComparison i B X)\n[GOAL]\ncase comm1.refine'_2\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB X\u271d Y\u271d : D\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (MonoidalCategory.tensorLeft B \u22d9 i).map f \u226b ((fun X => asIso (prodComparison i B X)) Y\u271d).hom =\n ((fun X => asIso (prodComparison i B X)) X\u271d).hom \u226b (i \u22d9 prod.functor.obj (i.obj B)).map f\n[PROOFSTEP]\ndsimp [asIso]\n[GOAL]\ncase comm1.refine'_2\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB X\u271d Y\u271d : D\nf : X\u271d \u27f6 Y\u271d\n\u22a2 i.map (prod.map (\ud835\udfd9 B) f) \u226b prodComparison i B Y\u271d = prodComparison i B X\u271d \u226b prod.map (\ud835\udfd9 (i.obj B)) (i.map f)\n[PROOFSTEP]\nrw [prodComparison_natural, Functor.map_id]\n[GOAL]\ncase comm2\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nsrc\u271d : MonoidalCategory D := monoidalOfHasFiniteProducts D\nB : D\n\u22a2 i \u22d9 exp (i.obj B) \u2245 (i \u22d9 exp (i.obj B) \u22d9 leftAdjoint i) \u22d9 i\n[PROOFSTEP]\napply (exponentialIdealReflective i _).symm\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\n\u22a2 \u2191(bijection i A B ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (\ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\nhave : PreservesLimits i := (Adjunction.ofRightAdjoint i).rightAdjointPreservesLimits\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis : PreservesLimits i\n\u22a2 \u2191(bijection i A B ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (\ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\nhave := preservesSmallestLimitsOfPreservesLimits i\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 \u2191(bijection i A B ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (\ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\ndsimp [bijection]\n -- Porting note: added\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction B) A (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))).symm\n (\u2191(unitCompPartialBijective A\n (_ : (B \u27f9 i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u2208 Functor.essImage i)).symm\n (\u2191(Adjunction.homEquiv (exp.adjunction B) (i.obj ((leftAdjoint i).obj A))\n (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)))\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A))) B\n (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))).symm\n (\u2191(unitCompPartialBijective B\n (_ :\n (i.obj ((leftAdjoint i).obj A) \u27f9\n i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u2208\n Functor.essImage i)).symm\n (\u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A)))\n (i.obj ((leftAdjoint i).obj B)) (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)))\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).inv \u226b\n i.map (\ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))))) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))))) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\nerw [homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective A\n (_ : (B \u27f9 i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u2208 Functor.essImage i)).symm\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective B\n (_ :\n (i.obj ((leftAdjoint i).obj A) \u27f9\n i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u2208\n Functor.essImage i)).symm\n (CartesianClosed.curry\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).inv \u226b\n i.map (\ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))))) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))))) \u226b\n \ud835\udfd9 (i.obj ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B))) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\nrw [comp_id, comp_id, comp_id, i.map_id, comp_id, unitCompPartialBijective_symm_apply,\n unitCompPartialBijective_symm_apply, uncurry_natural_left, uncurry_curry, uncurry_natural_left, uncurry_curry,\n prod.lift_map_assoc, comp_id, prod.lift_map_assoc, comp_id]\n -- Porting note: added\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (prod.lift prod.snd (prod.fst \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit A) \u226b\n prod.lift prod.snd (prod.fst \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit B) \u226b\n (PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).inv) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\ndsimp only [Functor.comp_obj]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)).symm\n (prod.lift prod.snd (prod.fst \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit A) \u226b\n prod.lift prod.snd (prod.fst \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit B) \u226b\n (PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).inv) =\n prodComparison (leftAdjoint i) A B\n[PROOFSTEP]\nrw [prod.comp_lift_assoc, prod.lift_snd, prod.lift_fst_assoc, prod.lift_fst_comp_snd_comp, \u2190\n Adjunction.eq_homEquiv_apply, Adjunction.homEquiv_unit, Iso.comp_inv_eq, assoc]\n -- Porting note: rw became erw\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.map (NatTrans.app (Adjunction.ofRightAdjoint i).unit A) (NatTrans.app (Adjunction.ofRightAdjoint i).unit B) =\n NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b\n i.map (prodComparison (leftAdjoint i) A B) \u226b\n (PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom\n[PROOFSTEP]\nerw [PreservesLimitPair.iso_hom i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.map (NatTrans.app (Adjunction.ofRightAdjoint i).unit A) (NatTrans.app (Adjunction.ofRightAdjoint i).unit B) =\n NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b\n i.map (prodComparison (leftAdjoint i) A B) \u226b prodComparison i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)\n[PROOFSTEP]\napply prod.hom_ext\n[GOAL]\ncase h\u2081\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.map (NatTrans.app (Adjunction.ofRightAdjoint i).unit A) (NatTrans.app (Adjunction.ofRightAdjoint i).unit B) \u226b\n prod.fst =\n (NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b\n i.map (prodComparison (leftAdjoint i) A B) \u226b prodComparison i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)) \u226b\n prod.fst\n[PROOFSTEP]\nrw [Limits.prod.map_fst, assoc, assoc, prodComparison_fst, \u2190 i.map_comp, prodComparison_fst]\n[GOAL]\ncase h\u2081\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.fst \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit A =\n NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b i.map ((leftAdjoint i).map prod.fst)\n[PROOFSTEP]\napply (Adjunction.ofRightAdjoint i).unit.naturality\n[GOAL]\ncase h\u2082\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.map (NatTrans.app (Adjunction.ofRightAdjoint i).unit A) (NatTrans.app (Adjunction.ofRightAdjoint i).unit B) \u226b\n prod.snd =\n (NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b\n i.map (prodComparison (leftAdjoint i) A B) \u226b prodComparison i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)) \u226b\n prod.snd\n[PROOFSTEP]\nrw [Limits.prod.map_snd, assoc, assoc, prodComparison_snd, \u2190 i.map_comp, prodComparison_snd]\n[GOAL]\ncase h\u2082\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nthis\u271d : PreservesLimits i\nthis : PreservesLimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2082, u\u2081} i\n\u22a2 prod.snd \u226b NatTrans.app (Adjunction.ofRightAdjoint i).unit B =\n NatTrans.app (Adjunction.ofRightAdjoint i).unit (A \u2a2f B) \u226b i.map ((leftAdjoint i).map prod.snd)\n[PROOFSTEP]\napply (Adjunction.ofRightAdjoint i).unit.naturality\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (leftAdjoint i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 \u2191(bijection i A B X') (f \u226b g) = \u2191(bijection i A B X) f \u226b g\n[PROOFSTEP]\ndsimp [bijection]\n -- Porting note: added\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (leftAdjoint i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A))) (i.obj ((leftAdjoint i).obj B))\n (i.obj X')).symm\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X') \u2208 Functor.essImage i))\n (\u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A))) B (i.obj X'))\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction B) (i.obj ((leftAdjoint i).obj A)) (i.obj X')).symm\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X') \u2208 Functor.essImage i))\n (\u2191(Adjunction.homEquiv (exp.adjunction B) A (i.obj X'))\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X')) =\n i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A))) (i.obj ((leftAdjoint i).obj B))\n (i.obj X)).symm\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X) \u2208 Functor.essImage i))\n (\u2191(Adjunction.homEquiv (exp.adjunction (i.obj ((leftAdjoint i).obj A))) B (i.obj X))\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (exp.adjunction B) (i.obj ((leftAdjoint i).obj A)) (i.obj X)).symm\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X) \u2208 Functor.essImage i))\n (\u2191(Adjunction.homEquiv (exp.adjunction B) A (i.obj X))\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)) \u226b\n g\n[PROOFSTEP]\nerw [homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq, homEquiv_symm_apply_eq,\n homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (leftAdjoint i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X') \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X') \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X')) =\n i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X) \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X) \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)) \u226b\n g\n[PROOFSTEP]\napply i.map_injective\n[GOAL]\ncase a\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (leftAdjoint i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.map\n (i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X') \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X') \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X') (f \u226b g) \u226b\n \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X')))) \u226b\n \ud835\udfd9 (i.obj X'))) =\n i.map\n (i.preimage\n ((PreservesLimitPair.iso i ((leftAdjoint i).obj A) ((leftAdjoint i).obj B)).hom \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective B (_ : (i.obj ((leftAdjoint i).obj A) \u27f9 i.obj X) \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n CartesianClosed.uncurry\n (\u2191(unitCompPartialBijective A (_ : (B \u27f9 i.obj X) \u2208 Functor.essImage i))\n (CartesianClosed.curry\n (prod.lift prod.snd prod.fst \u226b\n \u2191(Adjunction.homEquiv (Adjunction.ofRightAdjoint i) (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)))) \u226b\n \ud835\udfd9 (i.obj X)) \u226b\n g)\n[PROOFSTEP]\nrw [i.image_preimage, i.map_comp, i.image_preimage, comp_id, comp_id, comp_id, comp_id, comp_id, comp_id,\n Adjunction.homEquiv_naturality_right, \u2190 assoc, curry_natural_right _ (i.map g), unitCompPartialBijective_natural,\n uncurry_natural_right, \u2190 assoc, curry_natural_right, unitCompPartialBijective_natural, uncurry_natural_right, assoc]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\n\u22a2 prodComparison (leftAdjoint i) A B \u226b\n \u2191(bijection i A B ((leftAdjoint i).obj (A \u2a2f B))) (\ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B))) =\n \ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B))\n[PROOFSTEP]\nrw [\u2190 (bijection i _ _ _).injective.eq_iff, bijection_natural, \u2190 bijection_symm_apply_id, Equiv.apply_symm_apply,\n id_comp]\n[GOAL]\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\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\n\u22a2 \u2191(bijection i A B ((leftAdjoint i).obj (A \u2a2f B))) (\ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B))) \u226b\n prodComparison (leftAdjoint i) A B =\n \ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B)\n[PROOFSTEP]\nrw [\u2190 bijection_natural, id_comp, \u2190 bijection_symm_apply_id, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2076 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2075 : HasFiniteProducts C\ninst\u271d\u2074 : Reflective i\ninst\u271d\u00b3 : CartesianClosed C\ninst\u271d\u00b2 : HasFiniteProducts D\ninst\u271d\u00b9 : ExponentialIdeal i\nJ : Type\ninst\u271d : Fintype J\n\u22a2 PreservesLimitsOfShape (Discrete J) (leftAdjoint i)\n[PROOFSTEP]\nletI := preservesBinaryProductsOfExponentialIdeal i\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2076 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2075 : HasFiniteProducts C\ninst\u271d\u2074 : Reflective i\ninst\u271d\u00b3 : CartesianClosed C\ninst\u271d\u00b2 : HasFiniteProducts D\ninst\u271d\u00b9 : ExponentialIdeal i\nJ : Type\ninst\u271d : Fintype J\nthis : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i) := preservesBinaryProductsOfExponentialIdeal i\n\u22a2 PreservesLimitsOfShape (Discrete J) (leftAdjoint i)\n[PROOFSTEP]\nletI := leftAdjointPreservesTerminalOfReflective.{0} i\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2076 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2075 : HasFiniteProducts C\ninst\u271d\u2074 : Reflective i\ninst\u271d\u00b3 : CartesianClosed C\ninst\u271d\u00b2 : HasFiniteProducts D\ninst\u271d\u00b9 : ExponentialIdeal i\nJ : Type\ninst\u271d : Fintype J\nthis\u271d : PreservesLimitsOfShape (Discrete WalkingPair) (leftAdjoint i) := preservesBinaryProductsOfExponentialIdeal i\nthis : PreservesLimitsOfShape (Discrete PEmpty) (leftAdjoint i) := leftAdjointPreservesTerminalOfReflective i\n\u22a2 PreservesLimitsOfShape (Discrete J) (leftAdjoint i)\n[PROOFSTEP]\napply preservesFiniteProductsOfPreservesBinaryAndTerminal (leftAdjoint i) J\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Closed.Ideal", "llama_tokens": 20721, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.27884731080299546}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d : SmallCategory K\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : Cover J X\ni : Cover.Arrow W\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W))\n\u22a2 \u2200 \u2983X_1 Y : K\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const K).obj E.pt).map f \u226b\n (fun k => NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj k)) i) Y =\n (fun k => NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj k)) i) X_1 \u226b\n (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op i.Y)).map f\n[PROOFSTEP]\nintro a b f\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d : SmallCategory K\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : Cover J X\ni : Cover.Arrow W\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W))\na b : K\nf : a \u27f6 b\n\u22a2 ((Functor.const K).obj E.pt).map f \u226b (fun k => NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj k)) i) b =\n (fun k => NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj k)) i) a \u226b\n (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op i.Y)).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d : SmallCategory K\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : Cover J X\ni : Cover.Arrow W\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W))\na b : K\nf : a \u27f6 b\n\u22a2 \ud835\udfd9 E.pt \u226b NatTrans.app E.\u03c0 b \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj b)) i =\n (NatTrans.app E.\u03c0 a \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj a)) i) \u226b NatTrans.app (F.map f) (op i.Y)\n[PROOFSTEP]\nrw [Category.id_comp, Category.assoc, \u2190 E.w f]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d : SmallCategory K\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : Cover J X\ni : Cover.Arrow W\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W))\na b : K\nf : a \u27f6 b\n\u22a2 (NatTrans.app E.\u03c0 a \u226b (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W)).map f) \u226b\n Multiequalizer.\u03b9 (Cover.index W (F.obj b)) i =\n NatTrans.app E.\u03c0 a \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj a)) i \u226b NatTrans.app (F.map f) (op i.Y)\n[PROOFSTEP]\ndsimp [diagramNatTrans]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d : SmallCategory K\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : Cover J X\ni : Cover.Arrow W\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj (op W))\na b : K\nf : a \u27f6 b\n\u22a2 (NatTrans.app E.\u03c0 a \u226b\n Multiequalizer.lift (Cover.index W (F.obj b)) (multiequalizer (Cover.index W (F.obj a)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W (F.obj a)) i \u226b NatTrans.app (F.map f) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index (op W).unop (F.obj b)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index (op W).unop (F.obj a)) i \u226b NatTrans.app (F.map f) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index (op W).unop (F.obj b)) i) \u226b\n MulticospanIndex.fst (Cover.index (op W).unop (F.obj b)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index (op W).unop (F.obj a)) i \u226b NatTrans.app (F.map f) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index (op W).unop (F.obj b)) i) \u226b\n MulticospanIndex.snd (Cover.index (op W).unop (F.obj b)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W (F.obj b)) i =\n NatTrans.app E.\u03c0 a \u226b Multiequalizer.\u03b9 (Cover.index W (F.obj a)) i \u226b NatTrans.app (F.map f) (op i.Y)\n[PROOFSTEP]\nsimp only [Multiequalizer.lift_\u03b9, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\n\u22a2 \u2200 (b : (Cover.index W.unop (limit F)).R),\n (fun i =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (limit F)) b =\n (fun i =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (limit F)) b\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\n\u22a2 (fun i =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (limit F)) i =\n (fun i =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (limit F)) i\n[PROOFSTEP]\nchange (_ \u226b _) \u226b _ = (_ \u226b _) \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\n\u22a2 ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit\n (limit.isLimit\n (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))\n (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E)).Hom \u226b\n ((Cones.functoriality F\n ((evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n MulticospanIndex.fst (Cover.index W.unop (limit F)) i =\n ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit\n (limit.isLimit\n (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))\n (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E)).Hom \u226b\n ((Cones.functoriality F\n ((evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n MulticospanIndex.snd (Cover.index W.unop (limit F)) i\n[PROOFSTEP]\ndsimp [evaluateCombinedCones]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\n\u22a2 ((limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E) \u226b\n \ud835\udfd9\n (getLimitCone\n ((Functor.flip F).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))).cone.pt) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)))\n (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (limit F)) i =\n ((limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E) \u226b\n \ud835\udfd9\n (getLimitCone\n ((Functor.flip F).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))).cone.pt) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)))\n (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (limit F)) i\n[PROOFSTEP]\nerw [Category.comp_id, Category.comp_id, Category.assoc, Category.assoc, \u2190 (limit.lift F _).naturality, \u2190\n (limit.lift F _).naturality, \u2190 Category.assoc, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\n\u22a2 (limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2081.op) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k))) (op i.Z) =\n (limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2082.op) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k))) (op i.Z)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\n\u22a2 limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2081.op =\n limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2082.op\n[PROOFSTEP]\nrefine' limit.hom_ext (fun j => _)\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\nj : K\n\u22a2 (limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2081.op) \u226b\n limit.\u03c0 ((Functor.flip F).obj (op i.Z)) j =\n (limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E) \u226b\n (combineCones F fun k => getLimitCone ((Functor.flip F).obj k)).pt.map i.g\u2082.op) \u226b\n limit.\u03c0 ((Functor.flip F).obj (op i.Z)) j\n[PROOFSTEP]\nerw [Category.assoc, Category.assoc, limit.lift_\u03c0, limit.lift_\u03c0, limit.lift_\u03c0_assoc, limit.lift_\u03c0_assoc, Category.assoc,\n Category.assoc, Multiequalizer.condition]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nW : (Cover J X)\u1d52\u1d56\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\ni : (Cover.index W.unop (limit F)).R\nj : K\n\u22a2 NatTrans.app E.\u03c0 j \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (F.obj j)) (MulticospanIndex.sndTo (Cover.index W.unop (F.obj j)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (F.obj j)) i =\n NatTrans.app E.\u03c0 j \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (F.obj j)) (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) \u226b\n NatTrans.app ((Functor.flip F).map i.g\u2082.op) j\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\n\u22a2 \u2200 (s : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)) (j : K),\n (fun E => liftToDiagramLimitObj F E) s \u226b\n NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nintro E k\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\n\u22a2 (fun E => liftToDiagramLimitObj F E) E \u226b\n NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 k =\n NatTrans.app E.\u03c0 k\n[PROOFSTEP]\ndsimp [diagramNatTrans]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\n\u22a2 liftToDiagramLimitObj F E \u226b\n Multiequalizer.lift (Cover.index W.unop (F.obj k)) (multiequalizer (Cover.index W.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (F.obj k)) i) =\n NatTrans.app E.\u03c0 k\n[PROOFSTEP]\nrefine' Multiequalizer.hom_ext _ _ _ (fun a => _)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\na : (Cover.index W.unop (F.obj k)).L\n\u22a2 (liftToDiagramLimitObj F E \u226b\n Multiequalizer.lift (Cover.index W.unop (F.obj k)) (multiequalizer (Cover.index W.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (F.obj k)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a =\n NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a\n[PROOFSTEP]\nsimp only [Multiequalizer.lift_\u03b9, Multiequalizer.lift_\u03b9_assoc, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\na : (Cover.index W.unop (F.obj k)).L\n\u22a2 IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op a.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) \u226b\n NatTrans.app (limit.\u03c0 F k) (op a.Y) =\n NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a\n[PROOFSTEP]\nchange (_ \u226b _) \u226b _ = _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\na : (Cover.index W.unop (F.obj k)).L\n\u22a2 ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit (limit.isLimit (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op a.Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k)) (op a.Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E)).Hom \u226b\n ((Cones.functoriality F ((evaluation C\u1d52\u1d56 D).obj (op a.Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n NatTrans.app (limit.\u03c0 F k) (op a.Y) =\n NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a\n[PROOFSTEP]\ndsimp [evaluateCombinedCones]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\na : (Cover.index W.unop (F.obj k)).L\n\u22a2 ((limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op a.Y)) (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) \u226b\n \ud835\udfd9 (getLimitCone ((Functor.flip F).obj (op a.Y))).cone.pt) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k))) (op a.Y)) \u226b\n NatTrans.app (limit.\u03c0 F k) (op a.Y) =\n NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a\n[PROOFSTEP]\nerw [Category.comp_id, Category.assoc, \u2190 NatTrans.comp_app, limit.lift_\u03c0, limit.lift_\u03c0]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nk : K\na : (Cover.index W.unop (F.obj k)).L\n\u22a2 NatTrans.app (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E).\u03c0 k =\n NatTrans.app E.\u03c0 k \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj k)) a\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\n\u22a2 \u2200 (s : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W))\n (m : s.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt),\n (\u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app s.\u03c0 j) \u2192\n m = (fun E => liftToDiagramLimitObj F E) s\n[PROOFSTEP]\nintro E m hm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\n\u22a2 m = (fun E => liftToDiagramLimitObj F E) E\n[PROOFSTEP]\nrefine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 (m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a) \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n ((fun E => liftToDiagramLimitObj F E) E \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a) \u226b\n NatTrans.app (limit.\u03c0 F j) (op a.Y)\n[PROOFSTEP]\ndelta liftToDiagramLimitObj\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 (m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a) \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n ((fun E =>\n Multiequalizer.lift (Cover.index W.unop (limit F)) E.pt\n (fun i =>\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))\n (_ :\n \u2200 (i : (Cover.index W.unop (limit F)).R),\n ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit\n (limit.isLimit\n (F \u22d9\n (evaluation C\u1d52\u1d56 D).obj\n (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))\n (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E)).Hom \u226b\n ((Cones.functoriality F\n ((evaluation C\u1d52\u1d56 D).obj\n (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n MulticospanIndex.fst (Cover.index W.unop (limit F)) i =\n ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit\n (limit.isLimit\n (F \u22d9\n (evaluation C\u1d52\u1d56 D).obj\n (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))\n (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation\n (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E)).Hom \u226b\n ((Cones.functoriality F\n ((evaluation C\u1d52\u1d56 D).obj\n (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n MulticospanIndex.snd (Cover.index W.unop (limit F)) i))\n E \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a) \u226b\n NatTrans.app (limit.\u03c0 F j) (op a.Y)\n[PROOFSTEP]\nerw [Multiequalizer.lift_\u03b9, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n IsLimit.lift (isLimitOfPreserves ((evaluation C\u1d52\u1d56 D).obj (op a.Y)) (limit.isLimit F))\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) \u226b\n NatTrans.app (limit.\u03c0 F j) (op a.Y)\n[PROOFSTEP]\nchange _ = (_ \u226b _) \u226b _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit.cone F).pt) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n ((IsLimit.liftConeMorphism\n (IsLimit.ofIsoLimit (limit.isLimit (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op a.Y)))\n (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k)) (op a.Y)).symm)\n (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E)).Hom \u226b\n ((Cones.functoriality F ((evaluation C\u1d52\u1d56 D).obj (op a.Y))).mapIso\n (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))\n (limit.isLimit F))).hom.Hom) \u226b\n NatTrans.app (limit.\u03c0 F j) (op a.Y)\n[PROOFSTEP]\ndsimp [evaluateCombinedCones]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n ((limit.lift (F \u22d9 (evaluation C\u1d52\u1d56 D).obj (op a.Y)) (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E) \u226b\n \ud835\udfd9 (getLimitCone ((Functor.flip F).obj (op a.Y))).cone.pt) \u226b\n NatTrans.app (limit.lift F (combineCones F fun k => getLimitCone ((Functor.flip F).obj k))) (op a.Y)) \u226b\n NatTrans.app (limit.\u03c0 F j) (op a.Y)\n[PROOFSTEP]\nerw [Category.comp_id, Category.assoc, \u2190 NatTrans.comp_app, limit.lift_\u03c0, limit.lift_\u03c0]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n NatTrans.app (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation a E).\u03c0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n NatTrans.app E.\u03c0 j \u226b Multiequalizer.\u03b9 (Cover.index W.unop (F.obj j)) a\n[PROOFSTEP]\nrw [\u2190 hm]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n (m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (F.obj j)) a\n[PROOFSTEP]\ndsimp [diagramNatTrans]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nW : (Cover J X)\u1d52\u1d56\nE : Cone (F \u22d9 diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W)\nm : E.pt \u27f6 ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((diagramFunctor J D X \u22d9 (evaluation (Cover J X)\u1d52\u1d56 D).obj W).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app E.\u03c0 j\na : (Cover.index W.unop (limit.cone F).pt).L\nj : K\n\u22a2 m \u226b Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) a \u226b NatTrans.app (limit.\u03c0 F j) (op a.Y) =\n (m \u226b\n Multiequalizer.lift (Cover.index W.unop (F.obj j)) (multiequalizer (Cover.index W.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F j) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop (F.obj j)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F j) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (F.obj j)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (F.obj j)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F j) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (F.obj j)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (F.obj j)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (F.obj j)) a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nK : Type (max v u)\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\n\u22a2 {K_1 : K \u2964 C\u1d52\u1d56 \u2964 D} \u2192 PreservesLimit K_1 (diagramFunctor J D X)\n[PROOFSTEP]\napply preservesLimit_diagramFunctor.{w, v, u}\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\ninst\u271d : HasLimits D\n\u22a2 PreservesLimits (diagramFunctor J D X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase preservesLimitsOfShape\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\ninst\u271d : HasLimits D\n\u22a2 autoParam\n ({J_1 : Type (max u v)} \u2192\n [inst : Category.{max u v, max u v} J_1] \u2192 PreservesLimitsOfShape J_1 (diagramFunctor J D X))\n _auto\u271d\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase preservesLimitsOfShape\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\ninst\u271d\u00b9 : HasLimits D\nJ\u271d : Type (max u v)\ninst\u271d : Category.{max u v, max u v} J\u271d\n\u22a2 PreservesLimitsOfShape J\u271d (diagramFunctor J D X)\n[PROOFSTEP]\napply preservesLimitsOfShape_diagramFunctor.{w, v, u}\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\n\u22a2 \u2200 {X_1 Y : K} (f : X_1 \u27f6 Y),\n (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f \u226b\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) Y).hom =\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) X_1).hom \u226b\n (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f\n[PROOFSTEP]\nintro i j f\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\n\u22a2 (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f \u226b\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) j).hom =\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) i).hom \u226b\n (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f\n[PROOFSTEP]\nrw [\u2190 Iso.eq_comp_inv, Category.assoc, \u2190 Iso.inv_comp_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\n\u22a2 ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) i).inv \u226b\n (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f =\n (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f \u226b\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) j).inv\n[PROOFSTEP]\nrefine' colimit.hom_ext (fun w => _)\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\nw : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (F.obj i) (op X).unop) w \u226b\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) i).inv \u226b\n (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f =\n colimit.\u03b9 (diagram J (F.obj i) (op X).unop) w \u226b\n (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)).map f \u226b\n ((fun k => colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k) j).inv\n[PROOFSTEP]\ndsimp [plusMap]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\nw : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (F.obj i) X) w \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) i).inv \u226b\n (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f =\n colimit.\u03b9 (diagram J (F.obj i) X) w \u226b\n colimMap (diagramNatTrans J (F.map f) X) \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) j).inv\n[PROOFSTEP]\nerw [colimit.\u03b9_map_assoc, colimitObjIsoColimitCompEvaluation_\u03b9_inv (F \u22d9 J.diagramFunctor D X).flip w j,\n colimitObjIsoColimitCompEvaluation_\u03b9_inv_assoc (F \u22d9 J.diagramFunctor D X).flip w i]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\nw : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) w) i \u226b\n (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))).map f =\n NatTrans.app (diagramNatTrans J (F.map f) X) w \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) w) j\n[PROOFSTEP]\nrw [\u2190 (colimit.\u03b9 (F \u22d9 J.diagramFunctor D X).flip w).naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\ne : colimit (limit (F \u22d9 diagramFunctor J D X)) \u2245 limit (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) :=\n colimitLimitIso (F \u22d9 diagramFunctor J D X)\nt : diagram J (limit F) X \u2245 limit (F \u22d9 diagramFunctor J D X) :=\n IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))\np : (plusObj J (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 diagramFunctor J D X)) := HasColimit.isoOfNatIso t\ni j : K\nf : i \u27f6 j\nw : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 ((Functor.flip (F \u22d9 diagramFunctor J D X)).obj w).map f \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) w) j =\n NatTrans.app (diagramNatTrans J (F.map f) X) w \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) w) j\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 liftToPlusObjLimitObj F X S \u226b NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) = NatTrans.app S.\u03c0 k\n[PROOFSTEP]\ndsimp only [liftToPlusObjLimitObj]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 (limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) S \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).inv \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).inv) \u226b\n NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n NatTrans.app S.\u03c0 k\n[PROOFSTEP]\nrw [\u2190 (limit.isLimit (F \u22d9 J.plusFunctor D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))).fac S k, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) S \u226b\n ((HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).inv \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).inv) \u226b\n NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n IsLimit.lift (limit.isLimit (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))) S \u226b\n NatTrans.app (limit.cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))).\u03c0 k\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 ((HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).inv \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).inv) \u226b\n NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n NatTrans.app (limit.cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))).\u03c0 k\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 ((HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).inv \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).inv) \u226b\n NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc, \u2190 Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\n\u22a2 NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\nrefine' colimit.hom_ext (fun j => _)\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (limit F) (op X).unop) j \u226b NatTrans.app (plusMap J (limit.\u03c0 F k)) (op X) =\n colimit.\u03b9 (diagram J (limit F) (op X).unop) j \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\ndsimp [plusMap]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (limit F) X) j \u226b colimMap (diagramNatTrans J (limit.\u03c0 F k) X) =\n colimit.\u03b9 (diagram J (limit F) X) j \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\nsimp only [HasColimit.isoOfNatIso_\u03b9_hom_assoc, \u03b9_colimMap]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 NatTrans.app (diagramNatTrans J (limit.\u03c0 F k) X) j \u226b colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X))).hom\n j \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X)) j \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\ndsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (limit.lift (F \u22d9 diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))) j \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X)) j \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))))) \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) k\n[PROOFSTEP]\nrw [limit.lift_\u03c0]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (limit.lift (F \u22d9 diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))) j \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X)) j \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n NatTrans.app\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))))).\u03c0\n k\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (limit.lift (F \u22d9 diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))) j \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X)) j \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X)).hom \u226b\n limit.\u03c0 (colimit (Functor.flip (F \u22d9 diagramFunctor J D X))) k \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom\n[PROOFSTEP]\nrw [\u03b9_colimitLimitIso_limit_\u03c0_assoc]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (limit.lift (F \u22d9 diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))) j \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X) k) j \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc, \u2190 NatTrans.comp_app]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n (NatTrans.app\n (limit.lift (F \u22d9 diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F)) \u226b\n limit.\u03c0 (F \u22d9 diagramFunctor J D X) k)\n j \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k) \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom\n[PROOFSTEP]\nrw [limit.lift_\u03c0, Category.assoc]\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (_ :\n \u2200 (i : (Cover.index j.unop (F.obj k)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index j.unop (limit F)) i \u226b NatTrans.app (limit.\u03c0 F k) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) \u226b\n MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) \u226b\n colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (NatTrans.app ((diagramFunctor J D X).mapCone (limit.cone F)).\u03c0 k) j \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (F.obj k) X) j =\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).hom\n[PROOFSTEP]\nrw [\u2190 Iso.comp_inv_eq]\n[GOAL]\ncase e_a.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (F.obj k) X) j \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X)) k).inv =\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k\n[PROOFSTEP]\nerw [colimit.\u03b9_desc]\n[GOAL]\ncase e_a.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))\nk : K\nj : (Cover J (op X).unop)\u1d52\u1d56\n\u22a2 NatTrans.app (((evaluation K D).obj k).mapCocone (colimit.cocone (Functor.flip (F \u22d9 diagramFunctor J D X)))).\u03b9 j =\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X)) j) k\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\n\u22a2 PreservesLimitsOfShape K (plusFunctor J D)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase preservesLimit\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\n\u22a2 autoParam ({K_1 : K \u2964 C\u1d52\u1d56 \u2964 D} \u2192 PreservesLimit K_1 (plusFunctor J D)) _auto\u271d\n[PROOFSTEP]\nintro F\n[GOAL]\ncase preservesLimit\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\n\u22a2 PreservesLimit F (plusFunctor J D)\n[PROOFSTEP]\napply preservesLimitOfEvaluation\n[GOAL]\ncase preservesLimit.H\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\n\u22a2 (k : C\u1d52\u1d56) \u2192 PreservesLimit F (plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj k)\n[PROOFSTEP]\nintro X\n[GOAL]\ncase preservesLimit.H\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\n\u22a2 PreservesLimit F (plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\n[PROOFSTEP]\napply preservesLimitOfPreservesLimitCone (limit.isLimit F)\n[GOAL]\ncase preservesLimit.H\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\n\u22a2 IsLimit ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F))\n[PROOFSTEP]\nrefine' \u27e8fun S => liftToPlusObjLimitObj.{w, v, u} F X.unop S, _, _\u27e9\n[GOAL]\ncase preservesLimit.H.refine'_1\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\n\u22a2 \u2200 (s : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)) (j : K),\n (fun S => liftToPlusObjLimitObj F X.unop S) s \u226b\n NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nintro S k\n[GOAL]\ncase preservesLimit.H.refine'_1\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nk : K\n\u22a2 (fun S => liftToPlusObjLimitObj F X.unop S) S \u226b\n NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 k =\n NatTrans.app S.\u03c0 k\n[PROOFSTEP]\napply liftToPlusObjLimitObj_fac\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\n\u22a2 \u2200 (s : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X))\n (m : s.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt),\n (\u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j =\n NatTrans.app s.\u03c0 j) \u2192\n m = (fun S => liftToPlusObjLimitObj F X.unop S) s\n[PROOFSTEP]\nintro S m hm\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\n\u22a2 m = (fun S => liftToPlusObjLimitObj F X.unop S) S\n[PROOFSTEP]\ndsimp [liftToPlusObjLimitObj]\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\n\u22a2 m =\n limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) S \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).inv \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).inv\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc, Iso.eq_comp_inv, \u2190 Iso.comp_inv_eq]\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\n\u22a2 ((m \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom) \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom) \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv =\n limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) S\n[PROOFSTEP]\nrefine' limit.hom_ext (fun k => _)\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk : K\n\u22a2 (((m \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom) \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom) \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv) \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k =\n limit.lift (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) S \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, Category.assoc, \u2190 hm]\n[GOAL]\ncase preservesLimit.H.refine'_2\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk : K\n\u22a2 m \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k =\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 k\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk : K\n\u22a2 (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k =\n NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 k\n[PROOFSTEP]\nrefine' colimit.hom_ext (fun k => _)\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (limit.cone F).pt X.unop) k \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k\u271d =\n colimit.\u03b9 (diagram J (limit.cone F).pt X.unop) k \u226b\n NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 k\u271d\n[PROOFSTEP]\ndsimp [plusMap, plusObj]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J (limit F) X.unop) k \u226b\n (HasColimit.isoOfNatIso\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop)))).hom \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n (HasLimit.isoOfNatIso\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm).inv \u226b\n limit.\u03c0 (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X) k\u271d =\n colimit.\u03b9 (diagram J (limit F) X.unop) k \u226b colimMap (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop)\n[PROOFSTEP]\nerw [colimit.\u03b9_map, colimit.\u03b9_desc_assoc, limit.lift_\u03c0]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app\n ((Cocones.precompose\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom).obj\n (colimit.cocone (limit (F \u22d9 diagramFunctor J D X.unop)))).\u03b9\n k \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n NatTrans.app\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop))\n k).symm.inv).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))))).\u03c0\n k\u271d =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nconv_lhs => dsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n ((Cocones.precompose\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom).obj\n (colimit.cocone (limit (F \u22d9 diagramFunctor J D X.unop)))).\u03b9\n k \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n NatTrans.app\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm.inv).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))))).\u03c0\n k\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n ((Cocones.precompose\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom).obj\n (colimit.cocone (limit (F \u22d9 diagramFunctor J D X.unop)))).\u03b9\n k \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n NatTrans.app\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm.inv).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))))).\u03c0\n k\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n ((Cocones.precompose\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom).obj\n (colimit.cocone (limit (F \u22d9 diagramFunctor J D X.unop)))).\u03b9\n k \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n NatTrans.app\n ((Cones.postcompose\n (NatIso.ofComponents fun k =>\n colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k).symm.inv).obj\n (limit.cone (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))))).\u03c0\n k\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X.unop)) k) \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n limit.\u03c0 (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))) k\u271d \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k\u271d).hom =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n colimit.\u03b9 (limit (F \u22d9 diagramFunctor J D X.unop)) k \u226b\n (colimitLimitIso (F \u22d9 diagramFunctor J D X.unop)).hom \u226b\n limit.\u03c0 (colimit (Functor.flip (F \u22d9 diagramFunctor J D X.unop))) k\u271d \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k\u271d).hom =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nrw [\u03b9_colimitLimitIso_limit_\u03c0_assoc]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n NatTrans.app (colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k) k\u271d \u226b\n (colimitObjIsoColimitCompEvaluation (Functor.flip (F \u22d9 diagramFunctor J D X.unop)) k\u271d).hom =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nsimp only [NatIso.ofComponents_inv_app, colimitObjIsoColimitCompEvaluation_\u03b9_app_hom, Iso.symm_inv]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nconv_lhs => dsimp [IsLimit.conePointUniqueUpToIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k\n[PROOFSTEP]\ndsimp [IsLimit.conePointUniqueUpToIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k\n[PROOFSTEP]\ndsimp [IsLimit.conePointUniqueUpToIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n| NatTrans.app\n (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X.unop) (limit.isLimit F))\n (limit.isLimit (F \u22d9 diagramFunctor J D X.unop))).hom\n k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k\n[PROOFSTEP]\ndsimp [IsLimit.conePointUniqueUpToIso]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (limit.lift (F \u22d9 diagramFunctor J D X.unop) ((diagramFunctor J D X.unop).mapCone (limit.cone F))) k \u226b\n NatTrans.app (limit.\u03c0 (F \u22d9 diagramFunctor J D X.unop) k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 NatTrans.comp_app, limit.lift_\u03c0]\n[GOAL]\ncase preservesLimit.H.refine'_2.e_a\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nK : Type (max v u)\ninst\u271d\u2074 : SmallCategory K\ninst\u271d\u00b3 : FinCategory K\ninst\u271d\u00b2 : HasLimitsOfShape K D\ninst\u271d\u00b9 : PreservesLimitsOfShape K (forget D)\ninst\u271d : ReflectsLimitsOfShape K (forget D)\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nX : C\u1d52\u1d56\nS : Cone (F \u22d9 plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X)\nm : S.pt \u27f6 ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).pt\nhm :\n \u2200 (j : K),\n m \u226b NatTrans.app ((plusFunctor J D \u22d9 (evaluation C\u1d52\u1d56 D).obj X).mapCone (limit.cone F)).\u03c0 j = NatTrans.app S.\u03c0 j\nk\u271d : K\nk : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app ((diagramFunctor J D X.unop).mapCone (limit.cone F)).\u03c0 k\u271d) k \u226b\n colimit.\u03b9 (Functor.flip (F \u22d9 diagramFunctor J D X.unop) \u22d9 (evaluation K D).obj k\u271d) k =\n NatTrans.app (diagramNatTrans J (limit.\u03c0 F k\u271d) X.unop) k \u226b colimit.\u03b9 (diagram J (F.obj k\u271d) X.unop) k\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b2 : HasFiniteLimits D\ninst\u271d\u00b9 : PreservesFiniteLimits (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 PreservesFiniteLimits (plusFunctor J D)\n[PROOFSTEP]\napply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b2 : HasFiniteLimits D\ninst\u271d\u00b9 : PreservesFiniteLimits (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 (J_1 : Type (max v u)) \u2192 {\ud835\udca5 : SmallCategory J_1} \u2192 FinCategory J_1 \u2192 PreservesLimitsOfShape J_1 (plusFunctor J D)\n[PROOFSTEP]\nintro K _ _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b2 : HasFiniteLimits D\ninst\u271d\u00b9 : PreservesFiniteLimits (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\n\ud835\udca5\u271d : SmallCategory K\nx\u271d : FinCategory K\n\u22a2 PreservesLimitsOfShape K (plusFunctor J D)\n[PROOFSTEP]\nhave : ReflectsLimitsOfShape K (forget D) := reflectsLimitsOfShapeOfReflectsIsomorphisms\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b2 : HasFiniteLimits D\ninst\u271d\u00b9 : PreservesFiniteLimits (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\n\ud835\udca5\u271d : SmallCategory K\nx\u271d : FinCategory K\nthis : ReflectsLimitsOfShape K (forget D)\n\u22a2 PreservesLimitsOfShape K (plusFunctor J D)\n[PROOFSTEP]\napply preservesLimitsOfShape_plusFunctor.{w, v, u}\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\n\u22a2 PreservesLimitsOfShape K (presheafToSheaf J D)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase preservesLimit\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\n\u22a2 autoParam ({K_1 : K \u2964 C\u1d52\u1d56 \u2964 D} \u2192 PreservesLimit K_1 (presheafToSheaf J D)) _auto\u271d\n[PROOFSTEP]\nintro F\n[GOAL]\ncase preservesLimit\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\n\u22a2 PreservesLimit F (presheafToSheaf J D)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase preservesLimit.preserves\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\n\u22a2 {c : Cone F} \u2192 IsLimit c \u2192 IsLimit ((presheafToSheaf J D).mapCone c)\n[PROOFSTEP]\nintro S hS\n[GOAL]\ncase preservesLimit.preserves\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nS : Cone F\nhS : IsLimit S\n\u22a2 IsLimit ((presheafToSheaf J D).mapCone S)\n[PROOFSTEP]\napply isLimitOfReflects (sheafToPresheaf J D)\n[GOAL]\ncase preservesLimit.preserves.t\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nS : Cone F\nhS : IsLimit S\n\u22a2 IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))\n[PROOFSTEP]\nhave : ReflectsLimitsOfShape K (forget D) := reflectsLimitsOfShapeOfReflectsIsomorphisms\n[GOAL]\ncase preservesLimit.preserves.t\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nS : Cone F\nhS : IsLimit S\nthis : ReflectsLimitsOfShape K (forget D)\n\u22a2 IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))\n[PROOFSTEP]\nhave : PreservesLimitsOfShape K (presheafToSheaf J D \u22d9 sheafToPresheaf J D) :=\n preservesLimitsOfShapeOfNatIso (J.sheafificationIsoPresheafToSheafCompSheafToPreasheaf D)\n[GOAL]\ncase preservesLimit.preserves.t\nC : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2079 : Category.{max v u, w} D\ninst\u271d\u2078 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2077 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b2 : SmallCategory K\ninst\u271d\u00b9 : FinCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 C\u1d52\u1d56 \u2964 D\nS : Cone F\nhS : IsLimit S\nthis\u271d : ReflectsLimitsOfShape K (forget D)\nthis : PreservesLimitsOfShape K (presheafToSheaf J D \u22d9 sheafToPresheaf J D)\n\u22a2 IsLimit ((sheafToPresheaf J D).mapCone ((presheafToSheaf J D).mapCone S))\n[PROOFSTEP]\nexact isLimitOfPreserves (presheafToSheaf J D \u22d9 sheafToPresheaf J D) hS\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9\u2070 : Category.{max v u, w} D\ninst\u271d\u2079 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2078 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2077 : ConcreteCategory D\ninst\u271d\u2076 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b3 : SmallCategory K\ninst\u271d\u00b2 : FinCategory K\ninst\u271d\u00b9 : HasLimitsOfShape K D\ninst\u271d : HasFiniteLimits D\n\u22a2 PreservesFiniteLimits (presheafToSheaf J D)\n[PROOFSTEP]\napply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{max v u}\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9\u2070 : Category.{max v u, w} D\ninst\u271d\u2079 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2078 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2077 : ConcreteCategory D\ninst\u271d\u2076 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b3 : SmallCategory K\ninst\u271d\u00b2 : FinCategory K\ninst\u271d\u00b9 : HasLimitsOfShape K D\ninst\u271d : HasFiniteLimits D\n\u22a2 (J_1 : Type (max v u)) \u2192 {\ud835\udca5 : SmallCategory J_1} \u2192 FinCategory J_1 \u2192 PreservesLimitsOfShape J_1 (presheafToSheaf J D)\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9\u00b9 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9\u2070 : Category.{max v u, w} D\ninst\u271d\u2079 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u2078 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u2077 : ConcreteCategory D\ninst\u271d\u2076 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget D)\nK : Type (max v u)\ninst\u271d\u00b3 : SmallCategory K\ninst\u271d\u00b2 : FinCategory K\ninst\u271d\u00b9 : HasLimitsOfShape K D\ninst\u271d : HasFiniteLimits D\nJ\u271d : Type (max v u)\n\ud835\udca5\u271d : SmallCategory J\u271d\nx\u271d : FinCategory J\u271d\n\u22a2 PreservesLimitsOfShape J\u271d (presheafToSheaf J D)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.LeftExact", "llama_tokens": 59223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.4073334000459302, "lm_q1q2_score": 0.27803508002021976}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\ninst\u271d : HasFiniteLimits C\n\u22a2 HasLimitsOfShape J C\n[PROOFSTEP]\napply @hasLimitsOfShape_of_equivalence _ _ _ _ _ _ (FinCategory.equivAsType J) ?_\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\ninst\u271d : HasFiniteLimits C\n\u22a2 HasLimitsOfShape (FinCategory.AsType J) C\n[PROOFSTEP]\napply HasFiniteLimits.out\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasLimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\n\u22a2 HasLimitsOfShape J C\n[PROOFSTEP]\nhaveI := h (ULiftHom.{w} (ULift.{w} J)) <| @CategoryTheory.finCategoryUlift J hJ hhJ\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasLimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\nthis : HasLimitsOfShape (ULiftHom (ULift J)) C\n\u22a2 HasLimitsOfShape J C\n[PROOFSTEP]\nhave l : @Equivalence J (ULiftHom (ULift J)) hJ (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) :=\n @ULiftHomULiftCategory.equiv J hJ\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasLimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\nthis : HasLimitsOfShape (ULiftHom (ULift J)) C\nl : J \u224c ULiftHom (ULift J)\n\u22a2 HasLimitsOfShape J C\n[PROOFSTEP]\napply\n @hasLimitsOfShape_of_equivalence (ULiftHom (ULift J)) (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) C _ J hJ\n (@Equivalence.symm J hJ (ULiftHom (ULift J)) (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) l)\n _\n /- Porting note: tried to factor out (@instCategoryULiftHom (ULift J) (@uliftCategory J hJ)\n but when doing that would then find the instance and say it was not definitionally equal to\n the provided one (the same thing factored out) -/\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\ninst\u271d : HasFiniteColimits C\n\u22a2 HasColimitsOfShape J C\n[PROOFSTEP]\nrefine @hasColimitsOfShape_of_equivalence _ _ _ _ _ _ (FinCategory.equivAsType J) ?_\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : Type w\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\ninst\u271d : HasFiniteColimits C\n\u22a2 HasColimitsOfShape (FinCategory.AsType J) C\n[PROOFSTEP]\napply HasFiniteColimits.out\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasColimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\n\u22a2 HasColimitsOfShape J C\n[PROOFSTEP]\nhaveI := h (ULiftHom.{w} (ULift.{w} J)) <| @CategoryTheory.finCategoryUlift J hJ hhJ\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasColimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\nthis : HasColimitsOfShape (ULiftHom (ULift J)) C\n\u22a2 HasColimitsOfShape J C\n[PROOFSTEP]\nhave l : @Equivalence J (ULiftHom (ULift J)) hJ (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) :=\n @ULiftHomULiftCategory.equiv J hJ\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nh : \u2200 (J : Type w) {\ud835\udca5 : SmallCategory J}, FinCategory J \u2192 HasColimitsOfShape J C\nJ : Type\nhJ : SmallCategory J\nhhJ : FinCategory J\nthis : HasColimitsOfShape (ULiftHom (ULift J)) C\nl : J \u224c ULiftHom (ULift J)\n\u22a2 HasColimitsOfShape J C\n[PROOFSTEP]\napply\n @hasColimitsOfShape_of_equivalence (ULiftHom (ULift J)) (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) C _ J hJ\n (@Equivalence.symm J hJ (ULiftHom (ULift J)) (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) l) _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nx : WalkingParallelPair\n\u22a2 x \u2208 List.toFinset [zero, one]\n[PROOFSTEP]\ncases x\n[GOAL]\ncase zero\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 zero \u2208 List.toFinset [zero, one]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase one\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 one \u2208 List.toFinset [zero, one]\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nj j' : WalkingParallelPair\n\u22a2 \u2200 (x : WalkingParallelPairHom j j'),\n x \u2208\n WalkingParallelPair.recOn j\n (WalkingParallelPair.recOn j' (List.toFinset [WalkingParallelPairHom.id zero]) (List.toFinset [left, right]))\n (WalkingParallelPair.recOn j' \u2205 (List.toFinset [WalkingParallelPairHom.id one]))\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 left \u2208\n WalkingParallelPair.recOn zero\n (WalkingParallelPair.recOn one (List.toFinset [WalkingParallelPairHom.id zero]) (List.toFinset [left, right]))\n (WalkingParallelPair.recOn one \u2205 (List.toFinset [WalkingParallelPairHom.id one]))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 right \u2208\n WalkingParallelPair.recOn zero\n (WalkingParallelPair.recOn one (List.toFinset [WalkingParallelPairHom.id zero]) (List.toFinset [left, right]))\n (WalkingParallelPair.recOn one \u2205 (List.toFinset [WalkingParallelPairHom.id one]))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id\nC : Type u\ninst\u271d : Category.{v, u} C\nj : WalkingParallelPair\n\u22a2 WalkingParallelPairHom.id j \u2208\n WalkingParallelPair.recOn j\n (WalkingParallelPair.recOn j (List.toFinset [WalkingParallelPairHom.id zero]) (List.toFinset [left, right]))\n (WalkingParallelPair.recOn j \u2205 (List.toFinset [WalkingParallelPairHom.id one]))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id\nC : Type u\ninst\u271d : Category.{v, u} C\nj : WalkingParallelPair\n\u22a2 \ud835\udfd9 j \u2208\n WalkingParallelPair.rec (WalkingParallelPair.rec {\ud835\udfd9 zero} {left, right} j) (WalkingParallelPair.rec \u2205 {\ud835\udfd9 one} j) j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase id.zero\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 \ud835\udfd9 zero \u2208\n WalkingParallelPair.rec (WalkingParallelPair.rec {\ud835\udfd9 zero} {left, right} zero)\n (WalkingParallelPair.rec \u2205 {\ud835\udfd9 one} zero) zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id.one\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 \ud835\udfd9 one \u2208\n WalkingParallelPair.rec (WalkingParallelPair.rec {\ud835\udfd9 zero} {left, right} one) (WalkingParallelPair.rec \u2205 {\ud835\udfd9 one} one)\n one\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasFiniteLimits C\n\u22a2 HasEqualizers C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasFiniteColimits C\n\u22a2 HasCoequalizers C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : Fintype J\n\u22a2 Fintype (WidePullbackShape J)\n[PROOFSTEP]\nrw [WidePullbackShape]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : Fintype J\n\u22a2 Fintype (Option J)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj j' : WidePullbackShape J\n\u22a2 Finset (j \u27f6 j')\n[PROOFSTEP]\ncases' j' with j'\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\n\u22a2 Finset (j \u27f6 none)\n[PROOFSTEP]\ncases' j with j\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 Finset (none \u27f6 none)\n[PROOFSTEP]\nexact {Hom.id none}\n[GOAL]\ncase none.some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : J\n\u22a2 Finset (some j \u27f6 none)\n[PROOFSTEP]\nexact {Hom.term j}\n[GOAL]\ncase some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\nj' : J\n\u22a2 Finset (j \u27f6 some j')\n[PROOFSTEP]\nby_cases some j' = j\n[GOAL]\ncase some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\nj' : J\n\u22a2 Finset (j \u27f6 some j')\n[PROOFSTEP]\nby_cases some j' = j\n[GOAL]\ncase pos\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\nj' : J\nh : some j' = j\n\u22a2 Finset (j \u27f6 some j')\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\nj' : J\nh : some j' = j\n\u22a2 Finset (j \u27f6 j)\n[PROOFSTEP]\nexact {Hom.id j}\n[GOAL]\ncase neg\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\nj' : J\nh : \u00acsome j' = j\n\u22a2 Finset (j \u27f6 some j')\n[PROOFSTEP]\nexact \u2205\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj j' : WidePullbackShape J\n\u22a2 \u2200 (x : j \u27f6 j'),\n x \u2208\n Option.casesOn (motive := fun t => j' = t \u2192 Finset (j \u27f6 j')) j'\n (fun h =>\n (_ : none = j') \u25b8\n Option.casesOn (motive := fun t => j = t \u2192 Finset (j \u27f6 none)) j (fun h => (_ : none = j) \u25b8 {Hom.id none})\n (fun j_1 h => (_ : some j_1 = j) \u25b8 {Hom.term j_1}) (_ : j = j))\n (fun j'_1 h =>\n (_ : some j'_1 = j') \u25b8\n if h : some j'_1 = j then Eq.mpr (_ : Finset (j \u27f6 some j'_1) = Finset (j \u27f6 j)) {Hom.id j} else \u2205)\n (_ : j' = j')\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase id\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\n\u22a2 Hom.id j \u2208\n Option.casesOn (motive := fun t => j = t \u2192 Finset (j \u27f6 j)) j\n (fun h =>\n (_ : none = j) \u25b8\n Option.casesOn (motive := fun t => j = t \u2192 Finset (j \u27f6 none)) j (fun h => (_ : none = j) \u25b8 {Hom.id none})\n (fun j_1 h => (_ : some j_1 = j) \u25b8 {Hom.term j_1}) (_ : j = j))\n (fun j' h =>\n (_ : some j' = j) \u25b8\n if h : some j' = j then Eq.mpr (_ : Finset (j \u27f6 some j') = Finset (j \u27f6 j)) {Hom.id j} else \u2205)\n (_ : j = j)\n[PROOFSTEP]\ncases j\n[GOAL]\ncase id.none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 Hom.id none \u2208\n Option.casesOn (motive := fun t => none = t \u2192 Finset (none \u27f6 none)) none\n (fun h =>\n (_ : none = none) \u25b8\n Option.casesOn (motive := fun t => none = t \u2192 Finset (none \u27f6 none)) none\n (fun h => (_ : none = none) \u25b8 {Hom.id none}) (fun j h => (_ : some j = none) \u25b8 {Hom.term j})\n (_ : none = none))\n (fun j' h =>\n (_ : some j' = none) \u25b8\n if h : some j' = none then Eq.mpr (_ : Finset (none \u27f6 some j') = Finset (none \u27f6 none)) {Hom.id none} else \u2205)\n (_ : none = none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id.some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nval\u271d : J\n\u22a2 Hom.id (some val\u271d) \u2208\n Option.casesOn (motive := fun t => some val\u271d = t \u2192 Finset (some val\u271d \u27f6 some val\u271d)) (some val\u271d)\n (fun h =>\n (_ : none = some val\u271d) \u25b8\n Option.casesOn (motive := fun t => some val\u271d = t \u2192 Finset (some val\u271d \u27f6 none)) (some val\u271d)\n (fun h => (_ : none = some val\u271d) \u25b8 {Hom.id none}) (fun j h => (_ : some j = some val\u271d) \u25b8 {Hom.term j})\n (_ : some val\u271d = some val\u271d))\n (fun j' h =>\n (_ : some j' = some val\u271d) \u25b8\n if h : some j' = some val\u271d then\n Eq.mpr (_ : Finset (some val\u271d \u27f6 some j') = Finset (some val\u271d \u27f6 some val\u271d)) {Hom.id (some val\u271d)}\n else \u2205)\n (_ : some val\u271d = some val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase term\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj\u271d : J\n\u22a2 Hom.term j\u271d \u2208\n Option.casesOn (motive := fun t => none = t \u2192 Finset (some j\u271d \u27f6 none)) none\n (fun h =>\n (_ : none = none) \u25b8\n Option.casesOn (motive := fun t => some j\u271d = t \u2192 Finset (some j\u271d \u27f6 none)) (some j\u271d)\n (fun h => (_ : none = some j\u271d) \u25b8 {Hom.id none}) (fun j h => (_ : some j = some j\u271d) \u25b8 {Hom.term j})\n (_ : some j\u271d = some j\u271d))\n (fun j' h =>\n (_ : some j' = none) \u25b8\n if h : some j' = some j\u271d then\n Eq.mpr (_ : Finset (some j\u271d \u27f6 some j') = Finset (some j\u271d \u27f6 some j\u271d)) {Hom.id (some j\u271d)}\n else \u2205)\n (_ : none = none)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : Fintype J\n\u22a2 Fintype (WidePushoutShape J)\n[PROOFSTEP]\nrw [WidePushoutShape]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : Fintype J\n\u22a2 Fintype (Option J)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj j' : WidePushoutShape J\n\u22a2 Finset (j \u27f6 j')\n[PROOFSTEP]\ncases' j with j\n[GOAL]\ncase none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\n\u22a2 Finset (none \u27f6 j')\n[PROOFSTEP]\ncases' j' with j'\n[GOAL]\ncase none.none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 Finset (none \u27f6 none)\n[PROOFSTEP]\nexact {Hom.id none}\n[GOAL]\ncase none.some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : J\n\u22a2 Finset (none \u27f6 some j')\n[PROOFSTEP]\nexact {Hom.init j'}\n[GOAL]\ncase some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\nj : J\n\u22a2 Finset (some j \u27f6 j')\n[PROOFSTEP]\nby_cases some j = j'\n[GOAL]\ncase some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\nj : J\n\u22a2 Finset (some j \u27f6 j')\n[PROOFSTEP]\nby_cases some j = j'\n[GOAL]\ncase pos\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\nj : J\nh : some j = j'\n\u22a2 Finset (some j \u27f6 j')\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\nj : J\nh : some j = j'\n\u22a2 Finset (j' \u27f6 j')\n[PROOFSTEP]\nexact {Hom.id j'}\n[GOAL]\ncase neg\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj' : WidePushoutShape J\nj : J\nh : \u00acsome j = j'\n\u22a2 Finset (some j \u27f6 j')\n[PROOFSTEP]\nexact \u2205\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj j' : WidePushoutShape J\n\u22a2 \u2200 (x : j \u27f6 j'),\n x \u2208\n Option.casesOn (motive := fun t => j = t \u2192 Finset (j \u27f6 j')) j\n (fun h =>\n (_ : none = j) \u25b8\n Option.casesOn (motive := fun t => j' = t \u2192 Finset (none \u27f6 j')) j'\n (fun h => (_ : none = j') \u25b8 {Hom.id none}) (fun j'_1 h => (_ : some j'_1 = j') \u25b8 {Hom.init j'_1})\n (_ : j' = j'))\n (fun j_1 h =>\n (_ : some j_1 = j) \u25b8\n if h : some j_1 = j' then Eq.mpr (_ : Finset (some j_1 \u27f6 j') = Finset (j' \u27f6 j')) {Hom.id j'} else \u2205)\n (_ : j = j)\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase id\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj : WidePushoutShape J\n\u22a2 Hom.id j \u2208\n Option.casesOn (motive := fun t => j = t \u2192 Finset (j \u27f6 j)) j\n (fun h =>\n (_ : none = j) \u25b8\n Option.casesOn (motive := fun t => j = t \u2192 Finset (none \u27f6 j)) j (fun h => (_ : none = j) \u25b8 {Hom.id none})\n (fun j' h => (_ : some j' = j) \u25b8 {Hom.init j'}) (_ : j = j))\n (fun j_1 h =>\n (_ : some j_1 = j) \u25b8\n if h : some j_1 = j then Eq.mpr (_ : Finset (some j_1 \u27f6 j) = Finset (j \u27f6 j)) {Hom.id j} else \u2205)\n (_ : j = j)\n[PROOFSTEP]\ncases j\n[GOAL]\ncase id.none\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 Hom.id none \u2208\n Option.casesOn (motive := fun t => none = t \u2192 Finset (none \u27f6 none)) none\n (fun h =>\n (_ : none = none) \u25b8\n Option.casesOn (motive := fun t => none = t \u2192 Finset (none \u27f6 none)) none\n (fun h => (_ : none = none) \u25b8 {Hom.id none}) (fun j' h => (_ : some j' = none) \u25b8 {Hom.init j'})\n (_ : none = none))\n (fun j h =>\n (_ : some j = none) \u25b8\n if h : some j = none then Eq.mpr (_ : Finset (some j \u27f6 none) = Finset (none \u27f6 none)) {Hom.id none} else \u2205)\n (_ : none = none)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase id.some\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nval\u271d : J\n\u22a2 Hom.id (some val\u271d) \u2208\n Option.casesOn (motive := fun t => some val\u271d = t \u2192 Finset (some val\u271d \u27f6 some val\u271d)) (some val\u271d)\n (fun h =>\n (_ : none = some val\u271d) \u25b8\n Option.casesOn (motive := fun t => some val\u271d = t \u2192 Finset (none \u27f6 some val\u271d)) (some val\u271d)\n (fun h => (_ : none = some val\u271d) \u25b8 {Hom.id none}) (fun j' h => (_ : some j' = some val\u271d) \u25b8 {Hom.init j'})\n (_ : some val\u271d = some val\u271d))\n (fun j h =>\n (_ : some j = some val\u271d) \u25b8\n if h : some j = some val\u271d then\n Eq.mpr (_ : Finset (some j \u27f6 some val\u271d) = Finset (some val\u271d \u27f6 some val\u271d)) {Hom.id (some val\u271d)}\n else \u2205)\n (_ : some val\u271d = some val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase init\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nj\u271d : J\n\u22a2 Hom.init j\u271d \u2208\n Option.casesOn (motive := fun t => none = t \u2192 Finset (none \u27f6 some j\u271d)) none\n (fun h =>\n (_ : none = none) \u25b8\n Option.casesOn (motive := fun t => some j\u271d = t \u2192 Finset (none \u27f6 some j\u271d)) (some j\u271d)\n (fun h => (_ : none = some j\u271d) \u25b8 {Hom.id none}) (fun j' h => (_ : some j' = some j\u271d) \u25b8 {Hom.init j'})\n (_ : some j\u271d = some j\u271d))\n (fun j h =>\n (_ : some j = none) \u25b8\n if h : some j = some j\u271d then\n Eq.mpr (_ : Finset (some j \u27f6 some j\u271d) = Finset (some j\u271d \u27f6 some j\u271d)) {Hom.id (some j\u271d)}\n else \u2205)\n (_ : none = none)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePullbacks C\n\u22a2 HasLimitsOfShape (WidePullbackShape J) C\n[PROOFSTEP]\ncases nonempty_fintype J\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePullbacks C\nval\u271d : Fintype J\n\u22a2 HasLimitsOfShape (WidePullbackShape J) C\n[PROOFSTEP]\nhaveI := @HasFiniteWidePullbacks.out C _ _ J\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePullbacks C\nval\u271d : Fintype J\nthis : \u2200 [inst : Fintype J], HasLimitsOfShape (WidePullbackShape J) C\n\u22a2 HasLimitsOfShape (WidePullbackShape J) C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePushouts C\n\u22a2 HasColimitsOfShape (WidePushoutShape J) C\n[PROOFSTEP]\ncases nonempty_fintype J\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePushouts C\nval\u271d : Fintype J\n\u22a2 HasColimitsOfShape (WidePushoutShape J) C\n[PROOFSTEP]\nhaveI := @HasFiniteWidePushouts.out C _ _ J\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ\u271d : Type v\nJ : Type\ninst\u271d\u00b9 : Finite J\ninst\u271d : HasFiniteWidePushouts C\nval\u271d : Fintype J\nthis : \u2200 [inst : Fintype J], HasColimitsOfShape (WidePushoutShape J) C\n\u22a2 HasColimitsOfShape (WidePushoutShape J) C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\nx : WalkingPair\n\u22a2 x \u2208 {WalkingPair.left, WalkingPair.right}\n[PROOFSTEP]\ncases x\n[GOAL]\ncase left\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 WalkingPair.left \u2208 {WalkingPair.left, WalkingPair.right}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nC : Type u\ninst\u271d : Category.{v, u} C\nJ : Type v\n\u22a2 WalkingPair.right \u2208 {WalkingPair.left, WalkingPair.right}\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : HasFiniteWidePullbacks C\n\u22a2 HasPullbacks C\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nJ : Type v\ninst\u271d : HasFiniteWidePushouts C\n\u22a2 HasPushouts C\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits", "llama_tokens": 8052, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.2778365400273305}} {"text": "[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 : C\n\u22a2 (\u03bb_ (F.obj X)).inv \u226b (F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b \u03bc F (\ud835\udfd9_ C) X = F.map (\u03bb_ X).inv\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, F.left_unitality, Category.assoc, Category.assoc, \u2190 F.toFunctor.map_comp, Iso.hom_inv_id,\n F.toFunctor.map_id, comp_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\nX : C\n\u22a2 (\u03c1_ (F.obj X)).inv \u226b (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5) \u226b \u03bc F X (\ud835\udfd9_ C) = F.map (\u03c1_ X).inv\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, F.right_unitality, Category.assoc, Category.assoc, \u2190 F.toFunctor.map_comp, Iso.hom_inv_id,\n F.toFunctor.map_id, comp_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\nX Y Z : C\n\u22a2 (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z) \u226b \u03bc F X (Y \u2297 Z) \u226b F.map (\u03b1_ X Y Z).inv =\n (\u03b1_ (F.obj X) (F.obj Y) (F.obj Z)).inv \u226b (\u03bc F X Y \u2297 \ud835\udfd9 (F.obj Z)) \u226b \u03bc F (X \u2297 Y) Z\n[PROOFSTEP]\nrw [Iso.eq_inv_comp, \u2190 F.associativity_assoc, \u2190 F.toFunctor.map_comp, Iso.hom_inv_id, F.toFunctor.map_id, comp_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 : MonoidalFunctor C D\nX Y X' Y' : C\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 F.map (f \u2297 g) =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X X') \u226b\n (F.map f \u2297 F.map g) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y Y'\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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03bb_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X) \u226b (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b (\u03bb_ (F.obj X)).hom\n[PROOFSTEP]\nsimp only [LaxMonoidalFunctor.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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03bb_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X) \u226b\n (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b\n (F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X \u226b F.map (\u03bb_ X).hom\n[PROOFSTEP]\nslice_rhs 2 3 =>\n rw [\u2190 comp_tensor_id]\n simp\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 : MonoidalFunctor C D\nX : C\n| (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b (F.\u03b5 \u2297 \ud835\udfd9 (F.obj 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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) 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 : MonoidalFunctor C D\nX : C\n| F.map (\u03bb_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X)\n[PROOFSTEP]\n rw [\u2190 comp_tensor_id]\n simp\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 : MonoidalFunctor C D\nX : C\n| (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b (F.\u03b5 \u2297 \ud835\udfd9 (F.obj 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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) 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 : MonoidalFunctor C D\nX : C\n| F.map (\u03bb_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X)\n[PROOFSTEP]\n rw [\u2190 comp_tensor_id]\n simp\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 : MonoidalFunctor C D\nX : C\n| (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b (F.\u03b5 \u2297 \ud835\udfd9 (F.obj 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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) 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 : MonoidalFunctor C D\nX : C\n| F.map (\u03bb_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X)\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id]\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 : MonoidalFunctor C D\nX : C\n| inv F.\u03b5 \u226b F.\u03b5 \u2297 \ud835\udfd9 (F.obj 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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) 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 : MonoidalFunctor C D\nX : C\n| F.map (\u03bb_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X)\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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03bb_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X) \u226b\n (\ud835\udfd9 (F.obj (\ud835\udfd9_ C) \u2297 F.obj X) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) X) \u226b F.map (\u03bb_ X).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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b (\u03c1_ (F.obj X)).hom\n[PROOFSTEP]\nsimp only [LaxMonoidalFunctor.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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)) \u226b\n (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b\n (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C) \u226b F.map (\u03c1_ X).hom\n[PROOFSTEP]\nslice_rhs 2 3 =>\n rw [\u2190 id_tensor_comp]\n simp\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 : MonoidalFunctor C D\nX : C\n| (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5)\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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)\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 : MonoidalFunctor C D\nX : C\n| F.map (\u03c1_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C))\n[PROOFSTEP]\n rw [\u2190 id_tensor_comp]\n simp\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 : MonoidalFunctor C D\nX : C\n| (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5)\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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)\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 : MonoidalFunctor C D\nX : C\n| F.map (\u03c1_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C))\n[PROOFSTEP]\n rw [\u2190 id_tensor_comp]\n simp\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 : MonoidalFunctor C D\nX : C\n| (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5)\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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)\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 : MonoidalFunctor C D\nX : C\n| F.map (\u03c1_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C))\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp]\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 : MonoidalFunctor C D\nX : C\n| \ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5 \u226b F.\u03b5\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 : MonoidalFunctor C D\nX : C\n| LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)\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 : MonoidalFunctor C D\nX : C\n| F.map (\u03c1_ X).hom\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 : MonoidalFunctor C D\nX : C\n| inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C))\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 : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)) \u226b\n (\ud835\udfd9 (F.obj X \u2297 F.obj (\ud835\udfd9_ C)) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X (\ud835\udfd9_ C)) \u226b F.map (\u03c1_ X).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 : MonoidalFunctor C D\n\u22a2 (X : C \u00d7 C) \u2192 (Functor.prod F.toFunctor F.toFunctor \u22d9 tensor D).obj X \u2245 (tensor C \u22d9 F.toFunctor).obj X\n[PROOFSTEP]\nintros\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 : MonoidalFunctor C D\nX\u271d : C \u00d7 C\n\u22a2 (Functor.prod F.toFunctor F.toFunctor \u22d9 tensor D).obj X\u271d \u2245 (tensor C \u22d9 F.toFunctor).obj X\u271d\n[PROOFSTEP]\napply F.\u03bcIso\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 : MonoidalFunctor C D\n\u22a2 \u2200 {X Y : C \u00d7 C} (f : X \u27f6 Y),\n (Functor.prod F.toFunctor F.toFunctor \u22d9 tensor D).map f \u226b (\u03bcIso F Y.fst Y.snd).hom =\n (\u03bcIso F X.fst X.snd).hom \u226b (tensor C \u22d9 F.toFunctor).map f\n[PROOFSTEP]\nintros\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 : MonoidalFunctor C D\nX\u271d Y\u271d : C \u00d7 C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (Functor.prod F.toFunctor F.toFunctor \u22d9 tensor D).map f\u271d \u226b (\u03bcIso F Y\u271d.fst Y\u271d.snd).hom =\n (\u03bcIso F X\u271d.fst X\u271d.snd).hom \u226b (tensor C \u22d9 F.toFunctor).map f\u271d\n[PROOFSTEP]\napply F.toLaxMonoidalFunctor.\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 : MonoidalFunctor C D\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 (F.toFunctor \u22d9 tensorLeft (F.obj X)).map f \u226b ((fun Y => \u03bcIso F X Y) Z).hom =\n ((fun Y => \u03bcIso F X Y) Y).hom \u226b (tensorLeft X \u22d9 F.toFunctor).map f\n[PROOFSTEP]\nconvert F.\u03bc_natural (\ud835\udfd9 X) f using 2\n[GOAL]\ncase h.e'_2.h.h.e'_6.h\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 : MonoidalFunctor C D\nX Y Z : C\nf : Y \u27f6 Z\ne_1\u271d :\n ((F.toFunctor \u22d9 tensorLeft (F.obj X)).obj Y \u27f6 (tensorLeft X \u22d9 F.toFunctor).obj Z) =\n (F.obj X \u2297 F.obj Y \u27f6 F.obj (X \u2297 Z))\ne_3\u271d : (F.toFunctor \u22d9 tensorLeft (F.obj X)).obj Y = F.obj X \u2297 F.obj Y\ne_4\u271d : (F.toFunctor \u22d9 tensorLeft (F.obj X)).obj Z = F.obj X \u2297 F.obj Z\n\u22a2 (F.toFunctor \u22d9 tensorLeft (F.obj X)).map f = F.map (\ud835\udfd9 X) \u2297 F.map f\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 : MonoidalFunctor C D\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 (F.toFunctor \u22d9 tensorRight (F.obj X)).map f \u226b ((fun Y => \u03bcIso F Y X) Z).hom =\n ((fun Y => \u03bcIso F Y X) Y).hom \u226b (tensorRight X \u22d9 F.toFunctor).map f\n[PROOFSTEP]\nconvert F.\u03bc_natural f (\ud835\udfd9 X) using 2\n[GOAL]\ncase h.e'_2.h.h.e'_6.h\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 : MonoidalFunctor C D\nX Y Z : C\nf : Y \u27f6 Z\ne_1\u271d :\n ((F.toFunctor \u22d9 tensorRight (F.obj X)).obj Y \u27f6 (tensorRight X \u22d9 F.toFunctor).obj Z) =\n (F.obj Y \u2297 F.obj X \u27f6 F.obj (Z \u2297 X))\ne_3\u271d : (F.toFunctor \u22d9 tensorRight (F.obj X)).obj Y = F.obj Y \u2297 F.obj X\ne_4\u271d : (F.toFunctor \u22d9 tensorRight (F.obj X)).obj Z = F.obj Z \u2297 F.obj X\n\u22a2 (F.toFunctor \u22d9 tensorRight (F.obj X)).map f = F.map f \u2297 F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nx\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d : C\nf : x\u271d\u00b3 \u27f6 x\u271d\u00b2\ng : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 ((Functor.mk src\u271d.toPrefunctor).map f \u2297 (Functor.mk src\u271d.toPrefunctor).map g) \u226b\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) x\u271d\u00b2 x\u271d =\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) x\u271d\u00b3 x\u271d\u00b9 \u226b (Functor.mk src\u271d.toPrefunctor).map (f \u2297 g)\n[PROOFSTEP]\nsimp only [Functor.comp_map, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nx\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d : C\nf : x\u271d\u00b3 \u27f6 x\u271d\u00b2\ng : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (G.map (F.map f) \u2297 G.map (F.map g)) \u226b \u03bc G (F.obj x\u271d\u00b2) (F.obj x\u271d) \u226b G.map (\u03bc F x\u271d\u00b2 x\u271d) =\n \u03bc G (F.obj x\u271d\u00b3) (F.obj x\u271d\u00b9) \u226b G.map (\u03bc F x\u271d\u00b3 x\u271d\u00b9) \u226b G.map (F.map (f \u2297 g))\n[PROOFSTEP]\nrw [\u2190 Category.assoc, LaxMonoidalFunctor.\u03bc_natural, Category.assoc, \u2190 map_comp, \u2190 map_comp, \u2190\n LaxMonoidalFunctor.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 ((fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) X Y \u2297 \ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj Z)) \u226b\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) (X \u2297 Y) Z \u226b\n (Functor.mk src\u271d.toPrefunctor).map (\u03b1_ X Y Z).hom =\n (\u03b1_ ((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 (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X) \u2297 (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) Y Z) \u226b\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) X (Y \u2297 Z)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 (\u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n (\u03bc G (F.obj (X \u2297 Y)) (F.obj Z) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b G.map (F.map (\u03b1_ X Y Z).hom) =\n (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z) \u226b G.map (\u03bc F Y Z)) \u226b\n \u03bc G (F.obj X) (F.obj (Y \u2297 Z)) \u226b G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [id_tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 (\u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n (\u03bc G (F.obj (X \u2297 Y)) (F.obj Z) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b G.map (F.map (\u03b1_ X Y Z).hom) =\n (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n ((\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)) \u226b (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map (\u03bc F Y Z))) \u226b\n \u03bc G (F.obj X) (F.obj (Y \u2297 Z)) \u226b G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nslice_rhs 3 4 => rw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map (\u03bc F Y Z)) \u226b \u03bc G (F.obj X) (F.obj (Y \u2297 Z))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map (\u03bc F Y Z)) \u226b \u03bc G (F.obj X) (F.obj (Y \u2297 Z))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map (\u03bc F Y Z)) \u226b \u03bc G (F.obj X) (F.obj (Y \u2297 Z))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 (\u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n (\u03bc G (F.obj (X \u2297 Y)) (F.obj Z) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b G.map (F.map (\u03b1_ X Y Z).hom) =\n (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)) \u226b\n (\u03bc G (F.obj X) (F.obj Y \u2297 F.obj Z) \u226b G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)) \u226b G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nslice_rhs 1 3 => rw [\u2190 G.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)) \u226b \u03bc G (F.obj X) (F.obj Y \u2297 F.obj Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [\u2190 G.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)) \u226b \u03bc G (F.obj X) (F.obj Y \u2297 F.obj Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [\u2190 G.associativity]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (\u03b1_ (G.obj (F.obj X)) (G.obj (F.obj Y)) (G.obj (F.obj Z))).hom \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 \u03bc G (F.obj Y) (F.obj Z)) \u226b \u03bc G (F.obj X) (F.obj Y \u2297 F.obj Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [\u2190 G.associativity]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 (\u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n (\u03bc G (F.obj (X \u2297 Y)) (F.obj Z) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b G.map (F.map (\u03b1_ X Y Z).hom) =\n (((\u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n \u03bc G (F.obj X \u2297 F.obj Y) (F.obj Z) \u226b G.map (\u03b1_ (F.obj X) (F.obj Y) (F.obj Z)).hom) \u226b\n G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)) \u226b\n G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [comp_tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 ((\u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b (G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z)))) \u226b\n (\u03bc G (F.obj (X \u2297 Y)) (F.obj Z) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b G.map (F.map (\u03b1_ X Y Z).hom) =\n (((\u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n \u03bc G (F.obj X \u2297 F.obj Y) (F.obj Z) \u226b G.map (\u03b1_ (F.obj X) (F.obj Y) (F.obj Z)).hom) \u226b\n G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)) \u226b\n G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b \u03bc G (F.obj (X \u2297 Y)) (F.obj Z)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F (X \u2297 Y) Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (F.map (\u03b1_ X Y Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b \u03bc G (F.obj (X \u2297 Y)) (F.obj Z)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F (X \u2297 Y) Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (F.map (\u03b1_ X Y Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| (G.map (\u03bc F X Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b \u03bc G (F.obj (X \u2297 Y)) (F.obj Z)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (\u03bc F (X \u2297 Y) Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| G.map (F.map (\u03b1_ X Y Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n| \u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))\n[PROOFSTEP]\nrw [\u2190 G.toFunctor.map_id, G.\u03bc_natural]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX Y Z : C\n\u22a2 (\u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n ((\u03bc G (F.obj X \u2297 F.obj Y) (F.obj Z) \u226b G.map (\u03bc F X Y \u2297 \ud835\udfd9 (F.obj Z))) \u226b G.map (\u03bc F (X \u2297 Y) Z)) \u226b\n G.map (F.map (\u03b1_ X Y Z).hom) =\n (((\u03bc G (F.obj X) (F.obj Y) \u2297 \ud835\udfd9 (G.obj (F.obj Z))) \u226b\n \u03bc G (F.obj X \u2297 F.obj Y) (F.obj Z) \u226b G.map (\u03b1_ (F.obj X) (F.obj Y) (F.obj Z)).hom) \u226b\n G.map (\ud835\udfd9 (F.obj X) \u2297 \u03bc F Y Z)) \u226b\n G.map (\u03bc F X (Y \u2297 Z))\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc, Category.assoc, Category.assoc, Category.assoc, \u2190 G.toFunctor.map_comp, \u2190\n G.toFunctor.map_comp, \u2190 G.toFunctor.map_comp, \u2190 G.toFunctor.map_comp, F.associativity]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (\u03bb_ ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n (G.\u03b5 \u226b G.map F.\u03b5 \u2297 \ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X)) \u226b\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) (\ud835\udfd9_ C) X \u226b (Functor.mk src\u271d.toPrefunctor).map (\u03bb_ X).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (\u03bb_ (G.obj (F.obj X))).hom =\n (G.\u03b5 \u226b G.map F.\u03b5 \u2297 \ud835\udfd9 (G.obj (F.obj X))) \u226b\n (\u03bc G (F.obj (\ud835\udfd9_ C)) (F.obj X) \u226b G.map (\u03bc F (\ud835\udfd9_ C) X)) \u226b G.map (F.map (\u03bb_ X).hom)\n[PROOFSTEP]\nrw [G.left_unitality, comp_tensor_id, Category.assoc, Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (G.\u03b5 \u2297 \ud835\udfd9 (G.obj (F.obj X))) \u226b \u03bc G (\ud835\udfd9_ D) (F.obj X) \u226b G.map (\u03bb_ (F.obj X)).hom =\n (G.\u03b5 \u2297 \ud835\udfd9 (G.obj (F.obj X))) \u226b\n (G.map F.\u03b5 \u2297 \ud835\udfd9 (G.obj (F.obj X))) \u226b \u03bc G (F.obj (\ud835\udfd9_ C)) (F.obj X) \u226b G.map (\u03bc F (\ud835\udfd9_ C) X) \u226b G.map (F.map (\u03bb_ X).hom)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 \u03bc G (\ud835\udfd9_ D) (F.obj X) \u226b G.map (\u03bb_ (F.obj X)).hom =\n (G.map F.\u03b5 \u2297 \ud835\udfd9 (G.obj (F.obj X))) \u226b \u03bc G (F.obj (\ud835\udfd9_ C)) (F.obj X) \u226b G.map (\u03bc F (\ud835\udfd9_ C) X) \u226b G.map (F.map (\u03bb_ X).hom)\n[PROOFSTEP]\nrw [F.left_unitality, map_comp, \u2190 NatTrans.id_app, \u2190 Category.assoc, \u2190 LaxMonoidalFunctor.\u03bc_natural, NatTrans.id_app,\n map_id, \u2190 Category.assoc, map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (\u03c1_ ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X) \u2297 G.\u03b5 \u226b G.map F.\u03b5) \u226b\n (fun X Y => \u03bc G (F.obj X) (F.obj Y) \u226b G.map (\u03bc F X Y)) X (\ud835\udfd9_ C) \u226b (Functor.mk src\u271d.toPrefunctor).map (\u03c1_ X).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (\u03c1_ (G.obj (F.obj X))).hom =\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.\u03b5 \u226b G.map F.\u03b5) \u226b\n (\u03bc G (F.obj X) (F.obj (\ud835\udfd9_ C)) \u226b G.map (\u03bc F X (\ud835\udfd9_ C))) \u226b G.map (F.map (\u03c1_ X).hom)\n[PROOFSTEP]\nrw [G.right_unitality, id_tensor_comp, Category.assoc, Category.assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.\u03b5) \u226b \u03bc G (F.obj X) (\ud835\udfd9_ D) \u226b G.map (\u03c1_ (F.obj X)).hom =\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.\u03b5) \u226b\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map F.\u03b5) \u226b \u03bc G (F.obj X) (F.obj (\ud835\udfd9_ C)) \u226b G.map (\u03bc F X (\ud835\udfd9_ C)) \u226b G.map (F.map (\u03c1_ X).hom)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor D E\nsrc\u271d : C \u2964 E := F.toFunctor \u22d9 G.toFunctor\nX : C\n\u22a2 \u03bc G (F.obj X) (\ud835\udfd9_ D) \u226b G.map (\u03c1_ (F.obj X)).hom =\n (\ud835\udfd9 (G.obj (F.obj X)) \u2297 G.map F.\u03b5) \u226b \u03bc G (F.obj X) (F.obj (\ud835\udfd9_ C)) \u226b G.map (\u03bc F X (\ud835\udfd9_ C)) \u226b G.map (F.map (\u03c1_ X).hom)\n[PROOFSTEP]\nrw [F.right_unitality, map_comp, \u2190 NatTrans.id_app, \u2190 Category.assoc, \u2190 LaxMonoidalFunctor.\u03bc_natural, NatTrans.id_app,\n map_id, \u2190 Category.assoc, map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\n\u22a2 (prod' F G).\u03b5 = (F.\u03b5, G.\u03b5)\n[PROOFSTEP]\ndsimp [prod']\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\n\u22a2 (F.\u03b5 \u226b F.map (\ud835\udfd9 (\ud835\udfd9_ C)), G.\u03b5 \u226b G.map (\ud835\udfd9 (\ud835\udfd9_ C))) = (F.\u03b5, G.\u03b5)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\nX Y : C\n\u22a2 \u03bc (prod' F G) X Y = (\u03bc F X Y, \u03bc G X Y)\n[PROOFSTEP]\ndsimp [prod']\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\nX Y : C\n\u22a2 (\u03bc F X Y \u226b F.map (\ud835\udfd9 (X \u2297 Y)), \u03bc G X Y \u226b G.map (\ud835\udfd9 (X \u2297 Y))) = (\u03bc F X Y, \u03bc G X Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : MonoidalFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\n\u22a2 IsIso (LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc).\u03b5\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : MonoidalFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\n\u22a2 IsIso (G.\u03b5 \u226b G.map F.\u03b5)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : MonoidalFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\n\u22a2 \u2200 (X Y : C), IsIso (LaxMonoidalFunctor.\u03bc (LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc) X Y)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : MonoidalFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\n\u22a2 \u2200 (X Y : C),\n IsIso\n (LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b\n G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX Y X' Y' : D\nf : X \u27f6 Y\ng : X' \u27f6 Y'\n\u22a2 (G.map f \u2297 G.map g) \u226b\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n Y Y' =\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n X X' \u226b\n G.map (f \u2297 g)\n[PROOFSTEP]\nrw [\u2190 h.homEquiv_naturality_left, \u2190 h.homEquiv_naturality_right, Equiv.apply_eq_iff_eq, assoc, IsIso.eq_inv_comp, \u2190\n F.toLaxMonoidalFunctor.\u03bc_natural_assoc, IsIso.hom_inv_id_assoc, \u2190 tensor_comp, Adjunction.counit_naturality,\n Adjunction.counit_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX Y Z : D\n\u22a2 ((fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n X Y \u2297\n \ud835\udfd9 (G.obj Z)) \u226b\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n (X \u2297 Y) Z \u226b\n G.map (\u03b1_ X Y Z).hom =\n (\u03b1_ (G.obj X) (G.obj Y) (G.obj Z)).hom \u226b\n (\ud835\udfd9 (G.obj X) \u2297\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n Y Z) \u226b\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n X (Y \u2297 Z)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX Y Z : D\n\u22a2 (\u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)) \u2297\n \ud835\udfd9 (G.obj Z)) \u226b\n \u2191(Adjunction.homEquiv h (G.obj (X \u2297 Y) \u2297 G.obj Z) ((X \u2297 Y) \u2297 Z))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj (X \u2297 Y)) (G.obj Z)) \u226b\n (NatTrans.app h.counit (X \u2297 Y) \u2297 NatTrans.app h.counit Z)) \u226b\n G.map (\u03b1_ X Y Z).hom =\n (\u03b1_ (G.obj X) (G.obj Y) (G.obj Z)).hom \u226b\n (\ud835\udfd9 (G.obj X) \u2297\n \u2191(Adjunction.homEquiv h (G.obj Y \u2297 G.obj Z) (Y \u2297 Z))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj Y) (G.obj Z)) \u226b\n (NatTrans.app h.counit Y \u2297 NatTrans.app h.counit Z))) \u226b\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj (Y \u2297 Z)) (X \u2297 Y \u2297 Z))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj (Y \u2297 Z))) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit (Y \u2297 Z)))\n[PROOFSTEP]\nrw [\u2190 h.homEquiv_naturality_right, \u2190 h.homEquiv_naturality_left, \u2190 h.homEquiv_naturality_left, \u2190\n h.homEquiv_naturality_left, Equiv.apply_eq_iff_eq, \u2190\n cancel_epi (F.toLaxMonoidalFunctor.\u03bc (G.obj X \u2297 G.obj Y) (G.obj Z)), \u2190\n cancel_epi (F.toLaxMonoidalFunctor.\u03bc (G.obj X) (G.obj Y) \u2297 \ud835\udfd9 (F.obj (G.obj Z))),\n F.toLaxMonoidalFunctor.associativity_assoc (G.obj X) (G.obj Y) (G.obj Z), \u2190 F.toLaxMonoidalFunctor.\u03bc_natural_assoc,\n assoc, IsIso.hom_inv_id_assoc, \u2190 F.toLaxMonoidalFunctor.\u03bc_natural_assoc, IsIso.hom_inv_id_assoc, \u2190 tensor_comp, \u2190\n tensor_comp, id_comp, Functor.map_id, Functor.map_id, id_comp, \u2190 tensor_comp_assoc, \u2190 tensor_comp_assoc, id_comp,\n id_comp, h.homEquiv_unit, h.homEquiv_unit, Functor.map_comp, assoc, assoc, h.counit_naturality,\n h.left_triangle_components_assoc, Functor.map_comp, assoc, h.counit_naturality, h.left_triangle_components_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX Y Z : D\n\u22a2 (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y) \u226b\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y) \u2297\n NatTrans.app h.counit Z) \u226b\n (\u03b1_ X Y Z).hom =\n (\u03b1_ (F.obj (G.obj X)) (F.obj (G.obj Y)) (F.obj (G.obj Z))).hom \u226b\n (NatTrans.app h.counit X \u2297\n LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj Y) (G.obj Z) \u226b\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj Y) (G.obj Z)) \u226b\n (NatTrans.app h.counit Y \u2297 NatTrans.app h.counit Z))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX : D\n\u22a2 (\u03bb_ (G.obj X)).hom =\n (\u2191(Adjunction.homEquiv h (\ud835\udfd9_ C) (\ud835\udfd9_ D)) (inv F.\u03b5) \u2297 \ud835\udfd9 (G.obj X)) \u226b\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n (\ud835\udfd9_ D) X \u226b\n G.map (\u03bb_ X).hom\n[PROOFSTEP]\nrw [\u2190 h.homEquiv_naturality_right, \u2190 h.homEquiv_naturality_left, \u2190 Equiv.symm_apply_eq, h.homEquiv_counit,\n F.map_leftUnitor, h.homEquiv_unit, assoc, assoc, assoc, F.map_tensor, assoc, assoc, IsIso.hom_inv_id_assoc, \u2190\n tensor_comp_assoc, Functor.map_id, id_comp, Functor.map_comp, assoc, h.counit_naturality,\n h.left_triangle_components_assoc, \u2190 leftUnitor_naturality, \u2190 tensor_comp_assoc, id_comp, comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX : D\n\u22a2 inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) (G.obj X)) \u226b\n (inv F.\u03b5 \u2297 NatTrans.app h.counit X) \u226b (\u03bb_ ((\ud835\udfed D).obj X)).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ C) (G.obj X)) \u226b\n (inv F.\u03b5 \u2297 NatTrans.app h.counit X) \u226b (\u03bb_ X).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX : D\n\u22a2 (\u03c1_ (G.obj X)).hom =\n (\ud835\udfd9 (G.obj X) \u2297 \u2191(Adjunction.homEquiv h (\ud835\udfd9_ C) (\ud835\udfd9_ D)) (inv F.\u03b5)) \u226b\n (fun X Y =>\n \u2191(Adjunction.homEquiv h (G.obj X \u2297 G.obj Y) (X \u2297 Y))\n (inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (G.obj Y)) \u226b\n (NatTrans.app h.counit X \u2297 NatTrans.app h.counit Y)))\n X (\ud835\udfd9_ D) \u226b\n G.map (\u03c1_ X).hom\n[PROOFSTEP]\nrw [\u2190 h.homEquiv_naturality_right, \u2190 h.homEquiv_naturality_left, \u2190 Equiv.symm_apply_eq, h.homEquiv_counit,\n F.map_rightUnitor, assoc, assoc, \u2190 rightUnitor_naturality, \u2190 tensor_comp_assoc, comp_id, id_comp, h.homEquiv_unit,\n F.map_tensor, assoc, assoc, assoc, IsIso.hom_inv_id_assoc, Functor.map_comp, Functor.map_id, \u2190 tensor_comp_assoc,\n assoc, h.counit_naturality, h.left_triangle_components_assoc, id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF : MonoidalFunctor C D\nG : D \u2964 C\nh : F.toFunctor \u22a3 G\nX : D\n\u22a2 inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (\ud835\udfd9_ C)) \u226b\n (NatTrans.app h.counit X \u2297 inv F.\u03b5) \u226b (\u03c1_ ((\ud835\udfed D).obj X)).hom =\n inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (G.obj X) (\ud835\udfd9_ C)) \u226b\n (NatTrans.app h.counit X \u2297 inv F.\u03b5) \u226b (\u03c1_ X).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 IsIso (monoidalAdjoint F (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1))).\u03b5\n[PROOFSTEP]\ndsimp [Equivalence.toAdjunction]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 IsIso\n (NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (\ud835\udfd9_ C) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map (inv F.\u03b5))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nX Y : D\n\u22a2 IsIso\n (LaxMonoidalFunctor.\u03bc (monoidalAdjoint F (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1))) X Y)\n[PROOFSTEP]\ndsimp [Equivalence.toAdjunction]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nX Y : D\n\u22a2 IsIso\n (NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.counit (asEquivalence F.toLaxMonoidalFunctor.1)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toLaxMonoidalFunctor.1)) Y)))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Functor", "llama_tokens": 28169, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.2778365400273305}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr\u271d : R\nf\u271d : R[X]\nr c : R\nf : R[X]\n\u22a2 AddHom.toFun\n { toFun := fun f => comp f (X + \u2191C r),\n map_add' := (_ : \u2200 (f g : R[X]), comp (f + g) (X + \u2191C r) = comp f (X + \u2191C r) + comp g (X + \u2191C r)) }\n (c \u2022 f) =\n \u2191(RingHom.id R) c \u2022\n AddHom.toFun\n { toFun := fun f => comp f (X + \u2191C r),\n map_add' := (_ : \u2200 (f g : R[X]), comp (f + g) (X + \u2191C r) = comp f (X + \u2191C r) + comp g (X + \u2191C r)) }\n f\n[PROOFSTEP]\nsimp only [smul_eq_C_mul, C_mul_comp, RingHom.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\n\u22a2 \u2191(taylor r) X = X + \u2191C r\n[PROOFSTEP]\nsimp only [taylor_apply, X_comp]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\nx : R\n\u22a2 \u2191(taylor r) (\u2191C x) = \u2191C x\n[PROOFSTEP]\nsimp only [taylor_apply, C_comp]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\n\u22a2 taylor 0 = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\nn\u271d\u00b9 n\u271d : \u2115\n\u22a2 coeff (\u2191(LinearMap.comp (taylor 0) (monomial n\u271d\u00b9)) 1) n\u271d = coeff (\u2191(LinearMap.comp LinearMap.id (monomial n\u271d\u00b9)) 1) n\u271d\n[PROOFSTEP]\nsimp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp, Function.comp_apply, LinearMap.coe_comp]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf\u271d f : R[X]\n\u22a2 \u2191(taylor 0) f = f\n[PROOFSTEP]\nrw [taylor_zero', LinearMap.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\n\u22a2 \u2191(taylor r) 1 = \u2191C 1\n[PROOFSTEP]\nrw [\u2190 C_1, taylor_C]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\ni : \u2115\nk : R\n\u22a2 \u2191(taylor r) (\u2191(monomial i) k) = \u2191C k * (X + \u2191C r) ^ i\n[PROOFSTEP]\nsimp [taylor_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\nn : \u2115\n\u22a2 \u2191(LinearMap.comp (lcoeff R n) (taylor r)) f = \u2191(LinearMap.comp (leval r) (hasseDeriv n)) f\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\nn : \u2115\n\u22a2 LinearMap.comp (lcoeff R n) (taylor r) = LinearMap.comp (leval r) (hasseDeriv n)\n[PROOFSTEP]\nclear! f\n[GOAL]\ncase e_a\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn : \u2115\n\u22a2 LinearMap.comp (lcoeff R n) (taylor r) = LinearMap.comp (leval r) (hasseDeriv n)\n[PROOFSTEP]\next i\n[GOAL]\ncase e_a.h.h\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (lcoeff R n) (taylor r)) (monomial i)) 1 =\n \u2191(LinearMap.comp (LinearMap.comp (leval r) (hasseDeriv n)) (monomial i)) 1\n[PROOFSTEP]\nsimp only [leval_apply, mul_one, one_mul, eval_monomial, LinearMap.comp_apply, coeff_C_mul, hasseDeriv_monomial,\n taylor_apply, monomial_comp, C_1, (commute_X (C r)).add_pow i, LinearMap.map_sum]\n[GOAL]\ncase e_a.h.h\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\n\u22a2 (Finset.sum (Finset.range (i + 1)) fun x => \u2191(lcoeff R n) (X ^ x * \u2191C r ^ (i - x) * \u2191(Nat.choose i x))) =\n \u2191(Nat.choose i n) * r ^ (i - n)\n[PROOFSTEP]\nsimp only [lcoeff_apply, \u2190 C_eq_nat_cast, mul_assoc, \u2190 C_pow, \u2190 C_mul, coeff_mul_C, (Nat.cast_commute _ _).eq,\n coeff_X_pow, boole_mul, Finset.sum_ite_eq, Finset.mem_range]\n[GOAL]\ncase e_a.h.h\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\n\u22a2 (if n < i + 1 then r ^ (i - n) * \u2191(Nat.choose i n) else 0) = r ^ (i - n) * \u2191(Nat.choose i n)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\nh : n < i + 1\n\u22a2 r ^ (i - n) * \u2191(Nat.choose i n) = r ^ (i - n) * \u2191(Nat.choose i n)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\nh : \u00acn < i + 1\n\u22a2 0 = r ^ (i - n) * \u2191(Nat.choose i n)\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nn i : \u2115\nh : i + 1 \u2264 n\n\u22a2 0 = r ^ (i - n) * \u2191(Nat.choose i n)\n[PROOFSTEP]\nrw [Nat.choose_eq_zero_of_lt h, Nat.cast_zero, mul_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\n\u22a2 coeff (\u2191(taylor r) f) 0 = eval r f\n[PROOFSTEP]\nrw [taylor_coeff, hasseDeriv_zero, LinearMap.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr : R\nf : R[X]\n\u22a2 coeff (\u2191(taylor r) f) 1 = eval r (\u2191derivative f)\n[PROOFSTEP]\nrw [taylor_coeff, hasseDeriv_one]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr\u271d : R\nf p : R[X]\nr : R\n\u22a2 natDegree (\u2191(taylor r) p) = natDegree p\n[PROOFSTEP]\nrefine' map_natDegree_eq_natDegree _ _\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr\u271d : R\nf p : R[X]\nr : R\n\u22a2 \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191(taylor r) (\u2191(monomial n) c)) = n\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr\u271d : R\nf p : R[X]\nr : R\n\u271d : Nontrivial R\n\u22a2 \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191(taylor r) (\u2191(monomial n) c)) = n\n[PROOFSTEP]\nintro n c c0\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nr\u271d : R\nf p : R[X]\nr : R\n\u271d : Nontrivial R\nn : \u2115\nc : R\nc0 : c \u2260 0\n\u22a2 natDegree (\u2191(taylor r) (\u2191(monomial n) c)) = n\n[PROOFSTEP]\nsimp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf : R\u271d[X]\nR : Type u_2\ninst\u271d : CommSemiring R\nr : R\np q : R[X]\n\u22a2 \u2191(taylor r) (p * q) = \u2191(taylor r) p * \u2191(taylor r) q\n[PROOFSTEP]\nsimp only [taylor_apply, mul_comp]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommSemiring R\nf : R[X]\nr s : R\n\u22a2 \u2191(taylor r) (\u2191(taylor s) f) = \u2191(taylor (r + s)) f\n[PROOFSTEP]\nsimp only [taylor_apply, comp_assoc, map_add, add_comp, X_comp, C_comp, C_add, add_assoc]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommSemiring R\nr : R\nf : R[X]\ns : R\n\u22a2 eval s (\u2191(taylor r) f) = eval (s + r) f\n[PROOFSTEP]\nsimp only [taylor_apply, eval_comp, eval_C, eval_X, eval_add]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nr : R\nf : R[X]\ns : R\n\u22a2 eval (s - r) (\u2191(taylor r) f) = eval s f\n[PROOFSTEP]\nrw [taylor_eval, sub_add_cancel]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nr : R\n\u22a2 Function.Injective \u2191(taylor r)\n[PROOFSTEP]\nintro f g h\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nr : R\nf g : R[X]\nh : \u2191(taylor r) f = \u2191(taylor r) g\n\u22a2 f = g\n[PROOFSTEP]\napply_fun taylor (-r) at h \n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nr : R\nf g : R[X]\nh : \u2191(taylor (-r)) (\u2191(taylor r) f) = \u2191(taylor (-r)) (\u2191(taylor r) g)\n\u22a2 f = g\n[PROOFSTEP]\nsimpa only [taylor_apply, comp_assoc, add_comp, X_comp, C_comp, C_neg, neg_add_cancel_right, comp_X] using h\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nf : R[X]\nr : R\nh : \u2200 (k : \u2115), eval r (\u2191(hasseDeriv k) f) = 0\n\u22a2 f = 0\n[PROOFSTEP]\napply taylor_injective r\n[GOAL]\ncase a\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nf : R[X]\nr : R\nh : \u2200 (k : \u2115), eval r (\u2191(hasseDeriv k) f) = 0\n\u22a2 \u2191(taylor r) f = \u2191(taylor r) 0\n[PROOFSTEP]\nrw [LinearMap.map_zero]\n[GOAL]\ncase a\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nf : R[X]\nr : R\nh : \u2200 (k : \u2115), eval r (\u2191(hasseDeriv k) f) = 0\n\u22a2 \u2191(taylor r) f = 0\n[PROOFSTEP]\next k\n[GOAL]\ncase a.a\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nf : R[X]\nr : R\nh : \u2200 (k : \u2115), eval r (\u2191(hasseDeriv k) f) = 0\nk : \u2115\n\u22a2 coeff (\u2191(taylor r) f) k = coeff 0 k\n[PROOFSTEP]\nsimp only [taylor_coeff, h, coeff_zero]\n[GOAL]\nR\u271d : Type u_1\ninst\u271d\u00b9 : Semiring R\u271d\nr\u271d : R\u271d\nf\u271d : R\u271d[X]\nR : Type u_2\ninst\u271d : CommRing R\nf : R[X]\nr : R\n\u22a2 (sum (\u2191(taylor r) f) fun i a => \u2191C a * (X - \u2191C r) ^ i) = f\n[PROOFSTEP]\nrw [\u2190 comp_eq_sum_left, sub_eq_add_neg, \u2190 C_neg, \u2190 taylor_apply, taylor_taylor, neg_add_self, taylor_zero]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Taylor", "llama_tokens": 4170, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.2772827069175984}} {"text": "[GOAL]\nV : QuivCat\n\u22a2 { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V) =\n \ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)\n[PROOFSTEP]\nchange (show Paths V \u2964 _ from _) = _\n[GOAL]\nV : QuivCat\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this) =\n \ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)\n[PROOFSTEP]\next\n[GOAL]\ncase h_obj.h\nV : QuivCat\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n x\u271d =\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n x\u271d\ncase h\nV : QuivCat\na\u271d b\u271d : \u2191V\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n a\u271d =\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n a\u271d) \u226b\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n b\u271d =\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n b\u271d)\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nV : QuivCat\na\u271d b\u271d : \u2191V\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n a\u271d =\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n a\u271d) \u226b\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n b\u271d =\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n b\u271d)\ncase h_obj.h\nV : QuivCat\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n x\u271d =\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n x\u271d\n[PROOFSTEP]\napply eq_conj_eqToHom\n[GOAL]\ncase h_obj.h\nV : QuivCat\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (\ud835\udfd9 V);\n this).obj\n x\u271d =\n (\ud835\udfd9\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj\n V)).obj\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nU x\u271d\u00b9 x\u271d : QuivCat\nF : U \u27f6 x\u271d\u00b9\nG : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G) =\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G\n[PROOFSTEP]\nchange (show Paths U \u2964 _ from _) = _\n[GOAL]\nU x\u271d\u00b9 x\u271d : QuivCat\nF : U \u27f6 x\u271d\u00b9\nG : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this) =\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G\n[PROOFSTEP]\next\n[GOAL]\ncase h_obj.h\nU x\u271d\u00b2 x\u271d\u00b9 : QuivCat\nF : U \u27f6 x\u271d\u00b2\nG : x\u271d\u00b2 \u27f6 x\u271d\u00b9\nx\u271d : Paths \u2191U\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n x\u271d =\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n x\u271d\ncase h\nU x\u271d\u00b9 x\u271d : QuivCat\nF : U \u27f6 x\u271d\u00b9\nG : x\u271d\u00b9 \u27f6 x\u271d\na\u271d b\u271d : \u2191U\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n a\u271d =\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n a\u271d) \u226b\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n b\u271d =\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n b\u271d)\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nU x\u271d\u00b9 x\u271d : QuivCat\nF : U \u27f6 x\u271d\u00b9\nG : x\u271d\u00b9 \u27f6 x\u271d\na\u271d b\u271d : \u2191U\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n a\u271d =\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n a\u271d) \u226b\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n b\u271d =\n (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk\n { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n b\u271d)\ncase h_obj.h\nU x\u271d\u00b2 x\u271d\u00b9 : QuivCat\nF : U \u27f6 x\u271d\u00b2\nG : x\u271d\u00b2 \u27f6 x\u271d\u00b9\nx\u271d : Paths \u2191U\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n x\u271d =\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n x\u271d\n[PROOFSTEP]\napply eq_conj_eqToHom\n[GOAL]\ncase h_obj.h\nU x\u271d\u00b2 x\u271d\u00b9 : QuivCat\nF : U \u27f6 x\u271d\u00b2\nG : x\u271d\u00b2 \u27f6 x\u271d\u00b9\nx\u271d : Paths \u2191U\n\u22a2 (let_fun this :=\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n (F \u226b G);\n this).obj\n x\u271d =\n ({ obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n F \u226b\n { obj := fun V => of (Paths \u2191V),\n map := fun {X Y} F =>\n Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map\n G).obj\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nV : QuivCat\nC : Cat\nF : Cat.free.obj V \u27f6 C\n\u22a2 \u2200 (a b : \u2191V) (e : a \u27f6 b),\n ((fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F)).map (Quiver.Hom.toPath e) =\n eqToHom (_ : ((fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F)).obj a = F.obj a) \u226b\n F.map (Quiver.Hom.toPath e) \u226b\n eqToHom (_ : F.obj b = ((fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F)).obj b)\n[PROOFSTEP]\nsimp\n[GOAL]\nV : QuivCat\nC : Cat\n\u22a2 Function.RightInverse (fun F => lift F) fun F => Paths.of \u22d9q F.toPrefunctor\n[PROOFSTEP]\nrintro \u27e8obj, map\u27e9\n[GOAL]\ncase mk\nV : QuivCat\nC : Cat\nobj : \u2191V \u2192 \u2191(forget.obj C)\nmap : {X Y : \u2191V} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\n\u22a2 (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) { obj := obj, map := map }) = { obj := obj, map := map }\n[PROOFSTEP]\ndsimp only [Prefunctor.comp]\n[GOAL]\ncase mk\nV : QuivCat\nC : Cat\nobj : \u2191V \u2192 \u2191(forget.obj C)\nmap : {X Y : \u2191V} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\n\u22a2 { obj := fun X => (lift { obj := obj, map := map }).obj (Paths.of.obj X),\n map := fun {X Y} f => (lift { obj := obj, map := map }).map (Paths.of.map f) } =\n { obj := obj, map := map }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.e_map\nV : QuivCat\nC : Cat\nobj : \u2191V \u2192 \u2191(forget.obj C)\nmap : {X Y : \u2191V} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\n\u22a2 (fun {X Y} f => (lift { obj := obj, map := map }).map (Paths.of.map f)) = map\n[PROOFSTEP]\nfunext X Y f\n[GOAL]\ncase mk.e_map.h.h.h\nV : QuivCat\nC : Cat\nobj : \u2191V \u2192 \u2191(forget.obj C)\nmap : {X Y : \u2191V} \u2192 (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y)\nX Y : \u2191V\nf : X \u27f6 Y\n\u22a2 (lift { obj := obj, map := map }).map (Paths.of.map f) = map f\n[PROOFSTEP]\nexact Category.id_comp _\n[GOAL]\nV x\u271d\u00b9 : QuivCat\nx\u271d : Cat\nf : V \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 forget.obj x\u271d\n\u22a2 \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ : \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ : \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g) =\n Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ : \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ : \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g\n[PROOFSTEP]\nchange (show Paths V \u2964 _ from _) = _\n[GOAL]\nV x\u271d\u00b9 : QuivCat\nx\u271d : Cat\nf : V \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 forget.obj x\u271d\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ : \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ : \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this) =\n Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ : \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ : \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g\n[PROOFSTEP]\next\n[GOAL]\ncase h_obj.h\nV x\u271d\u00b2 : QuivCat\nx\u271d\u00b9 : Cat\nf : V \u27f6 x\u271d\u00b2\ng : x\u271d\u00b2 \u27f6 forget.obj x\u271d\u00b9\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d\u00b9).symm\n (f \u226b g);\n this).obj\n x\u271d =\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b2 x\u271d\u00b9).symm\n g).obj\n x\u271d\ncase h\nV x\u271d\u00b9 : QuivCat\nx\u271d : Cat\nf : V \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 forget.obj x\u271d\na\u271d b\u271d : \u2191V\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).obj\n a\u271d =\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).obj\n a\u271d) \u226b\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).obj\n b\u271d =\n (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).obj\n b\u271d)\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nV x\u271d\u00b9 : QuivCat\nx\u271d : Cat\nf : V \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 forget.obj x\u271d\na\u271d b\u271d : \u2191V\ne\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).map\n (Quiver.Hom.toPath e\u271d) =\n eqToHom\n (_ :\n (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).obj\n a\u271d =\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).obj\n a\u271d) \u226b\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).map\n (Quiver.Hom.toPath e\u271d) \u226b\n eqToHom\n (_ :\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b9 x\u271d).symm\n g).obj\n b\u271d =\n (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C),\n (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C),\n (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d).symm\n (f \u226b g);\n this).obj\n b\u271d)\ncase h_obj.h\nV x\u271d\u00b2 : QuivCat\nx\u271d\u00b9 : Cat\nf : V \u27f6 x\u271d\u00b2\ng : x\u271d\u00b2 \u27f6 forget.obj x\u271d\u00b9\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d\u00b9).symm\n (f \u226b g);\n this).obj\n x\u271d =\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b2 x\u271d\u00b9).symm\n g).obj\n x\u271d\n[PROOFSTEP]\napply eq_conj_eqToHom\n[GOAL]\ncase h_obj.h\nV x\u271d\u00b2 : QuivCat\nx\u271d\u00b9 : Cat\nf : V \u27f6 x\u271d\u00b2\ng : x\u271d\u00b2 \u27f6 forget.obj x\u271d\u00b9\nx\u271d : Paths \u2191V\n\u22a2 (let_fun this :=\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n V x\u271d\u00b9).symm\n (f \u226b g);\n this).obj\n x\u271d =\n (Cat.free.map f \u226b\n \u2191((fun V C =>\n { toFun := fun F => Paths.of \u22d9q F.toPrefunctor, invFun := fun F => lift F,\n left_inv :=\n (_ :\n \u2200 (F : Cat.free.obj V \u27f6 C), (fun F => lift F) ((fun F => Paths.of \u22d9q F.toPrefunctor) F) = F),\n right_inv :=\n (_ :\n \u2200 (x : V \u27f6 forget.obj C), (fun F => Paths.of \u22d9q F.toPrefunctor) ((fun F => lift F) x) = x) })\n x\u271d\u00b2 x\u271d\u00b9).symm\n g).obj\n x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Category.QuivCat", "llama_tokens": 12006, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7401743620390162, "lm_q2_score": 0.3738758227716966, "lm_q1q2_score": 0.27673329860185286}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 0\nf : X\u271d \u27f6 Y\u271d\ninst\u271d : Mono f\n\u22a2 f \u226b 0 = g\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q : C\ni : P \u2245 Q\nhP : Injective P\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nobtain \u27e8h, h_eq\u27e9 := @Injective.factors C _ P _ _ _ (g \u226b i.inv) f mono\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q : C\ni : P \u2245 Q\nhP : Injective P\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nh : Y\u271d \u27f6 P\nh_eq : f \u226b h = g \u226b i.inv\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nrefine' \u27e8h \u226b i.hom, _\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q : C\ni : P \u2245 Q\nhP : Injective P\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nh : Y\u271d \u27f6 P\nh_eq : f \u226b h = g \u226b i.inv\n\u22a2 f \u226b h \u226b i.hom = g\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h_eq, Category.assoc, Iso.inv_hom_id, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nz : Y\u271d\n\u22a2 X\n[PROOFSTEP]\nclassical exact if h : z \u2208 Set.range f then g (Classical.choose h) else Nonempty.some inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nz : Y\u271d\n\u22a2 X\n[PROOFSTEP]\nexact if h : z \u2208 Set.range f then g (Classical.choose h) else Nonempty.some inferInstance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b fun z => if h : z \u2208 Set.range f then g (Classical.choose h) else Nonempty.some (_ : Nonempty X)) = g\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\n\u22a2 (f \u226b fun z => if h : z \u2208 Set.range f then g (Classical.choose h) else Nonempty.some (_ : Nonempty X)) y = g y\n[PROOFSTEP]\nclassical\nchange dite (f y \u2208 Set.range f) (fun h => g (Classical.choose h)) _ = _\nsplit_ifs <;> rename_i h\n\u00b7 rw [mono_iff_injective] at mono \n erw [mono (Classical.choose_spec h)]\n\u00b7 exact False.elim (h \u27e8y, rfl\u27e9)\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\n\u22a2 (f \u226b fun z => if h : z \u2208 Set.range f then g (Classical.choose h) else Nonempty.some (_ : Nonempty X)) y = g y\n[PROOFSTEP]\nchange dite (f y \u2208 Set.range f) (fun h => g (Classical.choose h)) _ = _\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\n\u22a2 (if h : f y \u2208 Set.range f then g (Classical.choose h) else (fun h => Nonempty.some (_ : Nonempty X)) h) = g y\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\nh\u271d : f y \u2208 Set.range f\n\u22a2 g (Classical.choose h\u271d) = g y\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase neg\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\nh\u271d : \u00acf y \u2208 Set.range f\n\u22a2 (fun h => Nonempty.some (_ : Nonempty X)) h\u271d = g y\n[PROOFSTEP]\nrename_i h\n[GOAL]\ncase pos\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\nh : f y \u2208 Set.range f\n\u22a2 g (Classical.choose h) = g y\n[PROOFSTEP]\nrw [mono_iff_injective] at mono \n[GOAL]\ncase pos\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Function.Injective f\ny : X\u271d\nh : f y \u2208 Set.range f\n\u22a2 g (Classical.choose h) = g y\n[PROOFSTEP]\nerw [mono (Classical.choose_spec h)]\n[GOAL]\ncase neg\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\ninst\u271d : Nonempty X\nX\u271d Y\u271d : Type u\u2081\ng : X\u271d \u27f6 X\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\ny : X\u271d\nh : \u00acf y \u2208 Set.range f\n\u22a2 (fun h => Nonempty.some (_ : Nonempty X)) h = g y\n[PROOFSTEP]\nexact False.elim (h \u27e8y, rfl\u27e9)\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\n\u22a2 Mono some\n[PROOFSTEP]\nrw [mono_iff_injective]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : Type u\u2081\n\u22a2 Function.Injective some\n[PROOFSTEP]\nexact Option.some_injective X\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nuse Limits.prod.lift (factorThru (g \u226b Limits.prod.fst) f) (factorThru (g \u226b Limits.prod.snd) f)\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 f \u226b prod.lift (factorThru (g \u226b prod.fst) f) (factorThru (g \u226b prod.snd) f) = g\n[PROOFSTEP]\nsimp only [prod.comp_lift, comp_factorThru]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 prod.lift (g \u226b prod.fst) (g \u226b prod.snd) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\u2081\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 prod.lift (g \u226b prod.fst) (g \u226b prod.snd) \u226b prod.fst = g \u226b prod.fst\n[PROOFSTEP]\nsimp only [prod.lift_fst]\n[GOAL]\ncase h.h\u2082\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b2 : HasBinaryProduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u2a2f Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 prod.lift (g \u226b prod.fst) (g \u226b prod.snd) \u226b prod.snd = g \u226b prod.snd\n[PROOFSTEP]\nsimp only [prod.lift_snd]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u220f c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u220f c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nrefine' \u27e8Pi.lift fun b => factorThru (g \u226b Pi.\u03c0 c _) f, _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u220f c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b Pi.lift fun b => factorThru (g \u226b Pi.\u03c0 c b) f) = g\n[PROOFSTEP]\next b\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b9 : HasProduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u220f c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nb : \u03b2\n\u22a2 (f \u226b Pi.lift fun b => factorThru (g \u226b Pi.\u03c0 c b) f) \u226b Pi.\u03c0 c b = g \u226b Pi.\u03c0 c b\n[PROOFSTEP]\nsimp only [Category.assoc, limit.lift_\u03c0, Fan.mk_\u03c0_app, comp_factorThru]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u229e Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u229e Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nrefine' \u27e8biprod.lift (factorThru (g \u226b biprod.fst) f) (factorThru (g \u226b biprod.snd) f), _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u229e Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 f \u226b biprod.lift (factorThru (g \u226b biprod.fst) f) (factorThru (g \u226b biprod.snd) f) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u229e Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b biprod.lift (factorThru (g \u226b biprod.fst) f) (factorThru (g \u226b biprod.snd) f)) \u226b biprod.fst = g \u226b biprod.fst\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.lift_fst, comp_factorThru]\n[GOAL]\ncase h\u2081\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nP Q : C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasBinaryBiproduct P Q\ninst\u271d\u00b9 : Injective P\ninst\u271d : Injective Q\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P \u229e Q\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b biprod.lift (factorThru (g \u226b biprod.fst) f) (factorThru (g \u226b biprod.snd) f)) \u226b biprod.snd = g \u226b biprod.snd\n[PROOFSTEP]\nsimp only [Category.assoc, biprod.lift_snd, comp_factorThru]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBiproduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u2a01 c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBiproduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u2a01 c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 \u2203 h, f \u226b h = g\n[PROOFSTEP]\nrefine' \u27e8biproduct.lift fun b => factorThru (g \u226b biproduct.\u03c0 _ _) f, _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBiproduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u2a01 c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b biproduct.lift fun b => factorThru (g \u226b biproduct.\u03c0 c b) f) = g\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\n\u03b2 : Type v\nc : \u03b2 \u2192 C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasBiproduct c\ninst\u271d : \u2200 (b : \u03b2), Injective (c b)\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 \u2a01 c\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\nj\u271d : \u03b2\n\u22a2 (f \u226b biproduct.lift fun b => factorThru (g \u226b biproduct.\u03c0 c b) f) \u226b biproduct.\u03c0 c j\u271d = g \u226b biproduct.\u03c0 c j\u271d\n[PROOFSTEP]\nsimp only [Category.assoc, biproduct.lift_\u03c0, comp_factorThru]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\u1d52\u1d56\ninst\u271d : Projective P\nX\u271d Y\u271d : C\ng : X\u271d \u27f6 P.unop\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b (Projective.factorThru g.op f.op).unop).op = g.op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : C\u1d52\u1d56\ninst\u271d : Injective J\nE\u271d X\u271d : C\nf : J.unop \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nhe : Epi e\n\u22a2 ((factorThru f.op e.op).unop \u226b e).op = f.op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : C\ninst\u271d : Injective J\nE\u271d X\u271d : C\u1d52\u1d56\nf : op J \u27f6 X\u271d\ne : E\u271d \u27f6 X\u271d\nepi : Epi e\n\u22a2 ((factorThru f.unop e.unop).op \u226b e).unop = f.unop\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nP : C\ninst\u271d : Projective P\nX\u271d Y\u271d : C\u1d52\u1d56\ng : X\u271d \u27f6 op P\nf : X\u271d \u27f6 Y\u271d\nmono : Mono f\n\u22a2 (f \u226b (Projective.factorThru g.unop f.unop).op).unop = g.unop\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nJ : C\n\u22a2 Injective J \u2194 Functor.PreservesEpimorphisms (yoneda.obj J)\n[PROOFSTEP]\nrw [injective_iff_projective_op, Projective.projective_iff_preservesEpimorphisms_coyoneda_obj]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nJ : C\n\u22a2 Functor.PreservesEpimorphisms (coyoneda.obj (op (op J))) \u2194 Functor.PreservesEpimorphisms (yoneda.obj J)\n[PROOFSTEP]\nexact Functor.preservesEpimorphisms.iso_iff (Coyoneda.objOpOp _)\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\ninst\u271d\u00b9 : PreservesMonomorphisms L\nadj : L \u22a3 R\nJ : D\ninst\u271d : Injective J\nA x\u271d : C\ng : A \u27f6 R.obj J\nf : A \u27f6 x\u271d\nim : Mono f\n\u22a2 \u2191(Adjunction.homEquiv adj A J).symm\n (f \u226b \u2191(Adjunction.homEquiv adj x\u271d J) (factorThru (\u2191(Adjunction.homEquiv adj A J).symm g) (L.map f))) =\n \u2191(Adjunction.homEquiv adj A J).symm g\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasImages C\u1d52\u1d56\ninst\u271d\u00b9 : HasEqualizers C\u1d52\u1d56\nJ Q R S : C\ninst\u271d : Injective J\nh : R \u27f6 J\nf : Q \u27f6 R\ng : R \u27f6 S\nhgf : Exact g.op f.op\nw : f \u226b h = 0\n\u22a2 g \u226b desc h f g hgf w = h\n[PROOFSTEP]\nconvert congr_arg Quiver.Hom.unop (Exact.lift_comp h.op g.op f.op hgf (congrArg Quiver.Hom.op w))\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesMonomorphisms F\nI : D\nhI : Injective I\nX Y : C\nf : X \u27f6 G.obj I\ng : X \u27f6 Y\n\u22a2 \u2200 [inst : Mono g], \u2203 h, g \u226b h = f\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Functor.PreservesMonomorphisms F\nI : D\nhI : Injective I\nX Y : C\nf : X \u27f6 G.obj I\ng : X \u27f6 Y\ninst\u271d : Mono g\n\u22a2 \u2203 h, g \u226b h = f\n[PROOFSTEP]\nrcases hI.factors (F.map f \u226b adj.counit.app _) (F.map g) with \u27e8w, h\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Functor.PreservesMonomorphisms F\nI : D\nhI : Injective I\nX Y : C\nf : X \u27f6 G.obj I\ng : X \u27f6 Y\ninst\u271d : Mono g\nw : F.obj Y \u27f6 I\nh : F.map g \u226b w = F.map f \u226b NatTrans.app adj.counit I\n\u22a2 \u2203 h, g \u226b h = f\n[PROOFSTEP]\nuse adj.unit.app Y \u226b G.map w\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Functor.PreservesMonomorphisms F\nI : D\nhI : Injective I\nX Y : C\nf : X \u27f6 G.obj I\ng : X \u27f6 Y\ninst\u271d : Mono g\nw : F.obj Y \u27f6 I\nh : F.map g \u226b w = F.map f \u226b NatTrans.app adj.counit I\n\u22a2 g \u226b NatTrans.app adj.unit Y \u226b G.map w = f\n[PROOFSTEP]\nrw [\u2190 unit_naturality_assoc, \u2190 G.map_comp, h]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Functor.PreservesMonomorphisms F\nI : D\nhI : Injective I\nX Y : C\nf : X \u27f6 G.obj I\ng : X \u27f6 Y\ninst\u271d : Mono g\nw : F.obj Y \u27f6 I\nh : F.map g \u226b w = F.map f \u226b NatTrans.app adj.counit I\n\u22a2 NatTrans.app adj.unit X \u226b G.map (F.map f \u226b NatTrans.app adj.counit I) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\n\u22a2 \u2200 [inst : Mono g], \u2203 h, g \u226b h = f\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b2 : Full G\ninst\u271d\u00b9 : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\ninst\u271d : Mono g\n\u22a2 \u2203 h, g \u226b h = f\n[PROOFSTEP]\nhaveI : PreservesLimitsOfSize.{0, 0} G := adj.rightAdjointPreservesLimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b2 : Full G\ninst\u271d\u00b9 : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\ninst\u271d : Mono g\nthis : PreservesLimitsOfSize.{0, 0, u_2, v\u2081, u_1, u\u2081} G\n\u22a2 \u2203 h, g \u226b h = f\n[PROOFSTEP]\nrcases hI.factors (G.map f) (G.map g) with \u27e8w, h\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b2 : Full G\ninst\u271d\u00b9 : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\ninst\u271d : Mono g\nthis : PreservesLimitsOfSize.{0, 0, u_2, v\u2081, u_1, u\u2081} G\nw : G.obj Y \u27f6 G.obj I\nh : G.map g \u226b w = G.map f\n\u22a2 \u2203 h, g \u226b h = f\n[PROOFSTEP]\nuse inv (adj.counit.app _) \u226b F.map w \u226b adj.counit.app _\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b2 : Full G\ninst\u271d\u00b9 : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\ninst\u271d : Mono g\nthis : PreservesLimitsOfSize.{0, 0, u_2, v\u2081, u_1, u\u2081} G\nw : G.obj Y \u27f6 G.obj I\nh : G.map g \u226b w = G.map f\n\u22a2 g \u226b inv (NatTrans.app adj.counit Y) \u226b F.map w \u226b NatTrans.app adj.counit I = f\n[PROOFSTEP]\nrefine' Faithful.map_injective (F := G) _\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d\u00b2 : Full G\ninst\u271d\u00b9 : Faithful G\nI : D\nhI : Injective (G.obj I)\nX Y : D\nf : X \u27f6 I\ng : X \u27f6 Y\ninst\u271d : Mono g\nthis : PreservesLimitsOfSize.{0, 0, u_2, v\u2081, u_1, u\u2081} G\nw : G.obj Y \u27f6 G.obj I\nh : G.map g \u226b w = G.map f\n\u22a2 G.map (g \u226b inv (NatTrans.app adj.counit Y) \u226b F.map w \u226b NatTrans.app adj.counit I) = G.map f\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.56282, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesMonomorphisms F\nX : D\nI : InjectivePresentation X\n\u22a2 Mono (G.map I.f)\n[PROOFSTEP]\nhaveI : PreservesLimitsOfSize.{0, 0} G := adj.rightAdjointPreservesLimits\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.56282, u_1} D\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : Functor.PreservesMonomorphisms F\nX : D\nI : InjectivePresentation X\nthis : PreservesLimitsOfSize.{0, 0, ?u.56282, v\u2081, u_1, u\u2081} G\n\u22a2 Mono (G.map I.f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\n\u22a2 EnoughInjectives C \u2194 EnoughInjectives D\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\n\u22a2 EnoughInjectives C \u2192 EnoughInjectives D\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\n\u22a2 EnoughInjectives D \u2192 EnoughInjectives C\n[PROOFSTEP]\nall_goals intro H; constructor; intro X; constructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\n\u22a2 EnoughInjectives C \u2192 EnoughInjectives D\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives C\n\u22a2 EnoughInjectives D\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.presentation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives C\n\u22a2 \u2200 (X : D), Nonempty (InjectivePresentation X)\n[PROOFSTEP]\nintro X\n[GOAL]\ncase mp.presentation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives C\nX : D\n\u22a2 Nonempty (InjectivePresentation X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\n\u22a2 EnoughInjectives D \u2192 EnoughInjectives C\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives D\n\u22a2 EnoughInjectives C\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.presentation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives D\n\u22a2 \u2200 (X : C), Nonempty (InjectivePresentation X)\n[PROOFSTEP]\nintro X\n[GOAL]\ncase mpr.presentation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives D\nX : C\n\u22a2 Nonempty (InjectivePresentation X)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.presentation.val\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives C\nX : D\n\u22a2 InjectivePresentation X\n[PROOFSTEP]\nexact F.symm.injectivePresentationOfMapInjectivePresentation _ (Nonempty.some (H.presentation (F.inverse.obj X)))\n[GOAL]\ncase mpr.presentation.val\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u271d F : C \u224c D\nH : EnoughInjectives D\nX : C\n\u22a2 InjectivePresentation X\n[PROOFSTEP]\nexact F.injectivePresentationOfMapInjectivePresentation X (Nonempty.some (H.presentation (F.functor.obj X)))\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.Injective", "llama_tokens": 10900, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.2764595573137879}} {"text": "[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 OfLocalizationSpan P \u2194 OfLocalizationFiniteSpan P\n[PROOFSTEP]\ndelta RingHom.OfLocalizationSpan RingHom.OfLocalizationFiniteSpan\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 (\u2200 \u2983R S : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] (f : R \u2192+* S) (s : Set R),\n Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)) \u2192 P f) \u2194\n \u2200 \u2983R S : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] (f : R \u2192+* S) (s : Finset R),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap f \u2191r)) \u2192 P f\n[PROOFSTEP]\napply forall\u2085_congr\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 \u2200 (a b : Type u) (c : CommRing a) (d : CommRing b) (e : a \u2192+* b),\n (\u2200 (s : Set a), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap e \u2191r)) \u2192 P e) \u2194\n \u2200 (s : Finset a), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e \u2191r)) \u2192 P e\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Set a\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d) \u2194\n \u2200 (s : Finset a\u271d), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Set a\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d) \u2192\n \u2200 (s : Finset a\u271d), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\n[PROOFSTEP]\nintro h s\n[GOAL]\ncase h.mp\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh : \u2200 (s : Set a\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\ns : Finset a\u271d\n\u22a2 Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\n[PROOFSTEP]\nexact h s\n[GOAL]\ncase h.mpr\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Finset a\u271d), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d) \u2192\n \u2200 (s : Set a\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\n[PROOFSTEP]\nintro h s hs hs'\n[GOAL]\ncase h.mpr\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh : \u2200 (s : Finset a\u271d), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\ns : Set a\u271d\nhs : Ideal.span s = \u22a4\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)\n\u22a2 P e\u271d\n[PROOFSTEP]\nobtain \u27e8s', h\u2081, h\u2082\u27e9 := (Ideal.span_eq_top_iff_finite s).mp hs\n[GOAL]\ncase h.mpr.intro.intro\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh : \u2200 (s : Finset a\u271d), Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (Localization.awayMap e\u271d \u2191r)) \u2192 P e\u271d\ns : Set a\u271d\nhs : Ideal.span s = \u22a4\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap e\u271d \u2191r)\ns' : Finset a\u271d\nh\u2081 : \u2191s' \u2286 s\nh\u2082 : Ideal.span \u2191s' = \u22a4\n\u22a2 P e\u271d\n[PROOFSTEP]\nexact h s' h\u2082 fun x => hs' \u27e8_, h\u2081 x.prop\u27e9\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 OfLocalizationSpanTarget P \u2194 OfLocalizationFiniteSpanTarget P\n[PROOFSTEP]\ndelta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 (\u2200 \u2983R S : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] (f : R \u2192+* S) (s : Set S),\n Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap S (Localization.Away \u2191r)) f)) \u2192 P f) \u2194\n \u2200 \u2983R S : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] (f : R \u2192+* S) (s : Finset S),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap S (Localization.Away \u2191r)) f)) \u2192 P f\n[PROOFSTEP]\napply forall\u2085_congr\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\n\u22a2 \u2200 (a b : Type u) (c : CommRing a) (d : CommRing b) (e : a \u2192+* b),\n (\u2200 (s : Set b), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap b (Localization.Away \u2191r)) e)) \u2192 P e) \u2194\n \u2200 (s : Finset b),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b (Localization.Away \u2191r)) e)) \u2192 P e\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Set b\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d) \u2194\n \u2200 (s : Finset b\u271d),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Set b\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d) \u2192\n \u2200 (s : Finset b\u271d),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\n[PROOFSTEP]\nintro h s\n[GOAL]\ncase h.mp\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh : \u2200 (s : Set b\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\ns : Finset b\u271d\n\u22a2 Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\n[PROOFSTEP]\nexact h s\n[GOAL]\ncase h.mpr\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\n\u22a2 (\u2200 (s : Finset b\u271d),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d) \u2192\n \u2200 (s : Set b\u271d), Ideal.span s = \u22a4 \u2192 (\u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\n[PROOFSTEP]\nintro h s hs hs'\n[GOAL]\ncase h.mpr\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh :\n \u2200 (s : Finset b\u271d),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\ns : Set b\u271d\nhs : Ideal.span s = \u22a4\nhs' : \u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)\n\u22a2 P e\u271d\n[PROOFSTEP]\nobtain \u27e8s', h\u2081, h\u2082\u27e9 := (Ideal.span_eq_top_iff_finite s).mp hs\n[GOAL]\ncase h.mpr.intro.intro\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\na\u271d b\u271d : Type u\nc\u271d : CommRing a\u271d\nd\u271d : CommRing b\u271d\ne\u271d : a\u271d \u2192+* b\u271d\nh :\n \u2200 (s : Finset b\u271d),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)) \u2192 P e\u271d\ns : Set b\u271d\nhs : Ideal.span s = \u22a4\nhs' : \u2200 (r : \u2191s), P (comp (algebraMap b\u271d (Localization.Away \u2191r)) e\u271d)\ns' : Finset b\u271d\nh\u2081 : \u2191s' \u2286 s\nh\u2082 : Ideal.span \u2191s' = \u22a4\n\u22a2 P e\u271d\n[PROOFSTEP]\nexact h s' h\u2082 fun x => hs' \u27e8_, h\u2081 x.prop\u27e9\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\n\u22a2 RespectsIso P\n[PROOFSTEP]\napply hP.StableUnderComposition.respectsIso\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\n\u22a2 \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\n[PROOFSTEP]\nintrov\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\n\u22a2 P (RingEquiv.toRingHom e)\n[PROOFSTEP]\nletI := e.toRingHom.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\nthis : Algebra R S := toAlgebra (RingEquiv.toRingHom e)\n\u22a2 P (RingEquiv.toRingHom e)\n[PROOFSTEP]\nhave : IsLocalization.Away (1 : R) S := by apply IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\nthis : Algebra R S := toAlgebra (RingEquiv.toRingHom e)\n\u22a2 IsLocalization.Away 1 S\n[PROOFSTEP]\napply IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\nthis\u271d : Algebra R S := toAlgebra (RingEquiv.toRingHom e)\nthis : IsLocalization.Away 1 S\n\u22a2 P (RingEquiv.toRingHom e)\n[PROOFSTEP]\nexact RingHom.PropertyIsLocal.HoldsForLocalizationAway hP S (1 : R)\n[GOAL]\nR S : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nH : LocalizationPreserves P\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\nhf : P f\n\u22a2 P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nhave : IsLocalization ((Submonoid.powers r).map f) S' := by rw [Submonoid.map_powers]; assumption\n[GOAL]\nR S : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nH : LocalizationPreserves P\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\nhf : P f\n\u22a2 IsLocalization (Submonoid.map f (Submonoid.powers r)) S'\n[PROOFSTEP]\nrw [Submonoid.map_powers]\n[GOAL]\nR S : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nH : LocalizationPreserves P\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\nhf : P f\n\u22a2 IsLocalization (Submonoid.powers (\u2191f r)) S'\n[PROOFSTEP]\nassumption\n[GOAL]\nR S : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nH : LocalizationPreserves P\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\nhf : P f\nthis : IsLocalization (Submonoid.map f (Submonoid.powers r)) S'\n\u22a2 P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nexact H f (Submonoid.powers r) R' S' hf\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\n\u22a2 OfLocalizationSpan P\n[PROOFSTEP]\nintrov R hs hs'\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs : Ideal.span s = \u22a4\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\n\u22a2 P f\n[PROOFSTEP]\napply_fun Ideal.map f at hs \n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\nhs : Ideal.map f (Ideal.span s) = Ideal.map f \u22a4\n\u22a2 P f\n[PROOFSTEP]\nrw [Ideal.map_span, Ideal.map_top] at hs \n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\nhs : Ideal.span (\u2191f '' s) = \u22a4\n\u22a2 P f\n[PROOFSTEP]\napply hP.OfLocalizationSpanTarget _ _ hs\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\nhs : Ideal.span (\u2191f '' s) = \u22a4\n\u22a2 \u2200 (r : \u2191(\u2191f '' s)), P (comp (algebraMap S (Localization.Away \u2191r)) f)\n[PROOFSTEP]\nrintro \u27e8_, r, hr, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\nhs : Ideal.span (\u2191f '' s) = \u22a4\nr : R\nhr : r \u2208 s\n\u22a2 P (comp (algebraMap S (Localization.Away \u2191{ val := \u2191f r, property := (_ : \u2203 a, a \u2208 s \u2227 \u2191f a = \u2191f r) })) f)\n[PROOFSTEP]\nconvert hP.StableUnderComposition _ _ (hP.HoldsForLocalizationAway (Localization.Away r) r) (hs' \u27e8r, hr\u27e9) using 1\n[GOAL]\ncase h.e'_5\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nR S : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\nhs' : \u2200 (r : \u2191s), P (Localization.awayMap f \u2191r)\nhs : Ideal.span (\u2191f '' s) = \u22a4\nr : R\nhr : r \u2208 s\n\u22a2 comp (algebraMap S (Localization.Away \u2191{ val := \u2191f r, property := (_ : \u2203 a, a \u2208 s \u2227 \u2191f a = \u2191f r) })) f =\n comp (Localization.awayMap f \u2191{ val := r, property := hr }) (algebraMap R (Localization.Away r))\n[PROOFSTEP]\nexact (IsLocalization.map_comp _).symm\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\n\u22a2 I \u2264 J\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 J\n[PROOFSTEP]\nsuffices J.colon (Ideal.span { x }) = \u22a4 by\n simpa using\n Submodule.mem_colon.mp (show (1 : R) \u2208 J.colon (Ideal.span { x }) from this.symm \u25b8 Submodule.mem_top) x\n (Ideal.mem_span_singleton_self x)\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nthis : Submodule.colon J (span {x}) = \u22a4\n\u22a2 x \u2208 J\n[PROOFSTEP]\nsimpa using\n Submodule.mem_colon.mp (show (1 : R) \u2208 J.colon (Ideal.span { x }) from this.symm \u25b8 Submodule.mem_top) x\n (Ideal.mem_span_singleton_self x)\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\n\u22a2 Submodule.colon J (span {x}) = \u22a4\n[PROOFSTEP]\nrefine' Not.imp_symm (J.colon (Ideal.span { x })).exists_le_maximal _\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\n\u22a2 \u00ac\u2203 M, IsMaximal M \u2227 Submodule.colon J (span {x}) \u2264 M\n[PROOFSTEP]\npush_neg\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\n\u22a2 \u2200 (M : Ideal R), IsMaximal M \u2192 \u00acSubmodule.colon J (span {x}) \u2264 M\n[PROOFSTEP]\nintro P hP le\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8a, ha\u27e9, \u27e8s, hs\u27e9\u27e9, eq\u27e9 :=\n (IsLocalization.mem_map_algebraMap_iff P.primeCompl _).mp (h P hP (Ideal.mem_map_of_mem _ hx))\n[GOAL]\ncase intro.mk.mk.mk\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\na : R\nha : a \u2208 J\ns : R\nhs : s \u2208 primeCompl P\neq :\n \u2191(algebraMap R (Localization.AtPrime P)) x *\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).snd =\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).fst\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 _root_.map_mul, \u2190 sub_eq_zero, \u2190 map_sub] at eq \n[GOAL]\ncase intro.mk.mk.mk\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\na : R\nha : a \u2208 J\ns : R\nhs : s \u2208 primeCompl P\neq\u271d :\n \u2191(algebraMap R (Localization.AtPrime P)) (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd) =\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).fst\neq :\n \u2191(algebraMap R (Localization.AtPrime P))\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8m, hm\u27e9, eq\u27e9 := (IsLocalization.map_eq_zero_iff P.primeCompl _ _).mp eq\n[GOAL]\ncase intro.mk.mk.mk.intro.mk\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\na : R\nha : a \u2208 J\ns : R\nhs : s \u2208 primeCompl P\neq\u271d\u00b9 :\n \u2191(algebraMap R (Localization.AtPrime P)) (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd) =\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).fst\neq\u271d :\n \u2191(algebraMap R (Localization.AtPrime P))\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\nm : R\nhm : m \u2208 primeCompl P\neq :\n \u2191{ val := m, property := hm } *\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\n\u22a2 False\n[PROOFSTEP]\nrefine' hs ((hP.isPrime.mem_or_mem (le (Ideal.mem_colon_singleton.mpr _))).resolve_right hm)\n[GOAL]\ncase intro.mk.mk.mk.intro.mk\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\na : R\nha : a \u2208 J\ns : R\nhs : s \u2208 primeCompl P\neq\u271d\u00b9 :\n \u2191(algebraMap R (Localization.AtPrime P)) (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd) =\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).fst\neq\u271d :\n \u2191(algebraMap R (Localization.AtPrime P))\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\nm : R\nhm : m \u2208 primeCompl P\neq :\n \u2191{ val := m, property := hm } *\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\n\u22a2 s * m * x \u2208 J\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, mul_sub, sub_eq_zero, mul_comm x s, mul_left_comm] at eq \n[GOAL]\ncase intro.mk.mk.mk.intro.mk\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI J : Ideal R\nh :\n \u2200 (P : Ideal R) (hP : IsMaximal P),\n map (algebraMap R (Localization.AtPrime P)) I \u2264 map (algebraMap R (Localization.AtPrime P)) J\nx : R\nhx : x \u2208 I\nP : Ideal R\nhP : IsMaximal P\nle : Submodule.colon J (span {x}) \u2264 P\na : R\nha : a \u2208 J\ns : R\nhs : s \u2208 primeCompl P\neq\u271d\u00b9 :\n \u2191(algebraMap R (Localization.AtPrime P)) (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd) =\n \u2191(algebraMap R (Localization.AtPrime P)) \u2191({ val := a, property := ha }, { val := s, property := hs }).fst\neq\u271d :\n \u2191(algebraMap R (Localization.AtPrime P))\n (x * \u2191({ val := a, property := ha }, { val := s, property := hs }).snd -\n \u2191({ val := a, property := ha }, { val := s, property := hs }).fst) =\n 0\nm : R\nhm : m \u2208 primeCompl P\neq : s * (m * x) = m * a\n\u22a2 s * m * x \u2208 J\n[PROOFSTEP]\nsimpa only [mul_assoc, eq] using J.mul_mem_left m ha\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI : Ideal R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), Ideal.map (algebraMap R (Localization.AtPrime J)) I = \u22a5\nP : Ideal R\nhP : Ideal.IsMaximal P\n\u22a2 Ideal.map (algebraMap R (Localization.AtPrime P)) I = Ideal.map (algebraMap R (Localization.AtPrime P)) \u22a5\n[PROOFSTEP]\nsimpa using h P hP\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI : Ideal R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), IsLocalization.coeSubmodule (Localization.AtPrime J) I = \u22a5\nP : Ideal R\nhP : Ideal.IsMaximal P\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 RingHom.ker (algebraMap R (Localization.AtPrime P))\n[PROOFSTEP]\nrw [RingHom.mem_ker, \u2190 Submodule.mem_bot R, \u2190 h P hP, IsLocalization.mem_coeSubmodule]\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nI : Ideal R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), IsLocalization.coeSubmodule (Localization.AtPrime J) I = \u22a5\nP : Ideal R\nhP : Ideal.IsMaximal P\nx : R\nhx : x \u2208 I\n\u22a2 \u2203 y, y \u2208 I \u2227 \u2191(algebraMap R ((fun x => Localization.AtPrime P) x)) y = \u2191(algebraMap R (Localization.AtPrime P)) x\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nrw [\u2190 Ideal.span_singleton_eq_bot]\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\n\u22a2 Ideal.span {r} = \u22a5\n[PROOFSTEP]\napply ideal_eq_bot_of_localization\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\n\u22a2 \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), IsLocalization.coeSubmodule (Localization.AtPrime J) (Ideal.span {r}) = \u22a5\n[PROOFSTEP]\nintro J hJ\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\nJ : Ideal R\nhJ : Ideal.IsMaximal J\n\u22a2 IsLocalization.coeSubmodule (Localization.AtPrime J) (Ideal.span {r}) = \u22a5\n[PROOFSTEP]\ndelta IsLocalization.coeSubmodule\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\nJ : Ideal R\nhJ : Ideal.IsMaximal J\n\u22a2 Submodule.map (Algebra.linearMap R (Localization.AtPrime J)) (Ideal.span {r}) = \u22a5\n[PROOFSTEP]\nerw [Submodule.map_span, Submodule.span_eq_bot]\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\nJ : Ideal R\nhJ : Ideal.IsMaximal J\n\u22a2 \u2200 (x : Localization.AtPrime J), x \u2208 \u2191(Algebra.linearMap R (Localization.AtPrime J)) '' {r} \u2192 x = 0\n[PROOFSTEP]\nrintro _ \u27e8_, h', rfl\u27e9\n[GOAL]\ncase h.intro.intro\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\nJ : Ideal R\nhJ : Ideal.IsMaximal J\nw\u271d : R\nh' : w\u271d \u2208 {r}\n\u22a2 \u2191(Algebra.linearMap R (Localization.AtPrime J)) w\u271d = 0\n[PROOFSTEP]\ncases Set.mem_singleton_iff.mpr h'\n[GOAL]\ncase h.intro.intro.refl\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\nr : R\nh : \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) r = 0\nJ : Ideal R\nhJ : Ideal.IsMaximal J\nh' : r \u2208 {r}\n\u22a2 \u2191(Algebra.linearMap R (Localization.AtPrime J)) r = 0\n[PROOFSTEP]\nexact h J hJ\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 LocalizationPreserves fun R hR => IsReduced R\n[PROOFSTEP]\nintrov R _ _\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\n\u22a2 IsReduced S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eq_zero\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\n\u22a2 \u2200 (x : S), IsNilpotent x \u2192 x = 0\n[PROOFSTEP]\nrintro x \u27e8_ | n, e\u27e9\n[GOAL]\ncase eq_zero.intro.zero\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\ne : x ^ Nat.zero = 0\n\u22a2 x = 0\n[PROOFSTEP]\nsimpa using congr_arg (\u00b7 * x) e\n[GOAL]\ncase eq_zero.intro.succ\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\n\u22a2 x = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8y, m\u27e9, hx\u27e9 := IsLocalization.surj M x\n[GOAL]\ncase eq_zero.intro.succ.intro.mk\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191(y, m).snd = \u2191(algebraMap R S) (y, m).fst\n\u22a2 x = 0\n[PROOFSTEP]\ndsimp only at hx \n[GOAL]\ncase eq_zero.intro.succ.intro.mk\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\n\u22a2 x = 0\n[PROOFSTEP]\nlet hx' := congr_arg (\u00b7 ^ n.succ) hx\n[GOAL]\ncase eq_zero.intro.succ.intro.mk\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : (fun x => x ^ Nat.succ n) (x * \u2191(algebraMap R S) \u2191m) = (fun x => x ^ Nat.succ n) (\u2191(algebraMap R S) y) :=\n congr_arg (fun x => x ^ Nat.succ n) hx\n\u22a2 x = 0\n[PROOFSTEP]\nsimp only [mul_pow, e, zero_mul, \u2190 RingHom.map_pow] at hx' \n[GOAL]\ncase eq_zero.intro.succ.intro.mk\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 (algebraMap R S).map_zero] at hx' \n[GOAL]\ncase eq_zero.intro.succ.intro.mk\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\n\u22a2 x = 0\n[PROOFSTEP]\nobtain \u27e8m', hm'\u27e9 := (IsLocalization.eq_iff_exists M S).mp hx'\n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : \u2191m' * 0 = \u2191m' * y ^ Nat.succ n\n\u22a2 x = 0\n[PROOFSTEP]\napply_fun (\u00b7 * (m' : R) ^ n) at hm' \n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : \u2191m' * 0 * \u2191m' ^ n = \u2191m' * y ^ Nat.succ n * \u2191m' ^ n\n\u22a2 x = 0\n[PROOFSTEP]\nsimp only [mul_assoc, zero_mul, mul_zero] at hm' \n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : 0 = \u2191m' * (y ^ Nat.succ n * \u2191m' ^ n)\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 mul_left_comm, \u2190 pow_succ, \u2190 mul_pow] at hm' \n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : 0 = (y * \u2191m') ^ Nat.succ n\n\u22a2 x = 0\n[PROOFSTEP]\nreplace hm' := IsNilpotent.eq_zero \u27e8_, hm'.symm\u27e9\n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : y * \u2191m' = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 (IsLocalization.map_units S m).mul_left_inj, hx, zero_mul, IsLocalization.map_eq_zero_iff M]\n[GOAL]\ncase eq_zero.intro.succ.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : y * \u2191m' = 0\n\u22a2 \u2203 m, \u2191m * y = 0\n[PROOFSTEP]\nexact \u27e8m', by rw [\u2190 hm', mul_comm]\u27e9\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsLocalization M S\na\u271d : IsReduced R\nx : S\nn : \u2115\ne : x ^ Nat.succ n = 0\ny : R\nm : { x // x \u2208 M }\nhx : x * \u2191(algebraMap R S) \u2191m = \u2191(algebraMap R S) y\nhx' : \u2191(algebraMap R S) 0 = \u2191(algebraMap R S) (y ^ Nat.succ n)\nm' : { x // x \u2208 M }\nhm' : y * \u2191m' = 0\n\u22a2 \u2191m' * y = 0\n[PROOFSTEP]\nrw [\u2190 hm', mul_comm]\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 OfLocalizationMaximal fun R hR => IsReduced R\n[PROOFSTEP]\nintrov R h\n[GOAL]\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nh :\n \u2200 (J : Ideal R) (x : Ideal.IsMaximal J),\n (fun R hR => IsReduced R) (Localization.AtPrime J) Localization.instCommRingLocalizationToCommMonoid\n\u22a2 IsReduced R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eq_zero\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nh :\n \u2200 (J : Ideal R) (x : Ideal.IsMaximal J),\n (fun R hR => IsReduced R) (Localization.AtPrime J) Localization.instCommRingLocalizationToCommMonoid\n\u22a2 \u2200 (x : R), IsNilpotent x \u2192 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase eq_zero\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nh :\n \u2200 (J : Ideal R) (x : Ideal.IsMaximal J),\n (fun R hR => IsReduced R) (Localization.AtPrime J) Localization.instCommRingLocalizationToCommMonoid\nx : R\nhx : IsNilpotent x\n\u22a2 x = 0\n[PROOFSTEP]\napply eq_zero_of_localization\n[GOAL]\ncase eq_zero.h\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nh :\n \u2200 (J : Ideal R) (x : Ideal.IsMaximal J),\n (fun R hR => IsReduced R) (Localization.AtPrime J) Localization.instCommRingLocalizationToCommMonoid\nx : R\nhx : IsNilpotent x\n\u22a2 \u2200 (J : Ideal R) (hJ : Ideal.IsMaximal J), \u2191(algebraMap R (Localization.AtPrime J)) x = 0\n[PROOFSTEP]\nintro J hJ\n[GOAL]\ncase eq_zero.h\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nh :\n \u2200 (J : Ideal R) (x : Ideal.IsMaximal J),\n (fun R hR => IsReduced R) (Localization.AtPrime J) Localization.instCommRingLocalizationToCommMonoid\nx : R\nhx : IsNilpotent x\nJ : Ideal R\nhJ : Ideal.IsMaximal J\n\u22a2 \u2191(algebraMap R (Localization.AtPrime J)) x = 0\n[PROOFSTEP]\nspecialize h J hJ\n[GOAL]\ncase eq_zero.h\nR\u271d S : Type u\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\nM : Submonoid R\u271d\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2074 : CommRing R'\ninst\u271d\u00b3 : CommRing S'\nf : R\u271d \u2192+* S\ninst\u271d\u00b2 : Algebra R\u271d R'\ninst\u271d\u00b9 : Algebra S S'\nR : Type u_1\ninst\u271d : CommRing R\nx : R\nhx : IsNilpotent x\nJ : Ideal R\nhJ : Ideal.IsMaximal J\nh : IsReduced (Localization.AtPrime J)\n\u22a2 \u2191(algebraMap R (Localization.AtPrime J)) x = 0\n[PROOFSTEP]\nexact (hx.map <| algebraMap R <| Localization.AtPrime J).eq_zero\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.LocalizationPreserves fun {R S} x x_1 f => Function.Surjective \u2191f\n[PROOFSTEP]\nintrov R H x\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nH : Function.Surjective \u2191f\nx : S'\n\u22a2 \u2203 a, \u2191(IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))) a = x\n[PROOFSTEP]\nobtain \u27e8x, \u27e8_, s, hs, rfl\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective (M.map f) x\n[GOAL]\ncase intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nH : Function.Surjective \u2191f\nx : S\ns : R\nhs : s \u2208 \u2191M\n\u22a2 \u2203 a,\n \u2191(IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))) a =\n IsLocalization.mk' S' x { val := \u2191f s, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f s) }\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 := H x\n[GOAL]\ncase intro.intro.mk.intro.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nH : Function.Surjective \u2191f\ns : R\nhs : s \u2208 \u2191M\ny : R\n\u22a2 \u2203 a,\n \u2191(IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))) a =\n IsLocalization.mk' S' (\u2191f y) { val := \u2191f s, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f s) }\n[PROOFSTEP]\nuse IsLocalization.mk' R' y \u27e8s, hs\u27e9\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nH : Function.Surjective \u2191f\ns : R\nhs : s \u2208 \u2191M\ny : R\n\u22a2 \u2191(IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n (IsLocalization.mk' R' y { val := s, property := hs }) =\n IsLocalization.mk' S' (\u2191f y) { val := \u2191f s, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f s) }\n[PROOFSTEP]\nrw [IsLocalization.map_mk']\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.OfLocalizationSpan fun {R S} x x_1 f => Function.Surjective \u2191f\n[PROOFSTEP]\nintrov R e H\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\n\u22a2 Function.Surjective \u2191f\n[PROOFSTEP]\nrw [\u2190 Set.range_iff_surjective, Set.eq_univ_iff_forall]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\n\u22a2 \u2200 (x : S), x \u2208 Set.range \u2191f\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\n\u22a2 \u2200 (x : S), x \u2208 Set.range \u2191f\n[PROOFSTEP]\nintro x\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\n\u22a2 x \u2208 Set.range \u2191f\n[PROOFSTEP]\napply Submodule.mem_of_span_eq_top_of_smul_pow_mem (LinearMap.range (Algebra.linearMap R S)) s e\n[GOAL]\ncase H\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\n\u22a2 \u2200 (r : \u2191s), \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nintro r\n[GOAL]\ncase H\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nobtain \u27e8a, e'\u27e9 := H r (algebraMap _ _ x)\n[GOAL]\ncase H.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\na : Localization.Away \u2191r\ne' : \u2191(Localization.awayMap f \u2191r) a = \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nobtain \u27e8b, \u27e8_, n, rfl\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective (Submonoid.powers (r : R)) a\n[GOAL]\ncase H.intro.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n \u2191(Localization.awayMap f \u2191r)\n (IsLocalization.mk' (Localization.Away \u2191r) b\n { val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) }) =\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nerw [IsLocalization.map_mk'] at e' \n[GOAL]\ncase H.intro.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n IsLocalization.mk' (Localization.Away (\u2191f \u2191r)) (\u2191f b)\n {\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) } =\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nrw [eq_comm, IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, Subtype.coe_mk, \u2190 map_mul] at e' \n[GOAL]\ncase H.intro.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (x *\n \u2191{\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) }) =\n \u2191(algebraMap ((fun x => S) b) (Localization.Away (\u2191f \u2191r))) (\u2191f b)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n', rfl\u27e9, e''\u27e9 := (IsLocalization.eq_iff_exists (Submonoid.powers (f r)) _).mp e'\n[GOAL]\ncase H.intro.intro.intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (x *\n \u2191{\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) }) =\n \u2191(algebraMap ((fun x => S) b) (Localization.Away (\u2191f \u2191r))) (\u2191f b)\nn' : \u2115\ne'' :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191f \u2191r) n',\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191f \u2191r) y = (fun x x_1 => x ^ x_1) (\u2191f \u2191r) n') } *\n (x *\n \u2191{\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) }) =\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191f \u2191r) n',\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191f \u2191r) y = (fun x x_1 => x ^ x_1) (\u2191f \u2191r) n') } *\n \u2191f b\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\ndsimp only at e'' \n[GOAL]\ncase H.intro.intro.intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (x *\n \u2191{\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) }) =\n \u2191(algebraMap ((fun x => S) b) (Localization.Away (\u2191f \u2191r))) (\u2191f b)\nn' : \u2115\ne'' : \u2191f \u2191r ^ n' * (x * \u2191f (\u2191r ^ n)) = \u2191f \u2191r ^ n' * \u2191f b\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nrw [mul_comm x, \u2190 mul_assoc, \u2190 map_pow, \u2190 map_mul, \u2190 map_mul, \u2190 pow_add] at e'' \n[GOAL]\ncase H.intro.intro.intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Set R\ne : Ideal.span s = \u22a4\nH :\n \u2200 (r : \u2191s),\n (fun {R S} x x_1 f => Function.Surjective \u2191f) Localization.instCommRingLocalizationToCommMonoid\n Localization.instCommRingLocalizationToCommMonoid (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\nx : S\nr : \u2191s\nb : R\nn : \u2115\ne' :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (x *\n \u2191{\n val :=\n \u2191f\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) },\n property :=\n (_ :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n) } \u2208\n Submonoid.comap f (Submonoid.powers (\u2191f \u2191r))) }) =\n \u2191(algebraMap ((fun x => S) b) (Localization.Away (\u2191f \u2191r))) (\u2191f b)\nn' : \u2115\ne'' : \u2191f (\u2191r ^ (n' + n)) * x = \u2191f (\u2191r ^ n' * b)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 LinearMap.range (Algebra.linearMap R S)\n[PROOFSTEP]\nexact \u27e8n' + n, _, e''.symm\u27e9\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.LocalizationPreserves @RingHom.Finite\n[PROOFSTEP]\nintrov R hf\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis : Algebra R S := RingHom.toAlgebra f\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := ((algebraMap S S').comp f).toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nlet f' : R' \u2192+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := f'.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis\u271d\u00b9 : Algebra R S := RingHom.toAlgebra f\nthis\u271d : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis : Algebra R' S' := RingHom.toAlgebra f'\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nhaveI : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nlet f\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') fun c x =>\n RingHom.map_mul _ _\n _\n -- We claim that if `S` is generated by `T` as an `R`-module,\n -- then `S'` is generated by `T` as an `R'`-module.\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.Finite f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nobtain \u27e8T, hT\u27e9 := hf\n[GOAL]\ncase mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\n\u22a2 RingHom.Finite (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nuse T.image (algebraMap S S')\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\n\u22a2 Submodule.span R' \u2191(Finset.image (\u2191(algebraMap S S')) T) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\n\u22a2 \u22a4 \u2264 Submodule.span R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nrintro x\n -\n -- By the hypotheses, for each `x : S'`, we have `x = y / (f r)` for some `y : S` and `r : M`.\n -- Since `S` is generated by `T`, the image of `y` should fall in the span of the image of `T`.\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\nx : S'\n\u22a2 x \u2208 Submodule.span R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nobtain \u27e8y, \u27e8_, \u27e8r, hr, rfl\u27e9\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective (M.map f) x\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 IsLocalization.mk' S' y { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } \u2208\n Submodule.span R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nrw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nhave hy : y \u2208 Submodule.span R \u2191T := by rw [hT]; trivial\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 y \u2208 Submodule.span R \u2191T\n[PROOFSTEP]\nrw [hT]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 y \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : y \u2208 Submodule.span R \u2191T\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nreplace hy : algebraMap S S' y \u2208 Submodule.map f\u2090.toLinearMap (Submodule.span R (T : Set S)) :=\n Submodule.mem_map_of_mem hy\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.map (AlgHom.toLinearMap f\u2090) (Submodule.span R \u2191T)\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nrw [Submodule.map_span f\u2090.toLinearMap T] at hy \n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nhave H : Submodule.span R (algebraMap S S' '' T) \u2264 (Submodule.span R' (algebraMap S S' '' T)).restrictScalars R := by\n rw [Submodule.span_le]; exact Submodule.subset_span\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\n\u22a2 Submodule.span R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T))\n[PROOFSTEP]\nrw [Submodule.span_le]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\n\u22a2 \u2191(algebraMap S S') '' \u2191T \u2286 \u2191(Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)))\n[PROOFSTEP]\nexact Submodule.subset_span\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\nH :\n Submodule.span R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Submodule.span R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nconvert (Submodule.span R' (algebraMap S S' '' T)).smul_mem (IsLocalization.mk' R' (1 : R) \u27e8r, hr\u27e9) (H hy) using 1\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\nH :\n Submodule.span R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n IsLocalization.mk' R' 1 { val := r, property := hr } \u2022 \u2191(algebraMap S S') y\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\nH :\n Submodule.span R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n \u2191(algebraMap R' S') (IsLocalization.mk' R' 1 { val := r, property := hr }) * \u2191(algebraMap S S') y\n[PROOFSTEP]\nerw [IsLocalization.map_mk' M.le_comap_map]\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Submodule.span R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Submodule.span R (\u2191(AlgHom.toLinearMap f\u2090) '' \u2191T)\nH :\n Submodule.span R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Submodule.restrictScalars R (Submodule.span R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n IsLocalization.mk' S' (\u2191f 1)\n { val := \u2191f \u2191{ val := r, property := hr },\n property := (_ : \u2191{ val := r, property := hr } \u2208 Submonoid.comap f (Submonoid.map f M)) } *\n \u2191(algebraMap S S') y\n[PROOFSTEP]\nrw [map_one]\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nlet g : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') fun c x => by\n simp [Algebra.algebraMap_eq_smul_one]\n -- We first obtain the `y' \u2208 M` such that `s' = y' \u2022 s` is falls in the image of `S` in `S'`.\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx\u271d : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x\u271d \u2208 Submodule.span R \u2191s\nc : R\nx : S\n\u22a2 \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x\n[PROOFSTEP]\nsimp [Algebra.algebraMap_eq_smul_one]\n -- We first obtain the `y' \u2208 M` such that `s' = y' \u2022 s` is falls in the image of `S` in `S'`.\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nlet y := IsLocalization.commonDenomOfFinset (M.map (algebraMap R S)) s\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nhave hx\u2081 : (y : S) \u2022 (s : Set S') = g '' _ := (IsLocalization.finsetIntegerMultiple_image _ s).symm\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nobtain \u27e8y', hy', e : algebraMap R S y' = y\u27e9 := y.prop\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nhave : algebraMap R S y' \u2022 (s : Set S') = y' \u2022 (s : Set S') := by\n simp_rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\n\u22a2 \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\n[PROOFSTEP]\nsimp_rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nrw [\u2190 e, this] at hx\u2081 \n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhx\u2081 : y' \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nreplace hx\u2081 := congr_arg (Submodule.span R) hx\u2081\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081 :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nrw [Submodule.span_smul] at hx\u2081 \n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Submodule.span R \u2191s\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nreplace hx : _ \u2208 y' \u2022 Submodule.span R (s : Set S') := Set.smul_mem_smul_set hx\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx : y' \u2022 \u2191(algebraMap S S') x \u2208 y' \u2022 Submodule.span R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nrw [hx\u2081] at hx \n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx : y' \u2022 \u2191(algebraMap S S') x \u2208 Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nerw [\u2190 g.map_smul, \u2190 Submodule.map_span (g : S \u2192\u2097[R] S')] at hx \n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx : \u2191g (y' \u2022 x) \u2208 Submodule.map (\u2191\u2191g) (Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nobtain \u27e8x', hx', hx'' : algebraMap _ _ _ = _\u27e9 := hx\n[GOAL]\ncase intro.intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nobtain \u27e8\u27e8_, a, ha\u2081, rfl\u27e9, ha\u2082\u27e9 := (IsLocalization.eq_iff_exists (M.map (algebraMap R S)) S').mp hx''\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 \u2203 m, m \u2022 x \u2208 Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nuse(\u27e8a, ha\u2081\u27e9 : M) * (\u27e8y', hy'\u27e9 : M)\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 ({ val := a, property := ha\u2081 } * { val := y', property := hy' }) \u2022 x \u2208\n Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nconvert\n (Submodule.span R (IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s : Set S)).smul_mem a\n hx' using\n 1\n[GOAL]\ncase h.e'_4\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 ({ val := a, property := ha\u2081 } * { val := y', property := hy' }) \u2022 x = a \u2022 x'\n[PROOFSTEP]\nconvert ha\u2082.symm using 1\n[GOAL]\ncase h.e'_2\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 ({ val := a, property := ha\u2081 } * { val := y', property := hy' }) \u2022 x =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n[PROOFSTEP]\nrw [Subtype.coe_mk, Submonoid.smul_def, Submonoid.coe_mul, \u2190 smul_smul]\n[GOAL]\ncase h.e'_2\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 \u2191{ val := a, property := ha\u2081 } \u2022 \u2191{ val := y', property := hy' } \u2022 x = \u2191(algebraMap R S) a * y' \u2022 x\n[PROOFSTEP]\nexact Algebra.smul_def _ _\n[GOAL]\ncase h.e'_3\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\ng : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S') (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (c \u2022 x) = c \u2022 \u2191(algebraMap S S') x)\ny : { x // x \u2208 Submonoid.map (algebraMap R S) M } := commonDenomOfFinset (Submonoid.map (algebraMap R S) M) s\ny' : R\nhy' : y' \u2208 \u2191M\ne : \u2191(algebraMap R S) y' = \u2191y\nthis : \u2191(algebraMap R S) y' \u2022 \u2191s = y' \u2022 \u2191s\nhx\u2081\u271d :\n Submodule.span R (y' \u2022 \u2191s) = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx\u2081 : y' \u2022 Submodule.span R \u2191s = Submodule.span R (\u2191g '' \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nx' : S\nhx' : x' \u2208 \u2191(Submodule.span R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s))\nhx'' : \u2191(algebraMap S S') x' = \u2191g (y' \u2022 x)\na : R\nha\u2081 : a \u2208 \u2191M\nha\u2082 :\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n y' \u2022 x\n\u22a2 a \u2022 x' =\n \u2191{ val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } *\n x'\n[PROOFSTEP]\nexact Algebra.smul_def _ _\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Submodule.span R' s\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R s\n[PROOFSTEP]\nobtain \u27e8s', hss', hs'\u27e9 := Submodule.mem_span_finite_of_mem_span hx\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Submodule.span R' s\ns' : Finset S\nhss' : \u2191s' \u2286 s\nhs' : x \u2208 Submodule.span R' \u2191s'\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R s\n[PROOFSTEP]\nrsuffices \u27e8t, ht\u27e9 : \u2203 t : M, t \u2022 x \u2208 Submodule.span R (s' : Set S)\n[GOAL]\ncase intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Submodule.span R' s\ns' : Finset S\nhss' : \u2191s' \u2286 s\nhs' : x \u2208 Submodule.span R' \u2191s'\nt : { x // x \u2208 M }\nht : t \u2022 x \u2208 Submodule.span R \u2191s'\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R s\n[PROOFSTEP]\nexact \u27e8t, Submodule.span_mono hss' ht\u27e9\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Submodule.span R' s\ns' : Finset S\nhss' : \u2191s' \u2286 s\nhs' : x \u2208 Submodule.span R' \u2191s'\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s'\n[PROOFSTEP]\nclear hx hss' s\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\nx : S\ns' : Finset S\nhs' : x \u2208 Submodule.span R' \u2191s'\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s'\n[PROOFSTEP]\ninduction s' using Finset.induction_on generalizing x\n[GOAL]\ncase empty\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\nx : S\nhs' : x \u2208 Submodule.span R' \u2191\u2205\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191\u2205\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\nx : S\nhs' : x \u2208 Submodule.span R' \u2191\u2205\n\u22a2 1 \u2022 x \u2208 Submodule.span R \u2191\u2205\n[PROOFSTEP]\nsimpa using hs'\n[GOAL]\ncase insert\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na\u271d\u00b2 : S\ns\u271d : Finset S\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 s\u271d\na\u271d : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s\u271d \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\u271d\nx : S\nhs' : x \u2208 Submodule.span R' \u2191(insert a\u271d\u00b2 s\u271d)\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191(insert a\u271d\u00b2 s\u271d)\n[PROOFSTEP]\nrename_i a s _ hs\n[GOAL]\ncase insert\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\nx : S\nhs' : x \u2208 Submodule.span R' \u2191(insert a s)\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191(insert a s)\n[PROOFSTEP]\nsimp only [Finset.coe_insert, Finset.image_insert, Finset.coe_image, Subtype.coe_mk, Submodule.mem_span_insert] at hs' \u22a2\n[GOAL]\ncase insert\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\nx : S\nhs' : \u2203 a_1 z, z \u2208 Submodule.span R' \u2191s \u2227 x = a_1 \u2022 a + z\n\u22a2 \u2203 t a_1 z, z \u2208 Submodule.span R \u2191s \u2227 t \u2022 x = a_1 \u2022 a + z\n[PROOFSTEP]\nrcases hs' with \u27e8y, z, hz, rfl\u27e9\n[GOAL]\ncase insert.intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\n\u22a2 \u2203 t a_1 z_1, z_1 \u2208 Submodule.span R \u2191s \u2227 t \u2022 (y \u2022 a + z) = a_1 \u2022 a + z_1\n[PROOFSTEP]\nrcases IsLocalization.surj M y with \u27e8\u27e8y', s'\u27e9, e\u27e9\n[GOAL]\ncase insert.intro.intro.intro.intro.mk\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne : y * \u2191(algebraMap R R') \u2191(y', s').snd = \u2191(algebraMap R R') (y', s').fst\n\u22a2 \u2203 t a_1 z_1, z_1 \u2208 Submodule.span R \u2191s \u2227 t \u2022 (y \u2022 a + z) = a_1 \u2022 a + z_1\n[PROOFSTEP]\nreplace e : _ * a = _ * a := (congr_arg (fun x => algebraMap R' S x * a) e : _)\n[GOAL]\ncase insert.intro.intro.intro.intro.mk\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne :\n \u2191(algebraMap R' S) (y * \u2191(algebraMap R R') \u2191(y', s').snd) * a =\n \u2191(algebraMap R' S) (\u2191(algebraMap R R') (y', s').fst) * a\n\u22a2 \u2203 t a_1 z_1, z_1 \u2208 Submodule.span R \u2191s \u2227 t \u2022 (y \u2022 a + z) = a_1 \u2022 a + z_1\n[PROOFSTEP]\nsimp_rw [RingHom.map_mul, \u2190 IsScalarTower.algebraMap_apply, mul_comm (algebraMap R' S y), mul_assoc, \u2190\n Algebra.smul_def] at e \n[GOAL]\ncase insert.intro.intro.intro.intro.mk\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne : \u2191s' \u2022 y \u2022 a = y' \u2022 a\n\u22a2 \u2203 t a_1 z_1, z_1 \u2208 Submodule.span R \u2191s \u2227 t \u2022 (y \u2022 a + z) = a_1 \u2022 a + z_1\n[PROOFSTEP]\nrcases hs _ hz with \u27e8t, ht\u27e9\n[GOAL]\ncase insert.intro.intro.intro.intro.mk.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne : \u2191s' \u2022 y \u2022 a = y' \u2022 a\nt : { x // x \u2208 M }\nht : t \u2022 z \u2208 Submodule.span R \u2191s\n\u22a2 \u2203 t a_1 z_1, z_1 \u2208 Submodule.span R \u2191s \u2227 t \u2022 (y \u2022 a + z) = a_1 \u2022 a + z_1\n[PROOFSTEP]\nrefine' \u27e8t * s', t * y', _, (Submodule.span R (s : Set S)).smul_mem s' ht, _\u27e9\n[GOAL]\ncase insert.intro.intro.intro.intro.mk.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne : \u2191s' \u2022 y \u2022 a = y' \u2022 a\nt : { x // x \u2208 M }\nht : t \u2022 z \u2208 Submodule.span R \u2191s\n\u22a2 (t * s') \u2022 (y \u2022 a + z) = (\u2191t * y') \u2022 a + \u2191s' \u2022 t \u2022 z\n[PROOFSTEP]\nrw [smul_add, \u2190 smul_smul, mul_comm, \u2190 smul_smul, \u2190 smul_smul, \u2190 e]\n[GOAL]\ncase insert.intro.intro.intro.intro.mk.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\na : S\ns : Finset S\na\u271d : \u00aca \u2208 s\nhs : \u2200 (x : S), x \u2208 Submodule.span R' \u2191s \u2192 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191s\ny : R'\nz : S\nhz : z \u2208 Submodule.span R' \u2191s\ny' : R\ns' : { x // x \u2208 M }\ne : \u2191s' \u2022 y \u2022 a = y' \u2022 a\nt : { x // x \u2208 M }\nht : t \u2022 z \u2208 Submodule.span R \u2191s\n\u22a2 t \u2022 s' \u2022 y \u2022 a + s' \u2022 t \u2022 z = \u2191t \u2022 \u2191s' \u2022 y \u2022 a + \u2191s' \u2022 t \u2022 z\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Algebra.adjoin R' s\n\u22a2 \u2203 t, t \u2022 x \u2208 Algebra.adjoin R s\n[PROOFSTEP]\nchange \u2203 t : M, t \u2022 x \u2208 Subalgebra.toSubmodule (Algebra.adjoin R s)\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Algebra.adjoin R' s\n\u22a2 \u2203 t, t \u2022 x \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin R s)\n[PROOFSTEP]\nchange x \u2208 Subalgebra.toSubmodule (Algebra.adjoin R' s) at hx \n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin R' s)\n\u22a2 \u2203 t, t \u2022 x \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin R s)\n[PROOFSTEP]\nsimp_rw [Algebra.adjoin_eq_span] at hx \u22a2\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R' S\ninst\u271d\u00b2 : Algebra R S\ninst\u271d\u00b9 : IsScalarTower R R' S\ninst\u271d : IsLocalization M R'\ns : Set S\nx : S\nhx : x \u2208 Submodule.span R' \u2191(Submonoid.closure s)\n\u22a2 \u2203 t, t \u2022 x \u2208 Submodule.span R \u2191(Submonoid.closure s)\n[PROOFSTEP]\nexact multiple_mem_span_of_mem_localization_span M R' _ _ hx\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.OfLocalizationSpan @RingHom.Finite\n[PROOFSTEP]\nrw [RingHom.ofLocalizationSpan_iff_finite]\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.OfLocalizationFiniteSpan @RingHom.Finite\n[PROOFSTEP]\nintrov R hs H\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\n\u22a2 RingHom.Finite f\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\n\u22a2 RingHom.Finite f\n[PROOFSTEP]\nletI := fun r : s => (Localization.awayMap f r).toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\n\u22a2 RingHom.Finite f\n[PROOFSTEP]\nhave : \u2200 r : s, IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) := by\n intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\n\u22a2 \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nintro r\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nr : { x // x \u2208 s }\n\u22a2 IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nrw [Submonoid.map_powers]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nr : { x // x \u2208 s }\n\u22a2 IsLocalization (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nexact Localization.isLocalization\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d\u00b9 : Algebra R S := RingHom.toAlgebra f\nthis\u271d : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n\u22a2 RingHom.Finite f\n[PROOFSTEP]\nhaveI : \u2200 r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) := fun r =>\n IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\n\u22a2 RingHom.Finite f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.Finite (Localization.awayMap f \u2191r)\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\n\u22a2 Submodule.FG \u22a4\n[PROOFSTEP]\nreplace H := fun r => (H r).1\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\nH : \u2200 (r : { x // x \u2208 s }), Submodule.FG \u22a4\n\u22a2 Submodule.FG \u22a4\n[PROOFSTEP]\nchoose s\u2081 s\u2082 using H\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\n\u22a2 Submodule.FG \u22a4\n[PROOFSTEP]\nlet sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s\u2081 x)\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 Submodule.FG \u22a4\n[PROOFSTEP]\nuse s.attach.biUnion sf\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 Submodule.span R \u2191(Finset.biUnion (Finset.attach s) sf) = \u22a4\n[PROOFSTEP]\nrw [Submodule.span_attach_biUnion, eq_top_iff]\n -- It suffices to show that `r ^ n \u2022 x \u2208 span T` for each `r : s`, since `{ r ^ n }` spans `R`.\n -- This then follows from the fact that each `x : R` is a linear combination of the generating set\n -- of `S\u1d63`. By multiplying a sufficiently large power of `r`, we can cancel out the `r`s in the\n -- denominators of both the generating set and the coefficients.\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 \u22a4 \u2264 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nrintro x -\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\n\u22a2 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\napply Submodule.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set R) hs _ _\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\n\u22a2 \u2200 (r : \u2191\u2191s), \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nintro r\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n\u2081, rfl\u27e9, hn\u2081\u27e9 :=\n multiple_mem_span_of_mem_localization_span (Submonoid.powers (r : R)) (Localization.Away (r : R))\n (s\u2081 r : Set (Localization.Away (f r))) (algebraMap S _ x) (by rw [s\u2082 r]; trivial)\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208 Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r)\n[PROOFSTEP]\nrw [s\u2082 r]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n { val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } \u2022\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208\n Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\ndsimp only at hn\u2081 \n[GOAL]\ncase intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n { val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } \u2022 \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208\n Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nrw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, \u2190 map_mul] at hn\u2081 \n[GOAL]\ncase intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 :=\n IsLocalization.smul_mem_finsetIntegerMultiple_span (Submonoid.powers (r : R)) (Localization.Away (f r)) _ (s\u2081 r) hn\u2081\n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n { val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2082,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2082) } \u2022\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R\n \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nrw [Submonoid.smul_def, \u2190 Algebra.smul_def, smul_smul, Subtype.coe_mk, \u2190 pow_add] at hn\u2082 \n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Submodule.span R\n \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nsimp_rw [Submonoid.map_powers] at hn\u2082 \n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Submodule.span R \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nuse n\u2082 + n\u2081\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Submodule.span (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S) \u2191{ val := \u2191r ^ n\u2081, property := (_ : \u2203 y, \u2191r ^ y = \u2191r ^ n\u2081) } * x) \u2208\n Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Submodule.span R \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 \u2a06 (x : { x // x \u2208 s }), Submodule.span R \u2191(sf x)\n[PROOFSTEP]\nexact le_iSup (fun x : s => Submodule.span R (sf x : Set S)) r hn\u2082\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.LocalizationPreserves @RingHom.FiniteType\n[PROOFSTEP]\nintrov R hf\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis : Algebra R S := RingHom.toAlgebra f\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := ((algebraMap S S').comp f).toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nlet f' : R' \u2192+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nletI := f'.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b9 : Algebra R S := RingHom.toAlgebra f\nthis\u271d : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis : Algebra R' S' := RingHom.toAlgebra f'\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nhaveI : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nlet f\u2090 : S \u2192\u2090[R] S' := AlgHom.mk' (algebraMap S S') fun c x => RingHom.map_mul _ _ _\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nobtain \u27e8T, hT\u27e9 := id hf\n[GOAL]\ncase mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\n\u22a2 RingHom.FiniteType (IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M)))\n[PROOFSTEP]\nuse T.image (algebraMap S S')\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\n\u22a2 Algebra.adjoin R' \u2191(Finset.image (\u2191(algebraMap S S')) T) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\n\u22a2 \u22a4 \u2264 Algebra.adjoin R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nrintro x -\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\nx : S'\n\u22a2 x \u2208 Algebra.adjoin R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nobtain \u27e8y, \u27e8_, \u27e8r, hr, rfl\u27e9\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective (M.map f) x\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 IsLocalization.mk' S' y { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } \u2208\n Algebra.adjoin R' \u2191(Finset.image (\u2191(algebraMap S S')) T)\n[PROOFSTEP]\nrw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nhave hy : y \u2208 Algebra.adjoin R (T : Set S) := by rw [hT]; trivial\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 y \u2208 Algebra.adjoin R \u2191T\n[PROOFSTEP]\nrw [hT]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\n\u22a2 y \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : y \u2208 Algebra.adjoin R \u2191T\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nreplace hy : algebraMap S S' y \u2208 (Algebra.adjoin R (T : Set S)).map f\u2090 := Subalgebra.mem_map.mpr \u27e8_, hy, rfl\u27e9\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Subalgebra.map f\u2090 (Algebra.adjoin R \u2191T)\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nrw [f\u2090.map_adjoin T] at hy \n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nhave H : Algebra.adjoin R (algebraMap S S' '' T) \u2264 (Algebra.adjoin R' (algebraMap S S' '' T)).restrictScalars R := by\n rw [Algebra.adjoin_le_iff]; exact Algebra.subset_adjoin\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\n\u22a2 Algebra.adjoin R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T))\n[PROOFSTEP]\nrw [Algebra.adjoin_le_iff]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\n\u22a2 \u2191(algebraMap S S') '' \u2191T \u2286 \u2191(Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)))\n[PROOFSTEP]\nexact Algebra.subset_adjoin\n[GOAL]\ncase h.intro.intro.mk.intro.intro\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\nH :\n Algebra.adjoin R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y \u2208\n Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T)\n[PROOFSTEP]\nconvert (Algebra.adjoin R' (algebraMap S S' '' T)).smul_mem (H hy) (IsLocalization.mk' R' (1 : R) \u27e8r, hr\u27e9) using 1\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\nH :\n Algebra.adjoin R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n IsLocalization.mk' R' 1 { val := r, property := hr } \u2022 \u2191(algebraMap S S') y\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\nH :\n Algebra.adjoin R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n \u2191(algebraMap R' S') (IsLocalization.mk' R' 1 { val := r, property := hr }) * \u2191(algebraMap S S') y\n[PROOFSTEP]\nerw [IsLocalization.map_mk' M.le_comap_map]\n[GOAL]\ncase h.e'_4\nR\u271d S\u271d : Type u\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing S\u271d\nM\u271d : Submonoid R\u271d\nN : Submonoid S\u271d\nR'\u271d S'\u271d : Type u\ninst\u271d\u00b9\u00b9 : CommRing R'\u271d\ninst\u271d\u00b9\u2070 : CommRing S'\u271d\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u2079 : Algebra R\u271d R'\u271d\ninst\u271d\u2078 : Algebra S\u271d S'\u271d\nR S : Type u_1\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nf : R \u2192+* S\nM : Submonoid R\nR' S' : Type u_1\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\ninst\u271d\u00b9 : IsLocalization M R'\ninst\u271d : IsLocalization (Submonoid.map f M) S'\nhf : RingHom.FiniteType f\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : Algebra R S' := RingHom.toAlgebra (RingHom.comp (algebraMap S S') f)\nf' : R' \u2192+* S' := IsLocalization.map S' f (_ : M \u2264 Submonoid.comap f (Submonoid.map f M))\nthis\u271d : Algebra R' S' := RingHom.toAlgebra f'\nthis : IsScalarTower R R' S'\nf\u2090 : S \u2192\u2090[R] S' :=\n AlgHom.mk' (algebraMap S S')\n (_ : \u2200 (c : R) (x : S), \u2191(algebraMap S S') (\u2191f c * x) = \u2191(algebraMap S S') (\u2191f c) * \u2191(algebraMap S S') x)\nT : Finset S\nhT : Algebra.adjoin R \u2191T = \u22a4\ny : S\nr : R\nhr : r \u2208 \u2191M\nhy : \u2191(algebraMap S S') y \u2208 Algebra.adjoin R (\u2191f\u2090 '' \u2191T)\nH :\n Algebra.adjoin R (\u2191(algebraMap S S') '' \u2191T) \u2264\n Subalgebra.restrictScalars R (Algebra.adjoin R' (\u2191(algebraMap S S') '' \u2191T))\n\u22a2 IsLocalization.mk' S' 1 { val := \u2191f r, property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191f a = \u2191f r) } * \u2191(algebraMap S S') y =\n IsLocalization.mk' S' (\u2191f 1)\n { val := \u2191f \u2191{ val := r, property := hr },\n property := (_ : \u2191{ val := r, property := hr } \u2208 Submonoid.comap f (Submonoid.map f M)) } *\n \u2191(algebraMap S S') y\n[PROOFSTEP]\nrw [map_one]\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nlet g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nlet y := IsLocalization.commonDenomOfFinset M s\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nhave hx\u2081 : (y : S) \u2022 (s : Set S') = g '' _ := (IsLocalization.finsetIntegerMultiple_image _ s).symm\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 :=\n Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)\n (by rw [hx\u2081]; exact Set.image_subset _ hA\u2081) hx (Set.mem_image_of_mem _ (hA\u2082 y.2))\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2191y \u2022 \u2191s \u2286 \u2191(Subalgebra.map g A)\n[PROOFSTEP]\nrw [hx\u2081]\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2191g '' \u2191(finsetIntegerMultiple M s) \u2286 \u2191(Subalgebra.map g A)\n[PROOFSTEP]\nexact Set.image_subset _ hA\u2081\n[GOAL]\ncase intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nobtain \u27e8x', hx', hx''\u27e9 := hn n (le_of_eq rfl)\n[GOAL]\ncase intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191y ^ n \u2022 \u2191(algebraMap S S') x\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nrw [Algebra.smul_def, \u2190 _root_.map_mul] at hx'' \n[GOAL]\ncase intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nobtain \u27e8a, ha\u2082\u27e9 := (IsLocalization.eq_iff_exists M S').mp hx''\n[GOAL]\ncase intro.intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A\n[PROOFSTEP]\nuse a * y ^ n\n[GOAL]\ncase h\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x \u2208 A\n[PROOFSTEP]\nconvert A.mul_mem hx' (hA\u2082 a.prop) using 1\n[GOAL]\ncase h.e'_4\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x = x' * \u2191a\n[PROOFSTEP]\nrw [Submonoid.smul_def, smul_eq_mul, Submonoid.coe_mul, SubmonoidClass.coe_pow, mul_assoc, \u2190 ha\u2082, mul_comm]\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nobtain \u27e8\u27e8_, a, ha, rfl\u27e9, e\u27e9 :=\n IsLocalization.exists_smul_mem_of_mem_adjoin (M.map (algebraMap R S)) x s (Algebra.adjoin R _) Algebra.subset_adjoin\n (by rintro _ \u27e8a, _, rfl\u27e9; exact Subalgebra.algebraMap_mem _ a) hx\n[GOAL]\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 Submonoid.map (algebraMap R S) M \u2264\n (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid\n[PROOFSTEP]\nrintro _ \u27e8a, _, rfl\u27e9\n[GOAL]\ncase intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nleft\u271d : a \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) a \u2208\n (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid\n[PROOFSTEP]\nexact Subalgebra.algebraMap_mem _ a\n[GOAL]\ncase intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nrefine' \u27e8\u27e8a, ha\u27e9, _\u27e9\n[GOAL]\ncase intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 { val := a, property := ha } \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n[PROOFSTEP]\nsimpa only [Submonoid.smul_def, algebraMap_smul] using e\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.OfLocalizationSpan @RingHom.FiniteType\n[PROOFSTEP]\nrw [RingHom.ofLocalizationSpan_iff_finite]\n[GOAL]\nR S : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u00b3 : CommRing R'\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R R'\ninst\u271d : Algebra S S'\n\u22a2 RingHom.OfLocalizationFiniteSpan @RingHom.FiniteType\n[PROOFSTEP]\nintrov R hs H\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\n\u22a2 RingHom.FiniteType f\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis : Algebra R S := RingHom.toAlgebra f\n\u22a2 RingHom.FiniteType f\n[PROOFSTEP]\nletI := fun r : s => (Localization.awayMap f r).toAlgebra\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\n\u22a2 RingHom.FiniteType f\n[PROOFSTEP]\nhave : \u2200 r : s, IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) := by\n intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\n\u22a2 \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nintro r\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nr : { x // x \u2208 s }\n\u22a2 IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nrw [Submonoid.map_powers]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d : Algebra R S := RingHom.toAlgebra f\nthis : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nr : { x // x \u2208 s }\n\u22a2 IsLocalization (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (Localization.Away (\u2191f \u2191r))\n[PROOFSTEP]\nexact Localization.isLocalization\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d\u00b9 : Algebra R S := RingHom.toAlgebra f\nthis\u271d : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\n\u22a2 RingHom.FiniteType f\n[PROOFSTEP]\nhaveI : \u2200 r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) := fun r =>\n IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\n\u22a2 RingHom.FiniteType f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), RingHom.FiniteType (Localization.awayMap f \u2191r)\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nreplace H := fun r => (H r).1\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\nH : \u2200 (r : { x // x \u2208 s }), Subalgebra.FG \u22a4\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nchoose s\u2081 s\u2082 using H\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nlet sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s\u2081 x)\n[GOAL]\ncase out\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nuse s.attach.biUnion sf\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf) = \u22a4\n[PROOFSTEP]\nconvert (Algebra.adjoin_attach_biUnion (R := R) sf).trans _\n[GOAL]\ncase h.convert_2\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 \u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\ncase h.convert_2\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\n\u22a2 \u22a4 \u2264 \u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x)\n[PROOFSTEP]\nrintro x -\n[GOAL]\ncase h.convert_2\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\n\u22a2 x \u2208 \u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x)\n[PROOFSTEP]\napply (\u2a06 x : s, Algebra.adjoin R (sf x : Set S)).toSubmodule.mem_of_span_eq_top_of_smul_pow_mem _ hs _ _\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\n\u22a2 \u2200 (r : \u2191\u2191s), \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nintro r\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n\u2081, rfl\u27e9, hn\u2081\u27e9 :=\n multiple_mem_adjoin_of_mem_localization_adjoin (Submonoid.powers (r : R)) (Localization.Away (r : R))\n (s\u2081 r : Set (Localization.Away (f r))) (algebraMap S (Localization.Away (f r)) x) (by rw [s\u2082 r]; trivial)\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208 Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r)\n[PROOFSTEP]\nrw [s\u2082 r]\n[GOAL]\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n { val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } \u2022\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r))) x \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nrw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, \u2190 map_mul] at hn\u2081 \n[GOAL]\ncase intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 :=\n IsLocalization.lift_mem_adjoin_finsetIntegerMultiple (Submonoid.powers (r : R)) _ (s\u2081 r) hn\u2081\n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n { val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2082,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2082) } \u2022\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R\n \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nrw [Submonoid.smul_def, \u2190 Algebra.smul_def, smul_smul, Subtype.coe_mk, \u2190 pow_add] at hn\u2082 \n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Algebra.adjoin R\n \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nsimp_rw [Submonoid.map_powers] at hn\u2082 \n[GOAL]\ncase intro.mk.intro.intro.mk.intro\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Algebra.adjoin R \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nuse n\u2082 + n\u2081\n[GOAL]\ncase h\nR\u271d S\u271d : Type u\ninst\u271d\u2077 : CommRing R\u271d\ninst\u271d\u2076 : CommRing S\u271d\nM : Submonoid R\u271d\nN : Submonoid S\u271d\nR' S' : Type u\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\nf\u271d : R\u271d \u2192+* S\u271d\ninst\u271d\u00b3 : Algebra R\u271d R'\ninst\u271d\u00b2 : Algebra S\u271d S'\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset R\nhs : Ideal.span \u2191s = \u22a4\nthis\u271d\u00b2 : Algebra R S := RingHom.toAlgebra f\nthis\u271d\u00b9 : (r : { x // x \u2208 s }) \u2192 Algebra (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r)) :=\n fun r => RingHom.toAlgebra (Localization.awayMap f \u2191r)\nthis\u271d :\n \u2200 (r : { x // x \u2208 s }),\n IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers \u2191r)) (Localization.Away (\u2191f \u2191r))\nthis : \u2200 (r : { x // x \u2208 s }), IsScalarTower R (Localization.Away \u2191r) (Localization.Away (\u2191f \u2191r))\ns\u2081 : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away (\u2191f \u2191r))\ns\u2082 : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin (Localization.Away \u2191r) \u2191(s\u2081 r) = \u22a4\nsf : (x : { x // x \u2208 s }) \u2192 Finset ((fun x => S) \u2191x) :=\n fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191f \u2191x)) (s\u2081 x)\nx : S\nr : \u2191\u2191s\nn\u2081 : \u2115\nhn\u2081 :\n \u2191(algebraMap S (Localization.Away (\u2191f \u2191r)))\n (\u2191(algebraMap R S)\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191r) n\u2081,\n property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) (\u2191r) y = (fun x x_1 => x ^ x_1) (\u2191r) n\u2081) } *\n x) \u2208\n Algebra.adjoin R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208\n Algebra.adjoin R \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers (\u2191(algebraMap R S) \u2191r)) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 \u2191Subalgebra.toSubmodule (\u2a06 (x : { x // x \u2208 s }), Algebra.adjoin R \u2191(sf x))\n[PROOFSTEP]\nexact le_iSup (fun x : s => Algebra.adjoin R (sf x : Set S)) r hn\u2082\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.LocalProperties", "llama_tokens": 120518, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.2763774521659273}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\nf : Filter \u03b1\n\u22a2 (\u2200 (i' : \u03b9), p i' \u2192 \u2203 i, i \u2208 f \u2227 id i \u00d7\u02e2 id i \u2286 s i') \u2194\n \u2200 (i : \u03b9), p i \u2192 \u2203 t, t \u2208 f \u2227 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 (x, y) \u2208 s i\n[PROOFSTEP]\nsimp only [subset_def, Prod.forall, mem_prod_eq, and_imp, id, ball_mem_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\n\u22a2 (NeBot f \u2227 \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2200 (y : \u03b1), y \u2208 t \u2192 (x, y) \u2208 s) \u2194\n NeBot f \u2227 \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s\n[PROOFSTEP]\nsimp only [subset_def, Prod.forall, mem_prod_eq, and_imp, id, ball_mem_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nl : Filter \u03b1\nh : Cauchy l\n\u22a2 Cauchy \u2191(Ultrafilter.of l)\n[PROOFSTEP]\nhaveI := h.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nl : Filter \u03b1\nh : Cauchy l\nthis : NeBot l\n\u22a2 Cauchy \u2191(Ultrafilter.of l)\n[PROOFSTEP]\nhave := Ultrafilter.of_le l\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nl : Filter \u03b1\nh : Cauchy l\nthis\u271d : NeBot l\nthis : \u2191(Ultrafilter.of l) \u2264 l\n\u22a2 Cauchy \u2191(Ultrafilter.of l)\n[PROOFSTEP]\nexact \u27e8Ultrafilter.neBot _, (Filter.prod_mono this this).trans h.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 Cauchy (map f l) \u2194 NeBot l \u2227 Tendsto (fun p => (f p.fst, f p.snd)) (l \u00d7\u02e2 l) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrw [Cauchy, map_neBot_iff, prod_map_map_eq, Tendsto]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\nhf : Cauchy f\nhg : Cauchy g\n\u22a2 Cauchy (f \u00d7\u02e2 g)\n[PROOFSTEP]\nrefine' \u27e8hf.1.prod hg.1, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\nhf : Cauchy f\nhg : Cauchy g\n\u22a2 (f \u00d7\u02e2 g) \u00d7\u02e2 f \u00d7\u02e2 g \u2264 \ud835\udce4 (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nsimp only [uniformity_prod, le_inf_iff, \u2190 map_le_iff_le_comap, \u2190 prod_map_map_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\nhf : Cauchy f\nhg : Cauchy g\n\u22a2 map Prod.fst (f \u00d7\u02e2 g) \u00d7\u02e2 map Prod.fst (f \u00d7\u02e2 g) \u2264 \ud835\udce4 \u03b1 \u2227 map Prod.snd (f \u00d7\u02e2 g) \u00d7\u02e2 map Prod.snd (f \u00d7\u02e2 g) \u2264 \ud835\udce4 \u03b2\n[PROOFSTEP]\nexact \u27e8le_trans (prod_mono tendsto_fst tendsto_fst) hf.2, le_trans (prod_mono tendsto_snd tendsto_snd) hg.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\n\u22a2 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\n\u22a2 s \u2208 f\n[PROOFSTEP]\nrcases comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 hs) with\n \u27e8U, U_mem, hU\u27e9\n -- Take a set `t \u2208 f`, `t \u00d7 t \u2286 U`, and a point `y \u2208 t` such that `(x, y) \u2208 U`\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.fst = x \u2192 p.snd \u2208 s}\n\u22a2 s \u2208 f\n[PROOFSTEP]\nrcases adhs U U_mem with \u27e8t, t_mem, ht, y, hxy, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.fst = x \u2192 p.snd \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 s \u2208 f\n[PROOFSTEP]\napply mem_of_superset t_mem\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.fst = x \u2192 p.snd \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 t \u2286 s\n[PROOFSTEP]\nexact fun z hz => hU (prod_mk_mem_compRel hxy (ht <| mk_mem_prod hy hz)) rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nhf : Cauchy f\nadhs : ClusterPt x f\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nobtain \u27e8t, t_mem, ht\u27e9 : \u2203 t \u2208 f, t \u00d7\u02e2 t \u2286 s := (cauchy_iff.1 hf).2 s hs\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nhf : Cauchy f\nadhs : ClusterPt x f\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 s\n\u22a2 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nuse t, t_mem, ht\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nhf : Cauchy f\nadhs : ClusterPt x f\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 s\n\u22a2 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nexact forall_mem_nonempty_iff_neBot.2 adhs _ (inter_mem_inf (mem_nhds_left x hs) t_mem)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\nh : CauchySeq u\n\u22a2 Tendsto (Prod.map u u) atTop (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nsimpa only [Tendsto, prod_map_map_eq', prod_atTop_atTop_eq] using h.right\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\nh : CauchySeq u\nV : Set (\u03b1 \u00d7 \u03b1)\nhV : V \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 k\u2080, \u2200 (i j : \u03b2), k\u2080 \u2264 i \u2192 k\u2080 \u2264 j \u2192 (u i, u j) \u2208 V\n[PROOFSTEP]\nhaveI := h.nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\nh : CauchySeq u\nV : Set (\u03b1 \u00d7 \u03b1)\nhV : V \u2208 \ud835\udce4 \u03b1\nthis : Nonempty \u03b2\n\u22a2 \u2203 k\u2080, \u2200 (i j : \u03b2), k\u2080 \u2264 i \u2192 k\u2080 \u2264 j \u2192 (u i, u j) \u2208 V\n[PROOFSTEP]\nhave := h.tendsto_uniformity\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\nh : CauchySeq u\nV : Set (\u03b1 \u00d7 \u03b1)\nhV : V \u2208 \ud835\udce4 \u03b1\nthis\u271d : Nonempty \u03b2\nthis : Tendsto (Prod.map u u) atTop (\ud835\udce4 \u03b1)\n\u22a2 \u2203 k\u2080, \u2200 (i j : \u03b2), k\u2080 \u2264 i \u2192 k\u2080 \u2264 j \u2192 (u i, u j) \u2208 V\n[PROOFSTEP]\nrw [\u2190 prod_atTop_atTop_eq] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\n\u03b2 : Type u_1\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\nh : CauchySeq u\nV : Set (\u03b1 \u00d7 \u03b1)\nhV : V \u2208 \ud835\udce4 \u03b1\nthis\u271d : Nonempty \u03b2\nthis : Tendsto (Prod.map u u) (atTop \u00d7\u02e2 atTop) (\ud835\udce4 \u03b1)\n\u22a2 \u2203 k\u2080, \u2200 (i j : \u03b2), k\u2080 \u2264 i \u2192 k\u2080 \u2264 j \u2192 (u i, u j) \u2208 V\n[PROOFSTEP]\nsimpa [MapsTo] using atTop_basis.prod_self.tendsto_left_iff.1 this V hV\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 Tendsto (fun p => (u p.fst, u p.snd)) (atTop \u00d7\u02e2 atTop) (\ud835\udce4 \u03b1) \u2194 Tendsto (Prod.map u u) atTop (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nsimp only [prod_atTop_atTop_eq, Prod.map_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : \u2115 \u2192 \u2115\nhf : Bijective f\nu : \u2115 \u2192 \u03b1\n\u22a2 CauchySeq (u \u2218 f) \u2194 CauchySeq u\n[PROOFSTEP]\nrefine' \u27e8fun H => _, fun H => H.comp_injective hf.injective\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : \u2115 \u2192 \u2115\nhf : Bijective f\nu : \u2115 \u2192 \u03b1\nH : CauchySeq (u \u2218 f)\n\u22a2 CauchySeq u\n[PROOFSTEP]\nlift f to \u2115 \u2243 \u2115 using hf\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nu : \u2115 \u2192 \u03b1\nf : \u2115 \u2243 \u2115\nH : CauchySeq (u \u2218 \u2191f)\n\u22a2 CauchySeq u\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), f.apply_symm_apply] using H.comp_injective f.symm.injective\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nf g : \u2115 \u2192 \u2115\nhf : Tendsto f atTop atTop\nhg : Tendsto g atTop atTop\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), ((u \u2218 f \u2218 \u03c6) n, (u \u2218 g \u2218 \u03c6) n) \u2208 V n\n[PROOFSTEP]\nrw [cauchySeq_iff_tendsto] at hu \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : Tendsto (Prod.map u u) atTop (\ud835\udce4 \u03b1)\nf g : \u2115 \u2192 \u2115\nhf : Tendsto f atTop atTop\nhg : Tendsto g atTop atTop\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), ((u \u2218 f \u2218 \u03c6) n, (u \u2218 g \u2218 \u03c6) n) \u2208 V n\n[PROOFSTEP]\nexact ((hu.comp <| hf.prod_atTop hg).comp tendsto_atTop_diagonal).subseq_mem hV\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nu : \u2115 \u2192 \u03b1\n\u22a2 CauchySeq u \u2194 \u2200 (V : Set (\u03b1 \u00d7 \u03b1)), V \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 N \u2192 (u k, u l) \u2208 V\n[PROOFSTEP]\nsimp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : UniformSpace \u03b1\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b3\ninst\u271d : SemilatticeSup \u03b4\nu : \u03b3 \u2192 \u03b1\nv : \u03b4 \u2192 \u03b2\nhu : CauchySeq u\nhv : CauchySeq v\n\u22a2 CauchySeq (Prod.map u v)\n[PROOFSTEP]\nsimpa only [CauchySeq, prod_map_map_eq', prod_atTop_atTop_eq] using hu.prod hv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V n\n[PROOFSTEP]\nhave : \u2200 n, \u2203 N, \u2200 k \u2265 N, \u2200 l \u2265 k, (u l, u k) \u2208 V n := fun n =>\n by\n rw [cauchySeq_iff] at hu \n rcases hu _ (hV n) with \u27e8N, H\u27e9\n exact \u27e8N, fun k hk l hl => H _ (le_trans hk hl) _ hk\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nn : \u2115\n\u22a2 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 k \u2192 (u l, u k) \u2208 V n\n[PROOFSTEP]\nrw [cauchySeq_iff] at hu \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : \u2200 (V : Set (\u03b1 \u00d7 \u03b1)), V \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 N \u2192 (u k, u l) \u2208 V\nn : \u2115\n\u22a2 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 k \u2192 (u l, u k) \u2208 V n\n[PROOFSTEP]\nrcases hu _ (hV n) with \u27e8N, H\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : \u2200 (V : Set (\u03b1 \u00d7 \u03b1)), V \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 N \u2192 (u k, u l) \u2208 V\nn N : \u2115\nH : \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 N \u2192 (u k, u l) \u2208 V n\n\u22a2 \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 k \u2192 (u l, u k) \u2208 V n\n[PROOFSTEP]\nexact \u27e8N, fun k hk l hl => H _ (le_trans hk hl) _ hk\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nthis : \u2200 (n : \u2115), \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 k \u2192 (u l, u k) \u2208 V n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V n\n[PROOFSTEP]\nobtain \u27e8\u03c6 : \u2115 \u2192 \u2115, \u03c6_extr : StrictMono \u03c6, h\u03c6 : \u2200 n, \u2200 l \u2265 \u03c6 n, (u l, u <| \u03c6 n) \u2208 V n\u27e9 :=\n extraction_forall_of_eventually' this\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\nhu : CauchySeq u\nthis : \u2200 (n : \u2115), \u2203 N, \u2200 (k : \u2115), k \u2265 N \u2192 \u2200 (l : \u2115), l \u2265 k \u2192 (u l, u k) \u2208 V n\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n l : \u2115), l \u2265 \u03c6 n \u2192 (u l, u (\u03c6 n)) \u2208 V n\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V n\n[PROOFSTEP]\nexact \u27e8\u03c6, \u03c6_extr, fun n => h\u03c6 _ _ (\u03c6_extr <| lt_add_one n).le\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)\n[PROOFSTEP]\nrcases mem_atTop_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity <| hV 0))) with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)\n[PROOFSTEP]\nrcases(hu.comp (tendsto_add_atTop_nat n)).cauchySeq.subseq_mem fun n => hV (n + 1) with \u27e8\u03c6, \u03c6_mono, h\u03c6V\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6V : \u2200 (n_1 : \u2115), ((u \u2218 fun a => a + n) (\u03c6 (n_1 + 1)), (u \u2218 fun a => a + n) (\u03c6 n_1)) \u2208 V (n_1 + 1)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)\n[PROOFSTEP]\nexact \u27e8fun k => \u03c6 k + n, \u03c6_mono.add_const _, hn _ le_add_self, h\u03c6V\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 CauchySeq u \u2194 \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nrw [cauchySeq_iff_tendsto, \u2190 prod_atTop_atTop_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 Tendsto (Prod.map u u) (atTop \u00d7\u02e2 atTop) (\ud835\udce4 \u03b1) \u2194\n \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nrefine' (atTop_basis.prod_self.tendsto_iff h).trans _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 (\u2200 (ib : \u03b3), p ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 Ici ia \u00d7\u02e2 Ici ia \u2192 Prod.map u u x \u2208 s ib) \u2194\n \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nsimp only [exists_prop, true_and_iff, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq, mem_setOf_eq, mem_Ici,\n and_imp, Prod.map, ge_iff_le, @forall_swap (_ \u2264 _) \u03b2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 CauchySeq u \u2194 \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\n[PROOFSTEP]\nrefine' H.cauchySeq_iff.trans \u27e8fun h i hi => _, fun h i hi => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\ni : \u03b3\nhi : p i\n\u22a2 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\n[PROOFSTEP]\nexact (h i hi).imp fun N hN n hn => hN n hn N le_rfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\ni : \u03b3\nhi : p i\n\u22a2 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nrcases comp_symm_of_uniformity (H.mem_of_mem hi) with \u27e8t, ht, ht', hts\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\ni : \u03b3\nhi : p i\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nht' : \u2200 {a b : \u03b1}, (a, b) \u2208 t \u2192 (b, a) \u2208 t\nhts : t \u25cb t \u2286 s i\n\u22a2 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nrcases H.mem_iff.1 ht with \u27e8j, hj, hjt\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\ni : \u03b3\nhi : p i\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nht' : \u2200 {a b : \u03b1}, (a, b) \u2208 t \u2192 (b, a) \u2208 t\nhts : t \u25cb t \u2286 s i\nj : \u03b3\nhj : p j\nhjt : s j \u2286 t\n\u22a2 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i\n[PROOFSTEP]\nrefine' (h j hj).imp fun N hN m hm n hn => hts \u27e8u N, hjt _, ht' <| hjt _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\ni : \u03b3\nhi : p i\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nht' : \u2200 {a b : \u03b1}, (a, b) \u2208 t \u2192 (b, a) \u2208 t\nhts : t \u25cb t \u2286 s i\nj : \u03b3\nhj : p j\nhjt : s j \u2286 t\nN : \u03b2\nhN : \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s j\nm : \u03b2\nhm : N \u2264 m\nn : \u03b2\nhn : N \u2264 n\n\u22a2 ((u m, u n).fst, u N) \u2208 s j\ncase refine'_2.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nH : HasBasis (\ud835\udce4 \u03b1) p s\nh : \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s i\ni : \u03b3\nhi : p i\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nht' : \u2200 {a b : \u03b1}, (a, b) \u2208 t \u2192 (b, a) \u2208 t\nhts : t \u25cb t \u2286 s i\nj : \u03b3\nhj : p j\nhjt : s j \u2286 t\nN : \u03b2\nhN : \u2200 (n : \u03b2), n \u2265 N \u2192 (u n, u N) \u2208 s j\nm : \u03b2\nhm : N \u2264 m\nn : \u03b2\nhn : N \u2264 n\n\u22a2 ((u m, u n).snd, u N) \u2208 s j\n[PROOFSTEP]\nexacts [hN m hm, hN n hn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\n\u22a2 Tendsto (Prod.map f f) atTop (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 s \u2208 map (Prod.map f f) atTop\n[PROOFSTEP]\nrw [mem_map, mem_atTop_sets]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 a, \u2200 (b : \u03b2 \u00d7 \u03b2), b \u2265 a \u2192 b \u2208 Prod.map f f \u207b\u00b9' s\n[PROOFSTEP]\ncases' hU s hs with N hN\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nN : \u03b2\nhN : U N \u2286 s\n\u22a2 \u2203 a, \u2200 (b : \u03b2 \u00d7 \u03b2), b \u2265 a \u2192 b \u2208 Prod.map f f \u207b\u00b9' s\n[PROOFSTEP]\nrefine' \u27e8(N, N), fun mn hmn => _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nN : \u03b2\nhN : U N \u2286 s\nmn : \u03b2 \u00d7 \u03b2\nhmn : mn \u2265 (N, N)\n\u22a2 mn \u2208 Prod.map f f \u207b\u00b9' s\n[PROOFSTEP]\ncases' mn with m n\n[GOAL]\ncase intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : Nonempty \u03b2\nU : \u03b2 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nf : \u03b2 \u2192 \u03b1\nhf : \u2200 \u2983N m n : \u03b2\u2984, N \u2264 m \u2192 N \u2264 n \u2192 (f m, f n) \u2208 U N\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nN : \u03b2\nhN : U N \u2286 s\nm n : \u03b2\nhmn : (m, n) \u2265 (N, N)\n\u22a2 (m, n) \u2208 Prod.map f f \u207b\u00b9' s\n[PROOFSTEP]\nexact hN (hf hmn.1 hmn.2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 IsComplete s \u2194 \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrefine' \u27e8fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l\n[PROOFSTEP]\nhaveI := hl.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l\n[PROOFSTEP]\nrcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with \u27e8x, hxs, hxl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\nx : \u03b1\nhxs : x \u2208 s\nhxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l\n[PROOFSTEP]\nexact \u27e8x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x) \u2194\n \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 s \u2208 l \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nsimp only [le_principal_iff, Ultrafilter.mem_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns t : Set \u03b1\nhs : IsComplete s\nht : IsComplete t\n\u22a2 IsComplete (s \u222a t)\n[PROOFSTEP]\nsimp only [isComplete_iff_ultrafilter', Ultrafilter.union_mem_iff, or_imp] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns t : Set \u03b1\nhs : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 s \u2208 l \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nht : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 t \u2208 l \u2192 \u2203 x, x \u2208 t \u2227 \u2191l \u2264 \ud835\udcdd x\n\u22a2 \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 (s \u2208 l \u2192 \u2203 x, x \u2208 s \u222a t \u2227 \u2191l \u2264 \ud835\udcdd x) \u2227 (t \u2208 l \u2192 \u2203 x, x \u2208 s \u222a t \u2227 \u2191l \u2264 \ud835\udcdd x)\n[PROOFSTEP]\nexact fun l hl =>\n \u27e8fun hsl => (hs l hl hsl).imp fun x hx => \u27e8Or.inl hx.1, hx.2\u27e9, fun htl =>\n (ht l hl htl).imp fun x hx => \u27e8Or.inr hx.1, hx.2\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\n\u22a2 IsComplete (\u22c3 (i : \u03b9), s i)\n[PROOFSTEP]\nset S := \u22c3 i, s i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\n\u22a2 IsComplete S\n[PROOFSTEP]\nintro l hl hls\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf S\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [le_principal_iff] at hls \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\ncases' cauchy_iff.1 hl with hl_ne hl'\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nobtain \u27e8t, htS, htl, htU\u27e9 : \u2203 t, t \u2286 S \u2227 t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 U :=\n by\n rcases hl' U hU with \u27e8t, htl, htU\u27e9\n exact\n \u27e8t \u2229 S, inter_subset_right _ _, inter_mem htl hls,\n (Set.prod_mono (inter_subset_left _ _) (inter_subset_left _ _)).trans htU\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\n\u22a2 \u2203 t, t \u2286 S \u2227 t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 U\n[PROOFSTEP]\nrcases hl' U hU with \u27e8t, htl, htU\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\n\u22a2 \u2203 t, t \u2286 S \u2227 t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 U\n[PROOFSTEP]\nexact\n \u27e8t \u2229 S, inter_subset_right _ _, inter_mem htl hls,\n (Set.prod_mono (inter_subset_left _ _) (inter_subset_left _ _)).trans htU\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 : \u2203 i, t \u2286 s i := by\n rcases Filter.nonempty_of_mem htl with \u27e8x, hx\u27e9\n rcases mem_iUnion.1 (htS hx) with \u27e8i, hi\u27e9\n refine' \u27e8i, fun y hy => _\u27e9\n rcases mem_iUnion.1 (htS hy) with \u27e8j, hj\u27e9\n rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\n\u22a2 \u2203 i, t \u2286 s i\n[PROOFSTEP]\nrcases Filter.nonempty_of_mem htl with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 i, t \u2286 s i\n[PROOFSTEP]\nrcases mem_iUnion.1 (htS hx) with \u27e8i, hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 s i\n\u22a2 \u2203 i, t \u2286 s i\n[PROOFSTEP]\nrefine' \u27e8i, fun y hy => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 s i\ny : \u03b1\nhy : y \u2208 t\n\u22a2 y \u2208 s i\n[PROOFSTEP]\nrcases mem_iUnion.1 (htS hy) with \u27e8j, hj\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 s i\ny : \u03b1\nhy : y \u2208 t\nj : \u03b9\nhj : y \u2208 s j\n\u22a2 y \u2208 s i\n[PROOFSTEP]\nrwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\ni : \u03b9\nhi : t \u2286 s i\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrcases hs i l hl (le_principal_iff.2 <| mem_of_superset htl hi) with \u27e8x, hxs, hlx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), IsComplete (s i)\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhd : \u2200 (i j : \u03b9) (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s j \u2192 (x, y) \u2208 U \u2192 i = j\nS : Set \u03b1 := \u22c3 (i : \u03b9), s i\nl : Filter \u03b1\nhl : Cauchy l\nhls : S \u2208 l\nhl_ne : NeBot l\nhl' : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 l \u2227 t \u00d7\u02e2 t \u2286 s\nt : Set \u03b1\nhtS : t \u2286 S\nhtl : t \u2208 l\nhtU : t \u00d7\u02e2 t \u2286 U\ni : \u03b9\nhi : t \u2286 s i\nx : \u03b1\nhxs : x \u2208 s i\nhlx : l \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 S \u2227 l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8x, mem_iUnion.2 \u27e8i, hxs\u27e9, hlx\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nx\u271d : f \u2264 \ud835\udcdf univ\n\u22a2 \u2203 x, x \u2208 univ \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrcases CompleteSpace.complete hf with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nx\u271d : f \u2264 \ud835\udcdf univ\nx : \u03b1\nhx : f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 univ \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8x, mem_univ x, hx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : CompleteSpace \u03b2\nf\u271d : Filter (\u03b1 \u00d7 \u03b2)\nhf : Cauchy f\u271d\nx1 : \u03b1\nhx1 : map (fun p => p.fst) f\u271d \u2264 \ud835\udcdd x1\nx2 : \u03b2\nhx2 : map (fun p => p.snd) f\u271d \u2264 \ud835\udcdd x2\n\u22a2 f\u271d \u2264 \ud835\udcdd (x1, x2)\n[PROOFSTEP]\nrw [nhds_prod_eq, Filter.prod_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : CompleteSpace \u03b2\nf\u271d : Filter (\u03b1 \u00d7 \u03b2)\nhf : Cauchy f\u271d\nx1 : \u03b1\nhx1 : map (fun p => p.fst) f\u271d \u2264 \ud835\udcdd x1\nx2 : \u03b2\nhx2 : map (fun p => p.snd) f\u271d \u2264 \ud835\udcdd x2\n\u22a2 f\u271d \u2264 Filter.lift (\ud835\udcdd x1) fun s => Filter.lift' (\ud835\udcdd x2) fun t => s \u00d7\u02e2 t\n[PROOFSTEP]\nexact Filter.le_lift.2 fun s hs => Filter.le_lift'.2 fun t ht => inter_mem (hx1 hs) (hx2 ht)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\n\u22a2 CompleteSpace \u03b1 \u2194 \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2203 x, \u2191l \u2264 \ud835\udcdd x\n[PROOFSTEP]\nsimp [completeSpace_iff_isComplete_univ, isComplete_iff_ultrafilter]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : SemilatticeSup \u03b2\nK : Set \u03b1\nh\u2081 : IsComplete K\nu : \u03b2 \u2192 \u03b1\nh\u2082 : \u2200 (n : \u03b2), u n \u2208 K\nh\u2083 : CauchySeq u\n\u22a2 u '' univ \u2286 K\n[PROOFSTEP]\nrwa [image_univ, range_subset_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nrcases comp_symm_of_uniformity hU with \u27e8r, hr, rs, rU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nrcases hs r hr with \u27e8k, fk, ks\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nlet u := k \u2229 {y | \u2203 x \u2208 s, (x, y) \u2208 r}\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nchoose f hfs hfr using fun x : u => x.coe_prop.2\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nrefine' \u27e8range f, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\n\u22a2 range f \u2286 s\n[PROOFSTEP]\nexact range_subset_iff.2 hfs\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\n\u22a2 Set.Finite (range f)\n[PROOFSTEP]\nhaveI : Fintype u := (fk.inter_of_left _).fintype\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\nthis : Fintype \u2191u\n\u22a2 Set.Finite (range f)\n[PROOFSTEP]\nexact finite_range f\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\n\u22a2 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 range f), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nintro x xs\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\nx : \u03b1\nxs : x \u2208 s\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 range f), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nobtain \u27e8y, hy, xy\u27e9 := mem_iUnion\u2082.1 (ks xs)\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nhy : y \u2208 k\nxy : x \u2208 {x | (x, y) \u2208 r}\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 range f), {x | (x, y) \u2208 U}\n[PROOFSTEP]\nrw [biUnion_range, mem_iUnion]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nhy : y \u2208 k\nxy : x \u2208 {x | (x, y) \u2208 r}\n\u22a2 \u2203 i, x \u2208 {x | (x, f i) \u2208 U}\n[PROOFSTEP]\nset z : \u21a5u := \u27e8y, hy, \u27e8x, xs, xy\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nhs : TotallyBounded s\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nr : Set (\u03b1 \u00d7 \u03b1)\nhr : r \u2208 \ud835\udce4 \u03b1\nrs : \u2200 {a b : \u03b1}, (a, b) \u2208 r \u2192 (b, a) \u2208 r\nrU : r \u25cb r \u2286 U\nk : Set \u03b1\nfk : Set.Finite k\nks : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 k), {x | (x, y) \u2208 r}\nu : Set \u03b1 := k \u2229 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}\nf : \u2191u \u2192 \u03b1\nhfs : \u2200 (x : \u2191u), f x \u2208 s\nhfr : \u2200 (x : \u2191u), (f x, \u2191x) \u2208 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nhy : y \u2208 k\nxy : x \u2208 {x | (x, y) \u2208 r}\nz : \u2191u := { val := y, property := (_ : y \u2208 k \u2227 y \u2208 {y | \u2203 x, x \u2208 s \u2227 (x, y) \u2208 r}) }\n\u22a2 \u2203 i, x \u2208 {x | (x, f i) \u2208 U}\n[PROOFSTEP]\nexact \u27e8z, rU <| mem_compRel.2 \u27e8y, xy, rs (hfr z)\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nh : \u2200 (V : Set (\u03b1 \u00d7 \u03b1)), V \u2208 \ud835\udce4 \u03b1 \u2192 SymmetricRel V \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), ball y V\nV : Set (\u03b1 \u00d7 \u03b1)\nhV : V \u2208 \ud835\udce4 \u03b1 \u2227 SymmetricRel V\n\u22a2 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 id V}\n[PROOFSTEP]\nsimpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nthis : {p | (f p.fst, f p.snd) \u2208 t} \u2208 \ud835\udce4 \u03b1\nc : Set \u03b1\nhfc : Set.Finite c\nhct : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 c), {x | (x, y) \u2208 {p | (f p.fst, f p.snd) \u2208 t}}\n\u22a2 f '' s \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 f '' c), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nsimp [image_subset_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nthis : {p | (f p.fst, f p.snd) \u2208 t} \u2208 \ud835\udce4 \u03b1\nc : Set \u03b1\nhfc : Set.Finite c\nhct : s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 c), {x | (x, y) \u2208 {p | (f p.fst, f p.snd) \u2208 t}}\n\u22a2 s \u2286 \u22c3 (i : \u03b1) (_ : i \u2208 c), {a | (f a, f i) \u2208 t}\n[PROOFSTEP]\nsimp [subset_def] at hct \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nthis : {p | (f p.fst, f p.snd) \u2208 t} \u2208 \ud835\udce4 \u03b1\nc : Set \u03b1\nhfc : Set.Finite c\nhct : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 i, i \u2208 c \u2227 (f x, f i) \u2208 t\n\u22a2 s \u2286 \u22c3 (i : \u03b1) (_ : i \u2208 c), {a | (f a, f i) \u2208 t}\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nthis : {p | (f p.fst, f p.snd) \u2208 t} \u2208 \ud835\udce4 \u03b1\nc : Set \u03b1\nhfc : Set.Finite c\nhct : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 i, i \u2208 c \u2227 (f x, f i) \u2208 t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (i : \u03b1) (_ : i \u2208 c), {a | (f a, f i) \u2208 t}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : TotallyBounded s\nhf : UniformContinuous f\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nthis : {p | (f p.fst, f p.snd) \u2208 t} \u2208 \ud835\udce4 \u03b1\nc : Set \u03b1\nhfc : Set.Finite c\nhct : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 i, i \u2208 c \u2227 (f x, f i) \u2208 t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i, i \u2208 c \u2227 (f x, f i) \u2208 t\n[PROOFSTEP]\nexact hct x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 TotallyBounded s \u2194 \u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 \ud835\udcdf s \u2192 \u2203 c, c \u2264 f \u2227 Cauchy c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 TotallyBounded s \u2192 \u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 \ud835\udcdf s \u2192 \u2203 c, c \u2264 f \u2227 Cauchy c\n[PROOFSTEP]\nexact fun H f hf hfs =>\n \u27e8Ultrafilter.of f, Ultrafilter.of_le f,\n (Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 \ud835\udcdf s \u2192 \u2203 c, c \u2264 f \u2227 Cauchy c) \u2192 TotallyBounded s\n[PROOFSTEP]\nintro H d hd\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 \ud835\udcdf s \u2192 \u2203 c, c \u2264 f \u2227 Cauchy c\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\n[PROOFSTEP]\ncontrapose! H with hd_cover\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\n\u22a2 \u2203 f, NeBot f \u2227 f \u2264 \ud835\udcdf s \u2227 \u2200 (c : Filter \u03b1), c \u2264 f \u2192 \u00acCauchy c\n[PROOFSTEP]\nset f := \u2a05 t : Finset \u03b1, \ud835\udcdf (s \\ \u22c3 y \u2208 t, {x | (x, y) \u2208 d})\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\n\u22a2 \u2203 f, NeBot f \u2227 f \u2264 \ud835\udcdf s \u2227 \u2200 (c : Filter \u03b1), c \u2264 f \u2192 \u00acCauchy c\n[PROOFSTEP]\nhave : Filter.NeBot f := by\n refine' iInf_neBot_of_directed' (directed_of_sup _) _\n \u00b7 intro t\u2081 t\u2082 h\n exact principal_mono.2 (diff_subset_diff_right <| biUnion_subset_biUnion_left h)\n \u00b7 intro t\n simpa [nonempty_diff] using hd_cover t t.finite_toSet\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\n\u22a2 NeBot f\n[PROOFSTEP]\nrefine' iInf_neBot_of_directed' (directed_of_sup _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\n\u22a2 \u2200 \u2983i j : Finset \u03b1\u2984,\n i \u2264 j \u2192 \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 i), {x | (x, y) \u2208 d}) \u2265 \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 j), {x | (x, y) \u2208 d})\n[PROOFSTEP]\nintro t\u2081 t\u2082 h\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nt\u2081 t\u2082 : Finset \u03b1\nh : t\u2081 \u2264 t\u2082\n\u22a2 \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t\u2081), {x | (x, y) \u2208 d}) \u2265 \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t\u2082), {x | (x, y) \u2208 d})\n[PROOFSTEP]\nexact principal_mono.2 (diff_subset_diff_right <| biUnion_subset_biUnion_left h)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\n\u22a2 \u2200 (i : Finset \u03b1), NeBot (\ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 i), {x | (x, y) \u2208 d}))\n[PROOFSTEP]\nintro t\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nt : Finset \u03b1\n\u22a2 NeBot (\ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}))\n[PROOFSTEP]\nsimpa [nonempty_diff] using hd_cover t t.finite_toSet\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis : NeBot f\n\u22a2 \u2203 f, NeBot f \u2227 f \u2264 \ud835\udcdf s \u2227 \u2200 (c : Filter \u03b1), c \u2264 f \u2192 \u00acCauchy c\n[PROOFSTEP]\nhave : f \u2264 \ud835\udcdf s := iInf_le_of_le \u2205 (by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis : NeBot f\n\u22a2 \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 \u2205), {x | (x, y) \u2208 d}) \u2264 \ud835\udcdf s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d : NeBot f\nthis : f \u2264 \ud835\udcdf s\n\u22a2 \u2203 f, NeBot f \u2227 f \u2264 \ud835\udcdf s \u2227 \u2200 (c : Filter \u03b1), c \u2264 f \u2192 \u00acCauchy c\n[PROOFSTEP]\nrefine' \u27e8f, \u2039_\u203a, \u2039_\u203a, fun c hcf hc => _\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d : NeBot f\nthis : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\n\u22a2 False\n[PROOFSTEP]\nrcases mem_prod_same_iff.1 (hc.2 hd) with \u27e8m, hm, hmd\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d : NeBot f\nthis : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\n\u22a2 False\n[PROOFSTEP]\nrcases hc.1.nonempty_of_mem hm with \u27e8y, hym\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d : NeBot f\nthis : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\n\u22a2 False\n[PROOFSTEP]\nset ys := \u22c3 y' \u2208 ({ y } : Finset \u03b1), {x | (x, y') \u2208 d}\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d : NeBot f\nthis : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\nys : Set \u03b1 := \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\n\u22a2 False\n[PROOFSTEP]\nhave : c \u2264 \ud835\udcdf (s \\ ys) := hcf.trans (iInf_le_of_le { y } le_rfl)\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d\u00b9 : NeBot f\nthis\u271d : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\nys : Set \u03b1 := \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\nthis : c \u2264 \ud835\udcdf (s \\ ys)\n\u22a2 False\n[PROOFSTEP]\nrefine' hc.1.ne (empty_mem_iff_bot.mp _)\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d\u00b9 : NeBot f\nthis\u271d : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\nys : Set \u03b1 := \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\nthis : c \u2264 \ud835\udcdf (s \\ ys)\n\u22a2 \u2205 \u2208 c\n[PROOFSTEP]\nfilter_upwards [le_principal_iff.1 this, hm]\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d\u00b9 : NeBot f\nthis\u271d : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\nys : Set \u03b1 := \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\nthis : c \u2264 \ud835\udcdf (s \\ ys)\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \\ \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d} \u2192 a \u2208 m \u2192 a \u2208 \u2205\n[PROOFSTEP]\nrefine' fun x hx hxm => hx.2 _\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nd : Set (\u03b1 \u00d7 \u03b1)\nhd : d \u2208 \ud835\udce4 \u03b1\nhd_cover : \u2200 (t : Set \u03b1), Set.Finite t \u2192 \u00acs \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d}\nf : Filter \u03b1 := \u2a05 (t : Finset \u03b1), \ud835\udcdf (s \\ \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 d})\nthis\u271d\u00b9 : NeBot f\nthis\u271d : f \u2264 \ud835\udcdf s\nc : Filter \u03b1\nhcf : c \u2264 f\nhc : Cauchy c\nm : Set \u03b1\nhm : m \u2208 c\nhmd : m \u00d7\u02e2 m \u2286 d\ny : \u03b1\nhym : y \u2208 m\nys : Set \u03b1 := \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\nthis : c \u2264 \ud835\udcdf (s \\ ys)\nx : \u03b1\nhx : x \u2208 s \\ \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\nhxm : x \u2208 m\n\u22a2 x \u2208 \u22c3 (y' : \u03b1) (_ : y' \u2208 {y}), {x | (x, y') \u2208 d}\n[PROOFSTEP]\nsimpa using hmd (mk_mem_prod hxm hym)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 TotallyBounded s \u2194 \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 Cauchy \u2191f\n[PROOFSTEP]\nrefine' \u27e8fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 Cauchy \u2191f\n\u22a2 \u2200 (f : Filter \u03b1), NeBot f \u2192 f \u2264 \ud835\udcdf s \u2192 \u2203 c, c \u2264 f \u2227 Cauchy c\n[PROOFSTEP]\nintro f hf hfs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (f : Ultrafilter \u03b1), \u2191f \u2264 \ud835\udcdf s \u2192 Cauchy \u2191f\nf : Filter \u03b1\nhf : NeBot f\nhfs : f \u2264 \ud835\udcdf s\n\u22a2 \u2203 c, c \u2264 f \u2227 Cauchy c\n[PROOFSTEP]\nexact \u27e8Ultrafilter.of f, Ultrafilter.of_le f, H _ ((Ultrafilter.of_le f).trans hfs)\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nf\u271d : Filter \u03b1\nhf : Cauchy f\u271d\n\u22a2 \u2203 x, f\u271d \u2264 \ud835\udcdd x\n[PROOFSTEP]\nsimpa using (isCompact_iff_totallyBounded_isComplete.1 isCompact_univ).2 _ hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\n\u22a2 TotallyBounded (range s)\n[PROOFSTEP]\nrefine' totallyBounded_iff_subset.2 fun a ha => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t, t \u2286 range s \u2227 Set.Finite t \u2227 range s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 a}\n[PROOFSTEP]\ncases' cauchySeq_iff.1 hs a ha with n hn\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\n\u22a2 \u2203 t, t \u2286 range s \u2227 Set.Finite t \u2227 range s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 a}\n[PROOFSTEP]\nrefine' \u27e8s '' {k | k \u2264 n}, image_subset_range _ _, (finite_le_nat _).image _, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\n\u22a2 range s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 s '' {k | k \u2264 n}), {x | (x, y) \u2208 a}\n[PROOFSTEP]\nrw [range_subset_iff, biUnion_image]\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\n\u22a2 \u2200 (y : \u2115), s y \u2208 \u22c3 (y : \u2115) (_ : y \u2208 {k | k \u2264 n}), {x | (x, s y) \u2208 a}\n[PROOFSTEP]\nintro m\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\nm : \u2115\n\u22a2 s m \u2208 \u22c3 (y : \u2115) (_ : y \u2208 {k | k \u2264 n}), {x | (x, s y) \u2208 a}\n[PROOFSTEP]\nrw [mem_iUnion\u2082]\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\nm : \u2115\n\u22a2 \u2203 i j, s m \u2208 {x | (x, s i) \u2208 a}\n[PROOFSTEP]\ncases' le_total m n with hm hm\n[GOAL]\ncase intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\nm : \u2115\nhm : m \u2264 n\n\u22a2 \u2203 i j, s m \u2208 {x | (x, s i) \u2208 a}\ncase intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\ns : \u2115 \u2192 \u03b1\nhs : CauchySeq s\na : Set (\u03b1 \u00d7 \u03b1)\nha : a \u2208 \ud835\udce4 \u03b1\nn : \u2115\nhn : \u2200 (k : \u2115), k \u2265 n \u2192 \u2200 (l : \u2115), l \u2265 n \u2192 (s k, s l) \u2208 a\nm : \u2115\nhm : n \u2264 m\n\u22a2 \u2203 i j, s m \u2208 {x | (x, s i) \u2208 a}\n[PROOFSTEP]\nexacts [\u27e8m, hm, refl_mem_uniformity ha\u27e9, \u27e8n, le_refl n, hn m hm n le_rfl\u27e9]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p \u2208 U N\n[PROOFSTEP]\nrefine' (setSeqAux hf U_mem N).2.2 \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p.fst \u2208 \u2191(setSeqAux hf U_mem N)\n[PROOFSTEP]\napply setSeq_sub_aux\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p.snd \u2208 \u2191(setSeqAux hf U_mem N)\n[PROOFSTEP]\napply setSeq_sub_aux\n[GOAL]\ncase refine'_1.a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p.fst \u2208 setSeq hf U_mem N\ncase refine'_2.a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p.snd \u2208 setSeq hf U_mem N\n[PROOFSTEP]\nexact setSeq_mono hf U_mem hm hp.1\n[GOAL]\ncase refine'_2.a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\np : \u03b1 \u00d7 \u03b1\nhp : p \u2208 setSeq hf U_mem m \u00d7\u02e2 setSeq hf U_mem n\n\u22a2 p.snd \u2208 setSeq hf U_mem N\n[PROOFSTEP]\nexact setSeq_mono hf U_mem hn hp.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (a, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nrcases U_le s hs with \u27e8m, hm\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nm : \u2115\nhm : U m \u2286 s\n\u22a2 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (a, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nrcases tendsto_atTop'.1 ha _ (mem_nhds_left a (U_mem m)) with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nm : \u2115\nhm : U m \u2286 s\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 seq hf U_mem b \u2208 {y | (a, y) \u2208 U m}\n\u22a2 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (a, y) \u2208 s \u2227 y \u2208 t\n[PROOFSTEP]\nrefine' \u27e8setSeq hf U_mem (max m n), setSeq_mem hf U_mem _, _, seq hf U_mem (max m n), _, seq_mem hf U_mem _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nm : \u2115\nhm : U m \u2286 s\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 seq hf U_mem b \u2208 {y | (a, y) \u2208 U m}\n\u22a2 setSeq hf U_mem (max m n) \u00d7\u02e2 setSeq hf U_mem (max m n) \u2286 s\n[PROOFSTEP]\nhave := le_max_left m n\n[GOAL]\ncase intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nm : \u2115\nhm : U m \u2286 s\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 seq hf U_mem b \u2208 {y | (a, y) \u2208 U m}\nthis : m \u2264 max m n\n\u22a2 setSeq hf U_mem (max m n) \u00d7\u02e2 setSeq hf U_mem (max m n) \u2286 s\n[PROOFSTEP]\nexact Set.Subset.trans (setSeq_prod_subset hf U_mem this this) hm\n[GOAL]\ncase intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nU_le : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 n, U n \u2286 s\na : \u03b1\nha : Tendsto (seq hf U_mem) atTop (\ud835\udcdd a)\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nm : \u2115\nhm : U m \u2286 s\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 seq hf U_mem b \u2208 {y | (a, y) \u2208 U m}\n\u22a2 (a, seq hf U_mem (max m n)) \u2208 s\n[PROOFSTEP]\nexact hm (hn _ <| le_max_right m n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n\u22a2 CompleteSpace \u03b1\n[PROOFSTEP]\nobtain \u27e8U', -, hU'\u27e9 := (\ud835\udce4 \u03b1).exists_antitone_seq\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\nU' : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU' : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 i, U' i \u2286 s\n\u22a2 CompleteSpace \u03b1\n[PROOFSTEP]\nhave Hmem : \u2200 n, U n \u2229 U' n \u2208 \ud835\udce4 \u03b1 := fun n => inter_mem (U_mem n) (hU'.2 \u27e8n, Subset.refl _\u27e9)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\nU' : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU' : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 i, U' i \u2286 s\nHmem : \u2200 (n : \u2115), U n \u2229 U' n \u2208 \ud835\udce4 \u03b1\n\u22a2 CompleteSpace \u03b1\n[PROOFSTEP]\nrefine \u27e8fun hf => (HU (seq hf Hmem) fun N m n hm hn => ?_).imp <| le_nhds_of_seq_tendsto_nhds _ _ fun s hs => ?_\u27e9\n[GOAL]\ncase intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\nU' : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU' : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 i, U' i \u2286 s\nHmem : \u2200 (n : \u2115), U n \u2229 U' n \u2208 \ud835\udce4 \u03b1\nf\u271d : Filter \u03b1\nhf : Cauchy f\u271d\nN m n : \u2115\nhm : N \u2264 m\nhn : N \u2264 n\n\u22a2 (SequentiallyComplete.seq hf Hmem m, SequentiallyComplete.seq hf Hmem n) \u2208 U N\n[PROOFSTEP]\nexact inter_subset_left _ _ (seq_pair_mem hf Hmem hm hn)\n[GOAL]\ncase intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\nU' : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU' : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 i, U' i \u2286 s\nHmem : \u2200 (n : \u2115), U n \u2229 U' n \u2208 \ud835\udce4 \u03b1\nf\u271d : Filter \u03b1\nhf : Cauchy f\u271d\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 n, U n \u2229 U' n \u2286 s\n[PROOFSTEP]\nrcases hU'.1 hs with \u27e8N, hN\u27e9\n[GOAL]\ncase intro.intro.refine_2.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\nU : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nU_mem : \u2200 (n : \u2115), U n \u2208 \ud835\udce4 \u03b1\nHU : \u2200 (u : \u2115 \u2192 \u03b1), (\u2200 (N m n : \u2115), N \u2264 m \u2192 N \u2264 n \u2192 (u m, u n) \u2208 U N) \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\nU' : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhU' : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 i, U' i \u2286 s\nHmem : \u2200 (n : \u2115), U n \u2229 U' n \u2208 \ud835\udce4 \u03b1\nf\u271d : Filter \u03b1\nhf : Cauchy f\u271d\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nN : \u2115\nhN : U' N \u2286 s\n\u22a2 \u2203 n, U n \u2229 U' n \u2286 s\n[PROOFSTEP]\nexact \u27e8N, Subset.trans (inter_subset_right _ _) hN\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\na : \u03b1\n\u22a2 IsCountablyGenerated (\ud835\udcdd a)\n[PROOFSTEP]\nrw [nhds_eq_comap_uniformity]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udce4 \u03b1)\na : \u03b1\n\u22a2 IsCountablyGenerated (Filter.comap (Prod.mk a) (\ud835\udce4 \u03b1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrcases exists_countable_dense \u03b1 with \u27e8s, hsc, hsd\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nobtain\n \u27e8t : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1), hto : \u2200 i : \u2115, t i \u2208 (\ud835\udce4 \u03b1).sets \u2227 IsOpen (t i) \u2227 SymmetricRel (t i), h_basis :\n (\ud835\udce4 \u03b1).HasAntitoneBasis t\u27e9 :=\n (@uniformity_hasBasis_open_symmetric \u03b1 _).exists_antitone_subbasis\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhto : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets \u2227 IsOpen (t i) \u2227 SymmetricRel (t i)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nchoose ht_mem hto hts using hto\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\n\u22a2 SecondCountableTopology \u03b1\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u22c3 x \u2208 s, range fun k => ball x (t k), hsc.biUnion fun x _ => countable_range _, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\n\u22a2 toTopologicalSpace = generateFrom (\u22c3 (x : \u03b1) (_ : x \u2208 s), range fun k => ball x (t k))\n[PROOFSTEP]\nrefine' (isTopologicalBasis_of_open_of_nhds _ _).eq_generateFrom\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\n\u22a2 \u2200 (u : Set \u03b1), (u \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 s), range fun k => ball x (t k)) \u2192 IsOpen u\n[PROOFSTEP]\nsimp only [mem_iUnion\u2082, mem_range]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\n\u22a2 \u2200 (u : Set \u03b1), (\u2203 i h y, ball i (t y) = u) \u2192 IsOpen u\n[PROOFSTEP]\nrintro _ \u27e8x, _, k, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nw\u271d : x \u2208 s\nk : \u2115\n\u22a2 IsOpen (ball x (t k))\n[PROOFSTEP]\nexact isOpen_ball x (hto k)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1),\n a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, (v \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 s), range fun k => ball x (t k)) \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro x V hxV hVo\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\n\u22a2 \u2203 v, (v \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 s), range fun k => ball x (t k)) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nsimp only [mem_iUnion\u2082, mem_range, exists_prop]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\n\u22a2 \u2203 v, (\u2203 i, i \u2208 s \u2227 \u2203 y, ball i (t y) = v) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nrcases UniformSpace.mem_nhds_iff.1 (IsOpen.mem_nhds hVo hxV) with \u27e8U, hU, hUV\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhUV : ball x U \u2286 V\n\u22a2 \u2203 v, (\u2203 i, i \u2208 s \u2227 \u2203 y, ball i (t y) = v) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nrcases comp_symm_of_uniformity hU with \u27e8U', hU', _, hUU'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhUV : ball x U \u2286 V\nU' : Set (\u03b1 \u00d7 \u03b1)\nhU' : U' \u2208 \ud835\udce4 \u03b1\nleft\u271d : \u2200 {a b : \u03b1}, (a, b) \u2208 U' \u2192 (b, a) \u2208 U'\nhUU' : U' \u25cb U' \u2286 U\n\u22a2 \u2203 v, (\u2203 i, i \u2208 s \u2227 \u2203 y, ball i (t y) = v) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nrcases h_basis.toHasBasis.mem_iff.1 hU' with \u27e8k, -, hk\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhUV : ball x U \u2286 V\nU' : Set (\u03b1 \u00d7 \u03b1)\nhU' : U' \u2208 \ud835\udce4 \u03b1\nleft\u271d : \u2200 {a b : \u03b1}, (a, b) \u2208 U' \u2192 (b, a) \u2208 U'\nhUU' : U' \u25cb U' \u2286 U\nk : \u2115\nhk : t k \u2286 U'\n\u22a2 \u2203 v, (\u2203 i, i \u2208 s \u2227 \u2203 y, ball i (t y) = v) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nrcases hsd.inter_open_nonempty (ball x <| t k) (isOpen_ball x (hto k)) \u27e8x, UniformSpace.mem_ball_self _ (ht_mem k)\u27e9 with\n \u27e8y, hxy, hys\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhUV : ball x U \u2286 V\nU' : Set (\u03b1 \u00d7 \u03b1)\nhU' : U' \u2208 \ud835\udce4 \u03b1\nleft\u271d : \u2200 {a b : \u03b1}, (a, b) \u2208 U' \u2192 (b, a) \u2208 U'\nhUU' : U' \u25cb U' \u2286 U\nk : \u2115\nhk : t k \u2286 U'\ny : \u03b1\nhxy : y \u2208 ball x (t k)\nhys : y \u2208 s\n\u22a2 \u2203 v, (\u2203 i, i \u2208 s \u2227 \u2203 y, ball i (t y) = v) \u2227 x \u2208 v \u2227 v \u2286 V\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8y, hys, k, rfl\u27e9, (hts k).subset hxy, fun z hz => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated (\ud835\udce4 \u03b1)\ninst\u271d : SeparableSpace \u03b1\ns : Set \u03b1\nhsc : Set.Countable s\nhsd : Dense s\nt : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh_basis : HasAntitoneBasis (\ud835\udce4 \u03b1) t\nht_mem : \u2200 (i : \u2115), t i \u2208 (\ud835\udce4 \u03b1).sets\nhto : \u2200 (i : \u2115), IsOpen (t i)\nhts : \u2200 (i : \u2115), SymmetricRel (t i)\nx : \u03b1\nV : Set \u03b1\nhxV : x \u2208 V\nhVo : IsOpen V\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\nhUV : ball x U \u2286 V\nU' : Set (\u03b1 \u00d7 \u03b1)\nhU' : U' \u2208 \ud835\udce4 \u03b1\nleft\u271d : \u2200 {a b : \u03b1}, (a, b) \u2208 U' \u2192 (b, a) \u2208 U'\nhUU' : U' \u25cb U' \u2286 U\nk : \u2115\nhk : t k \u2286 U'\ny : \u03b1\nhxy : y \u2208 ball x (t k)\nhys : y \u2208 s\nz : \u03b1\nhz : z \u2208 ball y (t k)\n\u22a2 z \u2208 V\n[PROOFSTEP]\nexact hUV (ball_subset_of_comp_subset (hk hxy) hUU' (hk hz))\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.Cauchy", "llama_tokens": 39329, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5698526660244837, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.27602528239440366}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\n\u22a2 IsSeq (some a :: \u2191s)\n[PROOFSTEP]\nrintro (n | _) h\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\nh : (some a :: \u2191s) Nat.zero = none\n\u22a2 (some a :: \u2191s) (Nat.zero + 1) = none\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\nn\u271d : \u2115\nh : (some a :: \u2191s) (Nat.succ n\u271d) = none\n\u22a2 (some a :: \u2191s) (Nat.succ n\u271d + 1) = none\n[PROOFSTEP]\nexact s.2 h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx y : \u03b1\ns t : Seq \u03b1\nh : cons x s = cons y t\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Option.some_inj, \u2190 get?_cons_zero, h, get?_cons_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx y : \u03b1\ns t : Seq \u03b1\nh : cons x s = cons y t\nn : \u2115\n\u22a2 get? s n = get? t n\n[PROOFSTEP]\nsimp_rw [\u2190 get?_cons_succ x s n, h, get?_cons_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 TerminatedAt s n \u2194 Option.isNone (get? s n) = true\n[PROOFSTEP]\nunfold TerminatedAt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 get? s n = none \u2194 Option.isNone (get? s n) = true\n[PROOFSTEP]\ncases s.get? n\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 none = none \u2194 Option.isNone none = true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\nval\u271d : \u03b1\n\u22a2 some val\u271d = none \u2194 Option.isNone (some val\u271d) = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u00acTerminates s \u2194 \u2200 (n : \u2115), Option.isSome (get? s n) = true\n[PROOFSTEP]\nsimp only [Terminates, TerminatedAt, \u2190 Ne.def, Option.ne_none_iff_isSome, not_exists, iff_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn\u271d : \u2115\nn' : Stream'.tail (\u2191s) n\u271d = none\n\u22a2 Stream'.tail (\u2191s) (n\u271d + 1) = none\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nn\u271d : \u2115\nf : Stream' (Option \u03b1)\nal : IsSeq f\nn' : Stream'.tail (\u2191{ val := f, property := al }) n\u271d = none\n\u22a2 Stream'.tail (\u2191{ val := f, property := al }) (n\u271d + 1) = none\n[PROOFSTEP]\nexact al n'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nm n : \u2115\nh : m \u2264 n\n\u22a2 get? s m = none \u2192 get? s n = none\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nm n : \u2115\nh : m \u2264 n\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 get? { val := f, property := al } m = none \u2192 get? { val := f, property := al } n = none\n[PROOFSTEP]\ninduction' h with n _ IH\n[GOAL]\ncase mk.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nm n : \u2115\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 get? { val := f, property := al } m = none \u2192 get? { val := f, property := al } m = none\ncase mk.step\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nm n\u271d : \u2115\nf : Stream' (Option \u03b1)\nal : IsSeq f\nn : \u2115\na\u271d : Nat.le m n\nIH : get? { val := f, property := al } m = none \u2192 get? { val := f, property := al } n = none\n\u22a2 get? { val := f, property := al } m = none \u2192 get? { val := f, property := al } (Nat.succ n) = none\n[PROOFSTEP]\nexacts [id, fun h2 => al (IH h2)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na\u2099 : \u03b1\nn m : \u2115\nm_le_n : m \u2264 n\ns_nth_eq_some : get? s n = some a\u2099\n\u22a2 get? s n \u2260 none\n[PROOFSTEP]\nsimp [s_nth_eq_some]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nx\u271d : a \u2208 nil\nw\u271d : \u2115\nh : some a = none\n\u22a2 False\n[PROOFSTEP]\ninjection h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nh\u271d : a \u2208 cons b { val := f, property := al }\nh : some a = some b\n\u22a2 a = b\n[PROOFSTEP]\ninjection h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\n\u22a2 a = b \u2228 a \u2208 s \u2192 a \u2208 cons b s\n[PROOFSTEP]\nrintro (rfl | m) <;> [apply mem_cons; exact mem_cons_of_mem _ m]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\n\u22a2 a = b \u2228 a \u2208 s \u2192 a \u2208 cons b s\n[PROOFSTEP]\nrintro (rfl | m)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\n\u22a2 a \u2208 cons a s\n[PROOFSTEP]\napply mem_cons\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\nm : a \u2208 s\n\u22a2 a \u2208 cons b s\n[PROOFSTEP]\nexact mem_cons_of_mem _ m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 destruct s = none \u2192 s = nil\n[PROOFSTEP]\ndsimp [destruct]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 Option.map (fun a' => (a', tail s)) (get? s 0) = none \u2192 s = nil\n[PROOFSTEP]\ninduction' f0 : get? s 0\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\n\u22a2 Option.map (fun a' => (a', tail s)) none = none \u2192 s = nil\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nval\u271d : \u03b1\nf0 : get? s 0 = some val\u271d\n\u22a2 Option.map (fun a' => (a', tail s)) (some val\u271d) = none \u2192 s = nil\n[PROOFSTEP]\nintro h\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n\u22a2 s = nil\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase none.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n\u22a2 \u2191s = \u2191nil\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase none.a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : \u2115\n\u22a2 \u2191s n = \u2191nil n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase none.a.h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\n\u22a2 \u2191s Nat.zero = \u2191nil Nat.zero\ncase none.a.h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = none\nn : \u2115\nIH : \u2191s n = \u2191nil n\n\u22a2 \u2191s (Nat.succ n) = \u2191nil (Nat.succ n)\n[PROOFSTEP]\nexacts [f0, s.2 IH]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nval\u271d : \u03b1\nf0 : get? s 0 = some val\u271d\nh : Option.map (fun a' => (a', tail s)) (some val\u271d) = none\n\u22a2 s = nil\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\n\u22a2 destruct s = some (a, s') \u2192 s = cons a s'\n[PROOFSTEP]\ndsimp [destruct]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\n\u22a2 Option.map (fun a' => (a', tail s)) (get? s 0) = some (a, s') \u2192 s = cons a s'\n[PROOFSTEP]\ninduction' f0 : get? s 0 with a'\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\n\u22a2 Option.map (fun a' => (a', tail s)) none = some (a, s') \u2192 s = cons a s'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\na' : \u03b1\nf0 : get? s 0 = some a'\n\u22a2 Option.map (fun a' => (a', tail s)) (some a') = some (a, s') \u2192 s = cons a s'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\nf0 : get? s 0 = none\nh : Option.map (fun a' => (a', tail s)) none = some (a, s')\n\u22a2 s = cons a s'\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\nf0\u271d : get? s 0 = x\u271d\na' : \u03b1\nf0 : get? s 0 = some a'\nh : Option.map (fun a' => (a', tail s)) (some a') = some (a, s')\n\u22a2 s = cons a s'\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a'\nh : Option.map (fun a' => (a', tail { val := f, property := al })) (some a') = some (a, s')\n\u22a2 { val := f, property := al } = cons a s'\n[PROOFSTEP]\ninjections _ h1 h2\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a'\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 { val := f, property := al } = cons a s'\n[PROOFSTEP]\nrw [\u2190 h2]\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a'\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 { val := f, property := al } = cons a (tail { val := f, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a'\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 \u2191{ val := f, property := al } = \u2191(cons a (tail { val := f, property := al }))\n[PROOFSTEP]\ndsimp [tail, cons]\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a'\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 f = some a :: Stream'.tail f\n[PROOFSTEP]\nrw [h1] at f0 \n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 f = some a :: Stream'.tail f\n[PROOFSTEP]\nrw [\u2190 f0]\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns' : Seq \u03b1\nx\u271d : Option \u03b1\na' : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\nf0\u271d : get? { val := f, property := al } 0 = x\u271d\nf0 : get? { val := f, property := al } 0 = some a\nh1 : a' = a\nh2 : tail { val := f, property := al } = s'\n\u22a2 f = get? { val := f, property := al } 0 :: Stream'.tail f\n[PROOFSTEP]\nexact (Stream'.eta f).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 destruct (cons a { val := f, property := al }) = some (a, { val := f, property := al })\n[PROOFSTEP]\nunfold cons destruct Functor.map\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 instFunctorOption.1\n (fun a' =>\n (a',\n tail\n { val := some a :: \u2191{ val := f, property := al },\n property :=\n (_ :\n \u2200 {n : \u2115},\n (some a :: \u2191{ val := f, property := al }) n = none \u2192\n (some a :: \u2191{ val := f, property := al }) (n + 1) = none) }))\n (get?\n { val := some a :: \u2191{ val := f, property := al },\n property :=\n (_ :\n \u2200 {n : \u2115},\n (some a :: \u2191{ val := f, property := al }) n = none \u2192\n (some a :: \u2191{ val := f, property := al }) (n + 1) = none) }\n 0) =\n some (a, { val := f, property := al })\n[PROOFSTEP]\napply congr_arg fun s => some (a, s)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 tail\n { val := some a :: \u2191{ val := f, property := al },\n property :=\n (_ :\n \u2200 {n : \u2115},\n (some a :: \u2191{ val := f, property := al }) n = none \u2192\n (some a :: \u2191{ val := f, property := al }) (n + 1) = none) } =\n { val := f, property := al }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 \u2191(tail\n { val := some a :: \u2191{ val := f, property := al },\n property :=\n (_ :\n \u2200 {n : \u2115},\n (some a :: \u2191{ val := f, property := al }) n = none \u2192\n (some a :: \u2191{ val := f, property := al }) (n + 1) = none) }) =\n \u2191{ val := f, property := al }\n[PROOFSTEP]\ndsimp [tail]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 head s = Prod.fst <$> destruct s\n[PROOFSTEP]\nunfold destruct head\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 get? s 0 = Prod.fst <$> (fun a' => (a', tail s)) <$> get? s 0\n[PROOFSTEP]\ncases get? s 0\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 none = Prod.fst <$> (fun a' => (a', tail s)) <$> none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nval\u271d : \u03b1\n\u22a2 some val\u271d = Prod.fst <$> (fun a' => (a', tail s)) <$> some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\n\u22a2 head (cons a s) = some a\n[PROOFSTEP]\nrw [head_eq_destruct, destruct_cons, Option.map_eq_map, Option.map_some']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\n\u22a2 tail (cons a s) = s\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 tail (cons a { val := f, property := al }) = { val := f, property := al }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : Stream' (Option \u03b1)\nal : IsSeq f\n\u22a2 \u2191(tail (cons a { val := f, property := al })) = \u2191{ val := f, property := al }\n[PROOFSTEP]\ndsimp [tail, cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\n\u22a2 C s\n[PROOFSTEP]\ncases' H : destruct s with v\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\nH : destruct s = none\n\u22a2 C s\n[PROOFSTEP]\nrw [destruct_eq_nil H]\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\nH : destruct s = none\n\u22a2 C nil\n[PROOFSTEP]\napply h1\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\nv : Seq1 \u03b1\nH : destruct s = some v\n\u22a2 C s\n[PROOFSTEP]\ncases' v with a s'\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\na : \u03b1\ns' : Seq \u03b1\nH : destruct s = some (a, s')\n\u22a2 C s\n[PROOFSTEP]\nrw [destruct_eq_cons H]\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Sort v\ns : Seq \u03b1\nh1 : C nil\nh2 : (x : \u03b1) \u2192 (s : Seq \u03b1) \u2192 C (cons x s)\na : \u03b1\ns' : Seq \u03b1\nH : destruct s = some (a, s')\n\u22a2 C (cons a s')\n[PROOFSTEP]\napply h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns : Seq \u03b1\nM : a \u2208 s\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\n\u22a2 C s\n[PROOFSTEP]\ncases' M with k e\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne : (fun b => some a = b) (nth (\u2191s) k)\n\u22a2 C s\n[PROOFSTEP]\nunfold Stream'.nth at e \n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne : (fun b => some a = b) (\u2191s k)\n\u22a2 C s\n[PROOFSTEP]\ninduction' k with k IH generalizing s\n[GOAL]\ncase intro.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\n\u22a2 C s\n[PROOFSTEP]\nhave TH : s = cons a (tail s) := by\n apply destruct_eq_cons\n unfold destruct get? Functor.map\n rw [\u2190 e]\n rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\n\u22a2 s = cons a (tail s)\n[PROOFSTEP]\napply destruct_eq_cons\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\n\u22a2 destruct s = some (a, tail s)\n[PROOFSTEP]\nunfold destruct get? Functor.map\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\n\u22a2 instFunctorOption.1 (fun a' => (a', tail s)) (\u2191s 0) = some (a, tail s)\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\n\u22a2 instFunctorOption.1 (fun a' => (a', tail s)) (some a) = some (a, tail s)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\nTH : s = cons a (tail s)\n\u22a2 C s\n[PROOFSTEP]\nrw [TH]\n[GOAL]\ncase intro.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k)\ns : Seq \u03b1\ne : some a = \u2191s Nat.zero\nTH : s = cons a (tail s)\n\u22a2 C (cons a (tail s))\n[PROOFSTEP]\napply\n h1 _ _\n (Or.inl rfl)\n -- porting note: had to reshuffle `intro`\n[GOAL]\ncase intro.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\ne : some a = \u2191s (Nat.succ k)\n\u22a2 C s\n[PROOFSTEP]\nrevert e\n[GOAL]\ncase intro.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\n\u22a2 some a = \u2191s (Nat.succ k) \u2192 C s\n[PROOFSTEP]\napply s.recOn _ fun b s' => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\n\u22a2 some a = \u2191nil (Nat.succ k) \u2192 C nil\n[PROOFSTEP]\nintro e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\ne : some a = \u2191nil (Nat.succ k)\n\u22a2 C nil\n[PROOFSTEP]\ninjection e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\n\u22a2 \u2200 (b : \u03b1) (s' : Seq \u03b1), some a = \u2191(cons b s') (Nat.succ k) \u2192 C (cons b s')\n[PROOFSTEP]\nintro b s' e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\nb : \u03b1\ns' : Seq \u03b1\ne : some a = \u2191(cons b s') (Nat.succ k)\n\u22a2 C (cons b s')\n[PROOFSTEP]\nhave h_eq : (cons b s').val (Nat.succ k) = s'.val k := by cases s'; rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\nb : \u03b1\ns' : Seq \u03b1\ne : some a = \u2191(cons b s') (Nat.succ k)\n\u22a2 \u2191(cons b s') (Nat.succ k) = \u2191s' k\n[PROOFSTEP]\ncases s'\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\nb : \u03b1\nval\u271d : Stream' (Option \u03b1)\nproperty\u271d : IsSeq val\u271d\ne : some a = \u2191(cons b { val := val\u271d, property := property\u271d }) (Nat.succ k)\n\u22a2 \u2191(cons b { val := val\u271d, property := property\u271d }) (Nat.succ k) = \u2191{ val := val\u271d, property := property\u271d } k\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\nb : \u03b1\ns' : Seq \u03b1\ne : some a = \u2191(cons b s') (Nat.succ k)\nh_eq : \u2191(cons b s') (Nat.succ k) = \u2191s' k\n\u22a2 C (cons b s')\n[PROOFSTEP]\nrw [h_eq] at e \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Seq \u03b1 \u2192 Prop\na : \u03b1\ns\u271d : Seq \u03b1\nh1 : \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')\nk\u271d : \u2115\ne\u271d : (fun b => some a = b) (\u2191s\u271d k\u271d)\nk : \u2115\nIH : \u2200 {s : Seq \u03b1}, some a = \u2191s k \u2192 C s\ns : Seq \u03b1\nb : \u03b1\ns' : Seq \u03b1\ne : some a = \u2191s' k\nh_eq : \u2191(cons b s') (Nat.succ k) = \u2191s' k\n\u22a2 C (cons b s')\n[PROOFSTEP]\napply h1 _ _ (Or.inr (IH e))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\n\u22a2 Seq \u03b1\n[PROOFSTEP]\nrefine' \u27e8Stream'.corec' (Corec.f f) (some b), fun {n} h => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nh : corec' (Corec.f f) (some b) n = none\n\u22a2 corec' (Corec.f f) (some b) (n + 1) = none\n[PROOFSTEP]\nrw [Stream'.corec'_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nh : corec' (Corec.f f) (some b) n = none\n\u22a2 ((Corec.f f (some b)).fst :: corec' (Corec.f f) (Corec.f f (some b)).snd) (n + 1) = none\n[PROOFSTEP]\nchange Stream'.corec' (Corec.f f) (Corec.f f (some b)).2 n = none\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nh : corec' (Corec.f f) (some b) n = none\n\u22a2 corec' (Corec.f f) (Corec.f f (some b)).snd n = none\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\n\u22a2 corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (Corec.f f (some b)).snd n = none\n[PROOFSTEP]\ngeneralize some b = o\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\no : Option \u03b2\n\u22a2 corec' (Corec.f f) o n = none \u2192 corec' (Corec.f f) (Corec.f f o).snd n = none\n[PROOFSTEP]\nrevert o\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\n\u22a2 \u2200 (o : Option \u03b2), corec' (Corec.f f) o n = none \u2192 corec' (Corec.f f) (Corec.f f o).snd n = none\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\n\u22a2 \u2200 (o : Option \u03b2), corec' (Corec.f f) o Nat.zero = none \u2192 corec' (Corec.f f) (Corec.f f o).snd Nat.zero = none\n[PROOFSTEP]\nintro o\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nIH : \u2200 (o : Option \u03b2), corec' (Corec.f f) o n = none \u2192 corec' (Corec.f f) (Corec.f f o).snd n = none\n\u22a2 \u2200 (o : Option \u03b2), corec' (Corec.f f) o (Nat.succ n) = none \u2192 corec' (Corec.f f) (Corec.f f o).snd (Nat.succ n) = none\n[PROOFSTEP]\nintro o\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\no : Option \u03b2\n\u22a2 corec' (Corec.f f) o Nat.zero = none \u2192 corec' (Corec.f f) (Corec.f f o).snd Nat.zero = none\n[PROOFSTEP]\nchange (Corec.f f o).1 = none \u2192 (Corec.f f (Corec.f f o).2).1 = none\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\no : Option \u03b2\n\u22a2 (Corec.f f o).fst = none \u2192 (Corec.f f (Corec.f f o).snd).fst = none\n[PROOFSTEP]\ncases' o with b\n[GOAL]\ncase zero.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\n\u22a2 (Corec.f f none).fst = none \u2192 (Corec.f f (Corec.f f none).snd).fst = none\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\n\u22a2 (Corec.f f (some b)).fst = none \u2192 (Corec.f f (Corec.f f (some b)).snd).fst = none\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh : (Corec.f f none).fst = none\n\u22a2 (Corec.f f (Corec.f f none).snd).fst = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\nh : (Corec.f f (some b)).fst = none\n\u22a2 (Corec.f f (Corec.f f (some b)).snd).fst = none\n[PROOFSTEP]\ndsimp [Corec.f] at h \n[GOAL]\ncase zero.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\nh :\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n\u22a2 (Corec.f f (Corec.f f (some b)).snd).fst = none\n[PROOFSTEP]\ndsimp [Corec.f]\n[GOAL]\ncase zero.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\nh :\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n\u22a2 (match\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase zero.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\n\u22a2 (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none \u2192\n (match\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\ncases' h\u2081 : f b with s\n[GOAL]\ncase zero.some.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\nh\u2081 : f b = none\n\u22a2 (match none with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none \u2192\n (match\n (match none with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.some.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\ns : \u03b1 \u00d7 \u03b2\nh\u2081 : f b = some s\n\u22a2 (match some s with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none \u2192\n (match\n (match some s with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.some.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\nh\u2081 : f b = none\nh :\n (match none with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n\u22a2 (match\n (match none with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.some.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\ns : \u03b1 \u00d7 \u03b2\nh\u2081 : f b = some s\nh :\n (match some s with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n\u22a2 (match\n (match some s with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\ncases' s with a b'\n[GOAL]\ncase zero.some.some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb\u271d b : \u03b2\na : \u03b1\nb' : \u03b2\nh\u2081 : f b = some (a, b')\nh :\n (match some (a, b') with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n\u22a2 (match\n (match some (a, b') with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n none\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nIH : \u2200 (o : Option \u03b2), corec' (Corec.f f) o n = none \u2192 corec' (Corec.f f) (Corec.f f o).snd n = none\no : Option \u03b2\n\u22a2 corec' (Corec.f f) o (Nat.succ n) = none \u2192 corec' (Corec.f f) (Corec.f f o).snd (Nat.succ n) = none\n[PROOFSTEP]\nrw [Stream'.corec'_eq (Corec.f f) (Corec.f f o).2, Stream'.corec'_eq (Corec.f f) o]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nn : \u2115\nIH : \u2200 (o : Option \u03b2), corec' (Corec.f f) o n = none \u2192 corec' (Corec.f f) (Corec.f f o).snd n = none\no : Option \u03b2\n\u22a2 ((Corec.f f o).fst :: corec' (Corec.f f) (Corec.f f o).snd) (Nat.succ n) = none \u2192\n ((Corec.f f (Corec.f f o).snd).fst :: corec' (Corec.f f) (Corec.f f (Corec.f f o).snd).snd) (Nat.succ n) = none\n[PROOFSTEP]\nexact IH (Corec.f f o).2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\n\u22a2 destruct (corec f b) = omap (corec f) (f b)\n[PROOFSTEP]\ndsimp [corec, destruct, nth]\n -- porting note: next two lines were `change`...`with`...\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (corec' (Corec.f f) (some b) 0) =\n match f b with\n | none => none\n | some (a, b) =>\n some\n (a,\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\nhave h : Stream'.corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).1 := rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (corec' (Corec.f f) (some b) 0) =\n match f b with\n | none => none\n | some (a, b) =>\n some\n (a,\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (Corec.f f (some b)).fst =\n match f b with\n | none => none\n | some (a, b) =>\n some\n (a,\n { val := corec' (Corec.f f) (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\ndsimp [Corec.f]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n match f b with\n | none => none\n | some (a, b) =>\n some\n (a,\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\ninduction' h : f b with s\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\nh : f b = none\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (match none with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n match none with\n | none => none\n | some (a, b) =>\n some\n (a,\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\ns : \u03b1 \u00d7 \u03b2\nh : f b = some s\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (match some s with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n match some s with\n | none => none\n | some (a, b) =>\n some\n (a,\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\ncases' s with a b'\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 Option.map\n (fun a' =>\n (a',\n tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }))\n (match some (a, b') with\n | none => (none, none)\n | some (a, b') => (some a, some b')).fst =\n match some (a, b') with\n | none => none\n | some (a, b) =>\n some\n (a,\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) })\n[PROOFSTEP]\ndsimp [Corec.f]\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 some\n (a,\n tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }) =\n some\n (a,\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b'),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b') n = none \u2192 corec' (Corec.f f) (some b') (n + 1) = none) })\n[PROOFSTEP]\napply congr_arg fun b' => some (a, b')\n[GOAL]\ncase some.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) } =\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b'),\n property := (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b') n = none \u2192 corec' (Corec.f f) (some b') (n + 1) = none) }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 \u2191(tail\n {\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b) n = none \u2192 corec' (Corec.f f) (some b) (n + 1) = none) }) =\n \u2191{\n val :=\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b'),\n property :=\n (_ : \u2200 {n : \u2115}, corec' (Corec.f f) (some b') n = none \u2192 corec' (Corec.f f) (some b') (n + 1) = none) }\n[PROOFSTEP]\ndsimp [corec, tail]\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 Stream'.tail\n (corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b)) =\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b')\n[PROOFSTEP]\nrw [Stream'.corec'_eq, Stream'.tail_cons]\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (match some b with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd =\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b')\n[PROOFSTEP]\ndsimp [Corec.f]\n[GOAL]\ncase some.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\nb : \u03b2\nh\u271d\u00b9 : corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).fst\nx\u271d : Option (\u03b1 \u00d7 \u03b2)\nh\u271d : f b = x\u271d\na : \u03b1\nb' : \u03b2\nh : f b = some (a, b')\n\u22a2 corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b')).snd =\n corec'\n (fun x =>\n match x with\n | none => (none, none)\n | some b =>\n match f b with\n | none => (none, none)\n | some (a, b') => (some a, some b'))\n (some b')\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 s\u2081 = s\u2082\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2191s\u2081 = \u2191s\u2082\n[PROOFSTEP]\napply Stream'.eq_of_bisim fun x y => \u2203 s s' : Seq \u03b1, s.1 = x \u2227 s'.1 = y \u2227 R s s'\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 Stream'.IsBisimulation fun x y => \u2203 s s', \u2191s = x \u2227 \u2191s' = y \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\ndsimp [Stream'.IsBisimulation]\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Stream' (Option \u03b1)\u2984,\n (\u2203 s s', \u2191s = s\u2081 \u2227 \u2191s' = s\u2082 \u2227 R s s') \u2192\n Stream'.head s\u2081 = Stream'.head s\u2082 \u2227 \u2203 s s', \u2191s = Stream'.tail s\u2081 \u2227 \u2191s' = Stream'.tail s\u2082 \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nintro t\u2081 t\u2082 e\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\n\u22a2 Stream'.head t\u2081 = Stream'.head t\u2082 \u2227 \u2203 s s', \u2191s = Stream'.tail t\u2081 \u2227 \u2191s' = Stream'.tail t\u2082 \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nexact\n match t\u2081, t\u2082, e with\n | _, _, \u27e8s, s', rfl, rfl, r\u27e9 =>\n by\n suffices head s = head s' \u2227 R (tail s) (tail s') from\n And.imp id (fun r => \u27e8tail s, tail s', by cases s; rfl, by cases s'; rfl, r\u27e9) this\n have := bisim r; revert r this\n apply recOn s _ _ <;> apply recOn s' _ _\n \u00b7 intro r _\n constructor\n \u00b7 rfl\n \u00b7 assumption\n \u00b7 intro x s _ this\n rw [destruct_nil, destruct_cons] at this \n exact False.elim this\n \u00b7 intro x s _ this\n rw [destruct_nil, destruct_cons] at this \n exact False.elim this\n \u00b7 intro x s x' s' _ this\n rw [destruct_cons, destruct_cons] at this \n rw [head_cons, head_cons, tail_cons, tail_cons]\n cases' this with h1 h2\n constructor\n rw [h1]\n exact h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R s s'\n\u22a2 Stream'.head \u2191s = Stream'.head \u2191s' \u2227 \u2203 s_1 s'_1, \u2191s_1 = Stream'.tail \u2191s \u2227 \u2191s'_1 = Stream'.tail \u2191s' \u2227 R s_1 s'_1\n[PROOFSTEP]\nsuffices head s = head s' \u2227 R (tail s) (tail s') from\n And.imp id (fun r => \u27e8tail s, tail s', by cases s; rfl, by cases s'; rfl, r\u27e9) this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr\u271d : R s s'\nthis : head s = head s' \u2227 R (tail s) (tail s')\nr : R (tail s) (tail s')\n\u22a2 \u2191(tail s) = Stream'.tail \u2191s\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns' : Seq \u03b1\nval\u271d : Stream' (Option \u03b1)\nproperty\u271d : IsSeq val\u271d\nr\u271d : R { val := val\u271d, property := property\u271d } s'\nthis : head { val := val\u271d, property := property\u271d } = head s' \u2227 R (tail { val := val\u271d, property := property\u271d }) (tail s')\nr : R (tail { val := val\u271d, property := property\u271d }) (tail s')\n\u22a2 \u2191(tail { val := val\u271d, property := property\u271d }) = Stream'.tail \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr\u271d : R s s'\nthis : head s = head s' \u2227 R (tail s) (tail s')\nr : R (tail s) (tail s')\n\u22a2 \u2191(tail s') = Stream'.tail \u2191s'\n[PROOFSTEP]\ncases s'\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns : Seq \u03b1\nval\u271d : Stream' (Option \u03b1)\nproperty\u271d : IsSeq val\u271d\nr\u271d : R s { val := val\u271d, property := property\u271d }\nthis : head s = head { val := val\u271d, property := property\u271d } \u2227 R (tail s) (tail { val := val\u271d, property := property\u271d })\nr : R (tail s) (tail { val := val\u271d, property := property\u271d })\n\u22a2 \u2191(tail { val := val\u271d, property := property\u271d }) = Stream'.tail \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R s s'\n\u22a2 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\nhave := bisim r\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R s s'\nthis : BisimO R (destruct s) (destruct s')\n\u22a2 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\nrevert r this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 R s s' \u2192 BisimO R (destruct s) (destruct s') \u2192 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\napply recOn s _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 R nil s' \u2192 BisimO R (destruct nil) (destruct s') \u2192 head nil = head s' \u2227 R (tail nil) (tail s')\n[PROOFSTEP]\napply recOn s' _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n R (cons x s) s' \u2192\n BisimO R (destruct (cons x s)) (destruct s') \u2192 head (cons x s) = head s' \u2227 R (tail (cons x s)) (tail s')\n[PROOFSTEP]\napply recOn s' _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 R nil nil \u2192 BisimO R (destruct nil) (destruct nil) \u2192 head nil = head nil \u2227 R (tail nil) (tail nil)\n[PROOFSTEP]\nintro r _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R nil nil\nthis\u271d : BisimO R (destruct nil) (destruct nil)\n\u22a2 head nil = head nil \u2227 R (tail nil) (tail nil)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R nil nil\nthis\u271d : BisimO R (destruct nil) (destruct nil)\n\u22a2 head nil = head nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\nr : R nil nil\nthis\u271d : BisimO R (destruct nil) (destruct nil)\n\u22a2 R (tail nil) (tail nil)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n R nil (cons x s) \u2192\n BisimO R (destruct nil) (destruct (cons x s)) \u2192 head nil = head (cons x s) \u2227 R (tail nil) (tail (cons x s))\n[PROOFSTEP]\nintro x s _ this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s' : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nr\u271d : R nil (cons x s)\nthis : BisimO R (destruct nil) (destruct (cons x s))\n\u22a2 head nil = head (cons x s) \u2227 R (tail nil) (tail (cons x s))\n[PROOFSTEP]\nrw [destruct_nil, destruct_cons] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s' : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nr\u271d : R nil (cons x s)\nthis : BisimO R none (some (x, s))\n\u22a2 head nil = head (cons x s) \u2227 R (tail nil) (tail (cons x s))\n[PROOFSTEP]\nexact False.elim this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n R (cons x s) nil \u2192\n BisimO R (destruct (cons x s)) (destruct nil) \u2192 head (cons x s) = head nil \u2227 R (tail (cons x s)) (tail nil)\n[PROOFSTEP]\nintro x s _ this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s' : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nr\u271d : R (cons x s) nil\nthis : BisimO R (destruct (cons x s)) (destruct nil)\n\u22a2 head (cons x s) = head nil \u2227 R (tail (cons x s)) (tail nil)\n[PROOFSTEP]\nrw [destruct_nil, destruct_cons] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s' : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nr\u271d : R (cons x s) nil\nthis : BisimO R (some (x, s)) none\n\u22a2 head (cons x s) = head nil \u2227 R (tail (cons x s)) (tail nil)\n[PROOFSTEP]\nexact False.elim this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1) (x_1 : \u03b1) (s_1 : Seq \u03b1),\n R (cons x_1 s_1) (cons x s) \u2192\n BisimO R (destruct (cons x_1 s_1)) (destruct (cons x s)) \u2192\n head (cons x_1 s_1) = head (cons x s) \u2227 R (tail (cons x_1 s_1)) (tail (cons x s))\n[PROOFSTEP]\nintro x s x' s' _ this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nthis : BisimO R (destruct (cons x' s')) (destruct (cons x s))\n\u22a2 head (cons x' s') = head (cons x s) \u2227 R (tail (cons x' s')) (tail (cons x s))\n[PROOFSTEP]\nrw [destruct_cons, destruct_cons] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nthis : BisimO R (some (x', s')) (some (x, s))\n\u22a2 head (cons x' s') = head (cons x s) \u2227 R (tail (cons x' s')) (tail (cons x s))\n[PROOFSTEP]\nrw [head_cons, head_cons, tail_cons, tail_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nthis : BisimO R (some (x', s')) (some (x, s))\n\u22a2 some x' = some x \u2227 R s' s\n[PROOFSTEP]\ncases' this with h1 h2\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nh1 : x' = x\nh2 : R s' s\n\u22a2 some x' = some x \u2227 R s' s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nh1 : x' = x\nh2 : R s' s\n\u22a2 some x' = some x\ncase intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nh1 : x' = x\nh2 : R s' s\n\u22a2 R s' s\n[PROOFSTEP]\nrw [h1]\n[GOAL]\ncase intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns\u271d s'\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\nx' : \u03b1\ns' : Seq \u03b1\nr\u271d : R (cons x' s') (cons x s)\nh1 : x' = x\nh2 : R s' s\n\u22a2 R s' s\n[PROOFSTEP]\nexact h2\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Seq \u03b1 \u2192 Seq \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Seq \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nexact \u27e8s\u2081, s\u2082, rfl, rfl, r\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nf g : Seq \u03b1 \u2192 Seq \u03b2\nH : \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\n\u22a2 f s = g s\n[PROOFSTEP]\nrefine' eq_of_bisim (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) _ \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nf g : Seq \u03b1 \u2192 Seq \u03b2\nH : \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\n\u22a2 IsBisimulation fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nf g : Seq \u03b1 \u2192 Seq \u03b2\nH : \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq \u03b2\nh : \u2203 s, s1 = f s \u2227 s2 = g s\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct s1) (destruct s2)\n[PROOFSTEP]\nrcases h with \u27e8s, h1, h2\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Seq \u03b1\nf g : Seq \u03b1 \u2192 Seq \u03b2\nH : \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq \u03b2\ns : Seq \u03b1\nh1 : s1 = f s\nh2 : s2 = g s\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct s1) (destruct s2)\n[PROOFSTEP]\nrw [h1, h2]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Seq \u03b1\nf g : Seq \u03b1 \u2192 Seq \u03b2\nH : \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\ns1 s2 : Seq \u03b2\ns : Seq \u03b1\nh1 : s1 = f s\nh2 : s2 = g s\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = f s \u2227 s2 = g s) (destruct (f s)) (destruct (g s))\n[PROOFSTEP]\napply H\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nn : \u2115\nh : List.get? l n = none\n\u22a2 List.get? l (n + 1) = none\n[PROOFSTEP]\nrw [List.get?_eq_none] at h \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nn : \u2115\nh : List.length l \u2264 n\n\u22a2 List.length l \u2264 n + 1\n[PROOFSTEP]\nexact h.trans (Nat.le_succ n)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 \u2191(a :: l) = cons a \u2191l\n[PROOFSTEP]\next1 (_ | n)\n[GOAL]\ncase h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 get? (\u2191(a :: l)) Nat.zero = get? (cons a \u2191l) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nn : \u2115\n\u22a2 get? (\u2191(a :: l)) (Nat.succ n) = get? (cons a \u2191l) (Nat.succ n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' \u03b1\nn : \u2115\nh : map some s n = none\n\u22a2 map some s (n + 1) = none\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\n\u22a2 Stream'.map (Option.map f) s n = none \u2192 Stream'.map (Option.map f) s (n + 1) = none\n[PROOFSTEP]\ndsimp [Stream'.map, Stream'.nth]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\n\u22a2 Option.map f (s n) = none \u2192 Option.map f (s (n + 1)) = none\n[PROOFSTEP]\ninduction' e : s n with e\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\ne : s n = none\n\u22a2 Option.map f none = none \u2192 Option.map f (s (n + 1)) = none\n[PROOFSTEP]\nintro\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\nx\u271d : Option \u03b1\ne\u271d\u00b9 : s n = x\u271d\ne\u271d : \u03b1\ne : s n = some e\u271d\n\u22a2 Option.map f (some e\u271d) = none \u2192 Option.map f (s (n + 1)) = none\n[PROOFSTEP]\nintro\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\ne : s n = none\na\u271d : Option.map f none = none\n\u22a2 Option.map f (s (n + 1)) = none\n[PROOFSTEP]\nrw [al e]\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\ne : s n = none\na\u271d : Option.map f none = none\n\u22a2 Option.map f none = none\n[PROOFSTEP]\nassumption\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\nn : \u2115\nx\u271d : Option \u03b1\ne\u271d\u00b9 : s n = x\u271d\ne\u271d : \u03b1\ne : s n = some e\u271d\na\u271d : Option.map f (some e\u271d) = none\n\u22a2 Option.map f (s (n + 1)) = none\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 append nil s = s\n[PROOFSTEP]\napply coinduction2\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = append nil s \u2227 s2 = s) (destruct (append nil s)) (destruct s)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = append nil s \u2227 s2 = s) (destruct (append nil s)) (destruct s)\n[PROOFSTEP]\ndsimp [append]\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 match\n destruct\n (corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s)),\n destruct s with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 match\n omap\n (corec fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (match destruct (nil, s).fst with\n | none =>\n match destruct (nil, s).snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', (nil, s).snd)),\n destruct s with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\ndsimp [append]\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 match\n match\n match destruct s with\n | none => none\n | some (a, b) => some (a, nil, b) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b),\n destruct s with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\napply recOn s _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 match\n match\n match destruct nil with\n | none => none\n | some (a, b) => some (a, nil, b) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b),\n destruct nil with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n match\n match\n match destruct (cons x s) with\n | none => none\n | some (a, b) => some (a, nil, b) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b),\n destruct (cons x s) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\nintro x s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 match\n match\n match destruct (cons x s) with\n | none => none\n | some (a, b) => some (a, nil, b) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b),\n destruct (cons x s) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\nrw [destruct_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 match\n match\n match some (x, s) with\n | none => none\n | some (a, b) => some (a, nil, b) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b),\n some (x, s) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227\n \u2203 s_1,\n s =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s' = s_1\n | x, x_1 => False\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 x = x \u2227\n \u2203 s_1,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s) =\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (nil, s_1) \u2227\n s = s_1\n[PROOFSTEP]\nexact \u27e8rfl, s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns t : Seq \u03b1\n\u22a2 destruct (append (cons a s) t) = some (a, append s t)\n[PROOFSTEP]\ndsimp [append]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns t : Seq \u03b1\n\u22a2 destruct\n (corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (cons a s, t)) =\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (s, t))\n[PROOFSTEP]\nrw [corec_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns t : Seq \u03b1\n\u22a2 omap\n (corec fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (match destruct (cons a s, t).fst with\n | none =>\n match destruct (cons a s, t).snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a_1, s\u2081') => some (a_1, s\u2081', (cons a s, t).snd)) =\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (s, t))\n[PROOFSTEP]\ndsimp [append]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns t : Seq \u03b1\n\u22a2 (match\n match destruct (cons a s) with\n | none =>\n match destruct t with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', t) with\n | none => none\n | some (a, b) =>\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n b)) =\n some\n (a,\n corec\n (fun x =>\n match destruct x.fst with\n | none =>\n match destruct x.snd with\n | none => none\n | some (a, b) => some (a, nil, b)\n | some (a, s\u2081') => some (a, s\u2081', x.snd))\n (s, t))\n[PROOFSTEP]\nrw [destruct_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 append s nil = s\n[PROOFSTEP]\napply coinduction2 s\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u2200 (s : Seq \u03b1), BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (destruct (append s nil)) (destruct s)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (destruct (append s nil)) (destruct s)\n[PROOFSTEP]\napply recOn s _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (destruct (append nil nil)) (destruct nil)\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (destruct (append (cons x s) nil)) (destruct (cons x s))\n[PROOFSTEP]\nintro x s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (destruct (append (cons x s) nil)) (destruct (cons x s))\n[PROOFSTEP]\nrw [cons_append, destruct_cons, destruct_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = append s nil \u2227 s2 = s) (some (x, append s nil)) (some (x, s))\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 s\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 x = x \u2227 \u2203 s_1, append s nil = append s_1 nil \u2227 s = s_1\n[PROOFSTEP]\nexact \u27e8rfl, s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t u : Seq \u03b1\n\u22a2 append (append s t) u = append s (append t u)\n[PROOFSTEP]\napply eq_of_bisim fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t u : Seq \u03b1\n\u22a2 IsBisimulation fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t u s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\n\u22a2 BisimO (fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)) (destruct s1) (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | _, _, \u27e8s, t, u, rfl, rfl\u27e9 => by\n apply recOn s <;> simp\n \u00b7 apply recOn t <;> simp\n \u00b7 apply recOn u <;> simp\n \u00b7 intro _ u\n refine' \u27e8nil, nil, u, _, _\u27e9 <;> simp\n \u00b7 intro _ t\n refine' \u27e8nil, t, u, _, _\u27e9 <;> simp\n \u00b7 intro _ s\n exact \u27e8s, t, u, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u))\n (destruct (append (append s t) u)) (destruct (append s (append t u)))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u))\n (destruct (append (append nil t) u)) (destruct (append nil (append t u)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u))\n (destruct (append (append (cons x s) t) u)) (destruct (append (cons x s) (append t u)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 match destruct (append t u), destruct (append t u) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\napply recOn t\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 match destruct (append nil u), destruct (append nil u) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n match destruct (append (cons x s) u), destruct (append (cons x s) u) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 match destruct u, destruct u with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\napply recOn u\n[GOAL]\ncase h1.h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 match destruct nil, destruct nil with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n match destruct (cons x s), destruct (cons x s) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s' = append s_1 (append t u)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u03b1 \u2192 \u2200 (s : Seq \u03b1), \u2203 s_1 t u, s = append (append s_1 t) u \u2227 s = append s_1 (append t u)\n[PROOFSTEP]\nintro _ u\n[GOAL]\ncase h1.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d\u00b9 s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u\u271d : Seq \u03b1\nx\u271d : \u03b1\nu : Seq \u03b1\n\u22a2 \u2203 s t u_1, u = append (append s t) u_1 \u2227 u = append s (append t u_1)\n[PROOFSTEP]\nrefine' \u27e8nil, nil, u, _, _\u27e9\n[GOAL]\ncase h1.h1.h2.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d\u00b9 s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u\u271d : Seq \u03b1\nx\u271d : \u03b1\nu : Seq \u03b1\n\u22a2 u = append (append nil nil) u\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d\u00b9 s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u\u271d : Seq \u03b1\nx\u271d : \u03b1\nu : Seq \u03b1\n\u22a2 u = append nil (append nil u)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u03b1 \u2192 \u2200 (s : Seq \u03b1), \u2203 s_1 t u_1, append s u = append (append s_1 t) u_1 \u2227 append s u = append s_1 (append t u_1)\n[PROOFSTEP]\nintro _ t\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d\u00b9 u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t\u271d u : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 \u2203 s t_1 u_1, append t u = append (append s t_1) u_1 \u2227 append t u = append s (append t_1 u_1)\n[PROOFSTEP]\nrefine' \u27e8nil, t, u, _, _\u27e9\n[GOAL]\ncase h1.h2.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d\u00b9 u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t\u271d u : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 append t u = append (append nil t) u\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d\u00b9 u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t\u271d u : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 append t u = append nil (append t u)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns t u : Seq \u03b1\n\u22a2 \u03b1 \u2192\n \u2200 (s : Seq \u03b1),\n \u2203 s_1 t_1 u_1,\n append (append s t) u = append (append s_1 t_1) u_1 \u2227 append s (append t u) = append s_1 (append t_1 u_1)\n[PROOFSTEP]\nintro _ s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 t\u271d u\u271d s1 s2 : Seq \u03b1\nh : \u2203 s t u, s1 = append (append s t) u \u2227 s2 = append s (append t u)\ns\u271d t u : Seq \u03b1\nx\u271d : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 t_1 u_1,\n append (append s t) u = append (append s_1 t_1) u_1 \u2227 append s (append t u) = append s_1 (append t_1 u_1)\n[PROOFSTEP]\nexact \u27e8s, t, u, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t u : Seq \u03b1\n\u22a2 \u2203 s_1 t_1 u_1,\n append (append s t) u = append (append s_1 t_1) u_1 \u2227 append s (append t u) = append s_1 (append t_1 u_1)\n[PROOFSTEP]\nexact \u27e8s, t, u, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 map f (cons a { val := s, property := al }) = cons (f a) (map f { val := s, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 \u2191(map f (cons a { val := s, property := al })) = \u2191(cons (f a) (map f { val := s, property := al }))\n[PROOFSTEP]\ndsimp [cons, map]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 Stream'.map (Option.map f) (some a :: s) = some (f a) :: Stream'.map (Option.map f) s\n[PROOFSTEP]\nrw [Stream'.map_cons]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 Option.map f (some a) :: Stream'.map (Option.map f) s = some (f a) :: Stream'.map (Option.map f) s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 map id { val := s, property := al } = { val := s, property := al }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 \u2191(map id { val := s, property := al }) = \u2191{ val := s, property := al }\n[PROOFSTEP]\ndsimp [map]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 Stream'.map (Option.map id) s = s\n[PROOFSTEP]\nrw [Option.map_id, Stream'.map_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 map f (tail { val := s, property := al }) = tail (map f { val := s, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 \u2191(map f (tail { val := s, property := al })) = \u2191(tail (map f { val := s, property := al }))\n[PROOFSTEP]\ndsimp [tail, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 map (g \u2218 f) { val := s, property := al } = map g (map f { val := s, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 \u2191(map (g \u2218 f) { val := s, property := al }) = \u2191(map g (map f { val := s, property := al }))\n[PROOFSTEP]\ndsimp [map]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 Stream'.map (Option.map (g \u2218 f)) s = Stream'.map (Option.map g \u2218 Option.map f) s\n[PROOFSTEP]\napply congr_arg fun f : _ \u2192 Option \u03b3 => Stream'.map f s\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\n\u22a2 Option.map (g \u2218 f) = Option.map g \u2218 Option.map f\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase a.h.none.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.map (g \u2218 f) none \u2194 a\u271d \u2208 (Option.map g \u2218 Option.map f) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.some.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : IsSeq s\nval\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.map (g \u2218 f) (some val\u271d) \u2194 a\u271d \u2208 (Option.map g \u2218 Option.map f) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns t : Seq \u03b1\n\u22a2 map f (append s t) = append (map f s) (map f t)\n[PROOFSTEP]\napply eq_of_bisim (fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)) _ \u27e8s, t, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns t : Seq \u03b1\n\u22a2 IsBisimulation fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns t : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\n\u22a2 BisimO (fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)) (destruct s1) (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | _, _, \u27e8s, t, rfl, rfl\u27e9 => by\n apply recOn s <;> simp\n \u00b7 apply recOn t <;> simp\n \u00b7 intro _ t\n refine' \u27e8nil, t, _, _\u27e9 <;> simp\n \u00b7 intro _ s\n refine' \u27e8s, t, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)) (destruct (map f (append s t)))\n (destruct (append (map f s) (map f t)))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t))\n (destruct (map f (append nil t))) (destruct (append (map f nil) (map f t)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t))\n (destruct (map f (append (cons x s) t))) (destruct (append (map f (cons x s)) (map f t)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 match destruct (map f t), destruct (map f t) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t, s = map f (append s_1 t) \u2227 s' = append (map f s_1) (map f t)\n | x, x_1 => False\n[PROOFSTEP]\napply recOn t\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 match destruct (map f nil), destruct (map f nil) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t, s = map f (append s_1 t) \u2227 s' = append (map f s_1) (map f t)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n match destruct (map f (cons x s)), destruct (map f (cons x s)) with\n | none, none => True\n | some (a, s), some (a', s') => a = a' \u2227 \u2203 s_1 t, s = map f (append s_1 t) \u2227 s' = append (map f s_1) (map f t)\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 \u03b1 \u2192 \u2200 (s : Seq \u03b1), \u2203 s_1 t, map f s = map f (append s_1 t) \u2227 map f s = append (map f s_1) (map f t)\n[PROOFSTEP]\nintro _ t\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d\u00b9 : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t\u271d : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 \u2203 s t_1, map f t = map f (append s t_1) \u2227 map f t = append (map f s) (map f t_1)\n[PROOFSTEP]\nrefine' \u27e8nil, t, _, _\u27e9\n[GOAL]\ncase h1.h2.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d\u00b9 : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t\u271d : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 map f t = map f (append nil t)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d\u00b9 : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t\u271d : Seq \u03b1\nx\u271d : \u03b1\nt : Seq \u03b1\n\u22a2 map f t = append (map f nil) (map f t)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns t : Seq \u03b1\n\u22a2 \u03b1 \u2192\n \u2200 (s : Seq \u03b1),\n \u2203 s_1 t_1,\n map f (append s t) = map f (append s_1 t_1) \u2227 append (map f s) (map f t) = append (map f s_1) (map f t_1)\n[PROOFSTEP]\nintro _ s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d\u00b9 t\u271d : Seq \u03b1\ns1 s2 : Seq \u03b2\nh : \u2203 s t, s1 = map f (append s t) \u2227 s2 = append (map f s) (map f t)\ns\u271d t : Seq \u03b1\nx\u271d : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 t_1, map f (append s t) = map f (append s_1 t_1) \u2227 append (map f s) (map f t) = append (map f s_1) (map f t_1)\n[PROOFSTEP]\nrefine' \u27e8s, t, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 destruct (join (cons (a, nil) S)) = some (a, join S)\n[PROOFSTEP]\nsimp [join]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 destruct (join (cons (a, cons b s) S)) = some (a, join (cons (b, s) S))\n[PROOFSTEP]\nsimp [join]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 join (cons (a, s) S) = cons a (append s (join S))\n[PROOFSTEP]\napply\n eq_of_bisim (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))) _\n (Or.inr \u27e8a, s, S, rfl, rfl\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 IsBisimulation fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))) (destruct s1)\n (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | s, _, Or.inl <| Eq.refl s => by\n apply recOn s; \u00b7 trivial\n \u00b7 intro x s\n rw [destruct_cons]\n exact \u27e8rfl, Or.inl rfl\u27e9\n | _, _, Or.inr \u27e8a, s, S, rfl, rfl\u27e9 => by\n apply recOn s\n \u00b7 simp [join_cons_cons, join_cons_nil]\n \u00b7 intro x s\n simp [join_cons_cons, join_cons_nil]\n refine' Or.inr \u27e8x, s, S, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns\u271d : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))) (destruct s)\n (destruct s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns\u271d : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))) (destruct nil)\n (destruct nil)\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns\u271d : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\ns : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (cons x s)) (destruct (cons x s))\n[PROOFSTEP]\nintro x s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns\u271d\u00b9 : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\ns\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (cons x s)) (destruct (cons x s))\n[PROOFSTEP]\nrw [destruct_cons]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns\u271d\u00b9 : Seq \u03b1\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\ns\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))) (some (x, s))\n (some (x, s))\n[PROOFSTEP]\nexact \u27e8rfl, Or.inl rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns\u271d : Seq \u03b1\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (join (cons (a, s) S))) (destruct (cons a (append s (join S))))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns\u271d : Seq \u03b1\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (join (cons (a, nil) S))) (destruct (cons a (append nil (join S))))\n[PROOFSTEP]\nsimp [join_cons_cons, join_cons_nil]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns\u271d : Seq \u03b1\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (join (cons (a, cons x s) S))) (destruct (cons a (append (cons x s) (join S))))\n[PROOFSTEP]\nintro x s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns\u271d\u00b9 : Seq \u03b1\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\na : \u03b1\ns\u271d : Seq \u03b1\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S)))\n (destruct (join (cons (a, cons x s) S))) (destruct (cons a (append (cons x s) (join S))))\n[PROOFSTEP]\nsimp [join_cons_cons, join_cons_nil]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns\u271d\u00b9 : Seq \u03b1\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : s1 = s2 \u2228 \u2203 a s S, s1 = join (cons (a, s) S) \u2227 s2 = cons a (append s (join S))\na : \u03b1\ns\u271d : Seq \u03b1\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 join (cons (x, s) S) = cons x (append s (join S)) \u2228\n \u2203 a s_1 S_1,\n join (cons (x, s) S) = join (cons (a, s_1) S_1) \u2227 cons x (append s (join S)) = cons a (append s_1 (join S_1))\n[PROOFSTEP]\nrefine' Or.inr \u27e8x, s, S, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\n\u22a2 join (append S T) = append (join S) (join T)\n[PROOFSTEP]\napply eq_of_bisim fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\n\u22a2 IsBisimulation fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\n\u22a2 BisimO (fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T)))\n (destruct s1) (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | _, _, \u27e8s, S, T, rfl, rfl\u27e9 => by\n apply recOn s <;> simp\n \u00b7 apply recOn S <;> simp\n \u00b7 apply recOn T\n \u00b7 simp\n \u00b7 intro s T\n cases' s with a s; simp\n refine' \u27e8s, nil, T, _, _\u27e9 <;> simp\n \u00b7 intro s S\n cases' s with a s; simp\n exact \u27e8s, S, T, rfl, rfl\u27e9\n \u00b7 intro _ s\n exact \u27e8s, S, T, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T)))\n (destruct (append s (join (append S T)))) (destruct (append s (append (join S) (join T))))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T)))\n (destruct (append nil (join (append S T)))) (destruct (append nil (append (join S) (join T))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T)))\n (destruct (append (cons x s) (join (append S T)))) (destruct (append (cons x s) (append (join S) (join T))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join (append S T)), destruct (append (join S) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\napply recOn S\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join (append nil T)), destruct (append (join nil) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : Seq1 \u03b1) (s : Seq (Seq1 \u03b1)),\n match destruct (join (append (cons x s) T)), destruct (append (join (cons x s)) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join T), destruct (join T) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\napply recOn T\n[GOAL]\ncase h1.h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join nil), destruct (join nil) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : Seq1 \u03b1) (s : Seq (Seq1 \u03b1)),\n match destruct (join (cons x s)), destruct (join (cons x s)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nintro s T\n[GOAL]\ncase h1.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T\u271d : Seq (Seq1 \u03b1)\ns : Seq1 \u03b1\nT : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join (cons s T)), destruct (join (cons s T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\ncases' s with a s\n[GOAL]\ncase h1.h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T\u271d T : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 match destruct (join (cons (a, s) T)), destruct (join (cons (a, s) T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T\u271d T : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 S T_1,\n append s (join T) = append s_1 (join (append S T_1)) \u2227 append s (join T) = append s_1 (append (join S) (join T_1))\n[PROOFSTEP]\nrefine' \u27e8s, nil, T, _, _\u27e9\n[GOAL]\ncase h1.h1.h2.mk.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T\u271d T : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 append s (join T) = append s (join (append nil T))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.h2.mk.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T\u271d T : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 append s (join T) = append s (append (join nil) (join T))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : Seq1 \u03b1) (s : Seq (Seq1 \u03b1)),\n match destruct (join (cons x (append s T))), destruct (append (join (cons x s)) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nintro s S\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d\u00b9 T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS\u271d T : Seq (Seq1 \u03b1)\ns : Seq1 \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 match destruct (join (cons s (append S T))), destruct (append (join (cons s S)) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\ncases' s with a s\n[GOAL]\ncase h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d\u00b9 T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS\u271d T S : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 match destruct (join (cons (a, s) (append S T))), destruct (append (join (cons (a, s) S)) (join T)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S T, s = append s_1 (join (append S T)) \u2227 s' = append s_1 (append (join S) (join T))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d\u00b9 T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS\u271d T S : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 S_1 T_1,\n append s (join (append S T)) = append s_1 (join (append S_1 T_1)) \u2227\n append s (append (join S) (join T)) = append s_1 (append (join S_1) (join T_1))\n[PROOFSTEP]\nexact \u27e8s, S, T, rfl, rfl\u27e9\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u03b1 \u2192\n \u2200 (s : Seq \u03b1),\n \u2203 s_1 S_1 T_1,\n append s (join (append S T)) = append s_1 (join (append S_1 T_1)) \u2227\n append s (append (join S) (join T)) = append s_1 (append (join S_1) (join T_1))\n[PROOFSTEP]\nintro _ s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS\u271d T\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b1\nh : \u2203 s S T, s1 = append s (join (append S T)) \u2227 s2 = append s (append (join S) (join T))\ns\u271d : Seq \u03b1\nS T : Seq (Seq1 \u03b1)\nx\u271d : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 S_1 T_1,\n append s (join (append S T)) = append s_1 (join (append S_1 T_1)) \u2227\n append s (append (join S) (join T)) = append s_1 (append (join S_1) (join T_1))\n[PROOFSTEP]\nexact \u27e8s, S, T, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\n\u22a2 \u2203 s S_1 T_1,\n join (append S T) = append s (join (append S_1 T_1)) \u2227\n append (join S) (join T) = append s (append (join S_1) (join T_1))\n[PROOFSTEP]\nrefine' \u27e8nil, S, T, _, _\u27e9\n[GOAL]\ncase r.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\n\u22a2 join (append S T) = append nil (join (append S T))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase r.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nS T : Seq (Seq1 \u03b1)\n\u22a2 append (join S) (join T) = append nil (append (join S) (join T))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Stream' \u03b1\n\u22a2 \u2191(a :: s) = cons a \u2191s\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Stream' \u03b1\n\u22a2 \u2191\u2191(a :: s) = \u2191(cons a \u2191s)\n[PROOFSTEP]\nsimp [ofStream, cons]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Stream' \u03b1\n\u22a2 Stream'.map some (a :: s) = some a :: Stream'.map some s\n[PROOFSTEP]\nrw [Stream'.map_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl l' : List \u03b1\n\u22a2 \u2191(l ++ l') = append \u2191l \u2191l'\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\n\u22a2 \u2191([] ++ l') = append \u2191[] \u2191l'\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2191(tail\u271d ++ l') = append \u2191tail\u271d \u2191l'\n\u22a2 \u2191(head\u271d :: tail\u271d ++ l') = append \u2191(head\u271d :: tail\u271d) \u2191l'\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\ns : Stream' \u03b1\n\u22a2 \u2191(l ++\u209b s) = append \u2191l \u2191s\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' \u03b1\n\u22a2 \u2191([] ++\u209b s) = append \u2191[] \u2191s\n[PROOFSTEP]\nsimp [*, Stream'.nil_append_stream, Stream'.cons_append_stream]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Stream' \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2191(tail\u271d ++\u209b s) = append \u2191tail\u271d \u2191s\n\u22a2 \u2191(head\u271d :: tail\u271d ++\u209b s) = append \u2191(head\u271d :: tail\u271d) \u2191s\n[PROOFSTEP]\nsimp [*, Stream'.nil_append_stream, Stream'.cons_append_stream]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 drop (tail s) n = drop s (n + 1)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 drop (tail s) n = drop s (1 + n)\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 drop s (1 + n) = drop (tail s) n\n[PROOFSTEP]\napply dropn_add\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 head (drop s n) = get? s n\n[PROOFSTEP]\ninduction' n with n IH generalizing s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 head (drop s Nat.zero) = get? s Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Seq \u03b1\nn : \u2115\nIH : \u2200 (s : Seq \u03b1), head (drop s n) = get? s n\ns : Seq \u03b1\n\u22a2 head (drop s (Nat.succ n)) = get? s (Nat.succ n)\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, \u2190 get?_tail, \u2190 dropn_tail]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Seq \u03b1\nn : \u2115\nIH : \u2200 (s : Seq \u03b1), head (drop s n) = get? s n\ns : Seq \u03b1\n\u22a2 head (drop (tail s) n) = get? (tail s) n\n[PROOFSTEP]\napply IH\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\nh : b \u2208 map f s\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = b\n[PROOFSTEP]\nmatch s with\n| \u27e8g, al\u27e9 =>\n let \u27e8o, om, oe\u27e9 := @Stream'.exists_of_mem_map _ _ (Option.map f) (some b) g h\n cases' o with a\n \u00b7 injection oe\n \u00b7 injection oe with h'; exact \u27e8a, om, h'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\ng : Stream' (Option \u03b1)\nal : IsSeq g\nh : b \u2208 map f { val := g, property := al }\n\u22a2 \u2203 a, a \u2208 { val := g, property := al } \u2227 f a = b\n[PROOFSTEP]\nlet \u27e8o, om, oe\u27e9 := @Stream'.exists_of_mem_map _ _ (Option.map f) (some b) g h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\ng : Stream' (Option \u03b1)\nal : IsSeq g\nh : b \u2208 map f { val := g, property := al }\no : Option \u03b1\nom : o \u2208 g\noe : Option.map f o = some b\n\u22a2 \u2203 a, a \u2208 { val := g, property := al } \u2227 f a = b\n[PROOFSTEP]\ncases' o with a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\ng : Stream' (Option \u03b1)\nal : IsSeq g\nh : b \u2208 map f { val := g, property := al }\nom : none \u2208 g\noe : Option.map f none = some b\n\u22a2 \u2203 a, a \u2208 { val := g, property := al } \u2227 f a = b\n[PROOFSTEP]\ninjection oe\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\ng : Stream' (Option \u03b1)\nal : IsSeq g\nh : b \u2208 map f { val := g, property := al }\na : \u03b1\nom : some a \u2208 g\noe : Option.map f (some a) = some b\n\u22a2 \u2203 a, a \u2208 { val := g, property := al } \u2227 f a = b\n[PROOFSTEP]\ninjection oe with h'\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Seq \u03b1\ng : Stream' (Option \u03b1)\nal : IsSeq g\nh : b \u2208 map f { val := g, property := al }\na : \u03b1\nom : some a \u2208 g\nh' : f a = b\n\u22a2 \u2203 a, a \u2208 { val := g, property := al } \u2227 f a = b\n[PROOFSTEP]\nexact \u27e8a, om, h'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 append s\u2081 s\u2082\n\u22a2 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nhave := h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh this : a \u2208 append s\u2081 s\u2082\n\u22a2 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nrevert this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 append s\u2081 s\u2082\n\u22a2 a \u2208 append s\u2081 s\u2082 \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\ngeneralize e : append s\u2081 s\u2082 = ss\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 append s\u2081 s\u2082\nss : Seq \u03b1\ne : append s\u2081 s\u2082 = ss\n\u22a2 a \u2208 ss \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh\u271d : a \u2208 append s\u2081 s\u2082\nss : Seq \u03b1\ne : append s\u2081 s\u2082 = ss\nh : a \u2208 ss\n\u22a2 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nrevert s\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\n\u22a2 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = ss \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\napply mem_rec_on h _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\n\u22a2 \u2200 (b : \u03b1) (s' : Seq \u03b1),\n (a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082) \u2192\n \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = cons b s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nintro b s' o s\u2081\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\n\u22a2 a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = cons b s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\napply s\u2081.recOn _ fun c t\u2081 => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\n\u22a2 a \u2208 append nil s\u2082 \u2192 append nil s\u2082 = cons b s' \u2192 a \u2208 nil \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nintro m _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nm : a \u2208 append nil s\u2082\ne\u271d : append nil s\u2082 = cons b s'\n\u22a2 a \u2208 nil \u2228 a \u2208 s\u2082\n[PROOFSTEP]\napply Or.inr\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nm : a \u2208 append nil s\u2082\ne\u271d : append nil s\u2082 = cons b s'\n\u22a2 a \u2208 s\u2082\n[PROOFSTEP]\nsimpa using m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\n\u22a2 \u2200 (c : \u03b1) (t\u2081 : Seq \u03b1), a \u2208 append (cons c t\u2081) s\u2082 \u2192 append (cons c t\u2081) s\u2082 = cons b s' \u2192 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nintro c t\u2081 m e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nhave this := congr_arg destruct e\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\ncases' show a = c \u2228 a \u2208 append t\u2081 s\u2082 by simpa using m with e' m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\n\u22a2 a = c \u2228 a \u2208 append t\u2081 s\u2082\n[PROOFSTEP]\nsimpa using m\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\ne' : a = c\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nrw [e']\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\ne' : a = c\n\u22a2 c \u2208 cons c t\u2081 \u2228 c \u2208 s\u2082\n[PROOFSTEP]\nexact Or.inl (mem_cons _ _)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm\u271d : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\nm : a \u2208 append t\u2081 s\u2082\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\ncases' show c = b \u2227 append t\u2081 s\u2082 = s' by simpa with i1 i2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm\u271d : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\nm : a \u2208 append t\u2081 s\u2082\n\u22a2 c = b \u2227 append t\u2081 s\u2082 = s'\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' : Seq \u03b1\no : a = b \u2228 \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\ns\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm\u271d : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\nm : a \u2208 append t\u2081 s\u2082\ni1 : c = b\ni2 : append t\u2081 s\u2082 = s'\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\ncases' o with e' IH\n[GOAL]\ncase inr.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' s\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm\u271d : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\nm : a \u2208 append t\u2081 s\u2082\ni1 : c = b\ni2 : append t\u2081 s\u2082 = s'\ne' : a = b\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nsimp [i1, e']\n[GOAL]\ncase inr.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 : Seq \u03b1\na : \u03b1\nss : Seq \u03b1\nh : a \u2208 ss\nb : \u03b1\ns' s\u2081 : Seq \u03b1\nc : \u03b1\nt\u2081 : Seq \u03b1\nm\u271d : a \u2208 append (cons c t\u2081) s\u2082\ne : append (cons c t\u2081) s\u2082 = cons b s'\nthis : destruct (append (cons c t\u2081) s\u2082) = destruct (cons b s')\nm : a \u2208 append t\u2081 s\u2082\ni1 : c = b\ni2 : append t\u2081 s\u2082 = s'\nIH : \u2200 {s\u2081 : Seq \u03b1}, a \u2208 append s\u2081 s\u2082 \u2192 append s\u2081 s\u2082 = s' \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n\u22a2 a \u2208 cons c t\u2081 \u2228 a \u2208 s\u2082\n[PROOFSTEP]\nexact Or.imp_left (mem_cons_of_mem _) (IH m i2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 s\u2081\n\u22a2 a \u2208 append s\u2081 s\u2082\n[PROOFSTEP]\napply mem_rec_on h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 s\u2081\n\u22a2 \u2200 (b : \u03b1) (s' : Seq \u03b1), a = b \u2228 a \u2208 append s' s\u2082 \u2192 a \u2208 append (cons b s') s\u2082\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2081 s\u2082 : Seq \u03b1\na : \u03b1\nh : a \u2208 s\u2081\nb\u271d : \u03b1\ns'\u271d : Seq \u03b1\na\u271d : a = b\u271d \u2228 a \u2208 append s'\u271d s\u2082\n\u22a2 a \u2208 append (cons b\u271d s'\u271d) s\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx : \u03b1\n\u22a2 enum (cons x s) = cons (0, x) (map (Prod.map Nat.succ id) (enum s))\n[PROOFSTEP]\next \u27e8n\u27e9 : 1\n[GOAL]\ncase h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx : \u03b1\n\u22a2 get? (enum (cons x s)) Nat.zero = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx : \u03b1\nn\u271d : \u2115\n\u22a2 get? (enum (cons x s)) (Nat.succ n\u271d) = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [get?_enum, get?_cons_succ, map_get?, Option.map_map]\n[GOAL]\ncase h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nx : \u03b1\nn\u271d : \u2115\n\u22a2 Option.map (Prod.mk (Nat.succ n\u271d)) (get? s n\u271d) = Option.map (Prod.map Nat.succ id \u2218 Prod.mk n\u271d) (get? s n\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Seq \u03b1\n\u22a2 map id (a, s) = (a, s)\n[PROOFSTEP]\nsimp [map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 join ((a, Seq.cons b s), S) = (a, Seq.join (Seq.cons (b, s) S))\n[PROOFSTEP]\ndsimp [join]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na b : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 (match destruct (Seq.cons b s) with\n | none => (a, Seq.join S)\n | some s' => (a, Seq.join (Seq.cons s' S))) =\n (a, Seq.join (Seq.cons (b, s) S))\n[PROOFSTEP]\nrw [destruct_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 Seq.join (Seq.map ret s) = s\n[PROOFSTEP]\napply coinduction2 s\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u2200 (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s, s1 = Seq.join (Seq.map ret s) \u2227 s2 = s) (destruct (Seq.join (Seq.map ret s))) (destruct s)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = Seq.join (Seq.map ret s) \u2227 s2 = s) (destruct (Seq.join (Seq.map ret s))) (destruct s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase H.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 BisimO (fun s1 s2 => \u2203 s, s1 = Seq.join (Seq.map ret s) \u2227 s2 = s) (destruct (Seq.join (Seq.map ret nil)))\n (destruct nil)\n[PROOFSTEP]\nsimp [ret]\n[GOAL]\ncase H.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s, s1 = Seq.join (Seq.map ret s) \u2227 s2 = s) (destruct (Seq.join (Seq.map ret (Seq.cons x s))))\n (destruct (Seq.cons x s))\n[PROOFSTEP]\nsimp [ret]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\n\u22a2 bind (a, s) (ret \u2218 f) = map f (a, s)\n[PROOFSTEP]\ndsimp [bind, map]\n -- Porting note: Was `rw [map_comp]; simp [Function.comp, ret]`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\n\u22a2 join (ret (f a), Seq.map (ret \u2218 f) s) = (f a, Seq.map f s)\n[PROOFSTEP]\nrw [map_comp, ret]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\n\u22a2 join ((f a, nil), Seq.map ret (Seq.map f s)) = (f a, Seq.map f s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\n\u22a2 bind (ret a) f = f a\n[PROOFSTEP]\nsimp [ret, bind, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\n\u22a2 join (f a, nil) = f a\n[PROOFSTEP]\ncases' f a with a s\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\na : \u03b2\ns : Seq \u03b2\n\u22a2 join ((a, s), nil) = (a, s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\na : \u03b2\ns : Seq \u03b2\n\u22a2 join ((a, nil), nil) = (a, nil)\n[PROOFSTEP]\nintros\n[GOAL]\ncase mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\na : \u03b2\ns : Seq \u03b2\n\u22a2 \u2200 (x : \u03b2) (s : Seq \u03b2), join ((a, Seq.cons x s), nil) = (a, Seq.cons x s)\n[PROOFSTEP]\nintros\n[GOAL]\ncase mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\na : \u03b2\ns : Seq \u03b2\n\u22a2 join ((a, nil), nil) = (a, nil)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\na : \u03b2\ns : Seq \u03b2\nx\u271d : \u03b2\ns\u271d : Seq \u03b2\n\u22a2 join ((a, Seq.cons x\u271d s\u271d), nil) = (a, Seq.cons x\u271d s\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 Seq.map f (Seq.join S) = Seq.join (Seq.map (map f) S)\n[PROOFSTEP]\napply\n Seq.eq_of_bisim fun s1 s2 =>\n \u2203 s S, s1 = Seq.append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 Seq.IsBisimulation fun s1 s2 =>\n \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\n\u22a2 BisimO (fun s1 s2 => \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S)))\n (destruct s1) (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | _, _, \u27e8s, S, rfl, rfl\u27e9 => by\n apply recOn s <;> simp\n \u00b7 apply recOn S <;> simp\n \u00b7 intro x S\n cases' x with a s; simp [map]\n exact \u27e8_, _, rfl, rfl\u27e9\n \u00b7 intro _ s\n refine' \u27e8s, S, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S)))\n (destruct (append s (Seq.map f (Seq.join S)))) (destruct (append s (Seq.join (Seq.map (map f) S))))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 BisimO (fun s1 s2 => \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S)))\n (destruct (append nil (Seq.map f (Seq.join S)))) (destruct (append nil (Seq.join (Seq.map (map f) S))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : \u03b2) (s : Seq \u03b2),\n BisimO (fun s1 s2 => \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S)))\n (destruct (append (Seq.cons x s) (Seq.map f (Seq.join S))))\n (destruct (append (Seq.cons x s) (Seq.join (Seq.map (map f) S))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 match destruct (Seq.map f (Seq.join S)), destruct (Seq.join (Seq.map (map f) S)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\napply recOn S\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 match destruct (Seq.map f (Seq.join nil)), destruct (Seq.join (Seq.map (map f) nil)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : Seq1 \u03b1) (s : Seq (Seq1 \u03b1)),\n match destruct (Seq.map f (Seq.join (Seq.cons x s))), destruct (Seq.join (Seq.map (map f) (Seq.cons x s))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : Seq1 \u03b1) (s : Seq (Seq1 \u03b1)),\n match destruct (Seq.map f (Seq.join (Seq.cons x s))),\n destruct (Seq.join (Seq.cons (map f x) (Seq.map (map f) s))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\nintro x S\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS\u271d : Seq (Seq1 \u03b1)\nx : Seq1 \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 match destruct (Seq.map f (Seq.join (Seq.cons x S))),\n destruct (Seq.join (Seq.cons (map f x) (Seq.map (map f) S))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\ncases' x with a s\n[GOAL]\ncase h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns\u271d : Seq \u03b2\nS\u271d S : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 match destruct (Seq.map f (Seq.join (Seq.cons (a, s) S))),\n destruct (Seq.join (Seq.cons (map f (a, s)) (Seq.map (map f) S))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 S, s = append s_1 (Seq.map f (Seq.join S)) \u2227 s' = append s_1 (Seq.join (Seq.map (map f) S))\n | x, x_1 => False\n[PROOFSTEP]\nsimp [map]\n[GOAL]\ncase h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d\u00b9 : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns\u271d : Seq \u03b2\nS\u271d S : Seq (Seq1 \u03b1)\na : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 S_1,\n append (Seq.map f s) (Seq.map f (Seq.join S)) = append s_1 (Seq.map f (Seq.join S_1)) \u2227\n append (Seq.map f s)\n (Seq.join\n (Seq.map\n (fun x =>\n match x with\n | (a, s) => (f a, Seq.map f s))\n S)) =\n append s_1\n (Seq.join\n (Seq.map\n (fun x =>\n match x with\n | (a, s) => (f a, Seq.map f s))\n S_1))\n[PROOFSTEP]\nexact \u27e8_, _, rfl, rfl\u27e9\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns : Seq \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 \u03b2 \u2192\n \u2200 (s : Seq \u03b2),\n \u2203 s_1 S_1,\n append s (Seq.map f (Seq.join S)) = append s_1 (Seq.map f (Seq.join S_1)) \u2227\n append s (Seq.join (Seq.map (map f) S)) = append s_1 (Seq.join (Seq.map (map f) S_1))\n[PROOFSTEP]\nintro _ s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS\u271d : Seq (Seq1 \u03b1)\ns1 s2 : Seq \u03b2\nh : \u2203 s S, s1 = append s (Seq.map f (Seq.join S)) \u2227 s2 = append s (Seq.join (Seq.map (map f) S))\ns\u271d : Seq \u03b2\nS : Seq (Seq1 \u03b1)\nx\u271d : \u03b2\ns : Seq \u03b2\n\u22a2 \u2203 s_1 S_1,\n append s (Seq.map f (Seq.join S)) = append s_1 (Seq.map f (Seq.join S_1)) \u2227\n append s (Seq.join (Seq.map (map f) S)) = append s_1 (Seq.join (Seq.map (map f) S_1))\n[PROOFSTEP]\nrefine' \u27e8s, S, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2203 s S_1,\n Seq.map f (Seq.join S) = append s (Seq.map f (Seq.join S_1)) \u2227\n Seq.join (Seq.map (map f) S) = append s (Seq.join (Seq.map (map f) S_1))\n[PROOFSTEP]\nrefine' \u27e8nil, S, _, _\u27e9\n[GOAL]\ncase r.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 Seq.map f (Seq.join S) = append nil (Seq.map f (Seq.join S))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase r.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nS : Seq (Seq1 \u03b1)\n\u22a2 Seq.join (Seq.map (map f) S) = append nil (Seq.join (Seq.map (map f) S))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 map f (join ((a, s), S)) = join (map (map f) ((a, s), S))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 map f (join ((a, nil), S)) = join (map (map f) ((a, nil), S))\n[PROOFSTEP]\nintros\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1), map f (join ((a, Seq.cons x s), S)) = join (map (map f) ((a, Seq.cons x s), S))\n[PROOFSTEP]\nintros\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 map f (join ((a, nil), S)) = join (map (map f) ((a, nil), S))\n[PROOFSTEP]\nsimp [map]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\nx\u271d : \u03b1\ns\u271d : Seq \u03b1\n\u22a2 map f (join ((a, Seq.cons x\u271d s\u271d), S)) = join (map (map f) ((a, Seq.cons x\u271d s\u271d), S))\n[PROOFSTEP]\nsimp [map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 Seq.join (Seq.join SS) = Seq.join (Seq.map join SS)\n[PROOFSTEP]\napply\n Seq.eq_of_bisim fun s1 s2 =>\n \u2203 s SS, s1 = Seq.append s (Seq.join (Seq.join SS)) \u2227 s2 = Seq.append s (Seq.join (Seq.map join SS))\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 Seq.IsBisimulation fun s1 s2 =>\n \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\n[PROOFSTEP]\nintro s1 s2 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\n\u22a2 BisimO (fun s1 s2 => \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS)))\n (destruct s1) (destruct s2)\n[PROOFSTEP]\nexact\n match s1, s2, h with\n | _, _, \u27e8s, SS, rfl, rfl\u27e9 => by\n apply recOn s <;> simp\n \u00b7 apply recOn SS <;> simp\n \u00b7 intro S SS\n cases' S with s S; cases' s with x s; simp [map]\n apply recOn s <;> simp\n \u00b7 exact \u27e8_, _, rfl, rfl\u27e9\n \u00b7 intro x s\n refine' \u27e8Seq.cons x (append s (Seq.join S)), SS, _, _\u27e9 <;> simp\n \u00b7 intro _ s\n exact \u27e8s, SS, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 BisimO (fun s1 s2 => \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS)))\n (destruct (append s (Seq.join (Seq.join SS)))) (destruct (append s (Seq.join (Seq.map join SS))))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 BisimO (fun s1 s2 => \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS)))\n (destruct (append nil (Seq.join (Seq.join SS)))) (destruct (append nil (Seq.join (Seq.map join SS))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n BisimO (fun s1 s2 => \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS)))\n (destruct (append (Seq.cons x s) (Seq.join (Seq.join SS))))\n (destruct (append (Seq.cons x s) (Seq.join (Seq.map join SS))))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 match destruct (Seq.join (Seq.join SS)), destruct (Seq.join (Seq.map join SS)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\napply recOn SS\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 match destruct (Seq.join (Seq.join nil)), destruct (Seq.join (Seq.map join nil)) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 \u2200 (x : Seq1 (Seq1 \u03b1)) (s : Seq (Seq1 (Seq1 \u03b1))),\n match destruct (Seq.join (Seq.join (Seq.cons x s))), destruct (Seq.join (Seq.map join (Seq.cons x s))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 \u2200 (x : Seq1 (Seq1 \u03b1)) (s : Seq (Seq1 (Seq1 \u03b1))),\n match destruct (Seq.join (Seq.join (Seq.cons x s))), destruct (Seq.join (Seq.cons (join x) (Seq.map join s))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\nintro S SS\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\nS : Seq1 (Seq1 \u03b1)\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 match destruct (Seq.join (Seq.join (Seq.cons S SS))), destruct (Seq.join (Seq.cons (join S) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\ncases' S with s S\n[GOAL]\ncase h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\ns : Seq1 \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 match destruct (Seq.join (Seq.join (Seq.cons (s, S) SS))),\n destruct (Seq.join (Seq.cons (join (s, S)) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\ncases' s with x s\n[GOAL]\ncase h1.h2.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 match destruct (Seq.join (Seq.join (Seq.cons ((x, s), S) SS))),\n destruct (Seq.join (Seq.cons (join ((x, s), S)) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\nsimp [map]\n[GOAL]\ncase h1.h2.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 match some (x, append s (append (Seq.join S) (Seq.join (Seq.join SS)))),\n destruct (Seq.join (Seq.cons (join ((x, s), S)) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1.h2.mk.mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 match some (x, append nil (append (Seq.join S) (Seq.join (Seq.join SS)))),\n destruct (Seq.join (Seq.cons (join ((x, nil), S)) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.mk.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 \u2200 (x_1 : \u03b1) (s : Seq \u03b1),\n match some (x, append (Seq.cons x_1 s) (append (Seq.join S) (Seq.join (Seq.join SS)))),\n destruct (Seq.join (Seq.cons (join ((x, Seq.cons x_1 s), S)) (Seq.map join SS))) with\n | none, none => True\n | some (a, s), some (a', s') =>\n a = a' \u2227 \u2203 s_1 SS, s = append s_1 (Seq.join (Seq.join SS)) \u2227 s' = append s_1 (Seq.join (Seq.map join SS))\n | x, x_2 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.mk.mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s SS_1,\n append (Seq.join S) (Seq.join (Seq.join SS)) = append s (Seq.join (Seq.join SS_1)) \u2227\n append (Seq.join S) (Seq.join (Seq.map join SS)) = append s (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nexact \u27e8_, _, rfl, rfl\u27e9\n[GOAL]\ncase h1.h2.mk.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx : \u03b1\ns : Seq \u03b1\n\u22a2 \u2200 (x : \u03b1) (s : Seq \u03b1),\n \u2203 s_1 SS_1,\n Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.join SS)))) = append s_1 (Seq.join (Seq.join SS_1)) \u2227\n Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.map join SS)))) =\n append s_1 (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nintro x s\n[GOAL]\ncase h1.h2.mk.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d\u00b9 : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx\u271d : \u03b1\ns\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 SS_1,\n Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.join SS)))) = append s_1 (Seq.join (Seq.join SS_1)) \u2227\n Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.map join SS)))) =\n append s_1 (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nrefine' \u27e8Seq.cons x (append s (Seq.join S)), SS, _, _\u27e9\n[GOAL]\ncase h1.h2.mk.mk.h2.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d\u00b9 : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx\u271d : \u03b1\ns\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.join SS)))) =\n append (Seq.cons x (append s (Seq.join S))) (Seq.join (Seq.join SS))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2.mk.mk.h2.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d\u00b9 : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d\u00b9 : Seq \u03b1\nSS\u271d SS : Seq (Seq1 (Seq1 \u03b1))\nS : Seq (Seq1 \u03b1)\nx\u271d : \u03b1\ns\u271d : Seq \u03b1\nx : \u03b1\ns : Seq \u03b1\n\u22a2 Seq.cons x (append s (append (Seq.join S) (Seq.join (Seq.map join SS)))) =\n append (Seq.cons x (append s (Seq.join S))) (Seq.join (Seq.map join SS))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 \u03b1 \u2192\n \u2200 (s : Seq \u03b1),\n \u2203 s_1 SS_1,\n append s (Seq.join (Seq.join SS)) = append s_1 (Seq.join (Seq.join SS_1)) \u2227\n append s (Seq.join (Seq.map join SS)) = append s_1 (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nintro _ s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS\u271d : Seq (Seq1 (Seq1 \u03b1))\ns1 s2 : Seq \u03b1\nh : \u2203 s SS, s1 = append s (Seq.join (Seq.join SS)) \u2227 s2 = append s (Seq.join (Seq.map join SS))\ns\u271d : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b1))\nx\u271d : \u03b1\ns : Seq \u03b1\n\u22a2 \u2203 s_1 SS_1,\n append s (Seq.join (Seq.join SS)) = append s_1 (Seq.join (Seq.join SS_1)) \u2227\n append s (Seq.join (Seq.map join SS)) = append s_1 (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nexact \u27e8s, SS, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 \u2203 s SS_1,\n Seq.join (Seq.join SS) = append s (Seq.join (Seq.join SS_1)) \u2227\n Seq.join (Seq.map join SS) = append s (Seq.join (Seq.map join SS_1))\n[PROOFSTEP]\nrefine' \u27e8nil, SS, _, _\u27e9\n[GOAL]\ncase r.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 Seq.join (Seq.join SS) = append nil (Seq.join (Seq.join SS))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase r.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nSS : Seq (Seq1 (Seq1 \u03b1))\n\u22a2 Seq.join (Seq.map join SS) = append nil (Seq.join (Seq.map join SS))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq1 \u03b1\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\n\u22a2 bind (bind s f) g = bind s fun x => bind (f x) g\n[PROOFSTEP]\ncases' s with a s\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\n\u22a2 bind (bind (a, s) f) g = bind (a, s) fun x => bind (f x) g\n[PROOFSTEP]\nsimp only [bind, map_pair, map_join]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\n\u22a2 join (join (map g (f a), Seq.map (map g) (Seq.map f s))) =\n join (join (map g (f a)), Seq.map (fun x => join (map g (f x))) s)\n[PROOFSTEP]\nrw [\u2190 map_comp]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\n\u22a2 join (join (map g (f a), Seq.map (map g \u2218 f) s)) = join (join (map g (f a)), Seq.map (fun x => join (map g (f x))) s)\n[PROOFSTEP]\nsimp only [show (fun x => join (map g (f x))) = join \u2218 (map g \u2218 f) from rfl]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\n\u22a2 join (join (map g (f a), Seq.map (map g \u2218 f) s)) = join (join (map g (f a)), Seq.map (join \u2218 map g \u2218 f) s)\n[PROOFSTEP]\nrw [map_comp _ join]\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\n\u22a2 join (join (map g (f a), Seq.map (map g \u2218 f) s)) = join (join (map g (f a)), Seq.map join (Seq.map (map g \u2218 f) s))\n[PROOFSTEP]\ngeneralize Seq.map (map g \u2218 f) s = SS\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\n\u22a2 join (join (map g (f a), SS)) = join (join (map g (f a)), Seq.map join SS)\n[PROOFSTEP]\nrcases map g (f a) with\n \u27e8\u27e8a, s\u27e9, S\u27e9\n -- Porting note: Instead of `apply recOn s <;> intros`, `induction'` are used to\n -- give names to variables.\n[GOAL]\ncase mk.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns\u271d : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\nS : Seq (Seq1 \u03b3)\na : \u03b3\ns : Seq \u03b3\n\u22a2 join (join (((a, s), S), SS)) = join (join ((a, s), S), Seq.map join SS)\n[PROOFSTEP]\ninduction' s using recOn with x s_1\n[GOAL]\ncase mk.mk.mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\nS : Seq (Seq1 \u03b3)\na : \u03b3\n\u22a2 join (join (((a, nil), S), SS)) = join (join ((a, nil), S), Seq.map join SS)\n[PROOFSTEP]\ninduction' S using recOn with x_1 s_2\n[GOAL]\ncase mk.mk.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\nS : Seq (Seq1 \u03b3)\na x : \u03b3\ns_1 : Seq \u03b3\n\u22a2 join (join (((a, Seq.cons x s_1), S), SS)) = join (join ((a, Seq.cons x s_1), S), Seq.map join SS)\n[PROOFSTEP]\ninduction' S using recOn with x_1 s_2\n[GOAL]\ncase mk.mk.mk.h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\n\u22a2 join (join (((a, nil), nil), SS)) = join (join ((a, nil), nil), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\nx_1 : Seq1 \u03b3\ns_2 : Seq (Seq1 \u03b3)\n\u22a2 join (join (((a, nil), Seq.cons x_1 s_2), SS)) = join (join ((a, nil), Seq.cons x_1 s_2), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h2.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na x : \u03b3\ns_1 : Seq \u03b3\n\u22a2 join (join (((a, Seq.cons x s_1), nil), SS)) = join (join ((a, Seq.cons x s_1), nil), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h2.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na x : \u03b3\ns_1 : Seq \u03b3\nx_1 : Seq1 \u03b3\ns_2 : Seq (Seq1 \u03b3)\n\u22a2 join (join (((a, Seq.cons x s_1), Seq.cons x_1 s_2), SS)) =\n join (join ((a, Seq.cons x s_1), Seq.cons x_1 s_2), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\nx_1 : Seq1 \u03b3\ns_2 : Seq (Seq1 \u03b3)\n\u22a2 (a, Seq.join (Seq.cons x_1 (append s_2 (Seq.join SS)))) = join ((a, Seq.join (Seq.cons x_1 s_2)), Seq.map join SS)\n[PROOFSTEP]\ncases' x_1 with x t\n[GOAL]\ncase mk.mk.mk.h1.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\ns_2 : Seq (Seq1 \u03b3)\nx : \u03b3\nt : Seq \u03b3\n\u22a2 (a, Seq.join (Seq.cons (x, t) (append s_2 (Seq.join SS)))) =\n join ((a, Seq.join (Seq.cons (x, t) s_2)), Seq.map join SS)\n[PROOFSTEP]\napply recOn t\n[GOAL]\ncase mk.mk.mk.h1.h2.mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\ns_2 : Seq (Seq1 \u03b3)\nx : \u03b3\nt : Seq \u03b3\n\u22a2 (a, Seq.join (Seq.cons (x, nil) (append s_2 (Seq.join SS)))) =\n join ((a, Seq.join (Seq.cons (x, nil) s_2)), Seq.map join SS)\n[PROOFSTEP]\nintros\n[GOAL]\ncase mk.mk.mk.h1.h2.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\ns_2 : Seq (Seq1 \u03b3)\nx : \u03b3\nt : Seq \u03b3\n\u22a2 \u2200 (x_1 : \u03b3) (s : Seq \u03b3),\n (a, Seq.join (Seq.cons (x, Seq.cons x_1 s) (append s_2 (Seq.join SS)))) =\n join ((a, Seq.join (Seq.cons (x, Seq.cons x_1 s) s_2)), Seq.map join SS)\n[PROOFSTEP]\nintros\n[GOAL]\ncase mk.mk.mk.h1.h2.mk.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\ns_2 : Seq (Seq1 \u03b3)\nx : \u03b3\nt : Seq \u03b3\n\u22a2 (a, Seq.join (Seq.cons (x, nil) (append s_2 (Seq.join SS)))) =\n join ((a, Seq.join (Seq.cons (x, nil) s_2)), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h1.h2.mk.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na : \u03b3\ns_2 : Seq (Seq1 \u03b3)\nx : \u03b3\nt : Seq \u03b3\nx\u271d : \u03b3\ns\u271d : Seq \u03b3\n\u22a2 (a, Seq.join (Seq.cons (x, Seq.cons x\u271d s\u271d) (append s_2 (Seq.join SS)))) =\n join ((a, Seq.join (Seq.cons (x, Seq.cons x\u271d s\u271d) s_2)), Seq.map join SS)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.h2.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na x : \u03b3\ns_1 : Seq \u03b3\nx_1 : Seq1 \u03b3\ns_2 : Seq (Seq1 \u03b3)\n\u22a2 (a, Seq.cons x (append s_1 (Seq.join (Seq.cons x_1 (append s_2 (Seq.join SS)))))) =\n (a, Seq.cons x (append s_1 (append (Seq.join (Seq.cons x_1 s_2)) (Seq.join (Seq.map join SS)))))\n[PROOFSTEP]\ncases' x_1 with y t\n[GOAL]\ncase mk.mk.mk.h2.h2.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 Seq1 \u03b2\ng : \u03b2 \u2192 Seq1 \u03b3\na\u271d : \u03b1\ns : Seq \u03b1\nSS : Seq (Seq1 (Seq1 \u03b3))\na x : \u03b3\ns_1 : Seq \u03b3\ns_2 : Seq (Seq1 \u03b3)\ny : \u03b3\nt : Seq \u03b3\n\u22a2 (a, Seq.cons x (append s_1 (Seq.join (Seq.cons (y, t) (append s_2 (Seq.join SS)))))) =\n (a, Seq.cons x (append s_1 (append (Seq.join (Seq.cons (y, t) s_2)) (Seq.join (Seq.map join SS)))))\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Seq.Seq", "llama_tokens": 69499, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.2758066636828176}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 \u00acSupIrred a \u2194 IsMin a \u2228 \u2203 b c, b \u2294 c = a \u2227 b < a \u2227 c < a\n[PROOFSTEP]\nrw [SupIrred, not_and_or]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 (\u00ac\u00acIsMin a \u2228 \u00ac\u2200 \u2983b c : \u03b1\u2984, b \u2294 c = a \u2192 b = a \u2228 c = a) \u2194 IsMin a \u2228 \u2203 b c, b \u2294 c = a \u2227 b < a \u2227 c < a\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 (IsMin a \u2228 Exists fun \u2983b\u2984 => Exists fun \u2983c\u2984 => b \u2294 c = a \u2227 b \u2260 a \u2227 c \u2260 a) \u2194 IsMin a \u2228 \u2203 b c, b \u2294 c = a \u2227 b < a \u2227 c < a\n[PROOFSTEP]\nrw [exists\u2082_congr]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 \u2200 (a_1 b : \u03b1), a_1 \u2294 b = a \u2227 a_1 \u2260 a \u2227 b \u2260 a \u2194 a_1 \u2294 b = a \u2227 a_1 < a \u2227 b < a\n[PROOFSTEP]\nsimp (config := { contextual := true }) [@eq_comm _ _ a]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 \u00acSupPrime a \u2194 IsMin a \u2228 \u2203 b c, a \u2264 b \u2294 c \u2227 \u00aca \u2264 b \u2227 \u00aca \u2264 c\n[PROOFSTEP]\nrw [SupPrime, not_and_or]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 (\u00ac\u00acIsMin a \u2228 \u00ac\u2200 \u2983b c : \u03b1\u2984, a \u2264 b \u2294 c \u2192 a \u2264 b \u2228 a \u2264 c) \u2194 IsMin a \u2228 \u2203 b c, a \u2264 b \u2294 c \u2227 \u00aca \u2264 b \u2227 \u00aca \u2264 c\n[PROOFSTEP]\npush_neg\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b c : \u03b1\n\u22a2 (IsMin a \u2228 Exists fun \u2983b\u2984 => Exists fun \u2983c\u2984 => a \u2264 b \u2294 c \u2227 \u00aca \u2264 b \u2227 \u00aca \u2264 c) \u2194\n IsMin a \u2228 \u2203 b c, a \u2264 b \u2294 c \u2227 \u00aca \u2264 b \u2227 \u00aca \u2264 c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeSup \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, a \u2264 b \u2294 c \u2192 a \u2264 b \u2228 a \u2264 c\nb c : \u03b1\nha : b \u2294 c = a\n\u22a2 b = a \u2228 c = a\n[PROOFSTEP]\nsimpa [\u2190 ha] using h ha.ge\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\n\u22a2 a \u2260 \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\nb c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact not_supIrred_bot ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime a\n\u22a2 a \u2260 \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\nb c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact not_supPrime_bot ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh : sup s f = a\n\u22a2 \u2203 i, i \u2208 s \u2227 f i = a\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction with i s _ ih\n\u00b7 simpa [ha.ne_bot] using h.symm\nsimp only [exists_prop, exists_mem_insert] at ih \u22a2\nrw [sup_insert] at h \nexact (ha.2 h).imp_right ih\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh : sup s f = a\n\u22a2 \u2203 i, i \u2208 s \u2227 f i = a\n[PROOFSTEP]\ninduction' s using Finset.induction with i s _ ih\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh\u271d : sup s f = a\nh : sup \u2205 f = a\n\u22a2 \u2203 i, i \u2208 \u2205 \u2227 f i = a\n[PROOFSTEP]\nsimpa [ha.ne_bot] using h.symm\n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh\u271d : sup s\u271d f = a\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nih : sup s f = a \u2192 \u2203 i, i \u2208 s \u2227 f i = a\nh : sup (insert i s) f = a\n\u22a2 \u2203 i_1, i_1 \u2208 insert i s \u2227 f i_1 = a\n[PROOFSTEP]\nsimp only [exists_prop, exists_mem_insert] at ih \u22a2\n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh\u271d : sup s\u271d f = a\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nih : sup s f = a \u2192 \u2203 i, i \u2208 s \u2227 f i = a\nh : sup (insert i s) f = a\n\u22a2 f i = a \u2228 \u2203 x, x \u2208 s \u2227 f x = a\n[PROOFSTEP]\nrw [sup_insert] at h \n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupIrred a\nh\u271d : sup s\u271d f = a\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nih : sup s f = a \u2192 \u2203 i, i \u2208 s \u2227 f i = a\nh : f i \u2294 sup s f = a\n\u22a2 f i = a \u2228 \u2203 x, x \u2208 s \u2227 f x = a\n[PROOFSTEP]\nexact (ha.2 h).imp_right ih\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime a\n\u22a2 a \u2264 sup s f \u2194 \u2203 i, i \u2208 s \u2227 a \u2264 f i\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction with i s _ ih\n\u00b7 simp [ha.ne_bot]\n\u00b7 simp only [exists_prop, exists_mem_insert, sup_insert, ha.le_sup, ih]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime a\n\u22a2 a \u2264 sup s f \u2194 \u2203 i, i \u2208 s \u2227 a \u2264 f i\n[PROOFSTEP]\ninduction' s using Finset.induction with i s _ ih\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime a\n\u22a2 a \u2264 sup \u2205 f \u2194 \u2203 i, i \u2208 \u2205 \u2227 a \u2264 f i\n[PROOFSTEP]\nsimp [ha.ne_bot]\n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : SupPrime a\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nih : a \u2264 sup s f \u2194 \u2203 i, i \u2208 s \u2227 a \u2264 f i\n\u22a2 a \u2264 sup (insert i s) f \u2194 \u2203 i_1, i_1 \u2208 insert i s \u2227 a \u2264 f i_1\n[PROOFSTEP]\nsimp only [exists_prop, exists_mem_insert, sup_insert, ha.le_sup, ih]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nclassical\napply WellFoundedLT.induction a _\nclear a\nrintro a ih\nby_cases ha : SupIrred a\n\u00b7 exact \u27e8{ a }, by simp [ha]\u27e9\nrw [not_supIrred] at ha \nobtain ha | \u27e8b, c, rfl, hb, hc\u27e9 := ha\n\u00b7 exact \u27e8\u2205, by simp [ha.eq_bot]\u27e9\nobtain \u27e8s, rfl, hs\u27e9 := ih _ hb\nobtain \u27e8t, rfl, ht\u27e9 := ih _ hc\nexact \u27e8s \u222a t, sup_union, forall_mem_union.2 \u27e8hs, ht\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\napply WellFoundedLT.induction a _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1),\n (\u2200 (y : \u03b1), y < x \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b) \u2192\n \u2203 s, sup s id = x \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nclear a\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\n\u22a2 \u2200 (x : \u03b1),\n (\u2200 (y : \u03b1), y < x \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b) \u2192\n \u2203 s, sup s id = x \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nrintro a ih\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nby_cases ha : SupIrred a\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : SupIrred a\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nexact \u27e8{ a }, by simp [ha]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : SupIrred a\n\u22a2 sup {a} id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 {a} \u2192 SupIrred b\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : \u00acSupIrred a\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nrw [not_supIrred] at ha \n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : IsMin a \u2228 \u2203 b c, b \u2294 c = a \u2227 b < a \u2227 c < a\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nobtain ha | \u27e8b, c, rfl, hb, hc\u27e9 := ha\n[GOAL]\ncase neg.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : IsMin a\n\u22a2 \u2203 s, sup s id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nexact \u27e8\u2205, by simp [ha.eq_bot]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na\u271d b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\na : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nha : IsMin a\n\u22a2 sup \u2205 id = a \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 \u2205 \u2192 SupIrred b\n[PROOFSTEP]\nsimp [ha.eq_bot]\n[GOAL]\ncase neg.inr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na b\u271d c\u271d : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\nb c : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2294 c \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nhb : b < b \u2294 c\nhc : c < b \u2294 c\n\u22a2 \u2203 s, sup s id = b \u2294 c \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\n[PROOFSTEP]\nobtain \u27e8s, rfl, hs\u27e9 := ih _ hb\n[GOAL]\ncase neg.inr.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na b c\u271d : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\nc : \u03b1\ns : Finset \u03b1\nhs : \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nih : \u2200 (y : \u03b1), y < sup s id \u2294 c \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nhb : sup s id < sup s id \u2294 c\nhc : c < sup s id \u2294 c\n\u22a2 \u2203 s_1, sup s_1 id = sup s id \u2294 c \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s_1 \u2192 SupIrred b\n[PROOFSTEP]\nobtain \u27e8t, rfl, ht\u27e9 := ih _ hc\n[GOAL]\ncase neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : SemilatticeSup \u03b1\na b c : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\ninst\u271d : WellFoundedLT \u03b1\ns : Finset \u03b1\nhs : \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nt : Finset \u03b1\nht : \u2200 \u2983b : \u03b1\u2984, b \u2208 t \u2192 SupIrred b\nih : \u2200 (y : \u03b1), y < sup s id \u2294 sup t id \u2192 \u2203 s, sup s id = y \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s \u2192 SupIrred b\nhb : sup s id < sup s id \u2294 sup t id\nhc : sup t id < sup s id \u2294 sup t id\n\u22a2 \u2203 s_1, sup s_1 id = sup s id \u2294 sup t id \u2227 \u2200 \u2983b : \u03b1\u2984, b \u2208 s_1 \u2192 SupIrred b\n[PROOFSTEP]\nexact \u27e8s \u222a t, sup_union, forall_mem_union.2 \u27e8hs, ht\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : SemilatticeInf \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c \u2264 a \u2192 b \u2264 a \u2228 c \u2264 a\nb c : \u03b1\nha : b \u2293 c = a\n\u22a2 b = a \u2228 c = a\n[PROOFSTEP]\nsimpa [\u2190 ha] using h ha.le\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeInf \u03b1\na b c : \u03b1\ninst\u271d : OrderTop \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : InfIrred a\n\u22a2 a \u2260 \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeInf \u03b1\nb c : \u03b1\ninst\u271d : OrderTop \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : InfIrred \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact not_infIrred_top ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeInf \u03b1\na b c : \u03b1\ninst\u271d : OrderTop \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : InfPrime a\n\u22a2 a \u2260 \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SemilatticeInf \u03b1\nb c : \u03b1\ninst\u271d : OrderTop \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nha : InfPrime \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact not_infPrime_top ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2294 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 a \u2264 b \u2294 c \u2192 a \u2264 b \u2228 a \u2264 c\n[PROOFSTEP]\nsimp_rw [\u2190 inf_eq_left, inf_sup_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2294 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 a \u2293 b \u2294 a \u2293 c = a \u2192 a \u2293 b = a \u2228 a \u2293 c = a\n[PROOFSTEP]\nexact @h _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 b \u2293 c \u2264 a \u2192 b \u2264 a \u2228 c \u2264 a\n[PROOFSTEP]\nsimp_rw [\u2190 sup_eq_left, sup_inf_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : DistribLattice \u03b1\na b\u271d c\u271d : \u03b1\nh : \u2200 \u2983b c : \u03b1\u2984, b \u2293 c = a \u2192 b = a \u2228 c = a\nb c : \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) = a \u2192 a \u2294 b = a \u2228 a \u2294 c = a\n[PROOFSTEP]\nexact @h _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 \u2200 \u2983b c : \u03b1\u2984, a \u2264 b \u2294 c \u2192 a \u2264 b \u2228 a \u2264 c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 \u2200 \u2983b c : \u03b1\u2984, b \u2293 c \u2264 a \u2192 b \u2264 a \u2228 c \u2264 a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrder \u03b1\na x\u271d\u00b9 x\u271d : \u03b1\n\u22a2 x\u271d\u00b9 \u2294 x\u271d = a \u2192 x\u271d\u00b9 = a \u2228 x\u271d = a\n[PROOFSTEP]\nsimpa only [sup_eq_max, max_eq_iff] using Or.imp And.left And.left\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : LinearOrder \u03b1\na x\u271d\u00b9 x\u271d : \u03b1\n\u22a2 x\u271d\u00b9 \u2293 x\u271d = a \u2192 x\u271d\u00b9 = a \u2228 x\u271d = a\n[PROOFSTEP]\nsimpa only [inf_eq_min, min_eq_iff] using Or.imp And.left And.left\n", "meta": {"mathlib_filename": "Mathlib.Order.Irreducible", "llama_tokens": 7347, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832058771036, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.27580665610399757}} {"text": "[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 Finite (\u03b1 \u00d7 \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 Finite (\u03b1 \u00d7 \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 Finite (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\ninfer_instance\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 Finite (\u03b1 \u2295 \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 Finite (\u03b1 \u2295 \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 Finite (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\n\u22a2 Finite ((a : \u03b1) \u00d7 \u03b2 a)\n[PROOFSTEP]\nletI := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nthis : Fintype \u03b1 := Fintype.ofFinite \u03b1\n\u22a2 Finite ((a : \u03b1) \u00d7 \u03b2 a)\n[PROOFSTEP]\nletI := fun a => Fintype.ofFinite (\u03b2 a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nthis\u271d : Fintype \u03b1 := Fintype.ofFinite \u03b1\nthis : (a : \u03b1) \u2192 Fintype (\u03b2 a) := fun a => Fintype.ofFinite (\u03b2 a)\n\u22a2 Finite ((a : \u03b1) \u00d7 \u03b2 a)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 Finite (Set \u03b1)\n[PROOFSTEP]\ncases nonempty_fintype \u03b1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 Finite (Set \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : \u03b1 \u2192 Sort u_5\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\n\u22a2 Finite ((a : \u03b1) \u2192 \u03b2 a)\n[PROOFSTEP]\nhaveI := Fintype.ofFinite (PLift \u03b1)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : \u03b1 \u2192 Sort u_5\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nthis : Fintype (PLift \u03b1)\n\u22a2 Finite ((a : \u03b1) \u2192 \u03b2 a)\n[PROOFSTEP]\nhaveI := fun a => Fintype.ofFinite (PLift (\u03b2 a))\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : \u03b1 \u2192 Sort u_5\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nthis\u271d : Fintype (PLift \u03b1)\nthis : (a : \u03b1) \u2192 Fintype (PLift (\u03b2 a))\n\u22a2 Finite ((a : \u03b1) \u2192 \u03b2 a)\n[PROOFSTEP]\nexact Finite.of_equiv (\u2200 a : PLift \u03b1, PLift (\u03b2 (Equiv.plift a))) (Equiv.piCongr Equiv.plift fun _ => Equiv.plift)\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\nn : \u2115\n\u22a2 Finite (Vector \u03b1 n)\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b1\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\nn : \u2115\nthis : Fintype \u03b1\n\u22a2 Finite (Vector \u03b1 n)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\n\u22a2 Finite (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\ncases' isEmpty_or_nonempty (\u03b1 \u21aa \u03b2) with _ h\n[GOAL]\ncase inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\nh\u271d : IsEmpty (\u03b1 \u21aa \u03b2)\n\u22a2 Finite (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\napply Finite.of_subsingleton\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\nh : Nonempty (\u03b1 \u21aa \u03b2)\n\u22a2 Finite (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\nrefine' h.elim fun f => _\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\nh : Nonempty (\u03b1 \u21aa \u03b2)\nf : \u03b1 \u21aa \u03b2\n\u22a2 Finite (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\nhaveI : Finite \u03b1 := Finite.of_injective _ f.injective\n[GOAL]\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\nh : Nonempty (\u03b1 \u21aa \u03b2)\nf : \u03b1 \u21aa \u03b2\nthis : Finite \u03b1\n\u22a2 Finite (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\nexact Finite.of_injective _ FunLike.coe_injective\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\ninst\u271d : Finite \u03b2\ne\u2081 e\u2082 : \u03b1 \u2243 \u03b2\nh : Equiv.toEmbedding e\u2081 = Equiv.toEmbedding e\u2082\n\u22a2 \u2200 (x : \u03b1), \u2191e\u2081 x = \u2191e\u2082 x\n[PROOFSTEP]\nconvert FunLike.congr_fun h using 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nn : \u2115\n\u22a2 Finite (Sym \u03b1 n)\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\nn : \u2115\nthis : Fintype \u03b1\n\u22a2 Finite (Sym \u03b1 n)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Data.Finite.Basic", "llama_tokens": 2394, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.27564024283741045}} {"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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\n\u22a2 IsLimit (G.mapCone (Fan.mk P g)) \u2243 IsLimit (Fan.mk (G.obj P) fun j => G.map (g j))\n[PROOFSTEP]\nrefine' (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)\n[GOAL]\ncase refine'_1\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\n\u22a2 (Discrete.functor fun b => f b) \u22d9 G \u2245 Discrete.functor fun j => G.obj (f j)\ncase refine'_2\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\n\u22a2 (Cones.postcompose ?refine'_1.hom).obj (G.mapCone (Fan.mk P g)) \u2245 Fan.mk (G.obj P) fun j => G.map (g j)\n[PROOFSTEP]\nrefine' Discrete.natIso fun j => Iso.refl (G.obj (f j.as))\n[GOAL]\ncase refine'_2\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\n\u22a2 (Cones.postcompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj (G.mapCone (Fan.mk P g)) \u2245\n Fan.mk (G.obj P) fun j => G.map (g j)\n[PROOFSTEP]\nrefine' Cones.ext (Iso.refl _) fun j => by dsimp; cases j; simp\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\nj : Discrete J\n\u22a2 NatTrans.app\n ((Cones.postcompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj (G.mapCone (Fan.mk P g))).\u03c0 j =\n (Iso.refl\n ((Cones.postcompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj\n (G.mapCone (Fan.mk P g))).pt).hom \u226b\n NatTrans.app (Fan.mk (G.obj P) fun j => G.map (g j)).\u03c0 j\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\nj : Discrete J\n\u22a2 G.map (g j.as) \u226b \ud835\udfd9 (G.obj (f j.as)) = \ud835\udfd9 (G.obj P) \u226b G.map (g j.as)\n[PROOFSTEP]\ncases j\n[GOAL]\ncase mk\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 P \u27f6 f j\nas\u271d : J\n\u22a2 G.map (g { as := as\u271d }.as) \u226b \ud835\udfd9 (G.obj (f { as := as\u271d }.as)) = \ud835\udfd9 (G.obj P) \u226b G.map (g { as := as\u271d }.as)\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\nG : C \u2964 D\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun j => G.obj (f j)\ni : IsIso (piComparison G f)\n\u22a2 PreservesLimit (Discrete.functor f) G\n[PROOFSTEP]\napply preservesLimitOfPreservesLimitCone (productIsProduct 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\nG : C \u2964 D\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun j => G.obj (f j)\ni : IsIso (piComparison G f)\n\u22a2 IsLimit (G.mapCone (Fan.mk (\u220f f) (Pi.\u03c0 f)))\n[PROOFSTEP]\napply (isLimitMapConeFanMkEquiv _ _ _).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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun j => G.obj (f j)\ni : IsIso (piComparison G f)\n\u22a2 IsLimit (Fan.mk (G.obj (\u220f f)) fun j => G.map (Pi.\u03c0 f j))\n[PROOFSTEP]\nrefine @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (Discrete.functor fun j : J => G.obj (f j))) ?_\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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasProduct f\ninst\u271d : HasProduct fun j => G.obj (f j)\ni : IsIso (piComparison G f)\n\u22a2 IsIso\n (IsLimit.lift (limit.isLimit (Discrete.functor fun j => G.obj (f j)))\n (Fan.mk (G.obj (\u220f f)) fun j => G.map (Pi.\u03c0 f j)))\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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b2 : HasProduct f\ninst\u271d\u00b9 : HasProduct fun j => G.obj (f j)\ninst\u271d : PreservesLimit (Discrete.functor f) G\n\u22a2 IsIso (piComparison G f)\n[PROOFSTEP]\nrw [\u2190 PreservesProduct.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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b2 : HasProduct f\ninst\u271d\u00b9 : HasProduct fun j => G.obj (f j)\ninst\u271d : PreservesLimit (Discrete.functor f) G\n\u22a2 IsIso (PreservesProduct.iso G f).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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\n\u22a2 IsColimit (G.mapCocone (Cofan.mk P g)) \u2243 IsColimit (Cofan.mk (G.obj P) fun j => G.map (g j))\n[PROOFSTEP]\nrefine' (IsColimit.precomposeHomEquiv _ _).symm.trans (IsColimit.equivIsoColimit _)\n[GOAL]\ncase refine'_1\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\n\u22a2 (Discrete.functor fun j => G.obj (f j)) \u2245 (Discrete.functor fun b => f b) \u22d9 G\ncase refine'_2\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\n\u22a2 (Cocones.precompose ?refine'_1.hom).obj (G.mapCocone (Cofan.mk P g)) \u2245 Cofan.mk (G.obj P) fun j => G.map (g j)\n[PROOFSTEP]\nrefine' Discrete.natIso fun j => Iso.refl (G.obj (f j.as))\n[GOAL]\ncase refine'_2\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\n\u22a2 (Cocones.precompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj (G.mapCocone (Cofan.mk P g)) \u2245\n Cofan.mk (G.obj P) fun j => G.map (g j)\n[PROOFSTEP]\nrefine' Cocones.ext (Iso.refl _) fun j => by dsimp; cases j; simp\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\nj : Discrete J\n\u22a2 NatTrans.app\n ((Cocones.precompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj\n (G.mapCocone (Cofan.mk P g))).\u03b9\n j \u226b\n (Iso.refl\n ((Cocones.precompose (Discrete.natIso fun j => Iso.refl (G.obj (f j.as))).hom).obj\n (G.mapCocone (Cofan.mk P g))).pt).hom =\n NatTrans.app (Cofan.mk (G.obj P) fun j => G.map (g j)).\u03b9 j\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\nj : Discrete J\n\u22a2 (\ud835\udfd9 (G.obj (f j.as)) \u226b G.map (g j.as)) \u226b \ud835\udfd9 (G.obj P) = G.map (g j.as)\n[PROOFSTEP]\ncases j\n[GOAL]\ncase mk\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\nJ : Type w\nf : J \u2192 C\nP : C\ng : (j : J) \u2192 f j \u27f6 P\nas\u271d : J\n\u22a2 (\ud835\udfd9 (G.obj (f { as := as\u271d }.as)) \u226b G.map (g { as := as\u271d }.as)) \u226b \ud835\udfd9 (G.obj P) = G.map (g { as := as\u271d }.as)\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\nG : C \u2964 D\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun j => G.obj (f j)\ni : IsIso (sigmaComparison G f)\n\u22a2 PreservesColimit (Discrete.functor f) G\n[PROOFSTEP]\napply preservesColimitOfPreservesColimitCocone (coproductIsCoproduct 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\nG : C \u2964 D\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun j => G.obj (f j)\ni : IsIso (sigmaComparison G f)\n\u22a2 IsColimit (G.mapCocone (Cofan.mk (\u2210 f) (Sigma.\u03b9 f)))\n[PROOFSTEP]\napply (isColimitMapCoconeCofanMkEquiv _ _ _).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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun j => G.obj (f j)\ni : IsIso (sigmaComparison G f)\n\u22a2 IsColimit (Cofan.mk (G.obj (\u2210 f)) fun j => G.map (Sigma.\u03b9 f j))\n[PROOFSTEP]\nrefine @IsColimit.ofPointIso _ _ _ _ _ _ _ (colimit.isColimit (Discrete.functor fun j : J => G.obj (f j))) ?_\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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b9 : HasCoproduct f\ninst\u271d : HasCoproduct fun j => G.obj (f j)\ni : IsIso (sigmaComparison G f)\n\u22a2 IsIso\n (IsColimit.desc (colimit.isColimit (Discrete.functor fun j => G.obj (f j)))\n (Cofan.mk (G.obj (\u2210 f)) fun j => G.map (Sigma.\u03b9 f j)))\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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b2 : HasCoproduct f\ninst\u271d\u00b9 : HasCoproduct fun j => G.obj (f j)\ninst\u271d : PreservesColimit (Discrete.functor f) G\n\u22a2 IsIso (sigmaComparison G f)\n[PROOFSTEP]\nrw [\u2190 PreservesCoproduct.inv_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\nJ : Type w\nf : J \u2192 C\ninst\u271d\u00b2 : HasCoproduct f\ninst\u271d\u00b9 : HasCoproduct fun j => G.obj (f j)\ninst\u271d : PreservesColimit (Discrete.functor f) G\n\u22a2 IsIso (PreservesCoproduct.iso G f).inv\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products", "llama_tokens": 4565, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5506073655352403, "lm_q2_score": 0.5, "lm_q1q2_score": 0.27530368276762013}} {"text": "[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nx\u271d : QuasiCompact f\nh : \u2200 (U : Set \u2191\u2191Y.toPresheafedSpace), IsOpen U \u2192 IsCompact U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' U)\n\u22a2 Continuous \u2191f.val.base\n[PROOFSTEP]\ncontinuity\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 QuasiCompact f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase isCompact_preimage\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 \u2200 (U : Set \u2191\u2191Y.toPresheafedSpace), IsOpen U \u2192 IsCompact U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' U)\n[PROOFSTEP]\nintro U _ hU'\n[GOAL]\ncase isCompact_preimage\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact (\u2191f.val.base \u207b\u00b9' U)\n[PROOFSTEP]\nconvert hU'.image (inv f.1.base).continuous_toFun using 1\n[GOAL]\ncase h.e'_3.h\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\ne_1\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace = \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2191f.val.base \u207b\u00b9' U = (inv f.val.base).toFun '' U\n[PROOFSTEP]\nrw [Set.image_eq_preimage_of_inverse]\n[GOAL]\ncase h.e'_3.h.h\u2081\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\ne_1\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace = \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.LeftInverse (\u2191f.val.base) (inv f.val.base).toFun\ncase h.e'_3.h.h\u2082\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\ne_1\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace = \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.RightInverse (\u2191f.val.base) (inv f.val.base).toFun\n[PROOFSTEP]\ndelta Function.LeftInverse\n[GOAL]\ncase h.e'_3.h.h\u2081\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\ne_1\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace = \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 (x : \u2191\u2191Y.toPresheafedSpace), \u2191f.val.base (ContinuousMap.toFun (inv f.val.base) x) = x\ncase h.e'_3.h.h\u2082\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\ninst\u271d : IsIso f\nU : Set \u2191\u2191Y.toPresheafedSpace\na\u271d : IsOpen U\nhU' : IsCompact U\ne_1\u271d : (forget TopCat).obj \u2191X.toPresheafedSpace = \u2191\u2191X.toPresheafedSpace\n\u22a2 Function.RightInverse (\u2191f.val.base) (inv f.val.base).toFun\n[PROOFSTEP]\nexacts [IsIso.inv_hom_id_apply f.1.base, IsIso.hom_inv_id_apply f.1.base]\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\n\u22a2 QuasiCompact (f \u226b g)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase isCompact_preimage\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\n\u22a2 \u2200 (U : Set \u2191\u2191Z.toPresheafedSpace), IsOpen U \u2192 IsCompact U \u2192 IsCompact (\u2191(f \u226b g).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nintro U hU hU'\n[GOAL]\ncase isCompact_preimage\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact (\u2191(f \u226b g).val.base \u207b\u00b9' U)\n[PROOFSTEP]\nrw [Scheme.comp_val_base, coe_comp, Set.preimage_comp]\n[GOAL]\ncase isCompact_preimage\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact (\u2191f.val.base \u207b\u00b9' (\u2191g.val.base \u207b\u00b9' U))\n[PROOFSTEP]\napply QuasiCompact.isCompact_preimage\n[GOAL]\ncase isCompact_preimage.a\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsOpen (\u2191g.val.base \u207b\u00b9' U)\n[PROOFSTEP]\nexact\n Continuous.isOpen_preimage\n (by\n -- porting note: `continuity` failed\n -- see https://github.com/leanprover-community/mathlib4/issues/5030exact Scheme.Hom.continuous g)\n _ hU\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 Continuous \u2191g.val.base\n[PROOFSTEP]\nexact Scheme.Hom.continuous g\n[GOAL]\ncase isCompact_preimage.a\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact (\u2191g.val.base \u207b\u00b9' U)\n[PROOFSTEP]\napply QuasiCompact.isCompact_preimage\n[GOAL]\ncase isCompact_preimage.a.a\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsOpen U\n[PROOFSTEP]\nassumption\n[GOAL]\ncase isCompact_preimage.a.a\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : QuasiCompact f\ninst\u271d : QuasiCompact g\nU : Set \u2191\u2191Z.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact U\n[PROOFSTEP]\nassumption\n[GOAL]\nX\u271d Y : Scheme\nf : X\u271d \u27f6 Y\nX : Scheme\nU : Set \u2191\u2191X.toPresheafedSpace\n\u22a2 IsCompact U \u2227 IsOpen U \u2194 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\napply Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion (fun (U : X.affineOpens) => (U : Opens X.carrier))\n[GOAL]\ncase hb\nX\u271d Y : Scheme\nf : X\u271d \u27f6 Y\nX : Scheme\nU : Set \u2191\u2191X.toPresheafedSpace\n\u22a2 Opens.IsBasis (Set.range fun U => \u2191U)\n[PROOFSTEP]\nrw [Subtype.range_coe]\n[GOAL]\ncase hb\nX\u271d Y : Scheme\nf : X\u271d \u27f6 Y\nX : Scheme\nU : Set \u2191\u2191X.toPresheafedSpace\n\u22a2 Opens.IsBasis (Scheme.affineOpens X)\n[PROOFSTEP]\nexact isBasis_affine_open X\n[GOAL]\ncase hb'\nX\u271d Y : Scheme\nf : X\u271d \u27f6 Y\nX : Scheme\nU : Set \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2200 (i : \u2191(Scheme.affineOpens X)), IsCompact \u2191\u2191i\n[PROOFSTEP]\nexact fun i => i.2.isCompact\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 QuasiCompact f \u2194 \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrw [QuasiCompact_iff]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2200 (U : Set \u2191\u2191Y.toPresheafedSpace), IsOpen U \u2192 IsCompact U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' U)) \u2194\n \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)\n[PROOFSTEP]\nrefine' \u27e8fun H U hU => H U U.isOpen hU.isCompact, _\u27e9\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)) \u2192\n \u2200 (U : Set \u2191\u2191Y.toPresheafedSpace), IsOpen U \u2192 IsCompact U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' U)\n[PROOFSTEP]\nintro H U hU hU'\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nH : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)\nU : Set \u2191\u2191Y.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n\u22a2 IsCompact (\u2191f.val.base \u207b\u00b9' U)\n[PROOFSTEP]\nobtain \u27e8S, hS, rfl\u27e9 := (isCompact_open_iff_eq_finset_affine_union U).mp \u27e8hU', hU\u27e9\n[GOAL]\ncase intro.intro\nX Y : Scheme\nf : X \u27f6 Y\nH : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)\nS : Set \u2191(Scheme.affineOpens Y)\nhS : Set.Finite S\nhU : IsOpen (\u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191\u2191i)\nhU' : IsCompact (\u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191\u2191i)\n\u22a2 IsCompact (\u2191f.val.base \u207b\u00b9' \u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191\u2191i)\n[PROOFSTEP]\nsimp only [Set.preimage_iUnion]\n[GOAL]\ncase intro.intro\nX Y : Scheme\nf : X \u27f6 Y\nH : \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)\nS : Set \u2191(Scheme.affineOpens Y)\nhS : Set.Finite S\nhU : IsOpen (\u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191\u2191i)\nhU' : IsCompact (\u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191\u2191i)\n\u22a2 IsCompact (\u22c3 (i : \u2191(Scheme.affineOpens Y)) (_ : i \u2208 S), \u2191f.val.base \u207b\u00b9' \u2191\u2191i)\n[PROOFSTEP]\nexact Set.Finite.isCompact_biUnion hS (fun i _ => H i i.prop)\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 AffineTargetMorphismProperty.toProperty affineProperty f \u2194 IsAffine Y \u2227 CompactSpace \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\ndelta AffineTargetMorphismProperty.toProperty QuasiCompact.affineProperty\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2203 h, CompactSpace \u2191\u2191X.toPresheafedSpace) \u2194 IsAffine Y \u2227 CompactSpace \u2191\u2191X.toPresheafedSpace\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 QuasiCompact f \u2194 targetAffineLocally QuasiCompact.affineProperty f\n[PROOFSTEP]\nrw [quasiCompact_iff_forall_affine]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)) \u2194\n targetAffineLocally QuasiCompact.affineProperty f\n[PROOFSTEP]\ntrans \u2200 U : Y.affineOpens, IsCompact (f.1.base \u207b\u00b9' (U : Set Y.carrier))\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace), IsAffineOpen U \u2192 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191U)) \u2194\n \u2200 (U : \u2191(Scheme.affineOpens Y)), IsCompact (\u2191f.val.base \u207b\u00b9' \u2191\u2191U)\n[PROOFSTEP]\nexact \u27e8fun h U => h U U.prop, fun h U hU => h \u27e8U, hU\u27e9\u27e9\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 (\u2200 (U : \u2191(Scheme.affineOpens Y)), IsCompact (\u2191f.val.base \u207b\u00b9' \u2191\u2191U)) \u2194 targetAffineLocally QuasiCompact.affineProperty f\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 (a : \u2191(Scheme.affineOpens Y)), IsCompact (\u2191f.val.base \u207b\u00b9' \u2191\u2191a) \u2194 QuasiCompact.affineProperty (f \u2223_ \u2191a)\n[PROOFSTEP]\nexact fun _ => isCompact_iff_compactSpace\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 @QuasiCompact = targetAffineLocally QuasiCompact.affineProperty\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.h.a\nX Y : Scheme\nf : X \u27f6 Y\nx\u271d\u00b2 x\u271d\u00b9 : Scheme\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 QuasiCompact x\u271d \u2194 targetAffineLocally QuasiCompact.affineProperty x\u271d\n[PROOFSTEP]\nexact quasiCompact_iff_affineProperty _\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\n\u22a2 IsCompact \u2191(Scheme.basicOpen X f)\n[PROOFSTEP]\nclassical\nrefine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1\nobtain \u27e8s, hs, e\u27e9 := (isCompact_open_iff_eq_finset_affine_union _).mp \u27e8hU, U.isOpen\u27e9\nlet g : s \u2192 X.affineOpens := by\n intro V\n use V.1 \u2293 X.basicOpen f\n have : V.1.1 \u27f6 U := by\n apply homOfLE; change _ \u2286 (U : Set X.carrier); rw [e]\n convert @Set.subset_iUnion\u2082 _ _ _ (fun (U : X.affineOpens) (_ : U \u2208 s) => \u2191U) V V.prop using 1\n erw [\u2190 X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]\n exact IsAffineOpen.basicOpenIsAffine V.1.prop _\nhaveI : Finite s := hs.to_subtype\nrefine' \u27e8Set.range g, Set.finite_range g, _\u27e9\nrefine' (Set.inter_eq_right_iff_subset.mpr (SetLike.coe_subset_coe.2 <| RingedSpace.basicOpen_le _ _)).symm.trans _\nrw [e, Set.iUnion\u2082_inter]\napply le_antisymm <;> apply Set.iUnion\u2082_subset\n\u00b7 intro i hi\n exact\n Set.Subset.trans (Set.Subset.rfl : _ \u2264 g \u27e8i, hi\u27e9)\n (@Set.subset_iUnion\u2082 _ _ _ (fun (i : Scheme.affineOpens X) (_ : i \u2208 Set.range g) => (i : Set X.toPresheafedSpace))\n _ (Set.mem_range_self \u27e8i, hi\u27e9))\n\u00b7 rintro \u27e8i, hi\u27e9 \u27e8\u27e8j, hj\u27e9, hj'\u27e9\n rw [\u2190 hj']\n refine' Set.Subset.trans _ (Set.subset_iUnion\u2082 j hj)\n exact Set.Subset.rfl\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\n\u22a2 IsCompact \u2191(Scheme.basicOpen X f)\n[PROOFSTEP]\nrefine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\n\u22a2 \u2203 s, Set.Finite s \u2227 \u2191(Scheme.basicOpen X f) = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nobtain \u27e8s, hs, e\u27e9 := (isCompact_open_iff_eq_finset_affine_union _).mp \u27e8hU, U.isOpen\u27e9\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 \u2203 s, Set.Finite s \u2227 \u2191(Scheme.basicOpen X f) = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nlet g : s \u2192 X.affineOpens := by\n intro V\n use V.1 \u2293 X.basicOpen f\n have : V.1.1 \u27f6 U := by\n apply homOfLE; change _ \u2286 (U : Set X.carrier); rw [e]\n convert @Set.subset_iUnion\u2082 _ _ _ (fun (U : X.affineOpens) (_ : U \u2208 s) => \u2191U) V V.prop using 1\n erw [\u2190 X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]\n exact IsAffineOpen.basicOpenIsAffine V.1.prop _\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 \u2191s \u2192 \u2191(Scheme.affineOpens X)\n[PROOFSTEP]\nintro V\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191(Scheme.affineOpens X)\n[PROOFSTEP]\nuse V.1 \u2293 X.basicOpen f\n[GOAL]\ncase property\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X\n[PROOFSTEP]\nhave : V.1.1 \u27f6 U := by\n apply homOfLE; change _ \u2286 (U : Set X.carrier); rw [e]\n convert @Set.subset_iUnion\u2082 _ _ _ (fun (U : X.affineOpens) (_ : U \u2208 s) => \u2191U) V V.prop using 1\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191\u2191V \u27f6 U\n[PROOFSTEP]\napply homOfLE\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191\u2191V \u2264 U\n[PROOFSTEP]\nchange _ \u2286 (U : Set X.carrier)\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191\u2191\u2191V \u2286 \u2191U\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\n\u22a2 \u2191\u2191\u2191V \u2286 \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nconvert @Set.subset_iUnion\u2082 _ _ _ (fun (U : X.affineOpens) (_ : U \u2208 s) => \u2191U) V V.prop using 1\n[GOAL]\ncase property\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\nthis : \u2191\u2191V \u27f6 U\n\u22a2 \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X\n[PROOFSTEP]\nerw [\u2190 X.toLocallyRingedSpace.toRingedSpace.basicOpen_res this.op]\n[GOAL]\ncase property\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\nV : \u2191s\nthis : \u2191\u2191V \u27f6 U\n\u22a2 RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace)\n (\u2191((LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace).toPresheafedSpace.presheaf.map this.op) f) \u2208\n Scheme.affineOpens X\n[PROOFSTEP]\nexact IsAffineOpen.basicOpenIsAffine V.1.prop _\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\n\u22a2 \u2203 s, Set.Finite s \u2227 \u2191(Scheme.basicOpen X f) = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nhaveI : Finite s := hs.to_subtype\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u2203 s, Set.Finite s \u2227 \u2191(Scheme.basicOpen X f) = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nrefine' \u27e8Set.range g, Set.finite_range g, _\u27e9\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u2191(Scheme.basicOpen X f) = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\nrefine' (Set.inter_eq_right_iff_subset.mpr (SetLike.coe_subset_coe.2 <| RingedSpace.basicOpen_le _ _)).symm.trans _\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u2191U \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) =\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\nrw [e, Set.iUnion\u2082_inter]\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) =\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase intro.intro.a\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) \u2264\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\napply Set.iUnion\u2082_subset\n[GOAL]\ncase intro.intro.a\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i \u2264\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)\n[PROOFSTEP]\napply Set.iUnion\u2082_subset\n[GOAL]\ncase intro.intro.a.h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u2200 (i : \u2191(Scheme.affineOpens X)),\n i \u2208 s \u2192\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) \u2286\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase intro.intro.a.h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\ni : \u2191(Scheme.affineOpens X)\nhi : i \u2208 s\n\u22a2 \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f) \u2286\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 Set.range g), \u2191\u2191i\n[PROOFSTEP]\nexact\n Set.Subset.trans (Set.Subset.rfl : _ \u2264 g \u27e8i, hi\u27e9)\n (@Set.subset_iUnion\u2082 _ _ _ (fun (i : Scheme.affineOpens X) (_ : i \u2208 Set.range g) => (i : Set X.toPresheafedSpace)) _\n (Set.mem_range_self \u27e8i, hi\u27e9))\n[GOAL]\ncase intro.intro.a.h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\n\u22a2 \u2200 (i : \u2191(Scheme.affineOpens X)),\n i \u2208 Set.range g \u2192\n \u2191\u2191i \u2286\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9 \u27e8\u27e8j, hj\u27e9, hj'\u27e9\n[GOAL]\ncase intro.intro.a.h.mk.intro.mk\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\ni : Opens \u2191\u2191X.toPresheafedSpace\nhi : i \u2208 Scheme.affineOpens X\nj : \u2191(Scheme.affineOpens X)\nhj : j \u2208 s\nhj' : g { val := j, property := hj } = { val := i, property := hi }\n\u22a2 \u2191\u2191{ val := i, property := hi } \u2286\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)\n[PROOFSTEP]\nrw [\u2190 hj']\n[GOAL]\ncase intro.intro.a.h.mk.intro.mk\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\ni : Opens \u2191\u2191X.toPresheafedSpace\nhi : i \u2208 Scheme.affineOpens X\nj : \u2191(Scheme.affineOpens X)\nhj : j \u2208 s\nhj' : g { val := j, property := hj } = { val := i, property := hi }\n\u22a2 \u2191\u2191(g { val := j, property := hj }) \u2286\n \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s),\n \u2191\u2191i \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)\n[PROOFSTEP]\nrefine' Set.Subset.trans _ (Set.subset_iUnion\u2082 j hj)\n[GOAL]\ncase intro.intro.a.h.mk.intro.mk\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact \u2191U\nf : \u2191(X.presheaf.obj (op U))\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : \u2191U = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\ng : \u2191s \u2192 \u2191(Scheme.affineOpens X) :=\n fun V => { val := \u2191\u2191V \u2293 Scheme.basicOpen X f, property := (_ : \u2191\u2191V \u2293 Scheme.basicOpen X f \u2208 Scheme.affineOpens X) }\nthis : Finite \u2191s\ni : Opens \u2191\u2191X.toPresheafedSpace\nhi : i \u2208 Scheme.affineOpens X\nj : \u2191(Scheme.affineOpens X)\nhj : j \u2208 s\nhj' : g { val := j, property := hj } = { val := i, property := hi }\n\u22a2 \u2191\u2191(g { val := j, property := hj }) \u2286\n \u2191\u2191j \u2229 \u2191(RingedSpace.basicOpen (LocallyRingedSpace.toRingedSpace X.toLocallyRingedSpace) f)\n[PROOFSTEP]\nexact Set.Subset.rfl\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 AffineTargetMorphismProperty.IsLocal affineProperty\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase RespectsIso\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 MorphismProperty.RespectsIso (AffineTargetMorphismProperty.toProperty affineProperty)\n[PROOFSTEP]\napply AffineTargetMorphismProperty.respectsIso_mk\n[GOAL]\ncase RespectsIso.h\u2081\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 {X Y Z : Scheme} (e : X \u2245 Y) (f : Y \u27f6 Z) [inst : IsAffine Z], affineProperty f \u2192 affineProperty (e.hom \u226b f)\n[PROOFSTEP]\nrintro X Y Z e _ _ H\n[GOAL]\ncase RespectsIso.h\u2082\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 {X Y Z : Scheme} (e : Y \u2245 Z) (f : X \u27f6 Y) [h : IsAffine Y], affineProperty f \u2192 affineProperty (f \u226b e.hom)\n[PROOFSTEP]\nrintro X Y Z e _ _ H\n[GOAL]\ncase RespectsIso.h\u2081\nX\u271d Y\u271d : Scheme\nf : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\ne : X \u2245 Y\nf\u271d : Y \u27f6 Z\ninst\u271d : IsAffine Z\nH : affineProperty f\u271d\n\u22a2 affineProperty (e.hom \u226b f\u271d)\ncase RespectsIso.h\u2082\nX\u271d Y\u271d : Scheme\nf : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\ne : Y \u2245 Z\nf\u271d : X \u27f6 Y\nh\u271d : IsAffine Y\nH : affineProperty f\u271d\n\u22a2 affineProperty (f\u271d \u226b e.hom)\n[PROOFSTEP]\nexacts [@Homeomorph.compactSpace _ _ _ _ H (TopCat.homeoOfIso (asIso e.inv.1.base)), H]\n[GOAL]\ncase toBasicOpen\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 {X Y : Scheme} [inst : IsAffine Y] (f : X \u27f6 Y) (r : \u2191(Y.presheaf.obj (op \u22a4))),\n affineProperty f \u2192 affineProperty (f \u2223_ Scheme.basicOpen Y r)\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : affineProperty f\n\u22a2 affineProperty (f \u2223_ Scheme.basicOpen Y r)\n[PROOFSTEP]\ndsimp [affineProperty] at H \u22a2\n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 CompactSpace \u2191((Opens.toTopCat \u2191X.toPresheafedSpace).obj ((Opens.map f.val.base).obj (Scheme.basicOpen Y r)))\n[PROOFSTEP]\nchange CompactSpace ((Opens.map f.val.base).obj (Y.basicOpen r))\n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 CompactSpace { x // x \u2208 (Opens.map f.val.base).obj (Scheme.basicOpen Y r) }\n[PROOFSTEP]\nrw [Scheme.preimage_basicOpen f r]\n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 CompactSpace { x // x \u2208 Scheme.basicOpen X (\u2191(NatTrans.app f.val.c (op \u22a4)) r) }\n[PROOFSTEP]\nerw [\u2190 isCompact_iff_compactSpace]\n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : CompactSpace \u2191\u2191X.toPresheafedSpace\n\u22a2 IsCompact \u2191(Scheme.basicOpen X (\u2191(NatTrans.app f.val.c (op \u22a4)) r))\n[PROOFSTEP]\nrw [\u2190 isCompact_univ_iff] at H \n[GOAL]\ncase toBasicOpen\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : IsCompact Set.univ\n\u22a2 IsCompact \u2191(Scheme.basicOpen X (\u2191(NatTrans.app f.val.c (op \u22a4)) r))\n[PROOFSTEP]\napply isCompact_basicOpen\n[GOAL]\ncase toBasicOpen.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\nr : \u2191(Y.presheaf.obj (op \u22a4))\nH : IsCompact Set.univ\n\u22a2 IsCompact \u2191((Opens.map f.val.base).obj \u22a4)\n[PROOFSTEP]\nexact H\n[GOAL]\ncase ofBasicOpenCover\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 {X Y : Scheme} [inst : IsAffine Y] (f : X \u27f6 Y) (s : Finset \u2191(Y.presheaf.obj (op \u22a4))),\n Ideal.span \u2191s = \u22a4 \u2192 (\u2200 (r : { x // x \u2208 s }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)) \u2192 affineProperty f\n[PROOFSTEP]\nrintro X Y H f S hS hS'\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 affineProperty f\n[PROOFSTEP]\nrw [\u2190 IsAffineOpen.basicOpen_union_eq_self_iff] at hS \n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 affineProperty f\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\ndelta QuasiCompact.affineProperty\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 CompactSpace \u2191\u2191X.toPresheafedSpace\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\nrw [\u2190 isCompact_univ_iff]\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsCompact Set.univ\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\nchange IsCompact ((Opens.map f.val.base).obj \u22a4).1\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsCompact ((Opens.map f.val.base).obj \u22a4).carrier\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\nrw [\u2190 hS]\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsCompact ((Opens.map f.val.base).obj (\u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f)).carrier\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\ndsimp [Opens.map]\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsCompact (\u2191f.val.base \u207b\u00b9' \u2191(\u2a06 (f : { x // x \u2208 S }), Scheme.basicOpen Y \u2191f))\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\nsimp only [Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk, Set.preimage_iUnion]\n[GOAL]\ncase ofBasicOpenCover\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : \u2a06 (f : \u2191\u2191S), Scheme.basicOpen Y \u2191f = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsCompact (\u22c3 (i : { x // x \u2208 S }), \u2191f.val.base \u207b\u00b9' \u2191(Scheme.basicOpen Y \u2191i))\ncase ofBasicOpenCover.hU\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nH : IsAffine Y\nf : X \u27f6 Y\nS : Finset \u2191(Y.presheaf.obj (op \u22a4))\nhS : Ideal.span \u2191S = \u22a4\nhS' : \u2200 (r : { x // x \u2208 S }), affineProperty (f \u2223_ Scheme.basicOpen Y \u2191r)\n\u22a2 IsAffineOpen \u22a4\n[PROOFSTEP]\nexacts [isCompact_iUnion fun i => isCompact_iff_compactSpace.mpr (hS' i), topIsAffineOpen _]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 AffineTargetMorphismProperty.StableUnderBaseChange affineProperty\n[PROOFSTEP]\nintro X Y S _ _ f g h\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : affineProperty g\n\u22a2 affineProperty pullback.fst\n[PROOFSTEP]\nrw [QuasiCompact.affineProperty] at h \u22a2\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\u22a2 CompactSpace \u2191\u2191(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\nskip\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\u22a2 CompactSpace \u2191\u2191(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\nlet \ud835\udcb0 := Scheme.Pullback.openCoverOfRight Y.affineCover.finiteSubcover f g\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\n\u22a2 CompactSpace \u2191\u2191(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\nhave : Finite \ud835\udcb0.J := by dsimp; infer_instance\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\n\u22a2 Finite \ud835\udcb0.J\n[PROOFSTEP]\ndsimp\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\n\u22a2 Finite (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)).J\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\nthis : Finite \ud835\udcb0.J\n\u22a2 CompactSpace \u2191\u2191(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\nhave : \u2200 i, CompactSpace (\ud835\udcb0.obj i).carrier := by intro i; dsimp; infer_instance\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\nthis : Finite \ud835\udcb0.J\n\u22a2 \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(Scheme.OpenCover.obj \ud835\udcb0 i).toPresheafedSpace\n[PROOFSTEP]\nintro i\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\nthis : Finite \ud835\udcb0.J\ni : \ud835\udcb0.J\n\u22a2 CompactSpace \u2191\u2191(Scheme.OpenCover.obj \ud835\udcb0 i).toPresheafedSpace\n[PROOFSTEP]\ndsimp\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\nthis : Finite \ud835\udcb0.J\ni : \ud835\udcb0.J\n\u22a2 CompactSpace\n \u2191\u2191(pullback f\n (Scheme.OpenCover.map (Scheme.affineCover Y) (Scheme.OpenCover.f (Scheme.affineCover Y) \u2191i) \u226b\n g)).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y S : Scheme\ninst\u271d\u00b9 : IsAffine S\ninst\u271d : IsAffine X\nf : X \u27f6 S\ng : Y \u27f6 S\nh : CompactSpace \u2191\u2191Y.toPresheafedSpace\n\ud835\udcb0 : Scheme.OpenCover (pullback f g) :=\n Scheme.Pullback.openCoverOfRight (Scheme.OpenCover.finiteSubcover (Scheme.affineCover Y)) f g\nthis\u271d : Finite \ud835\udcb0.J\nthis : \u2200 (i : \ud835\udcb0.J), CompactSpace \u2191\u2191(Scheme.OpenCover.obj \ud835\udcb0 i).toPresheafedSpace\n\u22a2 CompactSpace \u2191\u2191(pullback f g).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\nexact \ud835\udcb0.compactSpace\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\n\u22a2 P S\n[PROOFSTEP]\nclassical\nobtain \u27e8s, hs, hs'\u27e9 := (isCompact_open_iff_eq_finset_affine_union S.1).mp \u27e8hS, S.2\u27e9\nreplace hs' : S = iSup fun i : s => (i : Opens X.carrier) := by ext1; simpa using hs'\nsubst hs'\napply @Set.Finite.induction_on _ _ _ hs\n\u00b7 convert h\u2081; rw [iSup_eq_bot]; rintro \u27e8_, h\u27e9; exact h.elim\n\u00b7 intro x s _ hs h\u2084\n have : IsCompact (\u2a06 i : s, (i : Opens X.carrier)).1 := by\n refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1; exact \u27e8s, hs, by simp\u27e9\n convert h\u2082 _ this x h\u2084\n rw [iSup_subtype, sup_comm]\n conv_rhs => rw [iSup_subtype]\n exact iSup_insert\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\n\u22a2 P S\n[PROOFSTEP]\nobtain \u27e8s, hs, hs'\u27e9 := (isCompact_open_iff_eq_finset_affine_union S.1).mp \u27e8hS, S.2\u27e9\n[GOAL]\ncase intro.intro\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhs' : S.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 P S\n[PROOFSTEP]\nreplace hs' : S = iSup fun i : s => (i : Opens X.carrier) := by ext1; simpa using hs'\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhs' : S.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 S = \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhs' : S.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 \u2191S = \u2191(\u2a06 (i : \u2191s), \u2191\u2191i)\n[PROOFSTEP]\nsimpa using hs'\n[GOAL]\ncase intro.intro\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nS : Opens \u2191\u2191X.toPresheafedSpace\nhS : IsCompact S.carrier\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhs' : S = \u2a06 (i : \u2191s), \u2191\u2191i\n\u22a2 P S\n[PROOFSTEP]\nsubst hs'\n[GOAL]\ncase intro.intro\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 P (\u2a06 (i : \u2191s), \u2191\u2191i)\n[PROOFSTEP]\napply @Set.Finite.induction_on _ _ _ hs\n[GOAL]\ncase intro.intro.H0\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 P (\u2a06 (i : \u2191\u2205), \u2191\u2191i)\n[PROOFSTEP]\nconvert h\u2081\n[GOAL]\ncase h.e'_1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2a06 (i : \u2191\u2205), \u2191\u2191i = \u22a5\n[PROOFSTEP]\nrw [iSup_eq_bot]\n[GOAL]\ncase h.e'_1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2200 (i : \u2191\u2205), \u2191\u2191i = \u22a5\n[PROOFSTEP]\nrintro \u27e8_, h\u27e9\n[GOAL]\ncase h.e'_1.mk\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\nval\u271d : \u2191(Scheme.affineOpens X)\nh : val\u271d \u2208 \u2205\n\u22a2 \u2191\u2191{ val := val\u271d, property := h } = \u22a5\n[PROOFSTEP]\nexact h.elim\n[GOAL]\ncase intro.intro.H1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nhS : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2200 {a : \u2191(Scheme.affineOpens X)} {s : Set \u2191(Scheme.affineOpens X)},\n \u00aca \u2208 s \u2192 Set.Finite s \u2192 P (\u2a06 (i : \u2191s), \u2191\u2191i) \u2192 P (\u2a06 (i : \u2191(insert a s)), \u2191\u2191i)\n[PROOFSTEP]\nintro x s _ hs h\u2084\n[GOAL]\ncase intro.intro.H1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\n\u22a2 P (\u2a06 (i : \u2191(insert x s)), \u2191\u2191i)\n[PROOFSTEP]\nhave : IsCompact (\u2a06 i : s, (i : Opens X.carrier)).1 := by\n refine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1; exact \u27e8s, hs, by simp\u27e9\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\n\u22a2 IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n[PROOFSTEP]\nrefine' ((isCompact_open_iff_eq_finset_affine_union _).mpr _).1\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\n\u22a2 \u2203 s_1, Set.Finite s_1 \u2227 (\u2a06 (i : \u2191s), \u2191\u2191i).carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s_1), \u2191\u2191i\n[PROOFSTEP]\nexact \u27e8s, hs, by simp\u27e9\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\n\u22a2 (\u2a06 (i : \u2191s), \u2191\u2191i).carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.H1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 P (\u2a06 (i : \u2191(insert x s)), \u2191\u2191i)\n[PROOFSTEP]\nconvert h\u2082 _ this x h\u2084\n[GOAL]\ncase h.e'_1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2a06 (i : \u2191(insert x s)), \u2191\u2191i = (\u2a06 (i : \u2191s), \u2191\u2191i) \u2294 \u2191x\n[PROOFSTEP]\nrw [iSup_subtype, sup_comm]\n[GOAL]\ncase h.e'_1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2a06 (i : \u2191(Scheme.affineOpens X)) (h : i \u2208 insert x s), \u2191\u2191{ val := i, property := h } = \u2191x \u2294 \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\nconv_rhs => rw [iSup_subtype]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n| \u2191x \u2294 \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\nrw [iSup_subtype]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n| \u2191x \u2294 \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\nrw [iSup_subtype]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n| \u2191x \u2294 \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\nrw [iSup_subtype]\n[GOAL]\ncase h.e'_1\nX Y : Scheme\nf : X \u27f6 Y\nZ : Scheme\nP : Opens \u2191\u2191X.toPresheafedSpace \u2192 Prop\nh\u2081 : P \u22a5\nh\u2082 : \u2200 (S : Opens \u2191\u2191X.toPresheafedSpace), IsCompact S.carrier \u2192 \u2200 (U : \u2191(Scheme.affineOpens X)), P S \u2192 P (S \u2294 \u2191U)\ns\u271d : Set \u2191(Scheme.affineOpens X)\nhs\u271d : Set.Finite s\u271d\nhS : IsCompact (\u2a06 (i : \u2191s\u271d), \u2191\u2191i).carrier\nx : \u2191(Scheme.affineOpens X)\ns : Set \u2191(Scheme.affineOpens X)\na\u271d : \u00acx \u2208 s\nhs : Set.Finite s\nh\u2084 : P (\u2a06 (i : \u2191s), \u2191\u2191i)\nthis : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\n\u22a2 \u2a06 (i : \u2191(Scheme.affineOpens X)) (h : i \u2208 insert x s), \u2191\u2191{ val := i, property := h } =\n \u2191x \u2294 \u2a06 (i : \u2191(Scheme.affineOpens X)) (h : i \u2208 s), \u2191\u2191{ val := i, property := h }\n[PROOFSTEP]\nexact iSup_insert\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ : Scheme\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nrw [\u2190 map_zero (X.presheaf.map (homOfLE <| X.basicOpen_le f : X.basicOpen f \u27f6 U).op)] at H \n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ : Scheme\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = \u2191(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f \u2264 U)).op) 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n, rfl\u27e9, e\u27e9 := (isLocalization_basicOpen hU f).eq_iff_exists'.mp H\n[GOAL]\ncase intro.mk.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ : Scheme\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = \u2191(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f \u2264 U)).op) 0\nn : \u2115\ne :\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) } *\n x =\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) } *\n 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nexact \u27e8n, by simpa [mul_comm x] using e\u27e9\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ : Scheme\nX : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = \u2191(X.presheaf.map (homOfLE (_ : Scheme.basicOpen X f \u2264 U)).op) 0\nn : \u2115\ne :\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) } *\n x =\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) } *\n 0\n\u22a2 f ^ n * x = 0\n[PROOFSTEP]\nsimpa [mul_comm x] using e\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nobtain \u27e8s, hs, e\u27e9 := (isCompact_open_iff_eq_finset_affine_union U.1).mp \u27e8hU, U.2\u27e9\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nreplace e : U = iSup fun i : s => (i : Opens X.carrier)\n[GOAL]\ncase e\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 U = \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\next1\n[GOAL]\ncase e.h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U.carrier = \u22c3 (i : \u2191(Scheme.affineOpens X)) (_ : i \u2208 s), \u2191\u2191i\n\u22a2 \u2191U = \u2191(\u2a06 (i : \u2191s), \u2191\u2191i)\n[PROOFSTEP]\nsimpa using e\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nhave h\u2081 : \u2200 i : s, i.1.1 \u2264 U := by\n intro i\n change (i : Opens X.carrier) \u2264 U\n rw [e]\n -- porting note: `exact le_iSup _ _` no longer works\n exact le_iSup (fun (i : s) => (i : Opens (X.toPresheafedSpace))) _\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\n\u22a2 \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\n[PROOFSTEP]\nintro i\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\ni : \u2191s\n\u22a2 \u2191\u2191i \u2264 U\n[PROOFSTEP]\nchange (i : Opens X.carrier) \u2264 U\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\ni : \u2191s\n\u22a2 \u2191\u2191i \u2264 U\n[PROOFSTEP]\nrw [e]\n -- porting note: `exact le_iSup _ _` no longer works\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\ni : \u2191s\n\u22a2 \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i\n[PROOFSTEP]\nexact le_iSup (fun (i : s) => (i : Opens (X.toPresheafedSpace))) _\n[GOAL]\ncase intro.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nhave H' := fun i : s =>\n exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen X i.1.2 (X.presheaf.map (homOfLE (h\u2081 i)).op x)\n (X.presheaf.map (homOfLE (h\u2081 i)).op f) ?_\n[GOAL]\ncase intro.intro.refine_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nH' :\n \u2200 (i : \u2191s),\n \u2203 n, \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\n\u22a2 \u2203 n, f ^ n * x = 0\ncase intro.intro.refine_1\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x |_\n Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f) =\n 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.refine_1\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x |_\n Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f) =\n 0\n[PROOFSTEP]\ndelta TopCat.Presheaf.restrictOpen TopCat.Presheaf.restrict at H \u22a2\n[GOAL]\ncase intro.intro.refine_1\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : \u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map\n (homOfLE\n (_ :\n \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984,\n a \u2208 \u2191(Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f)) \u2192 a \u2208 \u2191\u2191\u2191i)).op)\n (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x) =\n 0\n[PROOFSTEP]\nconvert congr_arg (X.presheaf.map (homOfLE _).op) H\n[GOAL]\ncase h.e'_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : \u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map\n (homOfLE\n (_ :\n \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984,\n a \u2208 \u2191(Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f)) \u2192 a \u2208 \u2191\u2191\u2191i)).op)\n (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x) =\n \u2191(X.presheaf.map (homOfLE ?intro.intro.refine_1.convert_1).op)\n (\u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x)\n[PROOFSTEP]\nsimp only [\u2190 comp_apply, \u2190 Functor.map_comp]\n[GOAL]\ncase h.e'_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : \u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map\n ((homOfLE (_ : \u2191\u2191i \u2264 U)).op \u226b\n (homOfLE\n (_ :\n \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984,\n a \u2208 \u2191(Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f)) \u2192 a \u2208 \u2191\u2191\u2191i)).op))\n x =\n \u2191(X.presheaf.map\n ((homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op \u226b\n (homOfLE ?intro.intro.refine_1.convert_1).op))\n x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : \u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 0 = \u2191(X.presheaf.map (homOfLE ?intro.intro.refine_1.convert_1).op) 0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase intro.intro.refine_1.convert_1\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : \u2191(X.presheaf.map (homOfLE (_ : \u2200 \u2983a : \u2191\u2191X.toPresheafedSpace\u2984, a \u2208 \u2191(Scheme.basicOpen X f) \u2192 a \u2208 \u2191U)).op) x = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\ni : \u2191s\n\u22a2 Scheme.basicOpen X (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f) \u2264 Scheme.basicOpen X f\n[PROOFSTEP]\nsimp only [Scheme.basicOpen_res, ge_iff_le, inf_le_right]\n[GOAL]\ncase intro.intro.refine_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nH' :\n \u2200 (i : \u2191s),\n \u2203 n, \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nchoose n hn using H'\n[GOAL]\ncase intro.intro.refine_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nhaveI := hs.to_subtype\n[GOAL]\ncase intro.intro.refine_2\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis : Finite \u2191s\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\ncases nonempty_fintype s\n[GOAL]\ncase intro.intro.refine_2.intro\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\n\u22a2 \u2203 n, f ^ n * x = 0\n[PROOFSTEP]\nuse Finset.univ.sup n\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\n\u22a2 f ^ Finset.sup Finset.univ n * x = 0\n[PROOFSTEP]\nsuffices \u2200 i : s, X.presheaf.map (homOfLE (h\u2081 i)).op (f ^ Finset.univ.sup n * x) = 0\n by\n subst e\n apply TopCat.Sheaf.eq_of_locally_eq X.sheaf fun i : s => (i : Opens X.carrier)\n intro i\n rw [map_zero]\n apply this\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis\u271d : Finite \u2191s\nval\u271d : Fintype \u2191s\nthis : \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n\u22a2 f ^ Finset.sup Finset.univ n * x = 0\n[PROOFSTEP]\nsubst e\n[GOAL]\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nn : \u2191s \u2192 \u2115\nthis\u271d : Finite \u2191s\nval\u271d : Fintype \u2191s\nhU : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\nx f : \u2191(X.presheaf.obj (op (\u2a06 (i : \u2191s), \u2191\u2191i)))\nH : x |_ Scheme.basicOpen X f = 0\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i\nhn :\n \u2200 (i : \u2191s),\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) f ^ n i *\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) x =\n 0\nthis : \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n\u22a2 f ^ Finset.sup Finset.univ n * x = 0\n[PROOFSTEP]\napply TopCat.Sheaf.eq_of_locally_eq X.sheaf fun i : s => (i : Opens X.carrier)\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nn : \u2191s \u2192 \u2115\nthis\u271d : Finite \u2191s\nval\u271d : Fintype \u2191s\nhU : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\nx f : \u2191(X.presheaf.obj (op (\u2a06 (i : \u2191s), \u2191\u2191i)))\nH : x |_ Scheme.basicOpen X f = 0\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i\nhn :\n \u2200 (i : \u2191s),\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) f ^ n i *\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) x =\n 0\nthis : \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n\u22a2 \u2200 (i : \u2191s),\n \u2191((Scheme.sheaf X).val.map (Opens.leSupr (fun i => \u2191\u2191i) i).op) (f ^ Finset.sup Finset.univ n * x) =\n \u2191((Scheme.sheaf X).val.map (Opens.leSupr (fun i => \u2191\u2191i) i).op) 0\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nn : \u2191s \u2192 \u2115\nthis\u271d : Finite \u2191s\nval\u271d : Fintype \u2191s\nhU : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\nx f : \u2191(X.presheaf.obj (op (\u2a06 (i : \u2191s), \u2191\u2191i)))\nH : x |_ Scheme.basicOpen X f = 0\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i\nhn :\n \u2200 (i : \u2191s),\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) f ^ n i *\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) x =\n 0\nthis : \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) (f ^ Finset.sup Finset.univ n * x) = 0\ni : \u2191s\n\u22a2 \u2191((Scheme.sheaf X).val.map (Opens.leSupr (fun i => \u2191\u2191i) i).op) (f ^ Finset.sup Finset.univ n * x) =\n \u2191((Scheme.sheaf X).val.map (Opens.leSupr (fun i => \u2191\u2191i) i).op) 0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\nn : \u2191s \u2192 \u2115\nthis\u271d : Finite \u2191s\nval\u271d : Fintype \u2191s\nhU : IsCompact (\u2a06 (i : \u2191s), \u2191\u2191i).carrier\nx f : \u2191(X.presheaf.obj (op (\u2a06 (i : \u2191s), \u2191\u2191i)))\nH : x |_ Scheme.basicOpen X f = 0\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i\nhn :\n \u2200 (i : \u2191s),\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) f ^ n i *\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) x =\n 0\nthis : \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 \u2a06 (i : \u2191s), \u2191\u2191i)).op) (f ^ Finset.sup Finset.univ n * x) = 0\ni : \u2191s\n\u22a2 \u2191((Scheme.sheaf X).val.map (Opens.leSupr (fun i => \u2191\u2191i) i).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\napply this\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\n\u22a2 \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nhn :\n \u2200 (i : \u2191s), \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x = 0\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\ni : \u2191s\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\nreplace hn := congr_arg (fun x => X.presheaf.map (homOfLE (h\u2081 i)).op (f ^ (Finset.univ.sup n - n i)) * x) (hn i)\n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\ni : \u2191s\nhn :\n (fun x => \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * x)\n (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x) =\n (fun x => \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * x) 0\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\ndsimp at hn \n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\ni : \u2191s\nhn :\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i)) *\n (\u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) f ^ n i * \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) x) =\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * 0\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\nsimp only [\u2190 map_mul, \u2190 map_pow] at hn \n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\ni : \u2191s\nhn :\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i) * (f ^ n i * x)) =\n \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i)) * 0\n\u22a2 \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ Finset.sup Finset.univ n * x) = 0\n[PROOFSTEP]\nrwa [mul_zero, \u2190 mul_assoc, \u2190 pow_add, tsub_add_cancel_of_le] at hn \n[GOAL]\ncase h\nX\u271d Y : Scheme\nf\u271d : X\u271d \u27f6 Y\nZ X : Scheme\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsCompact U.carrier\nx f : \u2191(X.presheaf.obj (op U))\nH : x |_ Scheme.basicOpen X f = 0\ns : Set \u2191(Scheme.affineOpens X)\nhs : Set.Finite s\ne : U = \u2a06 (i : \u2191s), \u2191\u2191i\nh\u2081 : \u2200 (i : \u2191s), \u2191\u2191i \u2264 U\nn : \u2191s \u2192 \u2115\nthis : Finite \u2191s\nval\u271d : Fintype \u2191s\ni : \u2191s\nhn : \u2191(X.presheaf.map (homOfLE (_ : \u2191\u2191i \u2264 U)).op) (f ^ (Finset.sup Finset.univ n - n i + n i) * x) = 0\n\u22a2 n i \u2264 Finset.sup Finset.univ n\n[PROOFSTEP]\napply Finset.le_sup (Finset.mem_univ i)\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact", "llama_tokens": 37467, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011397337391, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.274289929935509}} {"text": "[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\ninst\u271d\u2079 : CommSemiring A\ninst\u271d\u2078 : AddCommMonoid M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module A M\u2081\ninst\u271d\u00b3 : SMulCommClass R A M\u2081\ninst\u271d\u00b2 : SMulCommClass A R M\u2081\ninst\u271d\u00b9 : IsScalarTower R A M\u2081\ninst\u271d : Module R M\u2082\nB\u2081 : BilinForm A M\u2081\nB\u2082 : BilinForm R M\u2082\nhB\u2081 : IsSymm B\u2081\nhB\u2082 : IsSymm B\u2082\n\u22a2 IsSymm (BilinForm.tmul B\u2081 B\u2082)\n[PROOFSTEP]\nrw [isSymm_iff_flip R]\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\ninst\u271d\u2079 : CommSemiring A\ninst\u271d\u2078 : AddCommMonoid M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module A M\u2081\ninst\u271d\u00b3 : SMulCommClass R A M\u2081\ninst\u271d\u00b2 : SMulCommClass A R M\u2081\ninst\u271d\u00b9 : IsScalarTower R A M\u2081\ninst\u271d : Module R M\u2082\nB\u2081 : BilinForm A M\u2081\nB\u2082 : BilinForm R M\u2082\nhB\u2081 : IsSymm B\u2081\nhB\u2082 : IsSymm B\u2082\n\u22a2 \u2191(flipHom R) (BilinForm.tmul B\u2081 B\u2082) = BilinForm.tmul B\u2081 B\u2082\n[PROOFSTEP]\napply toLin.injective\n[GOAL]\ncase a\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\ninst\u271d\u2079 : CommSemiring A\ninst\u271d\u2078 : AddCommMonoid M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module A M\u2081\ninst\u271d\u00b3 : SMulCommClass R A M\u2081\ninst\u271d\u00b2 : SMulCommClass A R M\u2081\ninst\u271d\u00b9 : IsScalarTower R A M\u2081\ninst\u271d : Module R M\u2082\nB\u2081 : BilinForm A M\u2081\nB\u2082 : BilinForm R M\u2082\nhB\u2081 : IsSymm B\u2081\nhB\u2082 : IsSymm B\u2082\n\u22a2 \u2191toLin (\u2191(flipHom R) (BilinForm.tmul B\u2081 B\u2082)) = \u2191toLin (BilinForm.tmul B\u2081 B\u2082)\n[PROOFSTEP]\next x\u2081 x\u2082 y\u2081 y\u2082\n[GOAL]\ncase a.a.h.h.a.h.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\ninst\u271d\u2079 : CommSemiring A\ninst\u271d\u2078 : AddCommMonoid M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module A M\u2081\ninst\u271d\u00b3 : SMulCommClass R A M\u2081\ninst\u271d\u00b2 : SMulCommClass A R M\u2081\ninst\u271d\u00b9 : IsScalarTower R A M\u2081\ninst\u271d : Module R M\u2082\nB\u2081 : BilinForm A M\u2081\nB\u2082 : BilinForm R M\u2082\nhB\u2081 : IsSymm B\u2081\nhB\u2082 : IsSymm B\u2082\nx\u2081 : M\u2081\nx\u2082 : M\u2082\ny\u2081 : M\u2081\ny\u2082 : M\u2082\n\u22a2 \u2191(\u2191(AlgebraTensorModule.curry (\u2191(\u2191(AlgebraTensorModule.curry (\u2191toLin (\u2191(flipHom R) (BilinForm.tmul B\u2081 B\u2082)))) x\u2081) x\u2082))\n y\u2081)\n y\u2082 =\n \u2191(\u2191(AlgebraTensorModule.curry (\u2191(\u2191(AlgebraTensorModule.curry (\u2191toLin (BilinForm.tmul B\u2081 B\u2082))) x\u2081) x\u2082)) y\u2081) y\u2082\n[PROOFSTEP]\nexact (congr_arg\u2082 (HSMul.hSMul) (hB\u2082 x\u2082 y\u2082) (hB\u2081 x\u2081 y\u2081)).symm\n[GOAL]\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : Module R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2081\ninst\u271d\u00b3 : Module.Finite R M\u2081\ninst\u271d\u00b2 : Module.Free R M\u2082\ninst\u271d\u00b9 : Module.Finite R M\u2082\ninst\u271d : Nontrivial R\n\u22a2 \u2191(tensorDistribEquiv R) = tensorDistrib R R\n[PROOFSTEP]\next B\u2081 B\u2082 : 3\n[GOAL]\ncase a.h.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : Module R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2081\ninst\u271d\u00b3 : Module.Finite R M\u2081\ninst\u271d\u00b2 : Module.Free R M\u2082\ninst\u271d\u00b9 : Module.Finite R M\u2082\ninst\u271d : Nontrivial R\nB\u2081 : BilinForm R M\u2081\nB\u2082 : BilinForm R M\u2082\n\u22a2 \u2191(\u2191(AlgebraTensorModule.curry \u2191(tensorDistribEquiv R)) B\u2081) B\u2082 =\n \u2191(\u2191(AlgebraTensorModule.curry (tensorDistrib R R)) B\u2081) B\u2082\n[PROOFSTEP]\napply toLin.injective\n[GOAL]\ncase a.h.h.a\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : Module R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2081\ninst\u271d\u00b3 : Module.Finite R M\u2081\ninst\u271d\u00b2 : Module.Free R M\u2082\ninst\u271d\u00b9 : Module.Finite R M\u2082\ninst\u271d : Nontrivial R\nB\u2081 : BilinForm R M\u2081\nB\u2082 : BilinForm R M\u2082\n\u22a2 \u2191toLin (\u2191(\u2191(AlgebraTensorModule.curry \u2191(tensorDistribEquiv R)) B\u2081) B\u2082) =\n \u2191toLin (\u2191(\u2191(AlgebraTensorModule.curry (tensorDistrib R R)) B\u2081) B\u2082)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h.h.a.a.h.h.a.h.h\n\u03b9 : Type u\u03b9\nR : Type uR\nA : Type uA\nM\u2081 : Type uM\u2081\nM\u2082 : Type uM\u2082\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : Module R M\u2082\ninst\u271d\u2074 : Module.Free R M\u2081\ninst\u271d\u00b3 : Module.Finite R M\u2081\ninst\u271d\u00b2 : Module.Free R M\u2082\ninst\u271d\u00b9 : Module.Finite R M\u2082\ninst\u271d : Nontrivial R\nB\u2081 : BilinForm R M\u2081\nB\u2082 : BilinForm R M\u2082\nx\u271d\u00b3 : M\u2081\nx\u271d\u00b2 : M\u2082\nx\u271d\u00b9 : M\u2081\nx\u271d : M\u2082\n\u22a2 \u2191(\u2191(AlgebraTensorModule.curry\n (\u2191(\u2191(AlgebraTensorModule.curry (\u2191toLin (\u2191(\u2191(AlgebraTensorModule.curry \u2191(tensorDistribEquiv R)) B\u2081) B\u2082)))\n x\u271d\u00b3)\n x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d =\n \u2191(\u2191(AlgebraTensorModule.curry\n (\u2191(\u2191(AlgebraTensorModule.curry (\u2191toLin (\u2191(\u2191(AlgebraTensorModule.curry (tensorDistrib R R)) B\u2081) B\u2082))) x\u271d\u00b3)\n x\u271d\u00b2))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nexact mul_comm _ _\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.BilinearForm.TensorProduct", "llama_tokens": 2385, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.38861802670584894, "lm_q1q2_score": 0.274280784796552}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : SetLike \u03c3 R\ninst\u271d : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ni : \u03b9\n\u22a2 AddCommMonoid { x // x \u2208 A i }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : SetLike \u03c3 R\ninst\u271d : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ni : \u03b9\n\u22a2 AddCommGroup { x // x \u2208 A i }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Algebra S R\nA : \u03b9 \u2192 Submodule S R\ninst\u271d : GradedOne A\ns : S\n\u22a2 \u2191(algebraMap S R) s \u2208 A 0\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Algebra S R\nA : \u03b9 \u2192 Submodule S R\ninst\u271d : GradedOne A\ns : S\n\u22a2 s \u2022 1 \u2208 A 0\n[PROOFSTEP]\nexact (A 0).smul_mem s <| SetLike.one_mem_graded _\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\nn : \u2115\n\u22a2 \u2191n \u2208 A 0\n[PROOFSTEP]\ninduction' n with _ n_ih\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\n\u22a2 \u2191Nat.zero \u2208 A 0\n[PROOFSTEP]\nrw [Nat.cast_zero]\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\n\u22a2 0 \u2208 A 0\n[PROOFSTEP]\nexact zero_mem (A 0)\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\nn\u271d : \u2115\nn_ih : \u2191n\u271d \u2208 A 0\n\u22a2 \u2191(Nat.succ n\u271d) \u2208 A 0\n[PROOFSTEP]\nrw [Nat.cast_succ]\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddMonoidWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\nn\u271d : \u2115\nn_ih : \u2191n\u271d \u2208 A 0\n\u22a2 \u2191n\u271d + 1 \u2208 A 0\n[PROOFSTEP]\nexact add_mem n_ih (SetLike.one_mem_graded _)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddGroupWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\nz : \u2124\n\u22a2 \u2191z \u2208 A 0\n[PROOFSTEP]\ninduction z\n[GOAL]\ncase ofNat\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddGroupWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\na\u271d : \u2115\n\u22a2 \u2191(Int.ofNat a\u271d) \u2208 A 0\n[PROOFSTEP]\nrw [Int.ofNat_eq_coe, Int.cast_ofNat]\n[GOAL]\ncase ofNat\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddGroupWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\na\u271d : \u2115\n\u22a2 \u2191a\u271d \u2208 A 0\n[PROOFSTEP]\nexact SetLike.nat_cast_mem_graded _ _\n[GOAL]\ncase negSucc\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddGroupWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\na\u271d : \u2115\n\u22a2 \u2191(Int.negSucc a\u271d) \u2208 A 0\n[PROOFSTEP]\nrw [Int.cast_negSucc]\n[GOAL]\ncase negSucc\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2074 : Zero \u03b9\ninst\u271d\u00b3 : AddGroupWithOne R\ninst\u271d\u00b2 : SetLike \u03c3 R\ninst\u271d\u00b9 : AddSubgroupClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d : GradedOne A\na\u271d : \u2115\n\u22a2 -\u2191(a\u271d + 1) \u2208 A 0\n[PROOFSTEP]\nexact neg_mem (SetLike.nat_cast_mem_graded _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2191(\u2191(r * r') n) =\n \u2211 ij in Finset.filter (fun ij => ij.fst + ij.snd = n) (DFinsupp.support r \u00d7\u02e2 DFinsupp.support r'),\n \u2191(\u2191r ij.fst) * \u2191(\u2191r' ij.snd)\n[PROOFSTEP]\nrw [mul_eq_sum_support_ghas_mul, DFinsupp.finset_sum_apply, AddSubmonoidClass.coe_finset_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2211 i in DFinsupp.support r \u00d7\u02e2 DFinsupp.support r',\n \u2191(\u2191(\u2191(of (fun i => (fun i => { x // x \u2208 A i }) i) (i.fst + i.snd)) (GradedMonoid.GMul.mul (\u2191r i.fst) (\u2191r' i.snd)))\n n) =\n \u2211 ij in Finset.filter (fun ij => ij.fst + ij.snd = n) (DFinsupp.support r \u00d7\u02e2 DFinsupp.support r'),\n \u2191(\u2191r ij.fst) * \u2191(\u2191r' ij.snd)\n[PROOFSTEP]\nsimp_rw [coe_of_apply, apply_ite, ZeroMemClass.coe_zero, \u2190 Finset.sum_filter, SetLike.coe_gMul]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2191(\u2191(r * r') n) = DFinsupp.sum r fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0\n[PROOFSTEP]\nrw [mul_eq_dfinsupp_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2191(\u2191(DFinsupp.sum r fun i ai =>\n DFinsupp.sum r' fun j aj => \u2191(of (fun i => { x // x \u2208 A i }) (i + j)) (GradedMonoid.GMul.mul ai aj))\n n) =\n DFinsupp.sum r fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0\n[PROOFSTEP]\niterate 2 rw [DFinsupp.sum_apply, DFinsupp.sum, AddSubmonoidClass.coe_finset_sum]; congr; ext\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2191(\u2191(DFinsupp.sum r fun i ai =>\n DFinsupp.sum r' fun j aj => \u2191(of (fun i => { x // x \u2208 A i }) (i + j)) (GradedMonoid.GMul.mul ai aj))\n n) =\n DFinsupp.sum r fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0\n[PROOFSTEP]\nrw [DFinsupp.sum_apply, DFinsupp.sum, AddSubmonoidClass.coe_finset_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 \u2211 i in DFinsupp.support r,\n \u2191(\u2191(DFinsupp.sum r' fun j aj => \u2191(of (fun i => { x // x \u2208 A i }) (i + j)) (GradedMonoid.GMul.mul (\u2191r i) aj)) n) =\n DFinsupp.sum r fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\n\u22a2 (fun i =>\n \u2191(\u2191(DFinsupp.sum r' fun j aj => \u2191(of (fun i => { x // x \u2208 A i }) (i + j)) (GradedMonoid.GMul.mul (\u2191r i) aj)) n)) =\n fun i => (fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) i (\u2191r i)\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d : \u03b9\n\u22a2 \u2191(\u2191(DFinsupp.sum r' fun j aj => \u2191(of (fun i => { x // x \u2208 A i }) (x\u271d + j)) (GradedMonoid.GMul.mul (\u2191r x\u271d) aj)) n) =\n (fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) x\u271d (\u2191r x\u271d)\n[PROOFSTEP]\nrw [DFinsupp.sum_apply, DFinsupp.sum, AddSubmonoidClass.coe_finset_sum]\n[GOAL]\ncase e_f.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d : \u03b9\n\u22a2 \u2211 i in DFinsupp.support r',\n \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d + i)) (GradedMonoid.GMul.mul (\u2191r x\u271d) (\u2191r' i))) n) =\n (fun i ri => DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) x\u271d (\u2191r x\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f.h.e_f\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d : \u03b9\n\u22a2 (fun i => \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d + i)) (GradedMonoid.GMul.mul (\u2191r x\u271d) (\u2191r' i))) n)) = fun i =>\n (fun j rj => if x\u271d + j = n then \u2191(\u2191r x\u271d) * \u2191rj else 0) i (\u2191r' i)\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h.e_f.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d\u00b9 x\u271d : \u03b9\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d\u00b9 + x\u271d)) (GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d))) n) =\n (fun j rj => if x\u271d\u00b9 + j = n then \u2191(\u2191r x\u271d\u00b9) * \u2191rj else 0) x\u271d (\u2191r' x\u271d)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase e_f.h.e_f.h\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d\u00b9 x\u271d : \u03b9\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d\u00b9 + x\u271d)) (GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d))) n) =\n if x\u271d\u00b9 + x\u271d = n then \u2191(\u2191r x\u271d\u00b9) * \u2191(\u2191r' x\u271d) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d\u00b9 x\u271d : \u03b9\nh : x\u271d\u00b9 + x\u271d = n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d\u00b9 + x\u271d)) (GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d))) n) = \u2191(\u2191r x\u271d\u00b9) * \u2191(\u2191r' x\u271d)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nx\u271d\u00b9 x\u271d : \u03b9\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d\u00b9 + x\u271d)) (GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d))) (x\u271d\u00b9 + x\u271d)) =\n \u2191(\u2191r x\u271d\u00b9) * \u2191(\u2191r' x\u271d)\n[PROOFSTEP]\nrw [of_eq_same]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nx\u271d\u00b9 x\u271d : \u03b9\n\u22a2 \u2191(GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d)) = \u2191(\u2191r x\u271d\u00b9) * \u2191(\u2191r' x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d\u00b9 x\u271d : \u03b9\nh : \u00acx\u271d\u00b9 + x\u271d = n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) (x\u271d\u00b9 + x\u271d)) (GradedMonoid.GMul.mul (\u2191r x\u271d\u00b9) (\u2191r' x\u271d))) n) = 0\n[PROOFSTEP]\nrw [of_eq_of_ne _ _ _ _ h]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : SetLike \u03c3 R\ninst\u271d\u00b3 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b2 : AddMonoid \u03b9\ninst\u271d\u00b9 : SetLike.GradedMonoid A\ninst\u271d : (i : \u03b9) \u2192 (x : { x // x \u2208 A i }) \u2192 Decidable (x \u2260 0)\nr r' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn x\u271d\u00b9 x\u271d : \u03b9\nh : \u00acx\u271d\u00b9 + x\u271d = n\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nclassical\nrw [coe_mul_apply_eq_dfinsupp_sum]\napply (DFinsupp.sum_single_index _).trans\nswap\n\u00b7 simp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]\n exact DFinsupp.sum_zero\nsimp_rw [DFinsupp.sum, H, Finset.sum_ite_eq']\nsplit_ifs with h\nrfl\nrw [DFinsupp.not_mem_support_iff.mp h, ZeroMemClass.coe_zero, mul_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nrw [coe_mul_apply_eq_dfinsupp_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r) fun i ri =>\n DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) =\n \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\napply (DFinsupp.sum_single_index _).trans\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = \u2191r * \u2191(\u2191r' j)\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0\n[PROOFSTEP]\nsimp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum r' fun j rj => 0) = 0\n[PROOFSTEP]\nexact DFinsupp.sum_zero\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nsimp_rw [DFinsupp.sum, H, Finset.sum_ite_eq']\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\n\u22a2 (if j \u2208 DFinsupp.support r' then \u2191r * \u2191(\u2191r' j) else 0) = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\nh : j \u2208 DFinsupp.support r'\n\u22a2 \u2191r * \u2191(\u2191r' j) = \u2191r * \u2191(\u2191r' j)\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\nh : \u00acj \u2208 DFinsupp.support r'\n\u22a2 0 = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), i + x = n \u2194 x = j\nh : \u00acj \u2208 DFinsupp.support r'\n\u22a2 0 = \u2191r * \u2191(\u2191r' j)\n[PROOFSTEP]\nrw [DFinsupp.not_mem_support_iff.mp h, ZeroMemClass.coe_zero, mul_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nclassical\nrw [coe_mul_apply_eq_dfinsupp_sum, DFinsupp.sum_comm]\napply (DFinsupp.sum_single_index _).trans\nswap\n\u00b7 simp_rw [ZeroMemClass.coe_zero, mul_zero, ite_self]\n exact DFinsupp.sum_zero\nsimp_rw [DFinsupp.sum, H, Finset.sum_ite_eq']\nsplit_ifs with h\nrfl\nrw [DFinsupp.not_mem_support_iff.mp h, ZeroMemClass.coe_zero, zero_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nrw [coe_mul_apply_eq_dfinsupp_sum, DFinsupp.sum_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r') fun i\u2082 x\u2082 =>\n DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i\u2082 = n then \u2191x\u2081 * \u2191x\u2082 else 0) =\n \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\napply (DFinsupp.sum_single_index _).trans\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u2191r' else 0) = \u2191(\u2191r j) * \u2191r'\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u21910 else 0) = 0\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u21910 else 0) = 0\n[PROOFSTEP]\nsimp_rw [ZeroMemClass.coe_zero, mul_zero, ite_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => 0) = 0\n[PROOFSTEP]\nexact DFinsupp.sum_zero\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u2191r' else 0) = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nsimp_rw [DFinsupp.sum, H, Finset.sum_ite_eq']\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\n\u22a2 (if j \u2208 DFinsupp.support r then \u2191(\u2191r j) * \u2191r' else 0) = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\nh : j \u2208 DFinsupp.support r\n\u22a2 \u2191(\u2191r j) * \u2191r' = \u2191(\u2191r j) * \u2191r'\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\nh : \u00acj \u2208 DFinsupp.support r\n\u22a2 0 = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nj n : \u03b9\nH : \u2200 (x : \u03b9), x + i = n \u2194 x = j\nh : \u00acj \u2208 DFinsupp.support r\n\u22a2 0 = \u2191(\u2191r j) * \u2191r'\n[PROOFSTEP]\nrw [DFinsupp.not_mem_support_iff.mp h, ZeroMemClass.coe_zero, zero_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = 0\n[PROOFSTEP]\nclassical\nrw [coe_mul_apply_eq_dfinsupp_sum]\napply (DFinsupp.sum_single_index _).trans\nswap\n\u00b7 simp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]\n exact DFinsupp.sum_zero\n\u00b7 rw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]\n exact fun x _ H => h ((self_le_add_right i x).trans_eq H)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = 0\n[PROOFSTEP]\nrw [coe_mul_apply_eq_dfinsupp_sum]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r) fun i ri =>\n DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) =\n 0\n[PROOFSTEP]\napply (DFinsupp.sum_single_index _).trans\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0\n[PROOFSTEP]\nsimp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => 0) = 0\n[PROOFSTEP]\nexact DFinsupp.sum_zero\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0\n[PROOFSTEP]\nrw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2200 (x : \u03b9), x \u2208 DFinsupp.support r' \u2192 i + x = n \u2192 False\n[PROOFSTEP]\nexact fun x _ H => h ((self_le_add_right i x).trans_eq H)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = 0\n[PROOFSTEP]\nclassical\nrw [coe_mul_apply_eq_dfinsupp_sum, DFinsupp.sum_comm]\napply (DFinsupp.sum_single_index _).trans\nswap\n\u00b7 simp_rw [ZeroMemClass.coe_zero, mul_zero, ite_self]\n exact DFinsupp.sum_zero\n\u00b7 rw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]\n exact fun x _ H => h ((self_le_add_left i x).trans_eq H)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = 0\n[PROOFSTEP]\nrw [coe_mul_apply_eq_dfinsupp_sum, DFinsupp.sum_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r') fun i\u2082 x\u2082 =>\n DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i\u2082 = n then \u2191x\u2081 * \u2191x\u2082 else 0) =\n 0\n[PROOFSTEP]\napply (DFinsupp.sum_single_index _).trans\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u2191r' else 0) = 0\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u21910 else 0) = 0\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u21910 else 0) = 0\n[PROOFSTEP]\nsimp_rw [ZeroMemClass.coe_zero, mul_zero, ite_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => 0) = 0\n[PROOFSTEP]\nexact DFinsupp.sum_zero\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r fun i\u2081 x\u2081 => if i\u2081 + i = n then \u2191x\u2081 * \u2191r' else 0) = 0\n[PROOFSTEP]\nrw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2200 (x : \u03b9), x \u2208 DFinsupp.support r \u2192 x + i = n \u2192 False\n[PROOFSTEP]\nexact fun x _ H => h ((self_le_add_left i x).trans_eq H)\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2078 : DecidableEq \u03b9\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : SetLike \u03c3 R\ninst\u271d\u2075 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2074 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u00b3 : SetLike.GradedMonoid A\ninst\u271d\u00b2 : Sub \u03b9\ninst\u271d\u00b9 : OrderedSub \u03b9\ninst\u271d : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : i \u2264 n\nx : \u03b9\n\u22a2 i + x = n \u2194 x = n - i\n[PROOFSTEP]\nrw [eq_tsub_iff_add_eq_of_le h, add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = if i \u2264 n then \u2191(\u2191r (n - i)) * \u2191r' else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\nh : i \u2264 n\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = \u2191(\u2191r (n - i)) * \u2191r'\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nr : \u2a01 (i : \u03b9), { x // x \u2208 A i }\ni : \u03b9\nr' : { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(r * \u2191(of (fun i => { x // x \u2208 A i }) i) r') n) = 0\n[PROOFSTEP]\nexacts [coe_mul_of_apply_of_le _ _ _ n h, coe_mul_of_apply_of_not_le _ _ _ n h]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = if i \u2264 n then \u2191r * \u2191(\u2191r' (n - i)) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\nh : i \u2264 n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = \u2191r * \u2191(\u2191r' (n - i))\ncase neg\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2079 : DecidableEq \u03b9\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : SetLike \u03c3 R\ninst\u271d\u2076 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u2075 : CanonicallyOrderedAddMonoid \u03b9\ninst\u271d\u2074 : SetLike.GradedMonoid A\ninst\u271d\u00b3 : Sub \u03b9\ninst\u271d\u00b2 : OrderedSub \u03b9\ninst\u271d\u00b9 : ContravariantClass \u03b9 \u03b9 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\ninst\u271d : Decidable (i \u2264 n)\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = 0\n[PROOFSTEP]\nexacts [coe_of_mul_apply_of_le _ _ _ n h, coe_of_mul_apply_of_not_le _ _ _ n h]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Algebra S R\np : Submodule S R\n\u22a2 1 \u2208 p ^ 0\n[PROOFSTEP]\nrw [\u2190 one_le, pow_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Algebra S R\np\u271d : Submodule S R\ni j : \u2115\np q : R\nhp : p \u2208 p\u271d ^ i\nhq : q \u2208 p\u271d ^ j\n\u22a2 p * q \u2208 p\u271d ^ (i + j)\n[PROOFSTEP]\nrw [pow_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Algebra S R\np\u271d : Submodule S R\ni j : \u2115\np q : R\nhp : p \u2208 p\u271d ^ i\nhq : q \u2208 p\u271d ^ j\n\u22a2 p * q \u2208 p\u271d ^ i * p\u271d ^ j\n[PROOFSTEP]\nexact Submodule.mul_mem_mul hp hq\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Algebra S R\nA : \u03b9 \u2192 Submodule S R\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nx : { x // x \u2208 A i }\n\u22a2 \u2191((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (A i)) i)) 0) GradedMonoid.GOne.one = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : AddMonoid \u03b9\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Algebra S R\nA : \u03b9 \u2192 Submodule S R\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nx : { x // x \u2208 A i }\ni\u271d j\u271d : \u03b9\nx\u271d\u00b9 : { x // x \u2208 A i\u271d }\nx\u271d : { x // x \u2208 A j\u271d }\n\u22a2 \u2191((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (A i)) i)) (i\u271d + j\u271d))\n (GradedMonoid.GMul.mul x\u271d\u00b9 x\u271d) =\n \u2191((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (A i)) i)) i\u271d) x\u271d\u00b9 *\n \u2191((fun i => LinearMap.toAddMonoidHom ((fun i => Submodule.subtype (A i)) i)) j\u271d) x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.DirectSum.Internal", "llama_tokens": 21346, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.27413187815409645}} {"text": "[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 Linear k (Rep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 AddCommGroup (CoeSort.coe V)\n[PROOFSTEP]\nchange AddCommGroup ((forget\u2082 (Rep k G) (ModuleCat k)).obj V)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 AddCommGroup \u2191((forget\u2082 (Rep k G) (ModuleCat k)).obj V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 Module k (CoeSort.coe V)\n[PROOFSTEP]\nchange Module k ((forget\u2082 (Rep k G) (ModuleCat k)).obj V)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 Module k \u2191((forget\u2082 (Rep k G) (ModuleCat k)).obj V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G\u271d : Type u\ninst\u271d\u00b2 : CommRing k\ninst\u271d\u00b9 : Monoid G\u271d\nG : Type u\ninst\u271d : Group G\nA : Rep k G\ng : G\nx : CoeSort.coe A\n\u22a2 \u2191(\u2191(\u03c1 A) g\u207b\u00b9 * \u2191(\u03c1 A) g) x = x\n[PROOFSTEP]\nrw [\u2190 map_mul, inv_mul_self, map_one, LinearMap.one_apply]\n[GOAL]\nk G\u271d : Type u\ninst\u271d\u00b2 : CommRing k\ninst\u271d\u00b9 : Monoid G\u271d\nG : Type u\ninst\u271d : Group G\nA : Rep k G\ng : G\nx : CoeSort.coe A\n\u22a2 \u2191(\u2191(\u03c1 A) g * \u2191(\u03c1 A) g\u207b\u00b9) x = x\n[PROOFSTEP]\nrw [\u2190 map_mul, mul_inv_self, map_one, LinearMap.one_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX : Action (Type u) (MonCat.of G)\ng : G\nx : X.V\n\u22a2 \u2191(\u2191(\u03c1 ((linearization k G).toLaxMonoidalFunctor.toFunctor.obj X)) g) (Finsupp.single x 1) =\n Finsupp.single (\u2191X.\u03c1 g x) 1\n[PROOFSTEP]\nrw [linearization_obj_\u03c1, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX : Action (Type u) (MonCat.of G)\ng : G\nx : X.V\nr : k\n\u22a2 \u2191(\u2191(\u03c1 ((linearization k G).toLaxMonoidalFunctor.toFunctor.obj X)) g) (Finsupp.single x r) =\n Finsupp.single (\u2191X.\u03c1 g x) r\n[PROOFSTEP]\nrw [linearization_obj_\u03c1, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX\u271d Y\u271d : Action (Type u) (MonCat.of G)\nf : X\u271d \u27f6 Y\u271d\nX Y : Action (Type u) (MonCat.of G)\n\u22a2 (inv (LaxMonoidalFunctor.\u03bc (linearization k G).toLaxMonoidalFunctor X Y)).hom =\n \u2191(LinearEquiv.symm (finsuppTensorFinsupp' k X.V Y.V))\n[PROOFSTEP]\nrw [\u2190 Action.forget_map, Functor.map_inv]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX\u271d Y\u271d : Action (Type u) (MonCat.of G)\nf : X\u271d \u27f6 Y\u271d\nX Y : Action (Type u) (MonCat.of G)\n\u22a2 inv\n ((Action.forget (ModuleCat k) (MonCat.of G)).map\n (LaxMonoidalFunctor.\u03bc (linearization k G).toLaxMonoidalFunctor X Y)) =\n \u2191(LinearEquiv.symm (finsuppTensorFinsupp' k X.V Y.V))\n[PROOFSTEP]\napply IsIso.inv_eq_of_hom_inv_id\n[GOAL]\ncase hom_inv_id\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX\u271d Y\u271d : Action (Type u) (MonCat.of G)\nf : X\u271d \u27f6 Y\u271d\nX Y : Action (Type u) (MonCat.of G)\n\u22a2 (Action.forget (ModuleCat k) (MonCat.of G)).map (LaxMonoidalFunctor.\u03bc (linearization k G).toLaxMonoidalFunctor X Y) \u226b\n \u2191(LinearEquiv.symm (finsuppTensorFinsupp' k X.V Y.V)) =\n \ud835\udfd9\n ((Action.forget (ModuleCat k) (MonCat.of G)).obj\n (MonoidalCategory.tensorObj ((linearization k G).toLaxMonoidalFunctor.toFunctor.obj X)\n ((linearization k G).toLaxMonoidalFunctor.toFunctor.obj Y)))\n[PROOFSTEP]\nexact LinearMap.ext fun x => LinearEquiv.symm_apply_apply (finsuppTensorFinsupp' k X.V Y.V) x\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : \u2191(MonCat.of G)\n\u22a2 (\u2191(ofMulAction k G G).\u03c1 g \u226b \u2191(Finsupp.lift ((fun x => CoeSort.coe A) x) k G) fun g => \u2191(\u2191(\u03c1 A) g) x) =\n (\u2191(Finsupp.lift ((fun x => CoeSort.coe A) x) k G) fun g => \u2191(\u2191(\u03c1 A) g) x) \u226b \u2191A.\u03c1 g\n[PROOFSTEP]\nrefine'\n Finsupp.lhom_ext' fun y =>\n LinearMap.ext_ring\n _\n /- Porting note: rest of broken proof was\n simpa only [LinearMap.comp_apply, ModuleCat.comp_def, Finsupp.lsingle_apply, Finsupp.lift_apply,\n Action_\u03c1_eq_\u03c1, of_\u03c1_apply, Representation.ofMulAction_single, Finsupp.sum_single_index,\n zero_smul, one_smul, smul_eq_mul, A.\u03c1.map_mul] -/\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : \u2191(MonCat.of G)\ny : G\n\u22a2 \u2191(LinearMap.comp (\u2191(ofMulAction k G G).\u03c1 g \u226b \u2191(Finsupp.lift ((fun x => CoeSort.coe A) x) k G) fun g => \u2191(\u2191(\u03c1 A) g) x)\n (Finsupp.lsingle y))\n 1 =\n \u2191(LinearMap.comp ((\u2191(Finsupp.lift ((fun x => CoeSort.coe A) x) k G) fun g => \u2191(\u2191(\u03c1 A) g) x) \u226b \u2191A.\u03c1 g)\n (Finsupp.lsingle y))\n 1\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, ModuleCat.comp_def, Finsupp.lsingle_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : \u2191(MonCat.of G)\ny : G\n\u22a2 \u2191(\u2191(Finsupp.lift (CoeSort.coe A) k G) fun g => \u2191(\u2191(\u03c1 A) g) x) (\u2191(\u2191(ofMulAction k G G).\u03c1 g) (Finsupp.single y 1)) =\n \u2191(\u2191A.\u03c1 g) (\u2191(\u2191(Finsupp.lift (CoeSort.coe A) k G) fun g => \u2191(\u2191(\u03c1 A) g) x) (Finsupp.single y 1))\n[PROOFSTEP]\nerw [Finsupp.lift_apply, Finsupp.lift_apply, Representation.ofMulAction_single (G := G)]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : \u2191(MonCat.of G)\ny : G\n\u22a2 (Finsupp.sum (Finsupp.single (g \u2022 y) 1) fun x_1 r => r \u2022 \u2191(\u2191(\u03c1 A) x_1) x) =\n \u2191(\u2191A.\u03c1 g) (Finsupp.sum (Finsupp.single y 1) fun x_1 r => r \u2022 \u2191(\u2191(\u03c1 A) x_1) x)\n[PROOFSTEP]\nsimp only [Finsupp.sum_single_index, zero_smul, one_smul, smul_eq_mul, A.\u03c1.map_mul, of_\u03c1]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : \u2191(MonCat.of G)\ny : G\n\u22a2 \u2191(\u2191(\u03c1 A) g * \u2191(\u03c1 A) y) x = \u2191(\u2191A.\u03c1 g) (\u2191(\u2191(\u03c1 A) y) x)\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\n\u22a2 \u2191(leftRegularHom A x).hom (Finsupp.single 1 1) = x\n[PROOFSTEP]\nrw [leftRegularHom_hom, Finsupp.lift_apply, Finsupp.sum_single_index, one_smul, A.\u03c1.map_one, LinearMap.one_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\n\u22a2 0 \u2022 \u2191(\u2191(\u03c1 A) 1) x = 0\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\n\u22a2 (fun x => leftRegularHom A x)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : ofMulAction k G G \u27f6 A),\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x) =\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x)) }.toAddHom\n f) =\n f\n[PROOFSTEP]\nrefine' Action.Hom.ext _ _ (Finsupp.lhom_ext' fun x : G => LinearMap.ext_ring _)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\n\u22a2 \u2191(LinearMap.comp\n ((fun x => leftRegularHom A x)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : ofMulAction k G G \u27f6 A),\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x) =\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x)) }.toAddHom\n f)).hom\n (Finsupp.lsingle x))\n 1 =\n \u2191(LinearMap.comp f.hom (Finsupp.lsingle x)) 1\n[PROOFSTEP]\nhave :\n f.hom ((ofMulAction k G G).\u03c1 x (Finsupp.single (1 : G) (1 : k))) = A.\u03c1 x (f.hom (Finsupp.single (1 : G) (1 : k))) :=\n LinearMap.ext_iff.1 (f.comm x)\n (Finsupp.single 1 1)\n /- Porting note: rest of broken proof was\n simp only [LinearMap.comp_apply, Finsupp.lsingle_apply, left_regular_hom_hom,\n Finsupp.lift_apply, Finsupp.sum_single_index, one_smul, \u2190 this, zero_smul, of_\u03c1_apply,\n Representation.ofMulAction_single x (1 : G) (1 : k), smul_eq_mul, mul_one] -/\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 \u2191(LinearMap.comp\n ((fun x => leftRegularHom A x)\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : ofMulAction k G G \u27f6 A),\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x) =\n AddHom.toFun\n { toFun := fun f => \u2191f.hom (Finsupp.single 1 1),\n map_add' :=\n (_ :\n \u2200 (x y : ofMulAction k G G \u27f6 A),\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y) =\n (fun f => \u2191f.hom (Finsupp.single 1 1)) (x + y)) }\n (r \u2022 x)) }.toAddHom\n f)).hom\n (Finsupp.lsingle x))\n 1 =\n \u2191(LinearMap.comp f.hom (Finsupp.lsingle x)) 1\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, Finsupp.lsingle_apply, leftRegularHom_hom]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 \u2191(\u2191(Finsupp.lift (CoeSort.coe A) k G) fun g => \u2191(\u2191(\u03c1 A) g) (\u2191f.hom (Finsupp.single 1 1))) (Finsupp.single x 1) =\n \u2191f.hom (Finsupp.single x 1)\n[PROOFSTEP]\nerw [Finsupp.lift_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 (Finsupp.sum (Finsupp.single x 1) fun x r => r \u2022 \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))) =\n \u2191f.hom (Finsupp.single x 1)\n[PROOFSTEP]\nrw [Finsupp.sum_single_index, \u2190 this, of_\u03c1_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 1 \u2022 \u2191f.hom (\u2191(\u2191(Representation.ofMulAction k G G) x) (Finsupp.single 1 1)) = \u2191f.hom (Finsupp.single x 1)\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 0 \u2022 \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1)) = 0\n[PROOFSTEP]\nerw [Representation.ofMulAction_single x (1 : G) (1 : k)]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 1 \u2022 \u2191f.hom (Finsupp.single (x \u2022 1) 1) = \u2191f.hom (Finsupp.single x 1)\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 0 \u2022 \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1)) = 0\n[PROOFSTEP]\nsimp only [one_smul, smul_eq_mul, mul_one]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf\u271d : X \u27f6 Y\nA : Rep k G\nf : ofMulAction k G G \u27f6 A\nx : G\nthis : \u2191f.hom (\u2191(\u2191(\u03c1 (ofMulAction k G G)) x) (Finsupp.single 1 1)) = \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1))\n\u22a2 0 \u2022 \u2191(\u2191(\u03c1 A) x) (\u2191f.hom (Finsupp.single 1 1)) = 0\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : G\n\u22a2 \u2191(\u2191(LinearEquiv.symm (leftRegularHomEquiv A)) x).hom (Finsupp.single g 1) = \u2191(\u2191(\u03c1 A) g) x\n[PROOFSTEP]\nrw [leftRegularHomEquiv_symm_apply, leftRegularHom_hom, Finsupp.lift_apply, Finsupp.sum_single_index, one_smul]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX Y : Action (Type u) (MonCat.of G)\nf : X \u27f6 Y\nA : Rep k G\nx : CoeSort.coe A\ng : G\n\u22a2 0 \u2022 \u2191(\u2191(\u03c1 A) g) x = 0\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A X Y : Rep k G\nf : X \u27f6 Y\ng : \u2191(MonCat.of G)\nx : \u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) X).V\ny : CoeSort.coe A\n\u22a2 \u2191(\u2191(\u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) X).\u03c1 g \u226b\n ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom))\n x)\n y =\n \u2191(\u2191(ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom) \u226b\n \u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) Y).\u03c1 g)\n x)\n y\n[PROOFSTEP]\nshow f.hom (X.\u03c1 g _) = _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A X Y : Rep k G\nf : X \u27f6 Y\ng : \u2191(MonCat.of G)\nx : \u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) X).V\ny : CoeSort.coe A\n\u22a2 \u2191f.hom (\u2191(\u2191(\u03c1 X) g) (\u2191(LinearMap.comp x (\u2191(\u03c1 A) g\u207b\u00b9)) y)) =\n \u2191(\u2191(ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom) \u226b\n \u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) Y).\u03c1 g)\n x)\n y\n[PROOFSTEP]\nsimp only [hom_comm_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A X Y : Rep k G\nf : X \u27f6 Y\ng : \u2191(MonCat.of G)\nx : \u2191((fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B))) X).V\ny : CoeSort.coe A\n\u22a2 \u2191(\u2191(\u03c1 Y) g) (\u2191f.hom (\u2191(LinearMap.comp x (\u2191(\u03c1 A) g\u207b\u00b9)) y)) =\n \u2191(\u2191(ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom) \u226b\n \u2191(of (Representation.linHom (\u03c1 A) (\u03c1 Y))).\u03c1 g)\n x)\n y\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A x\u271d : Rep k G\n\u22a2 { obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n (\ud835\udfd9 x\u271d) =\n \ud835\udfd9\n ({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.obj\n x\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A x\u271d\u00b9 : Rep k G\nx\u271d :\n \u2191({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.obj\n x\u271d\u00b9).V\n\u22a2 \u2191({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f =>\n Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n (\ud835\udfd9 x\u271d\u00b9)).hom\n x\u271d =\n \u2191(\ud835\udfd9\n ({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f =>\n Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.obj\n x\u271d\u00b9)).hom\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A X\u271d Y\u271d Z\u271d : Rep k G\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n (x\u271d\u00b9 \u226b x\u271d) =\n { obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n x\u271d\u00b9 \u226b\n { obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B C A X\u271d Y\u271d Z\u271d : Rep k G\nx\u271d\u00b2 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 : Y\u271d \u27f6 Z\u271d\nx\u271d :\n \u2191({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f => Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.obj\n X\u271d).V\n\u22a2 \u2191({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f =>\n Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n (x\u271d\u00b2 \u226b x\u271d\u00b9)).hom\n x\u271d =\n \u2191({ obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f =>\n Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n x\u271d\u00b2 \u226b\n { obj := fun B => of (Representation.linHom (\u03c1 A) (\u03c1 B)),\n map := fun {X Y} f =>\n Hom.mk (ModuleCat.ofHom (\u2191(LinearMap.llcomp k (CoeSort.coe A) \u2191X.V \u2191Y.V) f.hom)) }.map\n x\u271d\u00b9).hom\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\n\u22a2 \u2191B.\u03c1 g \u226b LinearMap.flip (TensorProduct.curry f.hom) =\n LinearMap.flip (TensorProduct.curry f.hom) \u226b \u2191((Rep.ihom A).obj C).\u03c1 g\n[PROOFSTEP]\nrefine' LinearMap.ext fun x => LinearMap.ext fun y => _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\nx : \u2191B.V\ny : CoeSort.coe A\n\u22a2 \u2191(\u2191(\u2191B.\u03c1 g \u226b LinearMap.flip (TensorProduct.curry f.hom)) x) y =\n \u2191(\u2191(LinearMap.flip (TensorProduct.curry f.hom) \u226b \u2191((Rep.ihom A).obj C).\u03c1 g) x) y\n[PROOFSTEP]\nchange f.hom (_ \u2297\u209c[k] _) = C.\u03c1 g (f.hom (_ \u2297\u209c[k] _))\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\nx : \u2191B.V\ny : CoeSort.coe A\n\u22a2 \u2191f.hom (y \u2297\u209c[k] \u2191(\u2191B.\u03c1 g) x) = \u2191(\u2191(\u03c1 C) g) (\u2191f.hom (\u2191(\u2191(\u03c1 A) g\u207b\u00b9) y \u2297\u209c[k] x))\n[PROOFSTEP]\nrw [\u2190 hom_comm_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\nx : \u2191B.V\ny : CoeSort.coe A\n\u22a2 \u2191f.hom (y \u2297\u209c[k] \u2191(\u2191B.\u03c1 g) x) = \u2191f.hom (\u2191(\u2191(\u03c1 (A \u2297 B)) g) (\u2191(\u2191(\u03c1 A) g\u207b\u00b9) y \u2297\u209c[k] x))\n[PROOFSTEP]\nchange _ = f.hom ((A.\u03c1 g * A.\u03c1 g\u207b\u00b9) y \u2297\u209c[k] _)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\nx : \u2191B.V\ny : CoeSort.coe A\n\u22a2 \u2191f.hom (y \u2297\u209c[k] \u2191(\u2191B.\u03c1 g) x) =\n \u2191f.hom\n (\u2191(\u2191(\u03c1 A) g * \u2191(\u03c1 A) g\u207b\u00b9) y \u2297\u209c[k]\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj B).map\n g)\n (\u2191(\u2191(\u03c1 A) g\u207b\u00b9) y, x).snd)\n[PROOFSTEP]\nsimp only [\u2190 map_mul, mul_inv_self, map_one]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : A \u2297 B \u27f6 C\ng : \u2191(MonCat.of G)\nx : \u2191B.V\ny : CoeSort.coe A\n\u22a2 \u2191f.hom (y \u2297\u209c[k] \u2191(\u2191B.\u03c1 g) x) =\n \u2191f.hom\n (\u21911 y \u2297\u209c[k]\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj B).map\n g)\n x)\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : B \u27f6 (Rep.ihom A).obj C\ng : \u2191(MonCat.of G)\nx :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj A).obj\n PUnit.unit)\ny :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj B).obj\n PUnit.unit)\n\u22a2 \u2191(\u2191(A \u2297 B).\u03c1 g \u226b \u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom))\n (x \u2297\u209c[k] y) =\n \u2191(\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom) \u226b \u2191C.\u03c1 g) (x \u2297\u209c[k] y)\n[PROOFSTEP]\nchange\n TensorProduct.uncurry k _ _ _ f.hom.flip (A.\u03c1 g x \u2297\u209c[k] B.\u03c1 g y) =\n C.\u03c1 g (TensorProduct.uncurry k _ _ _ f.hom.flip (x \u2297\u209c[k] y))\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : B \u27f6 (Rep.ihom A).obj C\ng : \u2191(MonCat.of G)\nx :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj A).obj\n PUnit.unit)\ny :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj B).obj\n PUnit.unit)\n\u22a2 \u2191(\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom))\n (\u2191(\u2191(\u03c1 A) g) x \u2297\u209c[k] \u2191(\u2191(\u03c1 B) g) y) =\n \u2191(\u2191(\u03c1 C) g)\n (\u2191(\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom)) (x \u2297\u209c[k] y))\n[PROOFSTEP]\nrw [TensorProduct.uncurry_apply, LinearMap.flip_apply, hom_comm_apply, Rep.ihom_obj_\u03c1_apply, LinearMap.comp_apply,\n LinearMap.comp_apply, \u03c1_inv_self_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : B \u27f6 (Rep.ihom A).obj C\ng : \u2191(MonCat.of G)\nx :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj A).obj\n PUnit.unit)\ny :\n \u2191(((CategoryTheory.Equivalence.symm (functorCategoryEquivalence (ModuleCat k) (MonCat.of G))).inverse.obj B).obj\n PUnit.unit)\n\u22a2 \u2191(\u2191(\u03c1 C) g) (\u2191(\u2191f.hom y) x) =\n \u2191(\u2191(\u03c1 C) g)\n (\u2191(\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom)) (x \u2297\u209c[k] y))\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : B \u27f6 (Rep.ihom A).obj C\n\u22a2 (fun f => Hom.mk (LinearMap.flip (TensorProduct.curry f.hom)))\n ((fun f => Hom.mk (\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom))) f) =\n f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C\u271d A B C : Rep k G\nf : B \u27f6 (Rep.ihom A).obj C\nx\u271d : \u2191B.V\n\u22a2 \u2191((fun f => Hom.mk (LinearMap.flip (TensorProduct.curry f.hom)))\n ((fun f =>\n Hom.mk (\u2191(TensorProduct.uncurry k (CoeSort.coe A) (\u2191B.V) (CoeSort.coe C)) (LinearMap.flip f.hom)))\n f)).hom\n x\u271d =\n \u2191f.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C A B : Rep k G\n\u22a2 (NatTrans.app (ihom.ev A) B).hom =\n \u2191(TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe A \u2192\u2097[k] CoeSort.coe B) (CoeSort.coe B))\n (LinearMap.flip LinearMap.id)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Group G\nA\u271d B\u271d C A B : Rep k G\nx\u271d : \u2191((ihom A \u22d9 tensorLeft A).obj B).V\n\u22a2 \u2191(NatTrans.app (ihom.ev A) B).hom x\u271d =\n \u2191(\u2191(TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe A \u2192\u2097[k] CoeSort.coe B) (CoeSort.coe B))\n (LinearMap.flip LinearMap.id))\n x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 SymmetricCategory (Rep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 MonoidalPreadditive (Rep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 MonoidalLinear k (Rep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\n\u22a2 \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) r) x) = \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) r) (\u2191f x)\n[PROOFSTEP]\napply MonoidAlgebra.induction_on r\n[GOAL]\ncase hM\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\n\u22a2 \u2200 (g : G),\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (\u2191(MonoidAlgebra.of k G) g)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (\u2191(MonoidAlgebra.of k G) g)) (\u2191f x)\n[PROOFSTEP]\nintro g\n[GOAL]\ncase hM\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\ng : G\n\u22a2 \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (\u2191(MonoidAlgebra.of k G) g)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (\u2191(MonoidAlgebra.of k G) g)) (\u2191f x)\n[PROOFSTEP]\nsimp only [one_smul, MonoidAlgebra.lift_single, MonoidAlgebra.of_apply]\n[GOAL]\ncase hM\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\ng : G\n\u22a2 \u2191f (\u2191(\u2191\u03c1 g) x) = \u2191(\u2191\u03c3 g) (\u2191f x)\n[PROOFSTEP]\nexact LinearMap.congr_fun (w g) x\n[GOAL]\ncase hadd\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\n\u22a2 \u2200 (f_1 g : MonoidAlgebra k G),\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) f_1) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) f_1) (\u2191f x) \u2192\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) g) x) = \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) g) (\u2191f x) \u2192\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (f_1 + g)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (f_1 + g)) (\u2191f x)\n[PROOFSTEP]\nintro g h gw hw\n[GOAL]\ncase hadd\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\ng h : MonoidAlgebra k G\ngw : \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) g) x) = \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) g) (\u2191f x)\nhw : \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) h) x) = \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) h) (\u2191f x)\n\u22a2 \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (g + h)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (g + h)) (\u2191f x)\n[PROOFSTEP]\nsimp only [map_add, add_left_inj, LinearMap.add_apply, hw, gw]\n[GOAL]\ncase hsmul\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr : MonoidAlgebra k G\nx : V\n\u22a2 \u2200 (r : k) (f_1 : MonoidAlgebra k G),\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) f_1) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) f_1) (\u2191f x) \u2192\n \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (r \u2022 f_1)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (r \u2022 f_1)) (\u2191f x)\n[PROOFSTEP]\nintro r g w\n[GOAL]\ncase hsmul\nk\u271d G\u271d : Type u\ninst\u271d\u2077 : CommRing k\u271d\ninst\u271d\u2076 : Monoid G\u271d\nk : Type u_1\nG : Type u_2\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : Monoid G\nV : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module k V\ninst\u271d : Module k W\n\u03c1 : G \u2192* V \u2192\u2097[k] V\n\u03c3 : G \u2192* W \u2192\u2097[k] W\nf : V \u2192\u2097[k] W\nw\u271d : \u2200 (g : G), LinearMap.comp f (\u2191\u03c1 g) = LinearMap.comp (\u2191\u03c3 g) f\nr\u271d : MonoidAlgebra k G\nx : V\nr : k\ng : MonoidAlgebra k G\nw : \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) g) x) = \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) g) (\u2191f x)\n\u22a2 \u2191f (\u2191(\u2191(\u2191(MonoidAlgebra.lift k G (V \u2192\u2097[k] V)) \u03c1) (r \u2022 g)) x) =\n \u2191(\u2191(\u2191(MonoidAlgebra.lift k G (W \u2192\u2097[k] W)) \u03c3) (r \u2022 g)) (\u2191f x)\n[PROOFSTEP]\nsimp only [AlgHom.map_smul, w, RingHom.id_apply, LinearMap.smul_apply, LinearMap.map_smul\u209b\u2097]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX\u271d Y\u271d : ModuleCat (MonoidAlgebra k G)\nf : X\u271d \u27f6 Y\u271d\ng : \u2191(MonCat.of G)\n\u22a2 \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).\u03c1 g \u226b\n { toAddHom := f.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).V),\n \u2191f (\u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 x) = \u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 \u2191f x) } =\n { toAddHom := f.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).V),\n \u2191f (\u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 x) = \u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 \u2191f x) } \u226b\n \u2191((fun M => of (Representation.ofModule \u2191M)) Y\u271d).\u03c1 g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nX\u271d Y\u271d : ModuleCat (MonoidAlgebra k G)\nf : X\u271d \u27f6 Y\u271d\ng : \u2191(MonCat.of G)\nx\u271d : \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).V\n\u22a2 \u2191(\u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).\u03c1 g \u226b\n { toAddHom := f.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).V),\n \u2191f (\u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 x) = \u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 \u2191f x) })\n x\u271d =\n \u2191({ toAddHom := f.toAddHom,\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191((fun M => of (Representation.ofModule \u2191M)) X\u271d).V),\n \u2191f (\u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 x) = \u2191(algebraMap k (MonoidAlgebra k G)) r \u2022 \u2191f x) } \u226b\n \u2191((fun M => of (Representation.ofModule \u2191M)) Y\u271d).\u03c1 g)\n x\u271d\n[PROOFSTEP]\napply f.map_smul\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\n\u22a2 \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M) \u2243+ \u2191M\n[PROOFSTEP]\ndsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\n\u22a2 \u2191(ModuleCat.of (MonoidAlgebra k G) (Representation.asModule (Representation.ofModule \u2191M))) \u2243+ \u2191M\n[PROOFSTEP]\nrefine' (Representation.ofModule M).asModuleEquiv.trans (RestrictScalars.addEquiv k (MonoidAlgebra k G) _)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 CoeSort.coe V \u2243+ CoeSort.coe ((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V)\n[PROOFSTEP]\ndsimp [ofModuleMonoidAlgebra, toModuleMonoidAlgebra]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 CoeSort.coe V \u2243+\n RestrictScalars k (MonoidAlgebra k G) \u2191(ModuleCat.of (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))\n[PROOFSTEP]\nrefine' V.\u03c1.asModuleEquiv.symm.trans _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\n\u22a2 Representation.asModule (\u03c1 V) \u2243+\n RestrictScalars k (MonoidAlgebra k G) \u2191(ModuleCat.of (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))\n[PROOFSTEP]\nexact (RestrictScalars.addEquiv _ _ _).symm\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\nsrc\u271d : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M) \u2243+ \u2191M := counitIsoAddEquiv\nr : MonoidAlgebra k G\nx : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)),\n Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id (MonoidAlgebra k G)) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)),\n Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n x\n[PROOFSTEP]\ndsimp [counitIsoAddEquiv]\n /- Porting note: rest of broken proof was `simp`. -/\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\nsrc\u271d : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M) \u2243+ \u2191M := counitIsoAddEquiv\nr : MonoidAlgebra k G\nx : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)\n\u22a2 \u2191\u2191(AddEquiv.trans (Representation.asModuleEquiv (Representation.ofModule \u2191M))\n (RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M))\n (r \u2022 x) =\n r \u2022\n \u2191\u2191(AddEquiv.trans (Representation.asModuleEquiv (Representation.ofModule \u2191M))\n (RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M))\n x\n[PROOFSTEP]\nrw [AddEquiv.coe_toEquiv, AddEquiv.trans_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\nsrc\u271d : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M) \u2243+ \u2191M := counitIsoAddEquiv\nr : MonoidAlgebra k G\nx : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)\n\u22a2 \u2191(RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M)\n (\u2191(Representation.asModuleEquiv (Representation.ofModule \u2191M)) (r \u2022 x)) =\n r \u2022\n \u2191(AddEquiv.trans (Representation.asModuleEquiv (Representation.ofModule \u2191M))\n (RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M))\n x\n[PROOFSTEP]\nerw [Representation.ofModule_asAlgebraHom_apply_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nM : ModuleCat (MonoidAlgebra k G)\nsrc\u271d : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M) \u2243+ \u2191M := counitIsoAddEquiv\nr : MonoidAlgebra k G\nx : \u2191((ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).obj M)\n\u22a2 \u2191(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => \u2191M) x)))\n (r \u2022 \u2191(RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M) x) =\n r \u2022\n \u2191(AddEquiv.trans (Representation.asModuleEquiv (Representation.ofModule \u2191M))\n (RestrictScalars.addEquiv k (MonoidAlgebra k G) \u2191M))\n x\n[PROOFSTEP]\nexact AddEquiv.symm_apply_apply _ _\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\ng : G\nx : CoeSort.coe V\n\u22a2 \u2191unitIsoAddEquiv (AddHom.toFun (\u2191(\u03c1 V) g).toAddHom x) =\n AddHom.toFun (\u2191(\u03c1 (ofModuleMonoidAlgebra.obj (toModuleMonoidAlgebra.obj V))) g).toAddHom (\u2191unitIsoAddEquiv x)\n[PROOFSTEP]\ndsimp [unitIsoAddEquiv, ofModuleMonoidAlgebra, toModuleMonoidAlgebra]\n /- Porting note: rest of broken proof was\n simp only [AddEquiv.apply_eq_iff_eq, AddEquiv.apply_symm_apply,\n Representation.asModuleEquiv_symm_map_rho, Representation.ofModule_asModule_act] -/\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\ng : G\nx : CoeSort.coe V\n\u22a2 \u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n (\u2191(\u2191(\u03c1 V) g) x) =\n AddHom.toFun\n (\u2191(Representation.ofModule \u2191(ModuleCat.of (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))) g).toAddHom\n (\u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n x)\n[PROOFSTEP]\nerw [Representation.asModuleEquiv_symm_map_rho]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\ng : G\nx : CoeSort.coe V\n\u22a2 \u2191(MonoidAlgebra.of k G) g \u2022 \u2191(AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V))) x =\n AddHom.toFun\n (\u2191(Representation.ofModule \u2191(ModuleCat.of (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))) g).toAddHom\n (\u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n x)\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\nsrc\u271d : CoeSort.coe V \u2243+ CoeSort.coe ((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V) := unitIsoAddEquiv\nr : k\nx : \u2191V.V\n\u22a2 AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n { toFun := src\u271d.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n x\n[PROOFSTEP]\ndsimp [unitIsoAddEquiv]\n /- Porting note: rest of broken proof was\n simp only [Representation.asModuleEquiv_symm_map_smul,\n RestrictScalars.addEquiv_symm_map_algebraMap_smul] -/\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\nsrc\u271d : CoeSort.coe V \u2243+ CoeSort.coe ((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V) := unitIsoAddEquiv\nr : k\nx : \u2191V.V\n\u22a2 \u2191\u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n (r \u2022 x) =\n r \u2022\n \u2191\u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n x\n[PROOFSTEP]\nrw [AddEquiv.coe_toEquiv, AddEquiv.trans_apply, Representation.asModuleEquiv_symm_map_smul,\n RestrictScalars.addEquiv_symm_map_algebraMap_smul]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\nsrc\u271d : CoeSort.coe V \u2243+ CoeSort.coe ((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V) := unitIsoAddEquiv\nr : k\nx : \u2191V.V\n\u22a2 r \u2022\n \u2191(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V))))\n (\u2191(AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V))) x) =\n r \u2022\n \u2191(AddEquiv.trans (AddEquiv.symm (Representation.asModuleEquiv (\u03c1 V)))\n (AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) (Representation.asModule (\u03c1 V)))))\n x\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\ng : \u2191(MonCat.of G)\n\u22a2 \u2191V.\u03c1 g \u226b\n (LinearEquiv.toModuleIso'\n (let src := unitIsoAddEquiv;\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) = Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191V.V),\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n x) },\n invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n right_inv := (_ : Function.RightInverse src.invFun src.toFun) })).hom =\n (LinearEquiv.toModuleIso'\n (let src := unitIsoAddEquiv;\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) = Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191V.V),\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n x) },\n invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n right_inv := (_ : Function.RightInverse src.invFun src.toFun) })).hom \u226b\n \u2191((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V).\u03c1 g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\nV : Rep k G\ng : \u2191(MonCat.of G)\nx\u271d : \u2191V.V\n\u22a2 \u2191(\u2191V.\u03c1 g \u226b\n (LinearEquiv.toModuleIso'\n (let src := unitIsoAddEquiv;\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191V.V),\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n x) },\n invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n right_inv := (_ : Function.RightInverse src.invFun src.toFun) })).hom)\n x\u271d =\n \u2191((LinearEquiv.toModuleIso'\n (let src := unitIsoAddEquiv;\n {\n toLinearMap :=\n {\n toAddHom :=\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) },\n map_smul' :=\n (_ :\n \u2200 (r : k) (x : \u2191V.V),\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n (r \u2022 x) =\n \u2191(RingHom.id k) r \u2022\n AddHom.toFun\n { toFun := src.toFun,\n map_add' :=\n (_ :\n \u2200 (x y : CoeSort.coe V),\n Equiv.toFun src.toEquiv (x + y) =\n Equiv.toFun src.toEquiv x + Equiv.toFun src.toEquiv y) }\n x) },\n invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n right_inv := (_ : Function.RightInverse src.invFun src.toFun) })).hom \u226b\n \u2191((toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).obj V).\u03c1 g)\n x\u271d\n[PROOFSTEP]\napply unit_iso_comm\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 \u2200 {X Y : Rep k G} (f : X \u27f6 Y),\n (\ud835\udfed (Rep k G)).map f \u226b ((fun V => unitIso V) Y).hom =\n ((fun V => unitIso V) X).hom \u226b (toModuleMonoidAlgebra \u22d9 ofModuleMonoidAlgebra).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : CommRing k\ninst\u271d : Monoid G\n\u22a2 \u2200 {X Y : ModuleCat (MonoidAlgebra k G)} (f : X \u27f6 Y),\n (ofModuleMonoidAlgebra \u22d9 toModuleMonoidAlgebra).map f \u226b ((fun M => counitIso M) Y).hom =\n ((fun M => counitIso M) X).hom \u226b (\ud835\udfed (ModuleCat (MonoidAlgebra k G))).map f\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.RepresentationTheory.Rep", "llama_tokens": 21932, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.2734034621938684}} {"text": "[GOAL]\n\u22a2 ConcreteCategory SemiNormedGroupCat\n[PROOFSTEP]\ndsimp [SemiNormedGroupCat]\n[GOAL]\n\u22a2 ConcreteCategory (Bundled SeminormedAddCommGroup)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nV W : SemiNormedGroupCat\nf g : V \u27f6 W\nh : (forget SemiNormedGroupCat).map f = (forget SemiNormedGroupCat).map g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nV W : SemiNormedGroupCat\ng : V \u27f6 W\ntoFun\u271d : \u2191V \u2192 \u2191W\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : \u2191V), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : \u2191V), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nh :\n (forget SemiNormedGroupCat).map { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } =\n (forget SemiNormedGroupCat).map g\n\u22a2 { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nV W : SemiNormedGroupCat\ntoFun\u271d\u00b9 : \u2191V \u2192 \u2191W\nmap_add'\u271d\u00b9 : \u2200 (v\u2081 v\u2082 : \u2191V), toFun\u271d\u00b9 (v\u2081 + v\u2082) = toFun\u271d\u00b9 v\u2081 + toFun\u271d\u00b9 v\u2082\nbound'\u271d\u00b9 : \u2203 C, \u2200 (v : \u2191V), \u2016toFun\u271d\u00b9 v\u2016 \u2264 C * \u2016v\u2016\ntoFun\u271d : \u2191V \u2192 \u2191W\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : \u2191V), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : \u2191V), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nh :\n (forget SemiNormedGroupCat).map { toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n (forget SemiNormedGroupCat).map { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }\n\u22a2 { toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\n\u22a2 Limits.IsZero V\n[PROOFSTEP]\nrefine' \u27e8fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9\u27e9\n[GOAL]\ncase refine'_1\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\nf : V \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_1.h\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\nf : V \u27f6 X\nx : \u2191V\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nhave : x = 0 := Subsingleton.elim _ _\n[GOAL]\ncase refine'_1.h\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\nf : V \u27f6 X\nx : \u2191V\nthis : x = 0\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nsimp only [this, map_zero]\n[GOAL]\ncase refine'_2\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\nf : X \u27f6 V\n\u22a2 f = default\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2.h\nV : SemiNormedGroupCat\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\nf : X \u27f6 V\nx\u271d : \u2191X\n\u22a2 \u2191f x\u271d = \u2191default x\u271d\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nV W : SemiNormedGroupCat\ni : V \u2245 W\nh1 : NormedAddGroupHom.NormNoninc i.hom\nh2 : NormedAddGroupHom.NormNoninc i.inv\n\u22a2 Isometry \u2191i.hom\n[PROOFSTEP]\napply AddMonoidHomClass.isometry_of_norm\n[GOAL]\ncase a\nV W : SemiNormedGroupCat\ni : V \u2245 W\nh1 : NormedAddGroupHom.NormNoninc i.hom\nh2 : NormedAddGroupHom.NormNoninc i.inv\n\u22a2 \u2200 (x : \u2191V), \u2016\u2191i.hom x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nintro v\n[GOAL]\ncase a\nV W : SemiNormedGroupCat\ni : V \u2245 W\nh1 : NormedAddGroupHom.NormNoninc i.hom\nh2 : NormedAddGroupHom.NormNoninc i.inv\nv : \u2191V\n\u22a2 \u2016\u2191i.hom v\u2016 = \u2016v\u2016\n[PROOFSTEP]\napply le_antisymm (h1 v)\n[GOAL]\ncase a\nV W : SemiNormedGroupCat\ni : V \u2245 W\nh1 : NormedAddGroupHom.NormNoninc i.hom\nh2 : NormedAddGroupHom.NormNoninc i.inv\nv : \u2191V\n\u22a2 \u2016v\u2016 \u2264 \u2016\u2191i.hom v\u2016\n[PROOFSTEP]\ncalc\n \u2016v\u2016 = \u2016i.inv (i.hom v)\u2016 := by rw [Iso.hom_inv_id_apply]\n _ \u2264 \u2016i.hom v\u2016 := h2 _\n[GOAL]\nV W : SemiNormedGroupCat\ni : V \u2245 W\nh1 : NormedAddGroupHom.NormNoninc i.hom\nh2 : NormedAddGroupHom.NormNoninc i.inv\nv : \u2191V\n\u22a2 \u2016v\u2016 = \u2016\u2191i.inv (\u2191i.hom v)\u2016\n[PROOFSTEP]\nrw [Iso.hom_inv_id_apply]\n[GOAL]\nM N : SemiNormedGroupCat\nf : M \u2245 N\ni : NormedAddGroupHom.NormNoninc f.hom\ni' : NormedAddGroupHom.NormNoninc f.inv\n\u22a2 mkHom f.hom i \u226b mkHom f.inv i' = \ud835\udfd9 (of \u2191M)\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\nM N : SemiNormedGroupCat\nf : M \u2245 N\ni : NormedAddGroupHom.NormNoninc f.hom\ni' : NormedAddGroupHom.NormNoninc f.inv\n\u22a2 \u2191(mkHom f.hom i \u226b mkHom f.inv i') = \u2191(\ud835\udfd9 (of \u2191M))\n[PROOFSTEP]\nexact f.hom_inv_id\n[GOAL]\nM N : SemiNormedGroupCat\nf : M \u2245 N\ni : NormedAddGroupHom.NormNoninc f.hom\ni' : NormedAddGroupHom.NormNoninc f.inv\n\u22a2 mkHom f.inv i' \u226b mkHom f.hom i = \ud835\udfd9 (of \u2191N)\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\nM N : SemiNormedGroupCat\nf : M \u2245 N\ni : NormedAddGroupHom.NormNoninc f.hom\ni' : NormedAddGroupHom.NormNoninc f.inv\n\u22a2 \u2191(mkHom f.inv i' \u226b mkHom f.hom i) = \u2191(\ud835\udfd9 (of \u2191N))\n[PROOFSTEP]\nexact f.inv_hom_id\n[GOAL]\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\n\u22a2 Limits.IsZero V\n[PROOFSTEP]\nrefine' \u27e8fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9\u27e9\n[GOAL]\ncase refine'_1\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\u2081\nf : V \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\next x\n[GOAL]\ncase refine'_1.w.h\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\u2081\nf : V \u27f6 X\nx : \u2191V\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nhave : x = 0 := Subsingleton.elim _ _\n[GOAL]\ncase refine'_1.w.h\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\u2081\nf : V \u27f6 X\nx : \u2191V\nthis : x = 0\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nsimp only [this, map_zero]\n[GOAL]\ncase refine'_2\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\u2081\nf : X \u27f6 V\n\u22a2 f = default\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2.w.h\nV : SemiNormedGroupCat\u2081\ninst\u271d : Subsingleton \u2191V\nX : SemiNormedGroupCat\u2081\nf : X \u27f6 V\nx\u271d : \u2191X\n\u22a2 \u2191f x\u271d = \u2191default x\u271d\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\n\u22a2 Isometry \u2191i.hom\n[PROOFSTEP]\nchange Isometry (\u27e8\u27e8i.hom, map_zero _\u27e9, fun _ _ => map_add _ _ _\u27e9 : V \u2192+ W)\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\n\u22a2 Isometry\n \u2191{ toZeroHom := { toFun := \u2191i.hom, map_zero' := (_ : \u2191i.hom 0 = 0) },\n map_add' := (_ : \u2200 (x x_1 : \u2191V), \u2191i.hom (x + x_1) = \u2191i.hom x + \u2191i.hom x_1) }\n[PROOFSTEP]\nrefine' AddMonoidHomClass.isometry_of_norm _ _\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\n\u22a2 \u2200 (x : \u2191V),\n \u2016\u2191{ toZeroHom := { toFun := \u2191i.hom, map_zero' := (_ : \u2191i.hom 0 = 0) },\n map_add' := (_ : \u2200 (x x_1 : \u2191V), \u2191i.hom (x + x_1) = \u2191i.hom x + \u2191i.hom x_1) }\n x\u2016 =\n \u2016x\u2016\n[PROOFSTEP]\nintro v\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\nv : \u2191V\n\u22a2 \u2016\u2191{ toZeroHom := { toFun := \u2191i.hom, map_zero' := (_ : \u2191i.hom 0 = 0) },\n map_add' := (_ : \u2200 (x x_1 : \u2191V), \u2191i.hom (x + x_1) = \u2191i.hom x + \u2191i.hom x_1) }\n v\u2016 =\n \u2016v\u2016\n[PROOFSTEP]\napply le_antisymm (i.hom.2 v)\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\nv : \u2191V\n\u22a2 \u2016v\u2016 \u2264 \u2016\u2191\u2191i.hom v\u2016\n[PROOFSTEP]\ncalc\n \u2016v\u2016 = \u2016i.inv (i.hom v)\u2016 := by rw [Iso.hom_inv_id_apply]\n _ \u2264 \u2016i.hom v\u2016 := i.inv.2 _\n[GOAL]\nV W : SemiNormedGroupCat\u2081\ni : V \u2245 W\nv : \u2191V\n\u22a2 \u2016v\u2016 = \u2016\u2191i.inv (\u2191i.hom v)\u2016\n[PROOFSTEP]\nrw [Iso.hom_inv_id_apply]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.SemiNormedGroupCat", "llama_tokens": 3450, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5, "lm_q1q2_score": 0.2733690759923069}} {"text": "[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 Function.Injective fun A => \u2191A.toSubring\n[PROOFSTEP]\nintro \u27e8_, _\u27e9 \u27e8_, _\u27e9 h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\ntoSubring\u271d\u00b9 : Subring K\nmem_or_inv_mem'\u271d\u00b9 : \u2200 (x : K), x \u2208 toSubring\u271d\u00b9.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d\u00b9.carrier\ntoSubring\u271d : Subring K\nmem_or_inv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh :\n (fun A => \u2191A.toSubring) { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 } =\n (fun A => \u2191A.toSubring) { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }\n\u22a2 { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 } =\n { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }\n[PROOFSTEP]\nreplace h := SetLike.coe_injective' h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\ntoSubring\u271d\u00b9 : Subring K\nmem_or_inv_mem'\u271d\u00b9 : \u2200 (x : K), x \u2208 toSubring\u271d\u00b9.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d\u00b9.carrier\ntoSubring\u271d : Subring K\nmem_or_inv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh :\n { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 }.toSubring =\n { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }.toSubring\n\u22a2 { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 } =\n { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA x y : ValuationSubring K\nh : x.toSubring = y.toSubring\n\u22a2 x = y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nK : Type u\ninst\u271d : Field K\nA y : ValuationSubring K\ntoSubring\u271d : Subring K\nmem_or_inv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh : { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }.toSubring = y.toSubring\n\u22a2 { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d } = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\ntoSubring\u271d\u00b9 : Subring K\nmem_or_inv_mem'\u271d\u00b9 : \u2200 (x : K), x \u2208 toSubring\u271d\u00b9.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d\u00b9.carrier\ntoSubring\u271d : Subring K\nmem_or_inv_mem'\u271d : \u2200 (x : K), x \u2208 toSubring\u271d.carrier \u2228 x\u207b\u00b9 \u2208 toSubring\u271d.carrier\nh :\n { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 }.toSubring =\n { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }.toSubring\n\u22a2 { toSubring := toSubring\u271d\u00b9, mem_or_inv_mem' := mem_or_inv_mem'\u271d\u00b9 } =\n { toSubring := toSubring\u271d, mem_or_inv_mem' := mem_or_inv_mem'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 CommRing { x // x \u2208 A.toSubring }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 IsDomain { x // x \u2208 A.toSubring }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases (b : K) = 0\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases (b : K) = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191b = 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 : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191b = 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 : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191b = 0\n\u22a2 a * 0 = b\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191b = 0\n\u22a2 \u2191(a * 0) = \u2191b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u00ac\u2191b = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases (a : K) = 0\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u00ac\u2191b = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nby_cases (a : K) = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u2191a = 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 : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u2191a = 0\n\u22a2 a * 0 = b \u2228 b * 0 = a\n[PROOFSTEP]\nright\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u2191a = 0\n\u22a2 b * 0 = a\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u2191a = 0\n\u22a2 \u2191(b * 0) = \u2191a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\ncases' A.mem_or_inv_mem (a / b) with hh hh\n[GOAL]\ncase neg.inl\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191a / \u2191b \u2208 A\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse\u27e8a / b, hh\u27e9\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191a / \u2191b \u2208 A\n\u22a2 a * { val := \u2191a / \u2191b, property := hh } = b \u2228 b * { val := \u2191a / \u2191b, property := hh } = a\n[PROOFSTEP]\nright\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191a / \u2191b \u2208 A\n\u22a2 b * { val := \u2191a / \u2191b, property := hh } = a\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191a / \u2191b \u2208 A\n\u22a2 \u2191(b * { val := \u2191a / \u2191b, property := hh }) = \u2191a\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191a / \u2191b \u2208 A\n\u22a2 \u2191b * \u2191a = \u2191a * \u2191b\n[PROOFSTEP]\nring\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : (\u2191a / \u2191b)\u207b\u00b9 \u2208 A\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nrw [show (a / b : K)\u207b\u00b9 = b / a by field_simp] at hh \n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : (\u2191a / \u2191b)\u207b\u00b9 \u2208 A\n\u22a2 (\u2191a / \u2191b)\u207b\u00b9 = \u2191b / \u2191a\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191b / \u2191a \u2208 A\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n[PROOFSTEP]\nuse\u27e8b / a, hh\u27e9\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191b / \u2191a \u2208 A\n\u22a2 a * { val := \u2191b / \u2191a, property := hh } = b \u2228 b * { val := \u2191b / \u2191a, property := hh } = a\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191b / \u2191a \u2208 A\n\u22a2 a * { val := \u2191b / \u2191a, property := hh } = b\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191b / \u2191a \u2208 A\n\u22a2 \u2191(a * { val := \u2191b / \u2191a, property := hh }) = \u2191b\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.h.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh\u271d : \u00ac\u2191b = 0\nh : \u00ac\u2191a = 0\nhh : \u2191b / \u2191a \u2208 A\n\u22a2 \u2191a * \u2191b = \u2191b * \u2191a\n[PROOFSTEP]\nring\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 Algebra { x // x \u2208 A.toSubring } K\n[PROOFSTEP]\ninfer_instance\n -- Porting note: Somehow it cannot find this instance and I'm too lazy to debug. wrong prio?\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\nby_cases z = 0\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\nby_cases z = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : z = 0\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\nuse(0, 1)\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : z = 0\n\u22a2 z * \u2191(algebraMap { x // x \u2208 A } K) \u2191(0, 1).snd = \u2191(algebraMap { x // x \u2208 A } K) (0, 1).fst\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : \u00acz = 0\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\ncases' A.mem_or_inv_mem z with hh hh\n[GOAL]\ncase neg.inl\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : \u00acz = 0\nhh : z \u2208 A\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\nuse(\u27e8z, hh\u27e9, 1)\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : \u00acz = 0\nhh : z \u2208 A\n\u22a2 z * \u2191(algebraMap { x // x \u2208 A } K) \u2191({ val := z, property := hh }, 1).snd =\n \u2191(algebraMap { x // x \u2208 A } K) ({ val := z, property := hh }, 1).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : \u00acz = 0\nhh : z\u207b\u00b9 \u2208 A\n\u22a2 \u2203 x, z * \u2191(algebraMap { x // x \u2208 A } K) \u2191x.snd = \u2191(algebraMap { x // x \u2208 A } K) x.fst\n[PROOFSTEP]\nrefine \u27e8\u27e81, \u27e8\u27e8_, hh\u27e9, ?_\u27e9\u27e9, mul_inv_cancel h\u27e9\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nz : K\nh : \u00acz = 0\nhh : z\u207b\u00b9 \u2208 A\n\u22a2 { val := z\u207b\u00b9, property := hh } \u2208 nonZeroDivisors { x // x \u2208 A }\n[PROOFSTEP]\nexact mem_nonZeroDivisors_iff_ne_zero.2 fun c => h (inv_eq_zero.mp (congr_arg Subtype.val c))\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191(algebraMap { x // x \u2208 A } K) a = \u2191(algebraMap { x // x \u2208 A } K) b\n\u22a2 \u21911 * a = \u21911 * b\n[PROOFSTEP]\next\n[GOAL]\ncase a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 A }\nh : \u2191(algebraMap { x // x \u2208 A } K) a = \u2191(algebraMap { x // x \u2208 A } K) b\n\u22a2 \u2191(\u21911 * a) = \u2191(\u21911 * b)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 LinearOrderedCommGroupWithZero (ValueGroup A)\n[PROOFSTEP]\nunfold ValueGroup\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 LinearOrderedCommGroupWithZero (ValuationRing.ValueGroup { x // x \u2208 A } K)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\u02e3\n\u22a2 \u2191(valuation A) \u2191\u2191a = 1\n[PROOFSTEP]\nrw [\u2190 A.valuation.map_one, valuation_eq_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\u02e3\n\u22a2 \u2203 a_1, \u2191\u2191a_1 * 1 = \u2191\u2191a\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\u02e3\n\u22a2 \u2191\u2191a * 1 = \u2191\u2191a\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\n\u22a2 IsUnit a\n[PROOFSTEP]\nhave ha : (a : K) \u2260 0\n[GOAL]\ncase ha\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\n\u22a2 \u2191a \u2260 0\n[PROOFSTEP]\nintro c\n[GOAL]\ncase ha\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nc : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [c, A.valuation.map_zero] at h \n[GOAL]\ncase ha\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : 0 = 1\nc : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nha : \u2191a \u2260 0\n\u22a2 IsUnit a\n[PROOFSTEP]\nhave ha' : (a : K)\u207b\u00b9 \u2208 A := by rw [\u2190 valuation_le_one_iff, map_inv\u2080, h, inv_one]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nha : \u2191a \u2260 0\n\u22a2 (\u2191a)\u207b\u00b9 \u2208 A\n[PROOFSTEP]\nrw [\u2190 valuation_le_one_iff, map_inv\u2080, h, inv_one]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nha : \u2191a \u2260 0\nha' : (\u2191a)\u207b\u00b9 \u2208 A\n\u22a2 IsUnit a\n[PROOFSTEP]\napply isUnit_of_mul_eq_one a \u27e8a\u207b\u00b9, ha'\u27e9\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nha : \u2191a \u2260 0\nha' : (\u2191a)\u207b\u00b9 \u2208 A\n\u22a2 a * { val := (\u2191a)\u207b\u00b9, property := ha' } = 1\n[PROOFSTEP]\next\n[GOAL]\ncase a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\nh : \u2191(valuation A) \u2191a = 1\nha : \u2191a \u2260 0\nha' : (\u2191a)\u207b\u00b9 \u2208 A\n\u22a2 \u2191(a * { val := (\u2191a)\u207b\u00b9, property := ha' }) = \u21911\n[PROOFSTEP]\nfield_simp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\n\u22a2 a \u2208 LocalRing.maximalIdeal { x // x \u2208 A } \u2194 \u2191(valuation A) \u2191a < 1\n[PROOFSTEP]\nrw [LocalRing.mem_maximalIdeal]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\n\u22a2 a \u2208 nonunits { x // x \u2208 A } \u2194 \u2191(valuation A) \u2191a < 1\n[PROOFSTEP]\ndsimp [nonunits]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\n\u22a2 \u00acIsUnit a \u2194 \u2191(valuation A) \u2191a < 1\n[PROOFSTEP]\nrw [valuation_eq_one_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\n\u22a2 \u00ac\u2191(valuation A) \u2191a = 1 \u2194 \u2191(valuation A) \u2191a < 1\n[PROOFSTEP]\nexact (A.valuation_le_one a).lt_iff_ne.symm\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\n\u22a2 \u2200 (x y : ValueGroup R),\n ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n (x * y) =\n ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n x *\n ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n y\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase mk.mk\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx\u271d : ValueGroup R\na\u271d\u00b9 : K\ny\u271d : ValueGroup R\na\u271d : K\n\u22a2 ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n (Quot.mk Setoid.r a\u271d\u00b9 * Quot.mk Setoid.r a\u271d) =\n ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n (Quot.mk Setoid.r a\u271d\u00b9) *\n ZeroHom.toFun\n { toFun := Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)),\n map_zero' :=\n (_ :\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0 =\n Quotient.map' id (_ : \u2200 (x y : K), Setoid.r x y \u2192 Setoid.r (id x) (id y)) 0) }\n (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\n\u22a2 Monotone \u2191(mapOfLE R S h)\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 \u27e8a, ha\u27e9\n[GOAL]\ncase mk.mk.intro\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na\u271d\u00b2 : ValueGroup R\na\u271d\u00b9 : K\nb\u271d : ValueGroup R\na\u271d : K\na : { x // x \u2208 R }\nha : a \u2022 a\u271d = a\u271d\u00b9\n\u22a2 \u2191(mapOfLE R S h) (Quot.mk Setoid.r a\u271d\u00b9) \u2264 \u2191(mapOfLE R S h) (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\nexact \u27e8R.inclusion S h a, ha\u27e9\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\n\u22a2 \u2191(mapOfLE R S h) \u2218 \u2191(valuation R) = \u2191(valuation S)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx\u271d : K\n\u22a2 (\u2191(mapOfLE R S h) \u2218 \u2191(valuation R)) x\u271d = \u2191(valuation S) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\n\u22a2 IsLocalization.AtPrime { x // x \u2208 ofPrime A P } P\n[PROOFSTEP]\napply Localization.subalgebra.isLocalization_ofField K P.primeCompl P.primeCompl_le_nonZeroDivisors\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\nx : { x // x \u2208 A }\n\u22a2 \u2191(valuation (ofPrime A P)) \u2191x = 1 \u2194 x \u2208 Ideal.primeCompl P\n[PROOFSTEP]\nrw [\u2190 IsLocalization.AtPrime.isUnit_to_map_iff (A.ofPrime P) P x, valuation_eq_one_iff]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\nx : { x // x \u2208 A }\n\u22a2 \u2191(valuation (ofPrime A P)) \u2191x = 1 \u2194\n \u2191(valuation (ofPrime A P)) \u2191(\u2191(algebraMap { x // x \u2208 A } { x // x \u2208 ofPrime A P }) x) = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\n\u22a2 idealOfLE A (ofPrime A P) (_ : A \u2264 ofPrime A P) = P\n[PROOFSTEP]\nrefine Ideal.ext (fun x => ?_)\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\nx : { x // x \u2208 A }\n\u22a2 x \u2208 idealOfLE A (ofPrime A P) (_ : A \u2264 ofPrime A P) \u2194 x \u2208 P\n[PROOFSTEP]\napply IsLocalization.AtPrime.to_map_mem_maximal_iff\n[GOAL]\ncase h\nK : Type u\ninst\u271d\u00b9 : Field K\nA\u271d A : ValuationSubring K\nP : Ideal { x // x \u2208 A }\ninst\u271d : Ideal.IsPrime P\nx : { x // x \u2208 A }\n\u22a2 optParam (LocalRing { x // x \u2208 ofPrime A P }) (_ : LocalRing { x // x \u2208 ofPrime A P })\n[PROOFSTEP]\nexact localRing (ofPrime A P)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\n\u22a2 ofPrime R (idealOfLE R S h) = S\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h) \u2194 x \u2208 S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h) \u2192 x \u2208 S\n[PROOFSTEP]\nrintro \u27e8a, r, hr, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 \u2191(algebraMap { x // x \u2208 R } K) a * (\u2191(algebraMap { x // x \u2208 R } K) r)\u207b\u00b9 \u2208 S\n[PROOFSTEP]\napply mul_mem\n[GOAL]\ncase h.mp.intro.intro.intro.a\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 \u2191(algebraMap { x // x \u2208 R } K) a \u2208 S\n[PROOFSTEP]\nexact h a.2\n[GOAL]\ncase h.mp.intro.intro.intro.a\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 (\u2191(algebraMap { x // x \u2208 R } K) r)\u207b\u00b9 \u2208 S\n[PROOFSTEP]\nrw [\u2190 valuation_le_one_iff, map_inv\u2080, \u2190 inv_one, inv_le_inv\u2080]\n[GOAL]\ncase h.mp.intro.intro.intro.a\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 1 \u2264 \u2191(valuation S) (\u2191(algebraMap { x // x \u2208 R } K) r)\n[PROOFSTEP]\nexact not_lt.1 ((not_iff_not.2 <| valuation_lt_one_iff S _).1 hr)\n[GOAL]\ncase h.mp.intro.intro.intro.a.ha\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 \u2191(valuation S) (\u2191(algebraMap { x // x \u2208 R } K) r) \u2260 0\n[PROOFSTEP]\nintro hh\n[GOAL]\ncase h.mp.intro.intro.intro.a.ha\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\nhh : \u2191(valuation S) (\u2191(algebraMap { x // x \u2208 R } K) r) = 0\n\u22a2 False\n[PROOFSTEP]\nerw [Valuation.zero_iff, Subring.coe_eq_zero_iff] at hh \n[GOAL]\ncase h.mp.intro.intro.intro.a.ha\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\nhh : r = 0\n\u22a2 False\n[PROOFSTEP]\napply hr\n[GOAL]\ncase h.mp.intro.intro.intro.a.ha\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\nhh : r = 0\n\u22a2 r \u2208 \u2191(idealOfLE R S h)\n[PROOFSTEP]\nrw [hh]\n[GOAL]\ncase h.mp.intro.intro.intro.a.ha\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\nhh : r = 0\n\u22a2 0 \u2208 \u2191(idealOfLE R S h)\n[PROOFSTEP]\napply Ideal.zero_mem (R.idealOfLE S h)\n[GOAL]\ncase h.mp.intro.intro.intro.a.hb\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\na r : { x // x \u2208 R }\nhr : r \u2208 Ideal.primeCompl (idealOfLE R S h)\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase h.mpr\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\n\u22a2 x \u2208 S \u2192 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.mpr\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nby_cases hr : x \u2208 R\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nhr : x \u2208 R\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nexact R.le_ofPrime _ hr\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nhr : \u00acx \u2208 R\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nhave : x \u2260 0 := fun h => hr (by rw [h]; exact R.zero_mem)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh\u271d : R \u2264 S\nx : K\nhx : x \u2208 S\nhr : \u00acx \u2208 R\nh : x = 0\n\u22a2 x \u2208 R\n[PROOFSTEP]\nrw [h]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh\u271d : R \u2264 S\nx : K\nhx : x \u2208 S\nhr : \u00acx \u2208 R\nh : x = 0\n\u22a2 0 \u2208 R\n[PROOFSTEP]\nexact R.zero_mem\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nhr : \u00acx \u2208 R\nthis : x \u2260 0\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nreplace hr := (R.mem_or_inv_mem x).resolve_left hr\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 x \u2208 ofPrime R (idealOfLE R S h)\n[PROOFSTEP]\nuse 1, \u27e8x\u207b\u00b9, hr\u27e9\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 \u2203 x_1, x = \u2191(algebraMap { x // x \u2208 R } K) 1 * (\u2191(algebraMap { x // x \u2208 R } K) { val := x\u207b\u00b9, property := hr })\u207b\u00b9\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 x = \u2191(algebraMap { x // x \u2208 R } K) 1 * (\u2191(algebraMap { x // x \u2208 R } K) { val := x\u207b\u00b9, property := hr })\u207b\u00b9\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.w\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 { val := x\u207b\u00b9, property := hr } \u2208 Ideal.primeCompl (idealOfLE R S h)\n[PROOFSTEP]\nchange (\u27e8x\u207b\u00b9, h hr\u27e9 : S) \u2209 nonunits S\n[GOAL]\ncase h.w\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 \u00ac{ val := x\u207b\u00b9, property := (_ : x\u207b\u00b9 \u2208 S) } \u2208 nonunits { x // x \u2208 S }\n[PROOFSTEP]\nrw [mem_nonunits_iff, Classical.not_not]\n[GOAL]\ncase h.w\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 IsUnit { val := x\u207b\u00b9, property := (_ : x\u207b\u00b9 \u2208 S) }\n[PROOFSTEP]\napply isUnit_of_mul_eq_one _ (\u27e8x, hx\u27e9 : S)\n[GOAL]\ncase h.w\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 { val := x\u207b\u00b9, property := (_ : x\u207b\u00b9 \u2208 S) } * { val := x, property := hx } = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h.w.a\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nh : R \u2264 S\nx : K\nhx : x \u2208 S\nthis : x \u2260 0\nhr : x\u207b\u00b9 \u2208 R\n\u22a2 \u2191({ val := x\u207b\u00b9, property := (_ : x\u207b\u00b9 \u2208 S) } * { val := x, property := hx }) = \u21911\n[PROOFSTEP]\nfield_simp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nhR : A \u2264 R\nhS : A \u2264 S\nh : R \u2264 S\nx : { x // x \u2208 A }\nhx : x \u2208 idealOfLE A S hS\n\u22a2 \u2191(valuation R) \u2191(\u2191(inclusion A R hR) x) < 1\n[PROOFSTEP]\nby_contra c\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nhR : A \u2264 R\nhS : A \u2264 S\nh : R \u2264 S\nx : { x // x \u2208 A }\nhx : x \u2208 idealOfLE A S hS\nc : \u00ac\u2191(valuation R) \u2191(\u2191(inclusion A R hR) x) < 1\n\u22a2 False\n[PROOFSTEP]\npush_neg at c \n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nhR : A \u2264 R\nhS : A \u2264 S\nh : R \u2264 S\nx : { x // x \u2208 A }\nhx : x \u2208 idealOfLE A S hS\nc : 1 \u2264 \u2191(valuation R) \u2191(\u2191(inclusion A R hR) x)\n\u22a2 False\n[PROOFSTEP]\nreplace c := monotone_mapOfLE R S h c\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nhR : A \u2264 R\nhS : A \u2264 S\nh : R \u2264 S\nx : { x // x \u2208 A }\nhx : x \u2208 idealOfLE A S hS\nc : \u2191(mapOfLE R S h) 1 \u2264 \u2191(mapOfLE R S h) (\u2191(valuation R) \u2191(\u2191(inclusion A R hR) x))\n\u22a2 False\n[PROOFSTEP]\nrw [(mapOfLE _ _ _).map_one, mapOfLE_valuation_apply] at c \n[GOAL]\nK : Type u\ninst\u271d : Field K\nA R S : ValuationSubring K\nhR : A \u2264 R\nhS : A \u2264 S\nh : R \u2264 S\nx : { x // x \u2208 A }\nhx : x \u2208 idealOfLE A S hS\nc : 1 \u2264 \u2191(valuation S) \u2191(\u2191(inclusion A R hR) x)\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_lt ((valuation_lt_one_iff S _).1 hx) c\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nP : PrimeSpectrum { x // x \u2208 A }\n\u22a2 (fun S =>\n { asIdeal := idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}),\n IsPrime := (_ : Ideal.IsPrime (idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}))) })\n ((fun P => { val := ofPrime A P.asIdeal, property := (_ : A \u2264 ofPrime A P.asIdeal) }) P) =\n P\n[PROOFSTEP]\next1\n[GOAL]\ncase asIdeal\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nP : PrimeSpectrum { x // x \u2208 A }\n\u22a2 ((fun S =>\n { asIdeal := idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}),\n IsPrime := (_ : Ideal.IsPrime (idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}))) })\n ((fun P => { val := ofPrime A P.asIdeal, property := (_ : A \u2264 ofPrime A P.asIdeal) }) P)).asIdeal =\n P.asIdeal\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nS : \u2191{S | A \u2264 S}\n\u22a2 (fun P => { val := ofPrime A P.asIdeal, property := (_ : A \u2264 ofPrime A P.asIdeal) })\n ((fun S =>\n { asIdeal := idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}),\n IsPrime := (_ : Ideal.IsPrime (idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}))) })\n S) =\n S\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nS : \u2191{S | A \u2264 S}\n\u22a2 \u2191((fun P => { val := ofPrime A P.asIdeal, property := (_ : A \u2264 ofPrime A P.asIdeal) })\n ((fun S =>\n { asIdeal := idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}),\n IsPrime := (_ : Ideal.IsPrime (idealOfLE A \u2191S (_ : \u2191S \u2208 {S | A \u2264 S}))) })\n S)) =\n \u2191S\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh :\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\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\u271d\n\u22a2 a\u271d \u2264 b\u271d\n[PROOFSTEP]\ndsimp at h \n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 a\u271d \u2264 b\u271d\n[PROOFSTEP]\nhave := idealOfLE_le_of_le A _ _ ?_ ?_ h\n[GOAL]\ncase refine_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\nthis : idealOfLE A \u2191(\u2191(primeSpectrumEquiv A) b\u271d) ?refine_2 \u2264 idealOfLE A \u2191(\u2191(primeSpectrumEquiv A) a\u271d) ?refine_1\n\u22a2 a\u271d \u2264 b\u271d\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\niterate 2 erw [idealOfLE_ofPrime] at this \n[GOAL]\ncase refine_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\nthis : idealOfLE A \u2191(\u2191(primeSpectrumEquiv A) b\u271d) ?refine_2 \u2264 idealOfLE A \u2191(\u2191(primeSpectrumEquiv A) a\u271d) ?refine_1\n\u22a2 a\u271d \u2264 b\u271d\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\nerw [idealOfLE_ofPrime] at this \n[GOAL]\ncase refine_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\nthis : b\u271d.asIdeal \u2264 idealOfLE A \u2191(\u2191(primeSpectrumEquiv A) a\u271d) ?refine_1\n\u22a2 a\u271d \u2264 b\u271d\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\nerw [idealOfLE_ofPrime] at this \n[GOAL]\ncase refine_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\nthis : b\u271d.asIdeal \u2264 a\u271d.asIdeal\n\u22a2 a\u271d \u2264 b\u271d\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\nall_goals exact le_ofPrime A (PrimeSpectrum.asIdeal _)\n[GOAL]\ncase refine_1\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) a\u271d)\n[PROOFSTEP]\nexact le_ofPrime A (PrimeSpectrum.asIdeal _)\n[GOAL]\ncase refine_2\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : \u2191(primeSpectrumEquiv A) a\u271d \u2264 \u2191(primeSpectrumEquiv A) b\u271d\n\u22a2 A \u2264 \u2191(\u2191(primeSpectrumEquiv A) b\u271d)\n[PROOFSTEP]\nexact le_ofPrime A (PrimeSpectrum.asIdeal _)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : a\u271d \u2264 b\u271d\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\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\u271d\n[PROOFSTEP]\napply ofPrime_le_of_le\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nsrc\u271d : PrimeSpectrum { x // x \u2208 A } \u2243 \u2191{S | A \u2264 S} := primeSpectrumEquiv A\na\u271d b\u271d : (PrimeSpectrum { x // x \u2208 A })\u1d52\u1d48\nh : a\u271d \u2264 b\u271d\n\u22a2 b\u271d.asIdeal \u2264 a\u271d.asIdeal\n[PROOFSTEP]\nexact h\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\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 \u2228\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]\nintro x\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\n\u22a2 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 \u2228\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\ncases' le_or_lt (v x) 1 with h h\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : \u2191v x \u2264 1\n\u22a2 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 \u2228\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : \u2191v x \u2264 1\n\u22a2 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[PROOFSTEP]\nexact h\n[GOAL]\ncase inr\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\n\u22a2 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 \u2228\n x\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\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]\nchange v x\u207b\u00b9 \u2264 1\n[GOAL]\ncase inr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\n\u22a2 \u2191v x\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrw [map_inv\u2080 v, \u2190 inv_one, inv_le_inv\u2080]\n[GOAL]\ncase inr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\n\u22a2 1 \u2264 \u2191v x\n[PROOFSTEP]\nexact le_of_lt h\n[GOAL]\ncase inr.h.ha\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\n\u22a2 \u2191v x \u2260 0\n[PROOFSTEP]\nintro c\n[GOAL]\ncase inr.h.ha\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\nc : \u2191v x = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [c] at h \n[GOAL]\ncase inr.h.hb\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nsrc\u271d : Subring K := integer v\nx : K\nh : 1 < \u2191v x\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\n\u22a2 IsEquiv v\u2081 v\u2082 \u2194 valuationSubring v\u2081 = valuationSubring v\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\n\u22a2 IsEquiv v\u2081 v\u2082 \u2192 valuationSubring v\u2081 = valuationSubring v\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : IsEquiv v\u2081 v\u2082\n\u22a2 valuationSubring v\u2081 = valuationSubring v\u2082\n[PROOFSTEP]\next x\n[GOAL]\ncase mp.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : IsEquiv v\u2081 v\u2082\nx : K\n\u22a2 x \u2208 valuationSubring v\u2081 \u2194 x \u2208 valuationSubring v\u2082\n[PROOFSTEP]\nspecialize h x 1\n[GOAL]\ncase mp.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nx : K\nh : \u2191v\u2081 x \u2264 \u2191v\u2081 1 \u2194 \u2191v\u2082 x \u2264 \u2191v\u2082 1\n\u22a2 x \u2208 valuationSubring v\u2081 \u2194 x \u2208 valuationSubring v\u2082\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\n\u22a2 valuationSubring v\u2081 = valuationSubring v\u2082 \u2192 IsEquiv v\u2081 v\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : valuationSubring v\u2081 = valuationSubring v\u2082\n\u22a2 IsEquiv v\u2081 v\u2082\n[PROOFSTEP]\napply isEquiv_of_val_le_one\n[GOAL]\ncase mpr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : valuationSubring v\u2081 = valuationSubring v\u2082\n\u22a2 \u2200 {x : K}, \u2191v\u2081 x \u2264 1 \u2194 \u2191v\u2082 x \u2264 1\n[PROOFSTEP]\nintro x\n[GOAL]\ncase mpr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : valuationSubring v\u2081 = valuationSubring v\u2082\nx : K\n\u22a2 \u2191v\u2081 x \u2264 1 \u2194 \u2191v\u2082 x \u2264 1\n[PROOFSTEP]\nhave : x \u2208 v\u2081.valuationSubring \u2194 x \u2208 v\u2082.valuationSubring := by rw [h]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : valuationSubring v\u2081 = valuationSubring v\u2082\nx : K\n\u22a2 x \u2208 valuationSubring v\u2081 \u2194 x \u2208 valuationSubring v\u2082\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr.h\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nh : valuationSubring v\u2081 = valuationSubring v\u2082\nx : K\nthis : x \u2208 valuationSubring v\u2081 \u2194 x \u2208 valuationSubring v\u2082\n\u22a2 \u2191v\u2081 x \u2264 1 \u2194 \u2191v\u2082 x \u2264 1\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\n\u22a2 IsEquiv v (ValuationSubring.valuation (valuationSubring v))\n[PROOFSTEP]\nrw [isEquiv_iff_val_le_one]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\n\u22a2 \u2200 {x : K}, \u2191v x \u2264 1 \u2194 \u2191(ValuationSubring.valuation (valuationSubring v)) x \u2264 1\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nx : K\n\u22a2 \u2191v x \u2264 1 \u2194 \u2191(ValuationSubring.valuation (valuationSubring v)) x \u2264 1\n[PROOFSTEP]\nrw [ValuationSubring.valuation_le_one_iff]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\n\u0393 : Type u_1\n\u0393\u2081 : Type u_2\n\u0393\u2082 : Type u_3\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2081\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2082\nv : Valuation K \u0393\nv\u2081 : Valuation K \u0393\u2081\nv\u2082 : Valuation K \u0393\u2082\nx : K\n\u22a2 \u2191v x \u2264 1 \u2194 x \u2208 valuationSubring v\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 Valuation.valuationSubring (valuation A) = A\n[PROOFSTEP]\next\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx\u271d : K\n\u22a2 x\u271d \u2208 Valuation.valuationSubring (valuation A) \u2194 x\u271d \u2208 A\n[PROOFSTEP]\nrw [\u2190 A.valuation_le_one_iff]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx\u271d : K\n\u22a2 x\u271d \u2208 Valuation.valuationSubring (valuation A) \u2194 \u2191(valuation A) x\u271d \u2264 1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u22a2 \u2191({ val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) }) = \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u22a2 \u2191({ val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }) = \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 unitGroup A }\n\u22a2 (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) })\n a) =\n a\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 unitGroup A }\n\u22a2 \u2191\u2191((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) })\n a)) =\n \u2191\u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\u02e3\n\u22a2 (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a) =\n a\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : { x // x \u2208 A }\u02e3\n\u22a2 \u2191\u2191((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a)) =\n \u2191\u2191a\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 unitGroup A }\n\u22a2 Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a) =\n a) }\n (a * b) =\n Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a) =\n a) }\n a *\n Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a) =\n a) }\n b\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : { x // x \u2208 unitGroup A }\n\u22a2 \u2191\u2191(Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) }, inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ : { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) }) a) =\n a) }\n (a * b)) =\n \u2191\u2191(Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n a) =\n a) }\n a *\n Equiv.toFun\n {\n toFun := fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ : { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } * { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } = 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } * { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } = 1) },\n invFun := fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) },\n left_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 unitGroup A }),\n (fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n ((fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n a) =\n a),\n right_inv :=\n (_ :\n \u2200 (a : { x // x \u2208 A }\u02e3),\n (fun x =>\n { val := { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) },\n inv := { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } *\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191\u2191x\u207b\u00b9, property := (_ : \u2191\u2191x\u207b\u00b9 \u2208 A) } *\n { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2208 A) } =\n 1) })\n ((fun x => { val := \u2191(Units.map \u2191(subtype A)) x, property := (_ : \u2191(valuation A) \u2191\u2191x = 1) })\n a) =\n a) }\n b)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 unitGroup A \u2264 unitGroup B \u2194 A \u2264 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 unitGroup A \u2264 unitGroup B \u2192 A \u2264 B\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : x \u2208 A\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrw [\u2190 A.valuation_le_one_iff x, le_iff_lt_or_eq] at hx \n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : \u2191(valuation A) x < 1 \u2228 \u2191(valuation A) x = 1\n\u22a2 x \u2208 B\n[PROOFSTEP]\nby_cases h_1 : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : \u2191(valuation A) x < 1 \u2228 \u2191(valuation A) x = 1\nh_1 : x = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimp only [h_1, zero_mem]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : \u2191(valuation A) x < 1 \u2228 \u2191(valuation A) x = 1\nh_1 : \u00acx = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nby_cases h_2 : 1 + x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : \u2191(valuation A) x < 1 \u2228 \u2191(valuation A) x = 1\nh_1 : \u00acx = 0\nh_2 : 1 + x = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimp only [\u2190 add_eq_zero_iff_neg_eq.1 h_2, neg_mem _ _ (one_mem _)]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nhx : \u2191(valuation A) x < 1 \u2228 \u2191(valuation A) x = 1\nh_1 : \u00acx = 0\nh_2 : \u00ac1 + x = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase neg.inl\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nh_1 : \u00acx = 0\nh_2 : \u00ac1 + x = 0\nhx : \u2191(valuation A) x < 1\n\u22a2 x \u2208 B\n[PROOFSTEP]\nhave := h (show Units.mk0 _ h_2 \u2208 A.unitGroup from A.valuation.map_one_add_of_lt hx)\n[GOAL]\ncase neg.inl\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nh_1 : \u00acx = 0\nh_2 : \u00ac1 + x = 0\nhx : \u2191(valuation A) x < 1\nthis : Units.mk0 (1 + x) h_2 \u2208 unitGroup B\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimpa using\n B.add_mem _ _ (show 1 + x \u2208 B from SetLike.coe_mem (B.unitGroupMulEquiv \u27e8_, this\u27e9 : B)) (B.neg_mem _ B.one_mem)\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nh_1 : \u00acx = 0\nh_2 : \u00ac1 + x = 0\nhx : \u2191(valuation A) x = 1\n\u22a2 x \u2208 B\n[PROOFSTEP]\nhave := h (show Units.mk0 x h_1 \u2208 A.unitGroup from hx)\n[GOAL]\ncase neg.inr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A \u2264 unitGroup B\nx : K\nh_1 : \u00acx = 0\nh_2 : \u00ac1 + x = 0\nhx : \u2191(valuation A) x = 1\nthis : Units.mk0 x h_1 \u2208 unitGroup B\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrefine' SetLike.coe_mem (B.unitGroupMulEquiv \u27e8_, this\u27e9 : B)\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 A \u2264 B \u2192 unitGroup A \u2264 unitGroup B\n[PROOFSTEP]\nrintro h x (hx : A.valuation x = 1)\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\u02e3\nhx : \u2191(valuation A) \u2191x = 1\n\u22a2 x \u2208 unitGroup B\n[PROOFSTEP]\napply_fun A.mapOfLE B h at hx \n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\u02e3\nhx : \u2191(mapOfLE A B h) (\u2191(valuation A) \u2191x) = \u2191(mapOfLE A B h) 1\n\u22a2 x \u2208 unitGroup B\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : unitGroup A = unitGroup B\n\u22a2 A = B\n[PROOFSTEP]\nsimpa only [le_antisymm_iff, unitGroup_le_unitGroup] using h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 nonunits B \u2264 nonunits A \u2194 A \u2264 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 nonunits B \u2264 nonunits A \u2192 A \u2264 B\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits B \u2264 nonunits A\nx : K\nhx : x \u2208 A\n\u22a2 x \u2208 B\n[PROOFSTEP]\nby_cases h_1 : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits B \u2264 nonunits A\nx : K\nhx : x \u2208 A\nh_1 : x = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimp only [h_1, zero_mem]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits B \u2264 nonunits A\nx : K\nhx : x \u2208 A\nh_1 : \u00acx = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrw [\u2190 valuation_le_one_iff, \u2190 not_lt, Valuation.one_lt_val_iff _ h_1] at hx \u22a2\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits B \u2264 nonunits A\nx : K\nhx : \u00ac\u2191(valuation A) x\u207b\u00b9 < 1\nh_1 : \u00acx = 0\n\u22a2 \u00ac\u2191(valuation B) x\u207b\u00b9 < 1\n[PROOFSTEP]\nby_contra h_2\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits B \u2264 nonunits A\nx : K\nhx : \u00ac\u2191(valuation A) x\u207b\u00b9 < 1\nh_1 : \u00acx = 0\nh_2 : \u2191(valuation B) x\u207b\u00b9 < 1\n\u22a2 False\n[PROOFSTEP]\nexact hx (h h_2)\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 A \u2264 B \u2192 nonunits B \u2264 nonunits A\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\nhx : x \u2208 nonunits B\n\u22a2 x \u2208 nonunits A\n[PROOFSTEP]\nby_contra h_1\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\nhx : x \u2208 nonunits B\nh_1 : \u00acx \u2208 nonunits A\n\u22a2 False\n[PROOFSTEP]\nexact not_lt.2 (monotone_mapOfLE _ _ h (not_lt.1 h_1)) hx\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : nonunits A = nonunits B\n\u22a2 A = B\n[PROOFSTEP]\nsimpa only [le_antisymm_iff, nonunits_le_nonunits] using h.symm\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 Subtype.val '' \u2191(LocalRing.maximalIdeal { x // x \u2208 A }) = \u2191(nonunits A)\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\n\u22a2 a \u2208 Subtype.val '' \u2191(LocalRing.maximalIdeal { x // x \u2208 A }) \u2194 a \u2208 \u2191(nonunits A)\n[PROOFSTEP]\nsimp only [Set.mem_image, SetLike.mem_coe, mem_nonunits_iff_exists_mem_maximalIdeal]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\n\u22a2 (\u2203 x, x \u2208 LocalRing.maximalIdeal { x // x \u2208 A } \u2227 \u2191x = a) \u2194\n \u2203 ha, { val := a, property := ha } \u2208 LocalRing.maximalIdeal { x // x \u2208 A }\n[PROOFSTEP]\nerw [Subtype.exists]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\n\u22a2 (\u2203 a_1 b,\n { val := a_1, property := b } \u2208 LocalRing.maximalIdeal { x // x \u2208 A } \u2227 \u2191{ val := a_1, property := b } = a) \u2194\n \u2203 ha, { val := a, property := ha } \u2208 LocalRing.maximalIdeal { x // x \u2208 A }\n[PROOFSTEP]\nsimp_rw [exists_and_right, exists_eq_right]\n -- Porting note: added\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\n\u22a2 (\u2203 x, { val := a, property := (_ : a \u2208 \u2191A) } \u2208 LocalRing.maximalIdeal { x // x \u2208 A }) \u2194\n \u2203 ha, { val := a, property := ha } \u2208 LocalRing.maximalIdeal { x // x \u2208 A }\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 \u2200 {a b : K\u02e3},\n a \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192\n b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192 a * b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1}\n[PROOFSTEP]\nintro a b ha hb\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : K\u02e3\nha : a \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1}\nhb : b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1}\n\u22a2 a * b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1}\n[PROOFSTEP]\nrw [Set.mem_setOf] at ha hb \n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\nhb : \u2191(valuation A) (\u2191b - 1) < 1\n\u22a2 a * b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1}\n[PROOFSTEP]\nrefine'\n lt_of_le_of_lt _\n (max_lt hb ha)\n -- Porting note: `sub_add_sub_cancel` needed some help\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\nhb : \u2191(valuation A) (\u2191b - 1) < 1\n\u22a2 \u2191(valuation A) (\u2191(a * b) - 1) \u2264 max (\u2191(valuation A) (\u2191b - 1)) (\u2191(valuation A) (\u2191a - 1))\n[PROOFSTEP]\nrw [\u2190 one_mul (A.valuation (b - 1)), \u2190 A.valuation.map_one_add_of_lt ha, add_sub_cancel'_right, \u2190 Valuation.map_mul,\n mul_sub_one, \u2190 sub_add_sub_cancel (\u2191(a * b) : K) _ 1]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\nhb : \u2191(valuation A) (\u2191b - 1) < 1\n\u22a2 \u2191(valuation A) (\u2191(a * b) - ?m.1681005 + (?m.1681005 - 1)) \u2264\n max (\u2191(valuation A) (\u2191a * \u2191b - \u2191a)) (\u2191(valuation A) (\u2191a - 1))\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na b : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\nhb : \u2191(valuation A) (\u2191b - 1) < 1\n\u22a2 K\n[PROOFSTEP]\nexact A.valuation.map_add _ _\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 1 \u2208\n { carrier := {x | \u2191(valuation A) (\u2191x - 1) < 1},\n mul_mem' :=\n (_ :\n \u2200 {a b : K\u02e3},\n a \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192\n b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192 \u2191(valuation A) (\u2191(a * b) - 1) < 1) }.carrier\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 \u2200 {x : K\u02e3},\n x \u2208\n {\n toSubsemigroup :=\n { carrier := {x | \u2191(valuation A) (\u2191x - 1) < 1},\n mul_mem' :=\n (_ :\n \u2200 {a b : K\u02e3},\n a \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192\n b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192 \u2191(valuation A) (\u2191(a * b) - 1) < 1) },\n one_mem' := (_ : \u2191(valuation A) (1 - 1) < 1) }.toSubsemigroup.carrier \u2192\n x\u207b\u00b9 \u2208\n {\n toSubsemigroup :=\n { carrier := {x | \u2191(valuation A) (\u2191x - 1) < 1},\n mul_mem' :=\n (_ :\n \u2200 {a b : K\u02e3},\n a \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192\n b \u2208 {x | \u2191(valuation A) (\u2191x - 1) < 1} \u2192 \u2191(valuation A) (\u2191(a * b) - 1) < 1) },\n one_mem' := (_ : \u2191(valuation A) (1 - 1) < 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\ndsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 \u2200 {x : K\u02e3}, \u2191(valuation A) (\u2191x - 1) < 1 \u2192 \u2191(valuation A) (\u2191x\u207b\u00b9 - 1) < 1\n[PROOFSTEP]\nintro a ha\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n\u22a2 \u2191(valuation A) (\u2191a\u207b\u00b9 - 1) < 1\n[PROOFSTEP]\nconv =>\n lhs\n rw [\u2190 mul_one (A.valuation _), \u2190 A.valuation.map_one_add_of_lt ha]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n| \u2191(valuation A) (\u2191a\u207b\u00b9 - 1) < 1\n[PROOFSTEP]\n lhs\n rw [\u2190 mul_one (A.valuation _), \u2190 A.valuation.map_one_add_of_lt ha]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n| \u2191(valuation A) (\u2191a\u207b\u00b9 - 1) < 1\n[PROOFSTEP]\n lhs\n rw [\u2190 mul_one (A.valuation _), \u2190 A.valuation.map_one_add_of_lt ha]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n| \u2191(valuation A) (\u2191a\u207b\u00b9 - 1) < 1\n[PROOFSTEP]\nlhs\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n| \u2191(valuation A) (\u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nrw [\u2190 mul_one (A.valuation _), \u2190 A.valuation.map_one_add_of_lt ha]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nha : \u2191(valuation A) (\u2191a - 1) < 1\n\u22a2 \u2191(valuation A) (\u2191a\u207b\u00b9 - 1) * \u2191(valuation A) (1 + (\u2191a - 1)) < 1\n[PROOFSTEP]\nrwa [add_sub_cancel'_right, \u2190 Valuation.map_mul, sub_mul, Units.inv_mul, \u2190 neg_sub, one_mul, Valuation.map_neg]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\na : K\u02e3\nh : a \u2208 principalUnitGroup A\n\u22a2 a \u2208 unitGroup A\n[PROOFSTEP]\nsimpa only [add_sub_cancel'_right] using A.valuation.map_one_add_of_lt h\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 principalUnitGroup B \u2264 principalUnitGroup A \u2194 A \u2264 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 principalUnitGroup B \u2264 principalUnitGroup A \u2192 A \u2264 B\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\n\u22a2 x \u2208 B\n[PROOFSTEP]\nby_cases h_1 : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\nh_1 : x = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimp only [h_1, zero_mem]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\nh_1 : \u00acx = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nby_cases h_2 : x\u207b\u00b9 + 1 = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\nh_1 : \u00acx = 0\nh_2 : x\u207b\u00b9 + 1 = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrw [add_eq_zero_iff_eq_neg, inv_eq_iff_eq_inv, inv_neg, inv_one] at h_2 \n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\nh_1 : \u00acx = 0\nh_2 : x = -1\n\u22a2 x \u2208 B\n[PROOFSTEP]\nsimpa only [h_2] using B.neg_mem _ B.one_mem\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx : x \u2208 A\nh_1 : \u00acx = 0\nh_2 : \u00acx\u207b\u00b9 + 1 = 0\n\u22a2 x \u2208 B\n[PROOFSTEP]\nrw [\u2190 valuation_le_one_iff, \u2190 not_lt, Valuation.one_lt_val_iff _ h_1, \u2190 add_sub_cancel x\u207b\u00b9, \u2190 Units.val_mk0 h_2, \u2190\n mem_principalUnitGroup_iff] at hx \u22a2\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup B \u2264 principalUnitGroup A\nx : K\nhx\u271d\u00b9 : \u00ac\u2191(valuation A) x\u207b\u00b9 < 1\nhx\u271d : \u00ac\u2191(valuation A) (x\u207b\u00b9 + 1 - 1) < 1\nh_1 : \u00acx = 0\nh_2 : \u00acx\u207b\u00b9 + 1 = 0\nhx : \u00acUnits.mk0 (x\u207b\u00b9 + 1) h_2 \u2208 principalUnitGroup A\n\u22a2 \u00acUnits.mk0 (x\u207b\u00b9 + 1) h_2 \u2208 principalUnitGroup B\n[PROOFSTEP]\nsimpa only [hx] using @h (Units.mk0 (x\u207b\u00b9 + 1) h_2)\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\n\u22a2 A \u2264 B \u2192 principalUnitGroup B \u2264 principalUnitGroup A\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\u02e3\nhx : x \u2208 principalUnitGroup B\n\u22a2 x \u2208 principalUnitGroup A\n[PROOFSTEP]\nby_contra h_1\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : A \u2264 B\nx : K\u02e3\nhx : x \u2208 principalUnitGroup B\nh_1 : \u00acx \u2208 principalUnitGroup A\n\u22a2 False\n[PROOFSTEP]\nexact not_lt.2 (monotone_mapOfLE _ _ h (not_lt.1 h_1)) hx\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA\u271d A B : ValuationSubring K\nh : principalUnitGroup A = principalUnitGroup B\n\u22a2 A = B\n[PROOFSTEP]\nsimpa [le_antisymm_iff, principalUnitGroup_le_principalUnitGroup] using h.symm\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u22a2 \u2191x \u2208 principalUnitGroup A \u2194 \u2191(unitGroupMulEquiv A) x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))\n[PROOFSTEP]\nrw [MonoidHom.mem_ker, Units.ext_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u22a2 \u2191x \u2208 principalUnitGroup A \u2194 \u2191(\u2191(Units.map \u2191(LocalRing.residue { x // x \u2208 A })) (\u2191(unitGroupMulEquiv A) x)) = \u21911\n[PROOFSTEP]\nlet \u03c0 := Ideal.Quotient.mk (LocalRing.maximalIdeal A)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u03c0 : { x // x \u2208 A } \u2192+* { x // x \u2208 A } \u29f8 LocalRing.maximalIdeal { x // x \u2208 A } :=\n Ideal.Quotient.mk (LocalRing.maximalIdeal { x // x \u2208 A })\n\u22a2 \u2191x \u2208 principalUnitGroup A \u2194 \u2191(\u2191(Units.map \u2191(LocalRing.residue { x // x \u2208 A })) (\u2191(unitGroupMulEquiv A) x)) = \u21911\n[PROOFSTEP]\nconvert_to _ \u2194 \u03c0 _ = 1\n[GOAL]\ncase convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u03c0 : { x // x \u2208 A } \u2192+* { x // x \u2208 A } \u29f8 LocalRing.maximalIdeal { x // x \u2208 A } :=\n Ideal.Quotient.mk (LocalRing.maximalIdeal { x // x \u2208 A })\n\u22a2 \u2191x \u2208 principalUnitGroup A \u2194 \u2191\u03c0 \u2191(\u2191(unitGroupMulEquiv A) x) = 1\n[PROOFSTEP]\nrw [\u2190 \u03c0.map_one, \u2190 sub_eq_zero, \u2190 \u03c0.map_sub, Ideal.Quotient.eq_zero_iff_mem, valuation_lt_one_iff]\n[GOAL]\ncase convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 unitGroup A }\n\u03c0 : { x // x \u2208 A } \u2192+* { x // x \u2208 A } \u29f8 LocalRing.maximalIdeal { x // x \u2208 A } :=\n Ideal.Quotient.mk (LocalRing.maximalIdeal { x // x \u2208 A })\n\u22a2 \u2191x \u2208 principalUnitGroup A \u2194 \u2191(valuation A) \u2191(\u2191(\u2191(unitGroupMulEquiv A) x) - 1) < 1\n[PROOFSTEP]\nsimp [mem_principalUnitGroup_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }\n\u22a2 \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A\n[PROOFSTEP]\nrw [A.coe_mem_principalUnitGroup_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }\n\u22a2 \u2191(unitGroupMulEquiv A) (\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))\n[PROOFSTEP]\nsimpa using SetLike.coe_mem x\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 principalUnitGroup A }\n\u22a2 (fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property := (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n ((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x) =\n x\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }\n\u22a2 (fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n ((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property := (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) =\n x\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx y : { x // x \u2208 principalUnitGroup A }\n\u22a2 Equiv.toFun\n {\n toFun := fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) },\n invFun := fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property := (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 principalUnitGroup A }),\n {\n val :=\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)),\n property :=\n (_ :\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)) \u2208\n principalUnitGroup A) } =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }),\n {\n val :=\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) } =\n x) }\n (x * y) =\n Equiv.toFun\n {\n toFun := fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) },\n invFun := fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property := (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 principalUnitGroup A }),\n {\n val :=\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)),\n property :=\n (_ :\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)) \u2208\n principalUnitGroup A) } =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }),\n {\n val :=\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ :\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) } =\n x) }\n x *\n Equiv.toFun\n {\n toFun := fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) },\n invFun := fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property := (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 principalUnitGroup A }),\n {\n val :=\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)),\n property :=\n (_ :\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A))\n \u2191((fun x =>\n { val := \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A) { val := \u2191x, property := (_ : \u2191x \u2208 unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) })\n x)) \u2208\n principalUnitGroup A) } =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A })) }),\n {\n val :=\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) },\n property :=\n (_ :\n \u2191(unitGroupMulEquiv A)\n {\n val :=\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ : \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x),\n property :=\n (_ :\n \u2191((fun x =>\n { val := \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x),\n property :=\n (_ :\n \u2191(\u2191(MulEquiv.symm (unitGroupMulEquiv A)) \u2191x) \u2208 principalUnitGroup A) })\n x) \u2208\n unitGroup A) } \u2208\n MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))) } =\n x) }\n y\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\n\u22a2 MonoidHom.ker (unitGroupToResidueFieldUnits A) =\n Subgroup.comap (Subgroup.subtype (unitGroup A)) (principalUnitGroup A)\n[PROOFSTEP]\next\n -- Porting note: simp fails but rw works\n -- See https://github.com/leanprover-community/mathlib4/issues/5026\n -- simp [Subgroup.mem_comap, Subgroup.coeSubtype, coe_mem_principalUnitGroup_iff]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx\u271d : { x // x \u2208 unitGroup A }\n\u22a2 x\u271d \u2208 MonoidHom.ker (unitGroupToResidueFieldUnits A) \u2194\n x\u271d \u2208 Subgroup.comap (Subgroup.subtype (unitGroup A)) (principalUnitGroup A)\n[PROOFSTEP]\nrw [Subgroup.mem_comap, Subgroup.coeSubtype, coe_mem_principalUnitGroup_iff]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx\u271d : { x // x \u2208 unitGroup A }\n\u22a2 x\u271d \u2208 MonoidHom.ker (unitGroupToResidueFieldUnits A) \u2194\n \u2191(unitGroupMulEquiv A) x\u271d \u2208 MonoidHom.ker (Units.map \u2191(LocalRing.residue { x // x \u2208 A }))\n[PROOFSTEP]\nrfl\n -- simp [Subgroup.mem_comap, Subgroup.coeSubtype, coe_mem_principalUnitGroup_iff]\n[GOAL]\nK : Type u\ninst\u271d\u00b2 : Field K\nA : ValuationSubring K\nG : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : MulSemiringAction G K\ng : G\nS : ValuationSubring K\nsrc\u271d : Subring K := g \u2022 S.toSubring\nx : K\nh : (g\u207b\u00b9 \u2022 x)\u207b\u00b9 \u2208 S\n\u22a2 g\u207b\u00b9 \u2022 x\u207b\u00b9 \u2208 S.toSubring\n[PROOFSTEP]\nrwa [smul_inv'']\n[GOAL]\nK : Type u\ninst\u271d\u00b2 : Field K\nA\u271d : ValuationSubring K\nL : Type u_1\nJ : Type u_2\ninst\u271d\u00b9 : Field L\ninst\u271d : Field J\nA : ValuationSubring L\nf : K \u2192+* L\nsrc\u271d : Subring K := Subring.comap f A.toSubring\nk : K\n\u22a2 k \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 \u2228\n k\u207b\u00b9 \u2208\n { toSubsemiring := src\u271d.toSubsemiring,\n neg_mem' :=\n (_ :\n \u2200 {x : K}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp [ValuationSubring.mem_or_inv_mem]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Valuation.ValuationSubring", "llama_tokens": 43928, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.27297251694821595}} {"text": "[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\n\u03b1 : Type u\nx : t' \u03b1\n\u22a2 Equiv.map eqv id x = x\n[PROOFSTEP]\nsimp [Equiv.map, id_map]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b3\nx : t' \u03b1\n\u22a2 Equiv.map eqv (h \u2218 g) x = Equiv.map eqv h (Equiv.map eqv g x)\n[PROOFSTEP]\nsimp [Equiv.map]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b3\nx : t' \u03b1\n\u22a2 (h \u2218 g) <$> \u2191(eqv \u03b1).symm x = h <$> g <$> \u2191(eqv \u03b1).symm x\n[PROOFSTEP]\napply comp_map\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nF : Functor t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 LawfulFunctor t'\n[PROOFSTEP]\nhave : F = Equiv.functor eqv := by\n cases F\n dsimp [Equiv.functor]\n congr <;> ext <;> dsimp only <;> [rw [\u2190 h\u2080]; rw [\u2190 h\u2081]] <;> rfl\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nF : Functor t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 F = Equiv.functor eqv\n[PROOFSTEP]\ncases F\n[GOAL]\ncase mk\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 { map := map\u271d, mapConst := mapConst\u271d } = Equiv.functor eqv\n[PROOFSTEP]\ndsimp [Equiv.functor]\n[GOAL]\ncase mk\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 { map := map\u271d, mapConst := mapConst\u271d } =\n { map := fun {\u03b1 \u03b2} => Equiv.map eqv, mapConst := fun {\u03b1 \u03b2} => Equiv.map eqv \u2218 Function.const \u03b2 }\n[PROOFSTEP]\ncongr <;> ext <;> dsimp only <;> [rw [\u2190 h\u2080]; rw [\u2190 h\u2081]]\n[GOAL]\ncase mk\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 { map := map\u271d, mapConst := mapConst\u271d } =\n { map := fun {\u03b1 \u03b2} => Equiv.map eqv, mapConst := fun {\u03b1 \u03b2} => Equiv.map eqv \u2218 Function.const \u03b2 }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.e_map\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 map\u271d = fun {\u03b1 \u03b2} => Equiv.map eqv\n[PROOFSTEP]\next\n[GOAL]\ncase mk.e_mapConst\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 mapConst\u271d = fun {\u03b1 \u03b2} => Equiv.map eqv \u2218 Function.const \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase mk.e_map.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3 \u2192 x\u271d\u00b2\nx\u271d : t' x\u271d\u00b3\n\u22a2 map\u271d x\u271d\u00b9 x\u271d = Equiv.map eqv x\u271d\u00b9 x\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.e_mapConst.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3\nx\u271d : t' x\u271d\u00b2\n\u22a2 mapConst\u271d x\u271d\u00b9 x\u271d = (fun {\u03b1 \u03b2} => Equiv.map eqv \u2218 Function.const \u03b2) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.e_map.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3 \u2192 x\u271d\u00b2\nx\u271d : t' x\u271d\u00b3\n\u22a2 map\u271d x\u271d\u00b9 x\u271d = Equiv.map eqv x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrw [\u2190 h\u2080]\n[GOAL]\ncase mk.e_mapConst.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3\nx\u271d : t' x\u271d\u00b2\n\u22a2 mapConst\u271d x\u271d\u00b9 x\u271d = (Equiv.map eqv \u2218 Function.const x\u271d\u00b2) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrw [\u2190 h\u2081]\n[GOAL]\ncase mk.e_map.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3 \u2192 x\u271d\u00b2\nx\u271d : t' x\u271d\u00b3\n\u22a2 map\u271d x\u271d\u00b9 x\u271d = x\u271d\u00b9 <$> x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.e_mapConst.h.h.h.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nmap\u271d : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 t' \u03b1 \u2192 t' \u03b2\nmapConst\u271d : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2192 t' \u03b2 \u2192 t' \u03b1\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nx\u271d\u00b3 x\u271d\u00b2 : Type u\nx\u271d\u00b9 : x\u271d\u00b3\nx\u271d : t' x\u271d\u00b2\n\u22a2 mapConst\u271d x\u271d\u00b9 x\u271d = mapConst x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nF : Functor t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nthis : F = Equiv.functor eqv\n\u22a2 LawfulFunctor t'\n[PROOFSTEP]\nsubst this\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9 : Functor t\ninst\u271d : LawfulFunctor t\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\n\u22a2 LawfulFunctor t'\n[PROOFSTEP]\nexact Equiv.lawfulFunctor eqv\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nx : t' \u03b1\n\u22a2 Equiv.traverse eqv pure x = x\n[PROOFSTEP]\nsimp [Equiv.traverse]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\nx : t' \u03b1\n\u22a2 Equiv.traverse eqv (pure \u2218 f) x = pure (Equiv.map eqv f x)\n[PROOFSTEP]\nsimp [Equiv.traverse, traverse_eq_map_id, functor_norm]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\nx : t' \u03b1\n\u22a2 \u2191(eqv \u03b2) (id.mk (f <$> \u2191(eqv \u03b1).symm x)) = Equiv.map eqv f x\n[PROOFSTEP]\nrfl\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : t' \u03b1\n\u22a2 Equiv.traverse eqv (Comp.mk \u2218 map f \u2218 g) x = Comp.mk (Equiv.traverse eqv f <$> Equiv.traverse eqv g x)\n[PROOFSTEP]\nsimp [Equiv.traverse, comp_traverse, functor_norm]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : t' \u03b1\n\u22a2 Comp.mk (((fun x => \u2191(eqv \u03b3) <$> x) \u2218 traverse f) <$> traverse g (\u2191(eqv \u03b1).symm x)) =\n Comp.mk (((fun x => \u2191(eqv \u03b3) <$> traverse f (\u2191(eqv \u03b2).symm x)) \u2218 \u2191(eqv \u03b2)) <$> traverse g (\u2191(eqv \u03b1).symm x))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x.e_a\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : t' \u03b1\n\u22a2 (fun x => \u2191(eqv \u03b3) <$> x) \u2218 traverse f = (fun x => \u2191(eqv \u03b3) <$> traverse f (\u2191(eqv \u03b2).symm x)) \u2218 \u2191(eqv \u03b2)\n[PROOFSTEP]\next\n[GOAL]\ncase e_x.e_a.h\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : t' \u03b1\nx\u271d : t \u03b2\n\u22a2 ((fun x => \u2191(eqv \u03b3) <$> x) \u2218 traverse f) x\u271d = ((fun x => \u2191(eqv \u03b3) <$> traverse f (\u2191(eqv \u03b2).symm x)) \u2218 \u2191(eqv \u03b2)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2075 : Traversable t\ninst\u271d\u2074 : LawfulTraversable t\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\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : t' \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Equiv.traverse eqv f x) =\n Equiv.traverse eqv ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\nsimp only [Equiv.traverse, functor_norm]\n[GOAL]\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u22a2 LawfulTraversable t'\n[PROOFSTEP]\nrefine' { toLawfulFunctor := Equiv.lawfulFunctor' eqv @h\u2080 @h\u2081 .. }\n[GOAL]\ncase refine'_1\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u22a2 \u2200 {\u03b1 : Type u} (x : t' \u03b1), traverse pure x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u22a2 \u2200 {F G : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : Applicative G] [inst_2 : LawfulApplicative F]\n [inst_3 : LawfulApplicative G] {\u03b1 \u03b2 \u03b3 : Type u} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t' \u03b1),\n traverse (Comp.mk \u2218 map f \u2218 g) x = Comp.mk (traverse f <$> traverse g x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1), traverse (pure \u2218 f) x = id.mk (f <$> x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u22a2 \u2200 {F G : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : Applicative G] [inst_2 : LawfulApplicative F]\n [inst_3 : LawfulApplicative G] (\u03b7 : ApplicativeTransformation F G) {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2) (x : t' \u03b1),\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (traverse f x) =\n traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u03b1\u271d : Type u\nx\u271d : t' \u03b1\u271d\n\u22a2 traverse pure x\u271d = x\u271d\n[PROOFSTEP]\nrw [h\u2082, Equiv.id_traverse]\n[GOAL]\ncase refine'_2\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9\u2070 : Traversable t\ninst\u271d\u2079 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2078 : Applicative F\ninst\u271d\u2077 : Applicative G\ninst\u271d\u2076 : LawfulApplicative F\ninst\u271d\u2075 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d\u2074 : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\nF\u271d G\u271d : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\u271d\ninst\u271d\u00b2 : Applicative G\u271d\ninst\u271d\u00b9 : LawfulApplicative F\u271d\ninst\u271d : LawfulApplicative G\u271d\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nf\u271d : \u03b2\u271d \u2192 F\u271d \u03b3\u271d\ng\u271d : \u03b1\u271d \u2192 G\u271d \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 traverse (Comp.mk \u2218 map f\u271d \u2218 g\u271d) x\u271d = Comp.mk (traverse f\u271d <$> traverse g\u271d x\u271d)\n[PROOFSTEP]\nrw [h\u2082, Equiv.comp_traverse, h\u2082]\n[GOAL]\ncase refine'_2\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9\u2070 : Traversable t\ninst\u271d\u2079 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2078 : Applicative F\ninst\u271d\u2077 : Applicative G\ninst\u271d\u2076 : LawfulApplicative F\ninst\u271d\u2075 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d\u2074 : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\nF\u271d G\u271d : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\u271d\ninst\u271d\u00b2 : Applicative G\u271d\ninst\u271d\u00b9 : LawfulApplicative F\u271d\ninst\u271d : LawfulApplicative G\u271d\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nf\u271d : \u03b2\u271d \u2192 F\u271d \u03b3\u271d\ng\u271d : \u03b1\u271d \u2192 G\u271d \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 Comp.mk (Equiv.traverse eqv f\u271d <$> Equiv.traverse eqv g\u271d x\u271d) = Comp.mk (Equiv.traverse eqv f\u271d <$> traverse g\u271d x\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_2.e_x.e_a\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9\u2070 : Traversable t\ninst\u271d\u2079 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2078 : Applicative F\ninst\u271d\u2077 : Applicative G\ninst\u271d\u2076 : LawfulApplicative F\ninst\u271d\u2075 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d\u2074 : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\nF\u271d G\u271d : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\u271d\ninst\u271d\u00b2 : Applicative G\u271d\ninst\u271d\u00b9 : LawfulApplicative F\u271d\ninst\u271d : LawfulApplicative G\u271d\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type u\nf\u271d : \u03b2\u271d \u2192 F\u271d \u03b3\u271d\ng\u271d : \u03b1\u271d \u2192 G\u271d \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 Equiv.traverse eqv g\u271d x\u271d = traverse g\u271d x\u271d\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\ncase refine'_3\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 traverse (pure \u2218 f\u271d) x\u271d = id.mk (f\u271d <$> x\u271d)\n[PROOFSTEP]\nrw [h\u2082, Equiv.traverse_eq_map_id, h\u2080]\n[GOAL]\ncase refine'_3\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u2076 : Traversable t\ninst\u271d\u2075 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 pure (Equiv.map eqv f\u271d x\u271d) = id.mk (Equiv.map eqv f\u271d x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\nt t' : Type u \u2192 Type u\neqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1\ninst\u271d\u00b9\u2070 : Traversable t\ninst\u271d\u2079 : LawfulTraversable t\nF G : Type u \u2192 Type u\ninst\u271d\u2078 : Applicative F\ninst\u271d\u2077 : Applicative G\ninst\u271d\u2076 : LawfulApplicative F\ninst\u271d\u2075 : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 \u03b2 \u03b3 : Type u\ninst\u271d\u2074 : Traversable t'\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), map f = Equiv.map eqv f\nh\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), mapConst f = (Equiv.map eqv \u2218 Function.const \u03b1) f\nh\u2082 :\n \u2200 {F : Type u \u2192 Type u} [inst : Applicative F] [inst_1 : LawfulApplicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n traverse f = Equiv.traverse eqv f\nF\u271d G\u271d : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\u271d\ninst\u271d\u00b2 : Applicative G\u271d\ninst\u271d\u00b9 : LawfulApplicative F\u271d\ninst\u271d : LawfulApplicative G\u271d\n\u03b7\u271d : ApplicativeTransformation F\u271d G\u271d\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 F\u271d \u03b2\u271d\nx\u271d : t' \u03b1\u271d\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7\u271d \u03b1) (traverse f\u271d x\u271d) =\n traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7\u271d \u03b1) \u2218 f\u271d) x\u271d\n[PROOFSTEP]\nrw [h\u2082, Equiv.naturality, h\u2082]\n", "meta": {"mathlib_filename": "Mathlib.Control.Traversable.Equiv", "llama_tokens": 10730, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.27290378446111546}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : AddUnits \u211d\u22650\u221e\n\u22a2 \u2191a \u2264 0\n[PROOFSTEP]\nrw [\u2190 a.add_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : AddUnits \u211d\u22650\u221e\n\u22a2 \u2191a \u2264 \u2191a + \u2191(-a)\n[PROOFSTEP]\nexact le_self_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : a \u2260 \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal a) = a\n[PROOFSTEP]\nsimp [ENNReal.toReal, ENNReal.ofReal, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\n\u22a2 \u2191(ENNReal.toNNReal \u2191r) \u2264 \u2191r\n[PROOFSTEP]\nrw [toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\n\u22a2 \u2191r = ENNReal.ofReal \u2191r\n[PROOFSTEP]\nrw [ENNReal.ofReal, Real.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 ENNReal.ofReal 0 = 0\n[PROOFSTEP]\nsimp [ENNReal.ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 ENNReal.ofReal 1 = 1\n[PROOFSTEP]\nsimp [ENNReal.ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e \u2192 Prop\n\u22a2 (\u2203 a, a \u2260 \u22a4 \u2227 p a) \u2194 \u2203 r, p \u2191r\n[PROOFSTEP]\nsimp only [exists_ne_top', \u2190 exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal x = 0 \u2194 x = 0 \u2228 x = \u22a4\n[PROOFSTEP]\nsimp [ENNReal.toReal, toNNReal_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 x = \u21911 \u2228 x = \u22a4 \u2227 1 = 0 \u2194 x = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal x = 1 \u2194 x = 1\n[PROOFSTEP]\nrw [ENNReal.toReal, NNReal.coe_eq_one, ENNReal.toNNReal_eq_one_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : ENNReal.ofReal (ENNReal.toReal a) = a\n\u22a2 a \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : ENNReal.ofReal (ENNReal.toReal a) = a\n\u22a2 ENNReal.ofReal (ENNReal.toReal a) \u2260 \u22a4\n[PROOFSTEP]\nexact ofReal_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nh : ENNReal.toReal (ENNReal.ofReal a) = a\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nh : ENNReal.toReal (ENNReal.ofReal a) = a\n\u22a2 0 \u2264 ENNReal.toReal (ENNReal.ofReal a)\n[PROOFSTEP]\nexact toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal x = ENNReal.toReal y \u2194 x = y \u2228 x = 0 \u2227 y = \u22a4 \u2228 x = \u22a4 \u2227 y = 0\n[PROOFSTEP]\nsimp only [ENNReal.toReal, NNReal.coe_eq, toNNReal_eq_toNNReal_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 ENNReal.toNNReal x = ENNReal.toNNReal y \u2194 x = y\n[PROOFSTEP]\nsimp only [ENNReal.toNNReal_eq_toNNReal_iff x y, hx, hy, and_false, false_and, or_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 ENNReal.toReal x = ENNReal.toReal y \u2194 x = y\n[PROOFSTEP]\nsimp only [ENNReal.toReal, NNReal.coe_eq, toNNReal_eq_toNNReal_iff' hx hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u21911 < \u21912\n[PROOFSTEP]\nexact_mod_cast (one_lt_two : 1 < 2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ninst\u271d : CompleteLattice \u03b1\nf : \u211d\u22650\u221e \u2192 \u03b1\n\u22a2 \u2a05 (x : \u211d\u22650\u221e) (_ : x \u2260 \u22a4), f x = \u2a05 (x : \u211d\u22650), f \u2191x\n[PROOFSTEP]\nrw [iInf_subtype', cinfi_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d\u22650\u221e A\nr : \u211d\u22650\nx : (fun x => A) r\n\u22a2 \u2191(RingHom.comp (algebraMap \u211d\u22650\u221e A) ofNNRealHom) r * x = x * \u2191(RingHom.comp (algebraMap \u211d\u22650\u221e A) ofNNRealHom) r\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d\u22650\u221e A\nr : \u211d\u22650\nx : (fun x => A) r\n\u22a2 r \u2022 x = \u2191(RingHom.comp (algebraMap \u211d\u22650\u221e A) ofNNRealHom) r * x\n[PROOFSTEP]\nsimp [\u2190 Algebra.smul_def (r : \u211d\u22650\u221e) x, smul_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nR : Type u_3\nr : R\ns : \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\n\u22a2 \u2191(r \u2022 s) = r \u2022 \u2191s\n[PROOFSTEP]\nrw [\u2190 smul_one_smul \u211d\u22650 r (s : \u211d\u22650\u221e), smul_def, smul_eq_mul, \u2190 ENNReal.coe_mul, smul_mul_assoc, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nr\u2081 r\u2082 : \u211d\u22650\u221e\nh\u2081 : r\u2081 \u2260 \u22a4\nh\u2082 : r\u2082 \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (r\u2081 + r\u2082) = ENNReal.toNNReal r\u2081 + ENNReal.toNNReal r\u2082\n[PROOFSTEP]\nlift r\u2081 to \u211d\u22650 using h\u2081\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nr\u2082 : \u211d\u22650\u221e\nh\u2082 : r\u2082 \u2260 \u22a4\nr\u2081 : \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2191r\u2081 + r\u2082) = ENNReal.toNNReal \u2191r\u2081 + ENNReal.toNNReal r\u2082\n[PROOFSTEP]\nlift r\u2082 to \u211d\u22650 using h\u2082\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q r\u2081 r\u2082 : \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2191r\u2081 + \u2191r\u2082) = ENNReal.toNNReal \u2191r\u2081 + ENNReal.toNNReal \u2191r\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 \u00acx < \u22a4 \u2194 x = \u22a4\n[PROOFSTEP]\nrw [lt_top_iff_ne_top, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a + b \u2260 \u22a4 \u2194 a \u2260 \u22a4 \u2227 b \u2260 \u22a4\n[PROOFSTEP]\nsimpa only [lt_top_iff_ne_top] using add_lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a * \u22a4 = if a = 0 then 0 else \u22a4\n[PROOFSTEP]\nconvert WithTop.mul_top' a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22a4 * a = if a = 0 then 0 else \u22a4\n[PROOFSTEP]\nconvert WithTop.top_mul' a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nR : Type u_3\ninst\u271d\u2074 : Zero R\ninst\u271d\u00b3 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : NoZeroSMulDivisors R \u211d\u22650\u221e\ninst\u271d : DecidableEq R\nc : R\n\u22a2 c \u2022 \u22a4 = if c = 0 then 0 else \u22a4\n[PROOFSTEP]\nrw [\u2190 smul_one_mul, mul_top']\n -- porting note: need the primed version of `one_ne_zero` now\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nR : Type u_3\ninst\u271d\u2074 : Zero R\ninst\u271d\u00b3 : SMulWithZero R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : NoZeroSMulDivisors R \u211d\u22650\u221e\ninst\u271d : DecidableEq R\nc : R\n\u22a2 (if c \u2022 1 = 0 then 0 else \u22a4) = if c = 0 then 0 else \u22a4\n[PROOFSTEP]\nsimp_rw [smul_eq_zero, or_iff_left (one_ne_zero' \u211d\u22650\u221e)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\nh : 0 < n\nm : \u2115\nx\u271d : Nat.succ 0 \u2264 m\nhm : \u22a4 ^ m = \u22a4\n\u22a2 \u22a4 ^ (m + 1) = \u22a4\n[PROOFSTEP]\nrw [pow_succ, hm, top_mul_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a \u2260 \u22a4 \u2192 b \u2260 \u22a4 \u2192 a * b \u2260 \u22a4\n[PROOFSTEP]\nsimpa only [lt_top_iff_ne_top] using mul_lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a * b \u2260 \u22a4\nha : a \u2260 0\n\u22a2 b * a \u2260 \u22a4\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 a * b < \u22a4 \u2194 a < \u22a4 \u2227 b < \u22a4 \u2228 a = 0 \u2228 b = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 a * b < \u22a4 \u2192 a < \u22a4 \u2227 b < \u22a4 \u2228 a = 0 \u2228 b = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a * b < \u22a4\n\u22a2 a < \u22a4 \u2227 b < \u22a4 \u2228 a = 0 \u2228 b = 0\n[PROOFSTEP]\nrw [\u2190 or_assoc, or_iff_not_imp_right, or_iff_not_imp_right]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a * b < \u22a4\n\u22a2 \u00acb = 0 \u2192 \u00aca = 0 \u2192 a < \u22a4 \u2227 b < \u22a4\n[PROOFSTEP]\nintro hb ha\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a * b < \u22a4\nhb : \u00acb = 0\nha : \u00aca = 0\n\u22a2 a < \u22a4 \u2227 b < \u22a4\n[PROOFSTEP]\nexact \u27e8lt_top_of_mul_ne_top_left h.ne hb, lt_top_of_mul_ne_top_right h.ne ha\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 a < \u22a4 \u2227 b < \u22a4 \u2228 a = 0 \u2228 b = 0 \u2192 a * b < \u22a4\n[PROOFSTEP]\nrintro (\u27e8ha, hb\u27e9 | rfl | rfl) <;> [exact mul_lt_top ha.ne hb.ne; simp; simp]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 a < \u22a4 \u2227 b < \u22a4 \u2228 a = 0 \u2228 b = 0 \u2192 a * b < \u22a4\n[PROOFSTEP]\nrintro (\u27e8ha, hb\u27e9 | rfl | rfl)\n[GOAL]\ncase mpr.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nha : a < \u22a4\nhb : b < \u22a4\n\u22a2 a * b < \u22a4\n[PROOFSTEP]\nexact mul_lt_top ha.ne hb.ne\n[GOAL]\ncase mpr.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\u22650\u221e\n\u22a2 0 * b < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a * 0 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a * a < \u22a4 \u2194 a < \u22a4\n[PROOFSTEP]\nrw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a = 0 \u2192 a < \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 0 < \u22a4\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 a ^ n = \u22a4 \u2194 a = \u22a4 \u2227 n \u2260 0\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with rfl | (hn : 0 < n)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a ^ 0 = \u22a4 \u2194 a = \u22a4 \u2227 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\nhn : 0 < n\n\u22a2 a ^ n = \u22a4 \u2194 a = \u22a4 \u2227 n \u2260 0\n[PROOFSTEP]\ninduction a using recTopCoe\n[GOAL]\ncase inr.top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\nhn : 0 < n\n\u22a2 \u22a4 ^ n = \u22a4 \u2194 \u22a4 = \u22a4 \u2227 n \u2260 0\n[PROOFSTEP]\nsimp only [Ne.def, hn.ne', top_pow hn]\n[GOAL]\ncase inr.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\nhn : 0 < n\nx\u271d : \u211d\u22650\n\u22a2 \u2191x\u271d ^ n = \u22a4 \u2194 \u2191x\u271d = \u22a4 \u2227 n \u2260 0\n[PROOFSTEP]\nsimp only [\u2190 coe_pow, coe_ne_top, false_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a < \u22a4 \u2192 \u2200 (n : \u2115), a ^ n < \u22a4\n[PROOFSTEP]\nsimpa only [lt_top_iff_ne_top] using pow_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.ofReal \u2191n = \u2191n\n[PROOFSTEP]\nsimp [ENNReal.ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.toNNReal \u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_nat n, ENNReal.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.toReal \u2191n = \u2191n\n[PROOFSTEP]\nrw [\u2190 ENNReal.ofReal_coe_nat n, ENNReal.toReal_ofReal (Nat.cast_nonneg _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\u22650\nh : a \u2264 \u2191b\n\u22a2 ENNReal.toReal a \u2264 \u2191b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ne_top_of_le_ne_top coe_ne_top h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q b a : \u211d\u22650\nh : \u2191a \u2264 \u2191b\n\u22a2 ENNReal.toReal \u2191a \u2264 \u2191b\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a \u2260 0 \u2192 \u2200 (n : \u2115), a ^ n \u2260 0\n[PROOFSTEP]\nsimpa only [pos_iff_ne_zero] using ENNReal.pow_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u00aca < 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 a < a + b\n[PROOFSTEP]\nrwa [\u2190 pos_iff_ne_zero, \u2190 ENNReal.add_lt_add_iff_left ha, add_zero] at hb \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a < b\n\u22a2 \u2203 q, 0 \u2264 q \u2227 a < \u2191(Real.toNNReal \u2191q) \u2227 \u2191(Real.toNNReal \u2191q) < b\n[PROOFSTEP]\nrcases lt_iff_exists_coe.1 h with \u27e8p, rfl, _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p\u271d q p : \u211d\u22650\nright\u271d h : \u2191p < b\n\u22a2 \u2203 q, 0 \u2264 q \u2227 \u2191p < \u2191(Real.toNNReal \u2191q) \u2227 \u2191(Real.toNNReal \u2191q) < b\n[PROOFSTEP]\nrcases exists_between h with \u27e8c, pc, cb\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c\u271d d : \u211d\u22650\u221e\nr p\u271d q p : \u211d\u22650\nright\u271d h : \u2191p < b\nc : \u211d\u22650\u221e\npc : \u2191p < c\ncb : c < b\n\u22a2 \u2203 q, 0 \u2264 q \u2227 \u2191p < \u2191(Real.toNNReal \u2191q) \u2227 \u2191(Real.toNNReal \u2191q) < b\n[PROOFSTEP]\nrcases lt_iff_exists_coe.1 cb with \u27e8r, rfl, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr\u271d p\u271d q p : \u211d\u22650\nright\u271d\u00b9 h : \u2191p < b\nr : \u211d\u22650\nright\u271d : \u2191r < b\npc : \u2191p < \u2191r\ncb : \u2191r < b\n\u22a2 \u2203 q, 0 \u2264 q \u2227 \u2191p < \u2191(Real.toNNReal \u2191q) \u2227 \u2191(Real.toNNReal \u2191q) < b\n[PROOFSTEP]\nrcases(NNReal.lt_iff_exists_rat_btwn _ _).1 (coe_lt_coe.1 pc) with \u27e8q, hq0, pq, qr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr\u271d p\u271d q\u271d p : \u211d\u22650\nright\u271d\u00b9 h : \u2191p < b\nr : \u211d\u22650\nright\u271d : \u2191r < b\npc : \u2191p < \u2191r\ncb : \u2191r < b\nq : \u211a\nhq0 : 0 \u2264 q\npq : p < Real.toNNReal \u2191q\nqr : Real.toNNReal \u2191q < r\n\u22a2 \u2203 q, 0 \u2264 q \u2227 \u2191p < \u2191(Real.toNNReal \u2191q) \u2227 \u2191(Real.toNNReal \u2191q) < b\n[PROOFSTEP]\nexact \u27e8q, hq0, coe_lt_coe.2 pq, lt_trans (coe_lt_coe.2 qr) cb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a < b \u2194 \u2203 r, 0 < r \u2227 a + \u2191r < b\n[PROOFSTEP]\nrefine' \u27e8fun hab => _, fun \u27e8r, _, hr\u27e9 => lt_of_le_of_lt le_self_add hr\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhab : a < b\n\u22a2 \u2203 r, 0 < r \u2227 a + \u2191r < b\n[PROOFSTEP]\nrcases lt_iff_exists_nnreal_btwn.1 hab with \u27e8c, ac, cb\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhab : a < b\nc : \u211d\u22650\nac : a < \u2191c\ncb : \u2191c < b\n\u22a2 \u2203 r, 0 < r \u2227 a + \u2191r < b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ac.ne_top\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c\u271d d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\ncb : \u2191c < b\na : \u211d\u22650\nhab : \u2191a < b\nac : \u2191a < \u2191c\n\u22a2 \u2203 r, 0 < r \u2227 \u2191a + \u2191r < b\n[PROOFSTEP]\nrw [coe_lt_coe] at ac \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c\u271d d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\ncb : \u2191c < b\na : \u211d\u22650\nhab : \u2191a < b\nac : a < c\n\u22a2 \u2203 r, 0 < r \u2227 \u2191a + \u2191r < b\n[PROOFSTEP]\nrefine \u27e8c - a, tsub_pos_iff_lt.2 ac, ?_\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c\u271d d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\ncb : \u2191c < b\na : \u211d\u22650\nhab : \u2191a < b\nac : a < c\n\u22a2 \u2191a + \u2191(c - a) < b\n[PROOFSTEP]\nrwa [\u2190 coe_add, add_tsub_cancel_of_le ac.le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 b < \u22a4 \u2192 a \u2264 b + \u2191\u03b5\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : b < a\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 b < \u22a4 \u2227 b + \u2191\u03b5 < a\n[PROOFSTEP]\nrcases lt_iff_exists_add_pos_lt.1 h with \u27e8r, hr0, hr\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nh : b < a\nr : \u211d\u22650\nhr0 : 0 < r\nhr : b + \u2191r < a\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 b < \u22a4 \u2227 b + \u2191\u03b5 < a\n[PROOFSTEP]\nexact \u27e8r, hr0, h.trans_le le_top, hr\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nr : \u211d\u22650\u221e\nh : r \u2260 \u22a4\n\u22a2 \u2203 n, r < \u2191n\n[PROOFSTEP]\nlift r to \u211d\u22650 using h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\n\u22a2 \u2203 n, \u2191r < \u2191n\n[PROOFSTEP]\nrcases exists_nat_gt r with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\nn : \u2115\nhn : r < \u2191n\n\u22a2 \u2203 n, \u2191r < \u2191n\n[PROOFSTEP]\nexact \u27e8n, coe_lt_coe_nat.2 hn\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c3 (n : \u2115), Iio \u2191n = {\u22a4}\u1d9c\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 x \u2208 \u22c3 (n : \u2115), Iio \u2191n \u2194 x \u2208 {\u22a4}\u1d9c\n[PROOFSTEP]\nrw [mem_iUnion]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 (\u2203 i, x \u2208 Iio \u2191i) \u2194 x \u2208 {\u22a4}\u1d9c\n[PROOFSTEP]\nexact \u27e8fun \u27e8n, hn\u27e9 => ne_top_of_lt hn, ENNReal.exists_nat_gt\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c3 (n : \u2115), Ioc a \u2191n = Ioi a \\ {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 Ioi_inter_Iic, \u2190 inter_iUnion, iUnion_Iic_coe_nat, diff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c3 (n : \u2115), Ioo a \u2191n = Ioi a \\ {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 Ioi_inter_Iio, \u2190 inter_iUnion, iUnion_Iio_coe_nat, diff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c3 (n : \u2115), Icc a \u2191n = Ici a \\ {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 Ici_inter_Iic, \u2190 inter_iUnion, iUnion_Iic_coe_nat, diff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c3 (n : \u2115), Ico a \u2191n = Ici a \\ {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 Ici_inter_Iio, \u2190 inter_iUnion, iUnion_Iio_coe_nat, diff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c2 (n : \u2115), Ici \u2191n = {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 compl_Iio, \u2190 compl_iUnion, iUnion_Iio_coe_nat, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22c2 (n : \u2115), Ioi \u2191n = {\u22a4}\n[PROOFSTEP]\nsimp only [\u2190 compl_Iic, \u2190 compl_iUnion, iUnion_Iic_coe_nat, compl_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nac : a < c\nbd : b < d\n\u22a2 a + b < c + d\n[PROOFSTEP]\nlift a to \u211d\u22650 using ac.ne_top\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nbd : b < d\na : \u211d\u22650\nac : \u2191a < c\n\u22a2 \u2191a + b < c + d\n[PROOFSTEP]\nlift b to \u211d\u22650 using bd.ne_top\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nac : \u2191a < c\nb : \u211d\u22650\nbd : \u2191b < d\n\u22a2 \u2191a + \u2191b < c + d\n[PROOFSTEP]\ncases c\n[GOAL]\ncase intro.intro.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nd : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\nbd : \u2191b < d\nac : \u2191a < none\n\u22a2 \u2191a + \u2191b < none + d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nd : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\nbd : \u2191b < d\nval\u271d : \u211d\u22650\nac : \u2191a < Option.some val\u271d\n\u22a2 \u2191a + \u2191b < Option.some val\u271d + d\n[PROOFSTEP]\ncases d\n[GOAL]\ncase intro.intro.some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr p q a b val\u271d : \u211d\u22650\nac : \u2191a < Option.some val\u271d\nbd : \u2191b < none\n\u22a2 \u2191a + \u2191b < Option.some val\u271d + none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr p q a b val\u271d\u00b9 : \u211d\u22650\nac : \u2191a < Option.some val\u271d\u00b9\nval\u271d : \u211d\u22650\nbd : \u2191b < Option.some val\u271d\n\u22a2 \u2191a + \u2191b < Option.some val\u271d\u00b9 + Option.some val\u271d\n[PROOFSTEP]\nsimp only [\u2190 coe_add, some_eq_coe, coe_lt_coe] at *\n[GOAL]\ncase intro.intro.some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr p q a b val\u271d\u00b9 val\u271d : \u211d\u22650\nac : a < val\u271d\u00b9\nbd : b < val\u271d\n\u22a2 a + b < val\u271d\u00b9 + val\u271d\n[PROOFSTEP]\nexact add_lt_add ac bd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a = \u22a4 \u2192 b = \u22a4\nh_nnreal : a \u2260 \u22a4 \u2192 b \u2260 \u22a4 \u2192 ENNReal.toNNReal a \u2264 ENNReal.toNNReal b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nby_contra' hlt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a = \u22a4 \u2192 b = \u22a4\nh_nnreal : a \u2260 \u22a4 \u2192 b \u2260 \u22a4 \u2192 ENNReal.toNNReal a \u2264 ENNReal.toNNReal b\nhlt : b < a\n\u22a2 False\n[PROOFSTEP]\nlift b to \u211d\u22650 using hlt.ne_top\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\u22650\nh : a = \u22a4 \u2192 \u2191b = \u22a4\nh_nnreal : a \u2260 \u22a4 \u2192 \u2191b \u2260 \u22a4 \u2192 ENNReal.toNNReal a \u2264 ENNReal.toNNReal \u2191b\nhlt : \u2191b < a\n\u22a2 False\n[PROOFSTEP]\nlift a to \u211d\u22650 using mt h coe_ne_top\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q b a : \u211d\u22650\nh : \u2191a = \u22a4 \u2192 \u2191b = \u22a4\nh_nnreal : \u2191a \u2260 \u22a4 \u2192 \u2191b \u2260 \u22a4 \u2192 ENNReal.toNNReal \u2191a \u2264 ENNReal.toNNReal \u2191b\nhlt : \u2191b < \u2191a\n\u22a2 False\n[PROOFSTEP]\nrefine hlt.not_le ?_\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q b a : \u211d\u22650\nh : \u2191a = \u22a4 \u2192 \u2191b = \u22a4\nh_nnreal : \u2191a \u2260 \u22a4 \u2192 \u2191b \u2260 \u22a4 \u2192 ENNReal.toNNReal \u2191a \u2264 ENNReal.toNNReal \u2191b\nhlt : \u2191b < \u2191a\n\u22a2 \u2191a \u2264 \u2191b\n[PROOFSTEP]\nsimpa using h_nnreal\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 |ENNReal.toReal x| = ENNReal.toReal x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 |ENNReal.toReal none| = ENNReal.toReal none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q val\u271d : \u211d\u22650\n\u22a2 |ENNReal.toReal (Option.some val\u271d)| = ENNReal.toReal (Option.some val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Set \u211d\u22650\n\u22a2 \u2191r \u2208 upperBounds (some '' s) \u2194 r \u2208 upperBounds s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [upperBounds, ball_image_iff, -mem_image, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nac : a < c\nbd : b < d\n\u22a2 a * b < c * d\n[PROOFSTEP]\nrcases lt_iff_exists_nnreal_btwn.1 ac with \u27e8a', aa', a'c\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nac : a < c\nbd : b < d\na' : \u211d\u22650\naa' : a < \u2191a'\na'c : \u2191a' < c\n\u22a2 a * b < c * d\n[PROOFSTEP]\nlift a to \u211d\u22650 using ne_top_of_lt aa'\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nbd : b < d\na' : \u211d\u22650\na'c : \u2191a' < c\na : \u211d\u22650\nac : \u2191a < c\naa' : \u2191a < \u2191a'\n\u22a2 \u2191a * b < c * d\n[PROOFSTEP]\nrcases lt_iff_exists_nnreal_btwn.1 bd with \u27e8b', bb', b'd\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nbd : b < d\na' : \u211d\u22650\na'c : \u2191a' < c\na : \u211d\u22650\nac : \u2191a < c\naa' : \u2191a < \u2191a'\nb' : \u211d\u22650\nbb' : b < \u2191b'\nb'd : \u2191b' < d\n\u22a2 \u2191a * b < c * d\n[PROOFSTEP]\nlift b to \u211d\u22650 using ne_top_of_lt bb'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a' : \u211d\u22650\na'c : \u2191a' < c\na : \u211d\u22650\nac : \u2191a < c\naa' : \u2191a < \u2191a'\nb' : \u211d\u22650\nb'd : \u2191b' < d\nb : \u211d\u22650\nbd : \u2191b < d\nbb' : \u2191b < \u2191b'\n\u22a2 \u2191a * \u2191b < c * d\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a' : \u211d\u22650\na'c : \u2191a' < c\na : \u211d\u22650\nac : \u2191a < c\nb' : \u211d\u22650\nb'd : \u2191b' < d\nb : \u211d\u22650\nbd : \u2191b < d\naa' : a < a'\nbb' : b < b'\n\u22a2 \u2191(a * b) < c * d\n[PROOFSTEP]\ncalc\n \u2191(a * b) < \u2191(a' * b') := coe_lt_coe.2 (mul_lt_mul\u2080 aa' bb')\n _ = \u2191a' * \u2191b' := coe_mul\n _ \u2264 c * d := mul_le_mul' a'c.le b'd.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d : 1 \u2260 0\n\u22a2 StrictMono fun x => x ^ 1\n[PROOFSTEP]\nsimpa only [pow_one] using strictMono_id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0 : a \u2260 0\nhinf : a \u2260 \u22a4\n\u22a2 StrictMono fun x => a * x\n[PROOFSTEP]\nlift a to \u211d\u22650 using hinf\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : \u2191a \u2260 0\n\u22a2 StrictMono fun x => \u2191a * x\n[PROOFSTEP]\nrw [coe_ne_zero] at h0 \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : a \u2260 0\n\u22a2 StrictMono fun x => \u2191a * x\n[PROOFSTEP]\nintro x y h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : a \u2260 0\nx y : \u211d\u22650\u221e\nh : x < y\n\u22a2 (fun x => \u2191a * x) x < (fun x => \u2191a * x) y\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : a \u2260 0\nx y : \u211d\u22650\u221e\nh : \u2191a * y \u2264 \u2191a * x\n\u22a2 y \u2264 x\n[PROOFSTEP]\nsimpa only [\u2190 mul_assoc, \u2190 coe_mul, inv_mul_cancel h0, coe_one, one_mul] using mul_le_mul_left' h (\u2191a\u207b\u00b9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 AddLECancellable a \u2194 a \u2260 \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 AddLECancellable a \u2192 a \u2260 \u22a4\n[PROOFSTEP]\nrintro h rfl\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : AddLECancellable \u22a4\n\u22a2 False\n[PROOFSTEP]\nrefine' zero_lt_one.not_le (h _)\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : AddLECancellable \u22a4\n\u22a2 \u22a4 + 1 \u2264 \u22a4 + 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a \u2260 \u22a4 \u2192 AddLECancellable a\n[PROOFSTEP]\nrintro h b c hbc\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : a \u2260 \u22a4\nb c : \u211d\u22650\u221e\nhbc : a + b \u2264 a + c\n\u22a2 b \u2264 c\n[PROOFSTEP]\napply ENNReal.le_of_add_le_add_left h hbc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nm n : \u2115\n\u22a2 \u2191(m - n) = \u2191m - \u2191n\n[PROOFSTEP]\nrw [\u2190 coe_nat, Nat.cast_tsub, coe_sub, coe_nat, coe_nat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4 \u2228 b \u2260 \u22a4\n\u22a2 a - b < c \u2192 a < b + c\n[PROOFSTEP]\nobtain rfl | hb := eq_or_ne b \u221e\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 a - \u22a4 < c \u2192 a < \u22a4 + c\n[PROOFSTEP]\nrw [top_add, lt_top_iff_ne_top]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 a - \u22a4 < c \u2192 a \u2260 \u22a4\n[PROOFSTEP]\nexact fun _ => h.resolve_right (Classical.not_not.2 rfl)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4 \u2228 b \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 a - b < c \u2192 a < b + c\n[PROOFSTEP]\nexact (cancel_of_ne hb).lt_add_of_tsub_lt_left\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 \u22a4\n\u22a2 (a - b) * c = a * c - b * 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\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 \u22a4\nhab : a \u2264 b\n\u22a2 (a - b) * c = a * c - b * c\n[PROOFSTEP]\nsimp [hab, mul_right_mono hab]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 \u22a4\nhab : b < a\n\u22a2 (a - b) * c = a * c - b * c\n[PROOFSTEP]\nrcases eq_or_lt_of_le (zero_le b) with (rfl | hb)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < 0 \u2192 0 < a \u2192 c \u2260 \u22a4\nhab : 0 < a\n\u22a2 (a - 0) * c = a * c - 0 * c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 \u22a4\nhab : b < a\nhb : 0 < b\n\u22a2 (a - b) * c = a * c - b * c\n[PROOFSTEP]\nexact (cancel_of_ne <| mul_ne_top hab.ne_top (h hb hab)).tsub_mul\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < c \u2192 c < b \u2192 a \u2260 \u22a4\n\u22a2 a * (b - c) = a * b - a * c\n[PROOFSTEP]\nsimp only [mul_comm a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < c \u2192 c < b \u2192 a \u2260 \u22a4\n\u22a2 (b - c) * a = b * a - c * a\n[PROOFSTEP]\nexact sub_mul h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (\u2211 a in s, f a) = \u2211 a in s, ENNReal.toNNReal (f a)\n[PROOFSTEP]\nrw [\u2190 coe_eq_coe, coe_toNNReal, coe_finset_sum, sum_congr rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 f x = \u2191(ENNReal.toNNReal (f x))\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = \u2191(ENNReal.toNNReal (f x))\n[PROOFSTEP]\nexact (coe_toNNReal (hf x hx)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\n\u22a2 \u2211 a in s, f a \u2260 \u22a4\n[PROOFSTEP]\nexact (sum_lt_top hf).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2211 a in s, f a) = \u2211 a in s, ENNReal.toReal (f a)\n[PROOFSTEP]\nrw [ENNReal.toReal, toNNReal_sum hf, NNReal.coe_sum]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 \u22a4\n\u22a2 \u2211 a in s, \u2191(ENNReal.toNNReal (f a)) = \u2211 a in s, ENNReal.toReal (f a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (i : \u03b1), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 ENNReal.ofReal (\u2211 i in s, f i) = \u2211 i in s, ENNReal.ofReal (f i)\n[PROOFSTEP]\nsimp_rw [ENNReal.ofReal, \u2190 coe_finset_sum, coe_eq_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (i : \u03b1), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 Real.toNNReal (\u2211 i in s, f i) = \u2211 a in s, Real.toNNReal (f a)\n[PROOFSTEP]\nexact Real.toNNReal_sum_of_nonneg hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nhs : Finset.Nonempty s\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHlt : \u2200 (i : \u03b1), i \u2208 s \u2192 f i < g i\n\u22a2 \u2211 i in s, f i < \u2211 i in s, g i\n[PROOFSTEP]\ninduction' hs using Finset.Nonempty.cons_induction with a a s as _ IH\n[GOAL]\ncase h\u2080\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHlt\u271d : \u2200 (i : \u03b1), i \u2208 s \u2192 f i < g i\na : \u03b1\nHlt : \u2200 (i : \u03b1), i \u2208 {a} \u2192 f i < g i\n\u22a2 \u2211 i in {a}, f i < \u2211 i in {a}, g i\n[PROOFSTEP]\nsimp [Hlt _ (Finset.mem_singleton_self _)]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHlt\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 f i < g i\na : \u03b1\ns : Finset \u03b1\nas : \u00aca \u2208 s\nhs\u271d : Finset.Nonempty s\nIH : (\u2200 (i : \u03b1), i \u2208 s \u2192 f i < g i) \u2192 \u2211 i in s, f i < \u2211 i in s, g i\nHlt : \u2200 (i : \u03b1), i \u2208 cons a s as \u2192 f i < g i\n\u22a2 \u2211 i in cons a s as, f i < \u2211 i in cons a s as, g i\n[PROOFSTEP]\nsimp only [as, Finset.sum_cons, not_false_iff]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns\u271d : Finset \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHlt\u271d : \u2200 (i : \u03b1), i \u2208 s\u271d \u2192 f i < g i\na : \u03b1\ns : Finset \u03b1\nas : \u00aca \u2208 s\nhs\u271d : Finset.Nonempty s\nIH : (\u2200 (i : \u03b1), i \u2208 s \u2192 f i < g i) \u2192 \u2211 i in s, f i < \u2211 i in s, g i\nHlt : \u2200 (i : \u03b1), i \u2208 cons a s as \u2192 f i < g i\n\u22a2 f a + \u2211 i in s, f i < g a + \u2211 i in s, g i\n[PROOFSTEP]\nexact ENNReal.add_lt_add (Hlt _ (Finset.mem_cons_self _ _)) (IH fun i hi => Hlt _ (Finset.mem_cons.2 <| Or.inr hi))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nhs : Finset.Nonempty s\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHle : \u2211 i in s, f i \u2264 \u2211 i in s, g i\n\u22a2 \u2203 i, i \u2208 s \u2227 f i \u2264 g i\n[PROOFSTEP]\ncontrapose! Hle\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nhs : Finset.Nonempty s\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nHle : \u2200 (i : \u03b1), i \u2208 s \u2192 g i < f i\n\u22a2 \u2211 i in s, g i < \u2211 i in s, f i\n[PROOFSTEP]\napply ENNReal.sum_lt_sum_of_nonempty hs Hle\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a / b = b\u207b\u00b9 * a\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 sInf {b | 1 \u2264 0 * b} = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : 0 < a\n\u22a2 a \u2208 {b | 1 \u2264 \u22a4 * b}\n[PROOFSTEP]\nsimp [*, h.ne', top_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\u22650\u221e\nhb : 1 \u2264 \u2191r * b\n\u22a2 \u2200 (p : \u211d\u22650), b = \u2191p \u2192 r\u207b\u00b9 \u2264 p\n[PROOFSTEP]\nrintro b rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q b : \u211d\u22650\nhb : 1 \u2264 \u2191r * \u2191b\n\u22a2 r\u207b\u00b9 \u2264 b\n[PROOFSTEP]\napply NNReal.inv_le_of_le_mul\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q b : \u211d\u22650\nhb : 1 \u2264 \u2191r * \u2191b\n\u22a2 1 \u2264 r * b\n[PROOFSTEP]\nrwa [\u2190 coe_mul, \u2190 coe_one, coe_le_coe] at hb \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\n\u22a2 1 \u2264 \u2191r * \u2191r\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 coe_mul, mul_inv_cancel hr, coe_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u21912\u207b\u00b9 = 2\u207b\u00b9\n[PROOFSTEP]\nrw [coe_inv _root_.two_ne_zero, coe_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\n\u22a2 \u2191(p / r) = \u2191p / \u2191r\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, coe_mul, coe_inv hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 0\n\u22a2 a / 0 = \u22a4\n[PROOFSTEP]\nsimp [div_eq_mul_inv, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nsrc\u271d : DivInvMonoid \u211d\u22650\u221e := inferInstanceAs (DivInvMonoid \u211d\u22650\u221e)\n\u22a2 1\u207b\u00b9 = 1\n[PROOFSTEP]\nsimpa only [coe_inv one_ne_zero, coe_one] using coe_eq_coe.2 inv_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d : \u211d\u22650\u221e\n\u22a2 (x\u271d ^ 0)\u207b\u00b9 = x\u271d\u207b\u00b9 ^ 0\n[PROOFSTEP]\nsimp only [pow_zero, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 (\u22a4 ^ (n + 1))\u207b\u00b9 = \u22a4\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nsimp [top_pow]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nn : \u2115\n\u22a2 (\u2191a ^ (n + 1))\u207b\u00b9 = (\u2191a)\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 (\u21910 ^ (n + 1))\u207b\u00b9 = (\u21910)\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nsimp [top_pow]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nn : \u2115\nha : a \u2260 0\n\u22a2 (\u2191a ^ (n + 1))\u207b\u00b9 = (\u2191a)\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nhave := pow_ne_zero (n + 1) ha\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nn : \u2115\nha : a \u2260 0\nthis : a ^ (n + 1) \u2260 0\n\u22a2 (\u2191a ^ (n + 1))\u207b\u00b9 = (\u2191a)\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nn : \u2115\nha : a \u2260 0\nthis : a ^ (n + 1) \u2260 0\n\u22a2 (a ^ (n + 1))\u207b\u00b9 = a\u207b\u00b9 ^ (n + 1)\n[PROOFSTEP]\nrw [inv_pow]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0 : a \u2260 0\nht : a \u2260 \u22a4\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nlift a to \u211d\u22650 using ht\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : \u2191a \u2260 0\n\u22a2 \u2191a * (\u2191a)\u207b\u00b9 = 1\n[PROOFSTEP]\nnorm_cast at h0 \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : \u00aca = 0\n\u22a2 \u2191a * (\u2191a)\u207b\u00b9 = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh0 : \u00aca = 0\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nexact mul_inv_cancel h0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0 : a \u2260 0\nhI : a \u2260 \u22a4\n\u22a2 b / a * a = b\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel h0 hI, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0 : a \u2260 0\nhI : a \u2260 \u22a4\n\u22a2 a * (b / a) = b\n[PROOFSTEP]\nrw [mul_comm, ENNReal.div_mul_cancel h0 hI]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a / b * c = a * (c / b)\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_right_comm, \u2190 mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a * b / c = a / c * b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\nby_cases a = 0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\nby_cases a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : a = 0\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : \u00aca = 0\n\u22a2 a\u207b\u00b9\u207b\u00b9 = a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase pos.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : none = 0\n\u22a2 none\u207b\u00b9\u207b\u00b9 = none\n[PROOFSTEP]\nsimp_all [none_eq_top, some_eq_coe, -coe_inv, (coe_inv _).symm]\n[GOAL]\ncase pos.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q val\u271d : \u211d\u22650\nh : Option.some val\u271d = 0\n\u22a2 (Option.some val\u271d)\u207b\u00b9\u207b\u00b9 = Option.some val\u271d\n[PROOFSTEP]\nsimp_all [none_eq_top, some_eq_coe, -coe_inv, (coe_inv _).symm]\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : \u00acnone = 0\n\u22a2 none\u207b\u00b9\u207b\u00b9 = none\n[PROOFSTEP]\nsimp_all [none_eq_top, some_eq_coe, -coe_inv, (coe_inv _).symm]\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q val\u271d : \u211d\u22650\nh : \u00acOption.some val\u271d = 0\n\u22a2 (Option.some val\u271d)\u207b\u00b9\u207b\u00b9 = Option.some val\u271d\n[PROOFSTEP]\nsimp_all [none_eq_top, some_eq_coe, -coe_inv, (coe_inv _).symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 \u2260 \u22a4 \u2194 a \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\n\u22a2 x\u207b\u00b9 < \u22a4 \u2194 0 < x\n[PROOFSTEP]\nsimp only [lt_top_iff_ne_top, inv_ne_top, pos_iff_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 \u2260 0 \u2194 a \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b \u2260 \u22a4\nhb : a \u2260 \u22a4 \u2228 b \u2260 0\n\u22a2 (a * b)\u207b\u00b9 = a\u207b\u00b9 * b\u207b\u00b9\n[PROOFSTEP]\ninduction' b using recTopCoe with b\n[GOAL]\ncase top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nha\u271d : a \u2260 0 \u2228 b \u2260 \u22a4\nhb\u271d : a \u2260 \u22a4 \u2228 b \u2260 0\nha : a \u2260 0 \u2228 \u22a4 \u2260 \u22a4\nhb : a \u2260 \u22a4 \u2228 \u22a4 \u2260 0\n\u22a2 (a * \u22a4)\u207b\u00b9 = a\u207b\u00b9 * \u22a4\u207b\u00b9\n[PROOFSTEP]\nreplace ha : a \u2260 0 := ha.neg_resolve_right rfl\n[GOAL]\ncase top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nha\u271d : a \u2260 0 \u2228 b \u2260 \u22a4\nhb\u271d : a \u2260 \u22a4 \u2228 b \u2260 0\nhb : a \u2260 \u22a4 \u2228 \u22a4 \u2260 0\nha : a \u2260 0\n\u22a2 (a * \u22a4)\u207b\u00b9 = a\u207b\u00b9 * \u22a4\u207b\u00b9\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b\u271d : \u211d\u22650\u221e\nha\u271d : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha : a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : a \u2260 \u22a4 \u2228 \u2191b \u2260 0\n\u22a2 (a * \u2191b)\u207b\u00b9 = a\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\ninduction' a using recTopCoe with a\n[GOAL]\ncase coe.top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nha\u271d : \u22a4 \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u22a4 \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u22a4 \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 0\n\u22a2 (\u22a4 * \u2191b)\u207b\u00b9 = \u22a4\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nreplace hb : b \u2260 0 := coe_ne_zero.1 (hb.neg_resolve_left rfl)\n[GOAL]\ncase coe.top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nha\u271d : \u22a4 \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u22a4 \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u22a4 \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 (\u22a4 * \u2191b)\u207b\u00b9 = \u22a4\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase coe.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\n\u22a2 (\u2191a * \u2191b)\u207b\u00b9 = (\u2191a)\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nby_cases h'a : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nh'a : a = 0\n\u22a2 (\u2191a * \u2191b)\u207b\u00b9 = (\u2191a)\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nsimp only [h'a, top_mul, ENNReal.inv_zero, ENNReal.coe_ne_top, zero_mul, Ne.def, not_false_iff, ENNReal.coe_zero,\n ENNReal.inv_eq_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nh'a : \u00aca = 0\n\u22a2 (\u2191a * \u2191b)\u207b\u00b9 = (\u2191a)\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nby_cases h'b : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nh'a : \u00aca = 0\nh'b : b = 0\n\u22a2 (\u2191a * \u2191b)\u207b\u00b9 = (\u2191a)\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nsimp only [h'b, ENNReal.inv_zero, ENNReal.coe_ne_top, mul_top, Ne.def, not_false_iff, mul_zero, ENNReal.coe_zero,\n ENNReal.inv_eq_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nh'a : \u00aca = 0\nh'b : \u00acb = 0\n\u22a2 (\u2191a * \u2191b)\u207b\u00b9 = (\u2191a)\u207b\u00b9 * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_mul, \u2190 ENNReal.coe_inv, \u2190 ENNReal.coe_inv h'a, \u2190 ENNReal.coe_inv h'b, \u2190 ENNReal.coe_mul, mul_inv_rev,\n mul_comm]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na\u271d b\u271d : \u211d\u22650\u221e\nha\u271d\u00b2 : a\u271d \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d\u00b2 : a\u271d \u2260 \u22a4 \u2228 b\u271d \u2260 0\nb : \u211d\u22650\nha\u271d\u00b9 : a\u271d \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb\u271d\u00b9 : a\u271d \u2260 \u22a4 \u2228 \u2191b \u2260 0\na : \u211d\u22650\nha\u271d : \u2191a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : \u2191a \u2260 \u22a4 \u2228 b\u271d \u2260 0\nha : \u2191a \u2260 0 \u2228 \u2191b \u2260 \u22a4\nhb : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 0\nh'a : \u00aca = 0\nh'b : \u00acb = 0\n\u22a2 a * b \u2260 0\n[PROOFSTEP]\nsimp [h'a, h'b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhc : c \u2260 0\nhc' : c \u2260 \u22a4\n\u22a2 c * a / (c * b) = a / b\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inl hc) (Or.inl hc'), mul_mul_mul_comm,\n ENNReal.mul_inv_cancel hc hc', one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhc : c \u2260 0\nhc' : c \u2260 \u22a4\n\u22a2 a * c / (b * c) = a / b\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, ENNReal.mul_inv (Or.inr hc') (Or.inr hc), mul_mul_mul_comm,\n ENNReal.mul_inv_cancel hc hc', mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 0\n\u22a2 (a - b) / c = a / c - b / c\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 0\n\u22a2 (a - b) * c\u207b\u00b9 = a * c\u207b\u00b9 - b * c\u207b\u00b9\n[PROOFSTEP]\nexact ENNReal.sub_mul (by simpa using h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : 0 < b \u2192 b < a \u2192 c \u2260 0\n\u22a2 0 < b \u2192 b < a \u2192 c\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 StrictAnti Inv.inv\n[PROOFSTEP]\nintro a b h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : a < b\n\u22a2 b\u207b\u00b9 < a\u207b\u00b9\n[PROOFSTEP]\nlift a to \u211d\u22650 using h.ne_top\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\u22650\u221e\na : \u211d\u22650\nh : \u2191a < b\n\u22a2 b\u207b\u00b9 < (\u2191a)\u207b\u00b9\n[PROOFSTEP]\ninduction b using recTopCoe\n[GOAL]\ncase intro.top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nh : \u2191a < \u22a4\n\u22a2 \u22a4\u207b\u00b9 < (\u2191a)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a x\u271d : \u211d\u22650\nh : \u2191a < \u2191x\u271d\n\u22a2 (\u2191x\u271d)\u207b\u00b9 < (\u2191a)\u207b\u00b9\n[PROOFSTEP]\nrw [coe_lt_coe] at h \n[GOAL]\ncase intro.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a x\u271d : \u211d\u22650\nh : a < x\u271d\n\u22a2 (\u2191x\u271d)\u207b\u00b9 < (\u2191a)\u207b\u00b9\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase intro.coe.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x\u271d : \u211d\u22650\nh : 0 < x\u271d\n\u22a2 (\u2191x\u271d)\u207b\u00b9 < (\u21910)\u207b\u00b9\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase intro.coe.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a x\u271d : \u211d\u22650\nh : a < x\u271d\nha : a \u2260 0\n\u22a2 (\u2191x\u271d)\u207b\u00b9 < (\u2191a)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 coe_inv h.ne_bot, \u2190 coe_inv ha, coe_lt_coe]\n[GOAL]\ncase intro.coe.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a x\u271d : \u211d\u22650\nh : a < x\u271d\nha : a \u2260 0\n\u22a2 x\u271d\u207b\u00b9 < a\u207b\u00b9\n[PROOFSTEP]\nexact NNReal.inv_lt_inv ha h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 < b \u2194 b\u207b\u00b9 < a\n[PROOFSTEP]\nsimpa only [inv_inv] using @ENNReal.inv_lt_inv a b\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a < b\u207b\u00b9 \u2194 b < a\u207b\u00b9\n[PROOFSTEP]\nsimpa only [inv_inv] using @ENNReal.inv_lt_inv a\u207b\u00b9 b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 b\u207b\u00b9 \u2264 a\n[PROOFSTEP]\nsimpa only [inv_inv] using @ENNReal.inv_le_inv a b\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a \u2264 b\u207b\u00b9 \u2194 b \u2264 a\u207b\u00b9\n[PROOFSTEP]\nsimpa only [inv_inv] using @ENNReal.inv_le_inv a\u207b\u00b9 b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 \u2264 1 \u2194 1 \u2264 a\n[PROOFSTEP]\nrw [inv_le_iff_inv_le, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 1 \u2264 a\u207b\u00b9 \u2194 a \u2264 1\n[PROOFSTEP]\nrw [le_inv_iff_le_inv, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a\u207b\u00b9 < 1 \u2194 1 < a\n[PROOFSTEP]\nrw [inv_lt_iff_inv_lt, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 1 < a\u207b\u00b9 \u2194 a < 1\n[PROOFSTEP]\nrw [lt_inv_iff_lt_inv, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a / \u22a4 = 0\n[PROOFSTEP]\nrw [div_eq_mul_inv, inv_top, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u22a4 / a = if a = \u22a4 then 0 else \u22a4\n[PROOFSTEP]\nsimp [div_eq_mul_inv, top_mul']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4\n\u22a2 \u22a4 / a = \u22a4\n[PROOFSTEP]\nsimp [top_div, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a / b = \u22a4 \u2194 a \u2260 0 \u2227 b = 0 \u2228 a = \u22a4 \u2227 b \u2260 \u22a4\n[PROOFSTEP]\nsimp [div_eq_mul_inv, ENNReal.mul_eq_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0 : b \u2260 0 \u2228 c \u2260 0\nht : b \u2260 \u22a4 \u2228 c \u2260 \u22a4\n\u22a2 a \u2264 c / b \u2194 a * b \u2264 c\n[PROOFSTEP]\ninduction' b using recTopCoe with b\n[GOAL]\ncase top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 c \u2260 \u22a4\nh0 : \u22a4 \u2260 0 \u2228 c \u2260 0\nht : \u22a4 \u2260 \u22a4 \u2228 c \u2260 \u22a4\n\u22a2 a \u2264 c / \u22a4 \u2194 a * \u22a4 \u2264 c\n[PROOFSTEP]\nlift c to \u211d\u22650 using ht.neg_resolve_left rfl\n[GOAL]\ncase top.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 \u2191c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\nh0 : \u22a4 \u2260 0 \u2228 \u2191c \u2260 0\nht : \u22a4 \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\n\u22a2 a \u2264 \u2191c / \u22a4 \u2194 a * \u22a4 \u2264 \u2191c\n[PROOFSTEP]\nrw [div_top, nonpos_iff_eq_zero]\n[GOAL]\ncase top.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 \u2191c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\nh0 : \u22a4 \u2260 0 \u2228 \u2191c \u2260 0\nht : \u22a4 \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\n\u22a2 a = 0 \u2194 a * \u22a4 \u2264 \u2191c\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase top.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 \u2191c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\nh0 : \u22a4 \u2260 0 \u2228 \u2191c \u2260 0\nht : \u22a4 \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\n\u22a2 0 = 0 \u2194 0 * \u22a4 \u2264 \u2191c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase top.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b d : \u211d\u22650\u221e\nr p q c : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 \u2191c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\nh0 : \u22a4 \u2260 0 \u2228 \u2191c \u2260 0\nht : \u22a4 \u2260 \u22a4 \u2228 \u2191c \u2260 \u22a4\nha : a \u2260 0\n\u22a2 a = 0 \u2194 a * \u22a4 \u2264 \u2191c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b\u271d \u2260 0 \u2228 c \u2260 0\nht\u271d : b\u271d \u2260 \u22a4 \u2228 c \u2260 \u22a4\nb : \u211d\u22650\nh0 : \u2191b \u2260 0 \u2228 c \u2260 0\nht : \u2191b \u2260 \u22a4 \u2228 c \u2260 \u22a4\n\u22a2 a \u2264 c / \u2191b \u2194 a * \u2191b \u2264 c\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb)\n[GOAL]\ncase coe.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 c \u2260 \u22a4\nh0 : \u21910 \u2260 0 \u2228 c \u2260 0\nht : \u21910 \u2260 \u22a4 \u2228 c \u2260 \u22a4\n\u22a2 a \u2264 c / \u21910 \u2194 a * \u21910 \u2264 c\n[PROOFSTEP]\nhave hc : c \u2260 0 := h0.neg_resolve_left rfl\n[GOAL]\ncase coe.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b \u2260 0 \u2228 c \u2260 0\nht\u271d : b \u2260 \u22a4 \u2228 c \u2260 \u22a4\nh0 : \u21910 \u2260 0 \u2228 c \u2260 0\nht : \u21910 \u2260 \u22a4 \u2228 c \u2260 \u22a4\nhc : c \u2260 0\n\u22a2 a \u2264 c / \u21910 \u2194 a * \u21910 \u2264 c\n[PROOFSTEP]\nsimp [div_zero hc]\n[GOAL]\ncase coe.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b\u271d \u2260 0 \u2228 c \u2260 0\nht\u271d : b\u271d \u2260 \u22a4 \u2228 c \u2260 \u22a4\nb : \u211d\u22650\nh0 : \u2191b \u2260 0 \u2228 c \u2260 0\nht : \u2191b \u2260 \u22a4 \u2228 c \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 a \u2264 c / \u2191b \u2194 a * \u2191b \u2264 c\n[PROOFSTEP]\nrw [\u2190 coe_ne_zero] at hb \n[GOAL]\ncase coe.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh0\u271d : b\u271d \u2260 0 \u2228 c \u2260 0\nht\u271d : b\u271d \u2260 \u22a4 \u2228 c \u2260 \u22a4\nb : \u211d\u22650\nh0 : \u2191b \u2260 0 \u2228 c \u2260 0\nht : \u2191b \u2260 \u22a4 \u2228 c \u2260 \u22a4\nhb : \u2191b \u2260 0\n\u22a2 a \u2264 c / \u2191b \u2194 a * \u2191b \u2264 c\n[PROOFSTEP]\nrw [\u2190 ENNReal.mul_le_mul_right hb coe_ne_top, ENNReal.div_mul_cancel hb coe_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb0 : b \u2260 0 \u2228 c \u2260 \u22a4\nhbt : b \u2260 \u22a4 \u2228 c \u2260 0\n\u22a2 a / b \u2264 c \u2194 a \u2264 c * b\n[PROOFSTEP]\nsuffices a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 c / b\u207b\u00b9 by simpa [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb0 : b \u2260 0 \u2228 c \u2260 \u22a4\nhbt : b \u2260 \u22a4 \u2228 c \u2260 0\nthis : a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 c / b\u207b\u00b9\n\u22a2 a / b \u2264 c \u2194 a \u2264 c * b\n[PROOFSTEP]\nsimpa [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb0 : b \u2260 0 \u2228 c \u2260 \u22a4\nhbt : b \u2260 \u22a4 \u2228 c \u2260 0\n\u22a2 a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 c / b\u207b\u00b9\n[PROOFSTEP]\nrefine' (ENNReal.le_div_iff_mul_le _ _).symm\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb0 : b \u2260 0 \u2228 c \u2260 \u22a4\nhbt : b \u2260 \u22a4 \u2228 c \u2260 0\n\u22a2 b\u207b\u00b9 \u2260 0 \u2228 c \u2260 0\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb0 : b \u2260 0 \u2228 c \u2260 \u22a4\nhbt : b \u2260 \u22a4 \u2228 c \u2260 0\n\u22a2 b\u207b\u00b9 \u2260 \u22a4 \u2228 c \u2260 \u22a4\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nby_cases h0 : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : c = 0\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nhave : a = 0 := by simpa [h0] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : c = 0\n\u22a2 a = 0\n[PROOFSTEP]\nsimpa [h0] using h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : c = 0\nthis : a = 0\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : \u00acc = 0\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nby_cases hinf : c = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : \u00acc = 0\nhinf : c = \u22a4\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nsimp [hinf]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b * c\nh0 : \u00acc = 0\nhinf : \u00acc = \u22a4\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nexact (ENNReal.div_le_iff_le_mul (Or.inl h0) (Or.inl hinf)).2 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a \u2264 1 * a\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b / c\n\u22a2 a * c \u2264 b\n[PROOFSTEP]\nrw [\u2190 inv_inv c]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b / c\n\u22a2 a * c\u207b\u00b9\u207b\u00b9 \u2264 b\n[PROOFSTEP]\nexact div_le_of_le_mul h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a < b / c\n\u22a2 a * c < b\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : b \u2264 a * c\n\u22a2 b / c \u2264 a\n[PROOFSTEP]\nexact ENNReal.div_le_of_le_mul h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a < b * c\n\u22a2 a < b / c\u207b\u00b9\n[PROOFSTEP]\nrwa [div_eq_mul_inv, inv_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a < b * c\n\u22a2 a < c * b\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh\u2081 : b = \u22a4 \u2192 a \u2260 0\nh\u2082 : a = \u22a4 \u2192 b \u2260 0\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 1 \u2264 a * b\n[PROOFSTEP]\nrw [\u2190 one_div, ENNReal.div_le_iff_le_mul, mul_comm]\n[GOAL]\ncase hb0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh\u2081 : b = \u22a4 \u2192 a \u2260 0\nh\u2082 : a = \u22a4 \u2192 b \u2260 0\n\u22a2 a \u2260 0 \u2228 b \u2260 \u22a4\ncase hbt \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h\u2081 : b = \u22a4 \u2192 a \u2260 0 h\u2082 : a = \u22a4 \u2192 b \u2260 0 \u22a2 a \u2260 \u22a4 \u2228 b \u2260 0\n[PROOFSTEP]\nexacts [or_not_of_imp h\u2081, not_or_of_imp h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a \u2264 b\u207b\u00b9 \u2194 a * b \u2264 1\n[PROOFSTEP]\nrw [\u2190 one_div, ENNReal.le_div_iff_mul_le]\n[GOAL]\ncase h0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 b \u2260 0 \u2228 1 \u2260 0\n[PROOFSTEP]\nright\n[GOAL]\ncase h0.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 b \u2260 \u22a4 \u2228 1 \u2260 \u22a4\n[PROOFSTEP]\nright\n[GOAL]\ncase ht.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 1 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a * b = 1\n\u22a2 a = b\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_one a, \u2190 ENNReal.mul_inv_cancel (right_ne_zero_of_mul_eq_one h), \u2190 mul_assoc, h, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a * b = 1\n\u22a2 b \u2260 \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a * \u22a4 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp [left_ne_zero_of_mul_eq_one h] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\na b r : \u211d\u22650\u221e\nhr\u2080 : r \u2260 0\nhr\u2081 : r \u2260 \u22a4\n\u22a2 r * a \u2264 b \u2194 a \u2264 r\u207b\u00b9 * b\n[PROOFSTEP]\nrw [\u2190 @ENNReal.mul_le_mul_left _ a _ hr\u2080 hr\u2081, \u2190 mul_assoc, ENNReal.mul_inv_cancel hr\u2080 hr\u2081, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\na b : \u211d\u22650\u221e\nr : \u211d\u22650\nhr\u2080 : r \u2260 0\n\u22a2 a \u2264 r\u207b\u00b9 \u2022 b \u2194 r \u2022 a \u2264 b\n[PROOFSTEP]\nsimpa [hr\u2080, ENNReal.smul_def] using (mul_le_iff_le_inv (coe_ne_zero.mpr hr\u2080) coe_ne_top).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\na b : \u211d\u22650\u221e\nr : \u211d\u22650\nhr\u2080 : r \u2260 0\n\u22a2 r\u207b\u00b9 \u2022 a \u2264 b \u2194 a \u2264 r \u2022 b\n[PROOFSTEP]\nsimpa only [inv_inv] using (ENNReal.le_inv_smul_iff (inv_ne_zero hr\u2080)).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nh : \u2200 (r : \u211d\u22650), \u2191r < x \u2192 \u2191r \u2264 y\n\u22a2 x \u2264 y\n[PROOFSTEP]\nrefine' le_of_forall_ge_of_dense fun r hr => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nh : \u2200 (r : \u211d\u22650), \u2191r < x \u2192 \u2191r \u2264 y\nr : \u211d\u22650\u221e\nhr : r < x\n\u22a2 r \u2264 y\n[PROOFSTEP]\nlift r to \u211d\u22650 using ne_top_of_lt hr\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nh : \u2200 (r : \u211d\u22650), \u2191r < x \u2192 \u2191r \u2264 y\nr : \u211d\u22650\nhr : \u2191r < x\n\u22a2 \u2191r \u2264 y\n[PROOFSTEP]\nexact h r hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nh : b = c / a\n\u22a2 a * b = c\n[PROOFSTEP]\nrw [h, ENNReal.mul_div_cancel' ha ha']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nh : a * b = c\n\u22a2 b = c / a\n[PROOFSTEP]\nrw [\u2190 h, mul_div_assoc, ENNReal.mul_div_cancel' ha ha']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n\u22a2 c / b = d / a \u2194 a * c = b * d\n[PROOFSTEP]\nrw [eq_div_iff ha ha']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n\u22a2 a * (c / b) = d \u2194 a * c = b * d\n[PROOFSTEP]\nconv_rhs => rw [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n| a * c = b * d\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n| a * c = b * d\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n| a * c = b * d\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nha' : a \u2260 \u22a4\nhb : b \u2260 0\nhb' : b \u2260 \u22a4\n\u22a2 a * (c / b) = d \u2194 b * d = a * c\n[PROOFSTEP]\nrw [\u2190 eq_div_iff hb hb', mul_div_assoc, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhb\u2080 : b \u2260 0\nhb\u2081 : b \u2260 \u22a4\nh : a / b = 1\n\u22a2 a = b\n[PROOFSTEP]\nrw [\u2190 (eq_div_iff hb\u2080 hb\u2081).mp h.symm, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 2\u207b\u00b9 + 2\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 two_mul, \u2190 div_eq_mul_inv, ENNReal.div_self two_ne_zero two_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 3\u207b\u00b9 + 3\u207b\u00b9 + 3\u207b\u00b9 = 3 * 3\u207b\u00b9\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 3 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a / 2 + a / 2 = a\n[PROOFSTEP]\nrw [div_eq_mul_inv, \u2190 mul_add, inv_two_add_inv_two, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 a / 3 + a / 3 + a / 3 = a\n[PROOFSTEP]\nrw [div_eq_mul_inv, \u2190 mul_add, \u2190 mul_add, inv_three_add_inv_three, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 a / b = 0 \u2194 a = 0 \u2228 b = \u22a4\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 0 < a / b \u2194 a \u2260 0 \u2227 b \u2260 \u22a4\n[PROOFSTEP]\nsimp [pos_iff_ne_zero, not_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 0\n\u22a2 0 < a / 2\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhz : a \u2260 0\nht : a \u2260 \u22a4\n\u22a2 a / 2 < a\n[PROOFSTEP]\nlift a to \u211d\u22650 using ht\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nhz : \u2191a \u2260 0\n\u22a2 \u2191a / 2 < \u2191a\n[PROOFSTEP]\nrw [coe_ne_zero] at hz \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nhz : a \u2260 0\n\u22a2 \u2191a / 2 < \u2191a\n[PROOFSTEP]\nrw [\u2190 coe_two, \u2190 coe_div, coe_lt_coe]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nhz : a \u2260 0\n\u22a2 a / 2 < a\ncase intro \u03b1 : Type u_1 \u03b2 : Type u_2 b c d : \u211d\u22650\u221e r p q a : \u211d\u22650 hz : a \u2260 0 \u22a2 2 \u2260 0\n[PROOFSTEP]\nexacts [NNReal.half_lt_self hz, two_ne_zero' _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2260 \u22a4\n\u22a2 a - a / 2 = a / 2\n[PROOFSTEP]\nlift a to \u211d\u22650 using h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\n\u22a2 \u2191a - \u2191a / 2 = \u2191a / 2\n[PROOFSTEP]\nexact sub_eq_of_add_eq (mul_ne_top coe_ne_top <| by simp) (ENNReal.add_halves a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\n\u22a2 2\u207b\u00b9 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 1 - 2\u207b\u00b9 = 2\u207b\u00b9\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv, one_mul] using sub_half one_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u211d\u22650\u221e \u2243o \u2191(Iic 1)\n[PROOFSTEP]\nrefine\n StrictMono.orderIsoOfRightInverse (fun x => \u27e8(x\u207b\u00b9 + 1)\u207b\u00b9, ENNReal.inv_le_one.2 <| le_add_self\u27e9) (fun x y hxy => ?_)\n (fun x => (x.1\u207b\u00b9 - 1)\u207b\u00b9) fun x => Subtype.ext ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nhxy : x < y\n\u22a2 (fun x => { val := (x\u207b\u00b9 + 1)\u207b\u00b9, property := (_ : (x\u207b\u00b9 + 1)\u207b\u00b9 \u2264 1) }) x <\n (fun x => { val := (x\u207b\u00b9 + 1)\u207b\u00b9, property := (_ : (x\u207b\u00b9 + 1)\u207b\u00b9 \u2264 1) }) y\n[PROOFSTEP]\nsimpa only [Subtype.mk_lt_mk, ENNReal.inv_lt_inv, ENNReal.add_lt_add_iff_right one_ne_top]\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u2191(Iic 1)\n\u22a2 \u2191((fun x => { val := (x\u207b\u00b9 + 1)\u207b\u00b9, property := (_ : (x\u207b\u00b9 + 1)\u207b\u00b9 \u2264 1) }) ((fun x => ((\u2191x)\u207b\u00b9 - 1)\u207b\u00b9) x)) = \u2191x\n[PROOFSTEP]\nhave : (1 : \u211d\u22650\u221e) \u2264 x.1\u207b\u00b9 := ENNReal.one_le_inv.2 x.2\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u2191(Iic 1)\nthis : 1 \u2264 (\u2191x)\u207b\u00b9\n\u22a2 \u2191((fun x => { val := (x\u207b\u00b9 + 1)\u207b\u00b9, property := (_ : (x\u207b\u00b9 + 1)\u207b\u00b9 \u2264 1) }) ((fun x => ((\u2191x)\u207b\u00b9 - 1)\u207b\u00b9) x)) = \u2191x\n[PROOFSTEP]\nsimp only [inv_inv, Subtype.coe_mk, tsub_add_cancel_of_le this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nx\u271d\u00b9 x\u271d : \u2191(Iic a)\n\u22a2 \u2191{ toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) },\n invFun := fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iic a)),\n (fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) })\n ((fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) }) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Iic \u2191a)),\n (fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) })\n ((fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) }) x) =\n x) }\n x\u271d\u00b9 \u2264\n \u2191{ toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) },\n invFun := fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iic a)),\n (fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) })\n ((fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) }) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Iic \u2191a)),\n (fun x => { val := \u2191\u2191x, property := (_ : \u2191\u2191x \u2264 \u2191a) })\n ((fun x => { val := ENNReal.toNNReal \u2191x, property := (_ : ENNReal.toNNReal \u2191x \u2264 a) }) x) =\n x) }\n x\u271d \u2194\n x\u271d\u00b9 \u2264 x\u271d\n[PROOFSTEP]\nsimp only [Equiv.coe_fn_mk, Subtype.mk_le_mk, coe_le_coe, Subtype.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : a \u2260 0\n\u22a2 \u2203 n, (\u2191n)\u207b\u00b9 < a\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nsimp only [ENNReal.inv_lt_inv, ENNReal.exists_nat_gt (inv_ne_top.2 h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nhb : b \u2260 \u22a4\nn : \u2115\nhn : b / a < \u2191n\n\u22a2 b < \u2191n * a\n[PROOFSTEP]\nrwa [\u2190 ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 \u2203 n, n > 0 \u2227 (\u2191n)\u207b\u00b9 * a < b\n[PROOFSTEP]\nrcases exists_nat_pos_mul_gt hb ha with \u27e8n, npos, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : n > 0\nhn : a < \u2191n * b\n\u22a2 \u2203 n, n > 0 \u2227 (\u2191n)\u207b\u00b9 * a < b\n[PROOFSTEP]\nuse n, npos\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : n > 0\nhn : a < \u2191n * b\n\u22a2 (\u2191n)\u207b\u00b9 * a < b\n[PROOFSTEP]\nrw [\u2190 ENNReal.div_eq_inv_mul]\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : n > 0\nhn : a < \u2191n * b\n\u22a2 a / \u2191n < b\n[PROOFSTEP]\nexact div_lt_of_lt_mul' hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 \u2203 n, n > 0 \u2227 \u2191n * a < b\n[PROOFSTEP]\nrcases exists_nat_pos_inv_mul_lt ha hb with \u27e8n, npos : 0 < n, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 \u2203 n, n > 0 \u2227 \u2191n * a < b\n[PROOFSTEP]\nuse(n : \u211d\u22650)\u207b\u00b9\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 (\u2191n)\u207b\u00b9 > 0 \u2227 \u2191(\u2191n)\u207b\u00b9 * a < b\n[PROOFSTEP]\nsimp [*, npos.ne', zero_lt_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\n\u22a2 \u2203 n, 2\u207b\u00b9 ^ n < a\n[PROOFSTEP]\nrcases exists_inv_nat_lt ha with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2203 n, 2\u207b\u00b9 ^ n < a\n[PROOFSTEP]\nrefine' \u27e8n, lt_trans _ hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 2\u207b\u00b9 ^ n < (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 ENNReal.inv_pow, ENNReal.inv_lt_inv]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2191n < 2 ^ n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 n < 2 ^ n\n[PROOFSTEP]\nexact n.lt_two_pow\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\nn : \u2124\n\u22a2 \u2191(r ^ n) = \u2191r ^ n\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\nn : \u2115\n\u22a2 \u2191(r ^ Int.ofNat n) = \u2191r ^ Int.ofNat n\n[PROOFSTEP]\nsimp only [Int.ofNat_eq_coe, coe_pow, zpow_ofNat]\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\nn : \u2115\n\u22a2 \u2191(r ^ Int.negSucc n) = \u2191r ^ Int.negSucc n\n[PROOFSTEP]\nhave : r ^ n.succ \u2260 0 := pow_ne_zero (n + 1) hr\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : r \u2260 0\nn : \u2115\nthis : r ^ Nat.succ n \u2260 0\n\u22a2 \u2191(r ^ Int.negSucc n) = \u2191r ^ Int.negSucc n\n[PROOFSTEP]\nsimp only [zpow_negSucc, coe_inv this, coe_pow]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\nn : \u2124\n\u22a2 0 < a ^ n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\na\u271d : \u2115\n\u22a2 0 < a ^ Int.ofNat a\u271d\n[PROOFSTEP]\nexact ENNReal.pow_pos ha.bot_lt _\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\na\u271d : \u2115\n\u22a2 0 < a ^ Int.negSucc a\u271d\n[PROOFSTEP]\nsimp only [h'a, pow_eq_top_iff, zpow_negSucc, Ne.def, not_false, ENNReal.inv_pos, false_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\nn : \u2124\n\u22a2 a ^ n < \u22a4\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\na\u271d : \u2115\n\u22a2 a ^ Int.ofNat a\u271d < \u22a4\n[PROOFSTEP]\nexact ENNReal.pow_lt_top h'a.lt_top _\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nh'a : a \u2260 \u22a4\na\u271d : \u2115\n\u22a2 a ^ Int.negSucc a\u271d < \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.pow_pos ha.bot_lt, zpow_negSucc, inv_lt_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nhx : x \u2260 0\nh'x : x \u2260 \u22a4\nhy : 1 < y\nh'y : y \u2260 \u22a4\n\u22a2 \u2203 n, x \u2208 Ico (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\nlift x to \u211d\u22650 using h'x\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\nhx : \u2191x \u2260 0\n\u22a2 \u2203 n, \u2191x \u2208 Ico (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\nlift y to \u211d\u22650 using h'y\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 \u2203 n, \u2191x \u2208 Ico (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nhave A : y \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 y \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 \u2203 n, \u2191x \u2208 Ico (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nobtain \u27e8n, hn, h'n\u27e9 : \u2203 n : \u2124, y ^ n \u2264 x \u2227 x < y ^ (n + 1) :=\n by\n refine' NNReal.exists_mem_Ico_zpow _ (one_lt_coe_iff.1 hy)\n simpa only [Ne.def, coe_eq_zero] using hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 \u2203 n, y ^ n \u2264 x \u2227 x < y ^ (n + 1)\n[PROOFSTEP]\nrefine' NNReal.exists_mem_Ico_zpow _ (one_lt_coe_iff.1 hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 x \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using hx\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 \u2203 n, \u2191x \u2208 Ico (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nrefine' \u27e8n, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 \u2191y ^ n \u2264 \u2191x\n[PROOFSTEP]\nrwa [\u2190 ENNReal.coe_zpow A, ENNReal.coe_le_coe]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 \u2191x < \u2191y ^ (n + 1)\n[PROOFSTEP]\nrwa [\u2190 ENNReal.coe_zpow A, ENNReal.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\u22650\u221e\nhx : x \u2260 0\nh'x : x \u2260 \u22a4\nhy : 1 < y\nh'y : y \u2260 \u22a4\n\u22a2 \u2203 n, x \u2208 Ioc (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\nlift x to \u211d\u22650 using h'x\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\nhx : \u2191x \u2260 0\n\u22a2 \u2203 n, \u2191x \u2208 Ioc (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\nlift y to \u211d\u22650 using h'y\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 \u2203 n, \u2191x \u2208 Ioc (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nhave A : y \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 y \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using (zero_lt_one.trans hy).ne'\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 \u2203 n, \u2191x \u2208 Ioc (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nobtain \u27e8n, hn, h'n\u27e9 : \u2203 n : \u2124, y ^ n < x \u2227 x \u2264 y ^ (n + 1) :=\n by\n refine' NNReal.exists_mem_Ioc_zpow _ (one_lt_coe_iff.1 hy)\n simpa only [Ne.def, coe_eq_zero] using hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 \u2203 n, y ^ n < x \u2227 x \u2264 y ^ (n + 1)\n[PROOFSTEP]\nrefine' NNReal.exists_mem_Ioc_zpow _ (one_lt_coe_iff.1 hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\n\u22a2 x \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using hx\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n < x\nh'n : x \u2264 y ^ (n + 1)\n\u22a2 \u2203 n, \u2191x \u2208 Ioc (\u2191y ^ n) (\u2191y ^ (n + 1))\n[PROOFSTEP]\nrefine' \u27e8n, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n < x\nh'n : x \u2264 y ^ (n + 1)\n\u22a2 \u2191y ^ n < \u2191x\n[PROOFSTEP]\nrwa [\u2190 ENNReal.coe_zpow A, ENNReal.coe_lt_coe]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x : \u211d\u22650\nhx : \u2191x \u2260 0\ny : \u211d\u22650\nhy : 1 < \u2191y\nA : y \u2260 0\nn : \u2124\nhn : y ^ n < x\nh'n : x \u2264 y ^ (n + 1)\n\u22a2 \u2191x \u2264 \u2191y ^ (n + 1)\n[PROOFSTEP]\nrwa [\u2190 ENNReal.coe_zpow A, ENNReal.coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\n\u22a2 Ioo 0 \u22a4 = \u22c3 (n : \u2124), Ico (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\n\u22a2 x \u2208 Ioo 0 \u22a4 \u2194 x \u2208 \u22c3 (n : \u2124), Ico (y ^ n) (y ^ (n + 1))\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_Ioo, mem_Ico]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\n\u22a2 0 < x \u2227 x < \u22a4 \u2194 \u2203 i, y ^ i \u2264 x \u2227 x < y ^ (i + 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\n\u22a2 0 < x \u2227 x < \u22a4 \u2192 \u2203 i, y ^ i \u2264 x \u2227 x < y ^ (i + 1)\n[PROOFSTEP]\nrintro \u27e8hx, h'x\u27e9\n[GOAL]\ncase h.mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nhx : 0 < x\nh'x : x < \u22a4\n\u22a2 \u2203 i, y ^ i \u2264 x \u2227 x < y ^ (i + 1)\n[PROOFSTEP]\nexact exists_mem_Ico_zpow hx.ne' h'x.ne hy h'y\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\n\u22a2 (\u2203 i, y ^ i \u2264 x \u2227 x < y ^ (i + 1)) \u2192 0 < x \u2227 x < \u22a4\n[PROOFSTEP]\nrintro \u27e8n, hn, h'n\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 0 < x \u2227 x < \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mpr.intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 0 < x\n[PROOFSTEP]\napply lt_of_lt_of_le _ hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 0 < y ^ n\n[PROOFSTEP]\nexact ENNReal.zpow_pos (zero_lt_one.trans hy).ne' h'y _\n[GOAL]\ncase h.mpr.intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 x < \u22a4\n[PROOFSTEP]\napply lt_trans h'n _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ny : \u211d\u22650\u221e\nhy : 1 < y\nh'y : y \u2260 \u22a4\nx : \u211d\u22650\u221e\nn : \u2124\nhn : y ^ n \u2264 x\nh'n : x < y ^ (n + 1)\n\u22a2 y ^ (n + 1) < \u22a4\n[PROOFSTEP]\nexact ENNReal.zpow_lt_top (zero_lt_one.trans hy).ne' h'y _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na b : \u2124\nh : a \u2264 b\n\u22a2 x ^ a \u2264 x ^ b\n[PROOFSTEP]\ninduction' a with a a\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b : \u2124\nh\u271d : a\u271d \u2264 b\na : \u2115\nh : Int.ofNat a \u2264 b\n\u22a2 x ^ Int.ofNat a \u2264 x ^ b\n[PROOFSTEP]\ninduction' b with b b\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b : \u2124\nh\u271d : a\u271d \u2264 b\na : \u2115\nh : Int.negSucc a \u2264 b\n\u22a2 x ^ Int.negSucc a \u2264 x ^ b\n[PROOFSTEP]\ninduction' b with b b\n[GOAL]\ncase ofNat.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.ofNat a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.ofNat a \u2264 Int.ofNat b\n\u22a2 x ^ Int.ofNat a \u2264 x ^ Int.ofNat b\n[PROOFSTEP]\nsimp only [Int.ofNat_eq_coe, zpow_ofNat]\n[GOAL]\ncase ofNat.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.ofNat a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.ofNat a \u2264 Int.ofNat b\n\u22a2 x ^ a \u2264 x ^ b\n[PROOFSTEP]\nexact pow_le_pow hx (Int.le_of_ofNat_le_ofNat h)\n[GOAL]\ncase ofNat.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.ofNat a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.negSucc b\nh : Int.ofNat a \u2264 Int.negSucc b\n\u22a2 x ^ Int.ofNat a \u2264 x ^ Int.negSucc b\n[PROOFSTEP]\napply absurd h (not_le_of_gt _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.ofNat a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.negSucc b\nh : Int.ofNat a \u2264 Int.negSucc b\n\u22a2 Int.ofNat a > Int.negSucc b\n[PROOFSTEP]\nexact lt_of_lt_of_le (Int.negSucc_lt_zero _) (Int.ofNat_nonneg _)\n[GOAL]\ncase negSucc.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.negSucc a \u2264 Int.ofNat b\n\u22a2 x ^ Int.negSucc a \u2264 x ^ Int.ofNat b\n[PROOFSTEP]\nsimp only [zpow_negSucc, Int.ofNat_eq_coe, zpow_ofNat]\n[GOAL]\ncase negSucc.ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.negSucc a \u2264 Int.ofNat b\n\u22a2 (x ^ (a + 1))\u207b\u00b9 \u2264 x ^ b\n[PROOFSTEP]\nrefine' (ENNReal.inv_le_one.2 _).trans _\n[GOAL]\ncase negSucc.ofNat.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.negSucc a \u2264 Int.ofNat b\n\u22a2 1 \u2264 x ^ (a + 1)\n[PROOFSTEP]\nexact one_le_pow_of_one_le' hx _\n[GOAL]\ncase negSucc.ofNat.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.ofNat b\nh : Int.negSucc a \u2264 Int.ofNat b\n\u22a2 1 \u2264 x ^ b\n[PROOFSTEP]\nexact one_le_pow_of_one_le' hx _\n[GOAL]\ncase negSucc.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.negSucc b\nh : Int.negSucc a \u2264 Int.negSucc b\n\u22a2 x ^ Int.negSucc a \u2264 x ^ Int.negSucc b\n[PROOFSTEP]\nsimp only [zpow_negSucc, ENNReal.inv_le_inv]\n[GOAL]\ncase negSucc.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.negSucc b\nh : Int.negSucc a \u2264 Int.negSucc b\n\u22a2 x ^ (b + 1) \u2264 x ^ (a + 1)\n[PROOFSTEP]\napply pow_le_pow hx\n[GOAL]\ncase negSucc.negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d\u00b9 b\u271d\u00b9 c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : 1 \u2264 x\na\u271d b\u271d : \u2124\nh\u271d\u00b2 : a\u271d \u2264 b\u271d\na : \u2115\nh\u271d\u00b9 : Int.negSucc a \u2264 b\u271d\nb : \u2115\nh\u271d : a\u271d \u2264 Int.negSucc b\nh : Int.negSucc a \u2264 Int.negSucc b\n\u22a2 b + 1 \u2264 a + 1\n[PROOFSTEP]\nsimpa only [\u2190 Int.ofNat_le, neg_le_neg_iff, Int.ofNat_add, Int.ofNat_one, Int.negSucc_eq] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\u22650\u221e\nhx : x \u2260 0\nh'x : x \u2260 \u22a4\nm n : \u2124\n\u22a2 x ^ (m + n) = x ^ m * x ^ n\n[PROOFSTEP]\nlift x to \u211d\u22650 using h'x\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nm n : \u2124\nx : \u211d\u22650\nhx : \u2191x \u2260 0\n\u22a2 \u2191x ^ (m + n) = \u2191x ^ m * \u2191x ^ n\n[PROOFSTEP]\nreplace hx : x \u2260 0\n[GOAL]\ncase hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nm n : \u2124\nx : \u211d\u22650\nhx : \u2191x \u2260 0\n\u22a2 x \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, coe_eq_zero] using hx\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nm n : \u2124\nx : \u211d\u22650\nhx : x \u2260 0\n\u22a2 \u2191x ^ (m + n) = \u2191x ^ m * \u2191x ^ n\n[PROOFSTEP]\nsimp only [\u2190 coe_zpow hx, zpow_add\u2080 hx, coe_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.toReal (a + b) = ENNReal.toReal a + ENNReal.toReal b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 ENNReal.toReal (\u2191a + b) = ENNReal.toReal \u2191a + ENNReal.toReal b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 ENNReal.toReal (\u2191a + \u2191b) = ENNReal.toReal \u2191a + ENNReal.toReal \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nh : b \u2264 a\nha : a \u2260 \u22a4\n\u22a2 ENNReal.toReal (a - b) = ENNReal.toReal a - ENNReal.toReal b\n[PROOFSTEP]\nlift b to \u211d\u22650 using ne_top_of_le_ne_top ha h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nha : a \u2260 \u22a4\nb : \u211d\u22650\nh : \u2191b \u2264 a\n\u22a2 ENNReal.toReal (a - \u2191b) = ENNReal.toReal a - ENNReal.toReal \u2191b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q b a : \u211d\u22650\nh : \u2191b \u2264 \u2191a\n\u22a2 ENNReal.toReal (\u2191a - \u2191b) = ENNReal.toReal \u2191a - ENNReal.toReal \u2191b\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.coe_sub, ENNReal.coe_toReal, NNReal.coe_sub (ENNReal.coe_le_coe.mp h)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.toReal a - ENNReal.toReal b \u2264 ENNReal.toReal (a - b)\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\u22650\n\u22a2 ENNReal.toReal a - ENNReal.toReal \u2191b \u2264 ENNReal.toReal (a - \u2191b)\n[PROOFSTEP]\ninduction a using recTopCoe\n[GOAL]\ncase intro.top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q b : \u211d\u22650\n\u22a2 ENNReal.toReal \u22a4 - ENNReal.toReal \u2191b \u2264 ENNReal.toReal (\u22a4 - \u2191b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q b x\u271d : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191x\u271d - ENNReal.toReal \u2191b \u2264 ENNReal.toReal (\u2191x\u271d - \u2191b)\n[PROOFSTEP]\nsimp only [\u2190 coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]\n[GOAL]\ncase intro.coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b\u271d c d : \u211d\u22650\u221e\nr p q b x\u271d : \u211d\u22650\n\u22a2 \u2191x\u271d - \u2191b \u2264 max (\u2191x\u271d - \u2191b) 0\n[PROOFSTEP]\nexact le_max_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a = \u22a4\n\u22a2 ENNReal.toReal (a + b) \u2264 ENNReal.toReal a + ENNReal.toReal b\n[PROOFSTEP]\nsimp only [ha, top_add, top_toReal, zero_add, toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : \u00aca = \u22a4\nhb : b = \u22a4\n\u22a2 ENNReal.toReal (a + b) \u2264 ENNReal.toReal a + ENNReal.toReal b\n[PROOFSTEP]\nsimp only [hb, add_top, top_toReal, add_zero, toReal_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhp : 0 \u2264 p\nhq : 0 \u2264 q\n\u22a2 ENNReal.ofReal (p + q) = ENNReal.ofReal p + ENNReal.ofReal q\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, ENNReal.ofReal, \u2190 coe_add, coe_eq_coe, Real.toNNReal_add hp hq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b \u2194 a \u2264 b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a \u2264 ENNReal.toReal b \u2194 \u2191a \u2264 b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a \u2264 ENNReal.toReal \u2191b \u2194 \u2191a \u2264 \u2191b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b\nht : b = \u22a4 \u2192 a = \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b\n[PROOFSTEP]\nrcases eq_or_ne a \u221e with rfl | ha\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : \u22a4 \u2264 b\nht : b = \u22a4 \u2192 \u22a4 = \u22a4\n\u22a2 ENNReal.toReal \u22a4 \u2264 ENNReal.toReal b\n[PROOFSTEP]\nexact toReal_nonneg\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b\nht : b = \u22a4 \u2192 a = \u22a4\nha : a \u2260 \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b\n[PROOFSTEP]\nexact toReal_mono (mt ht ha) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.toReal a < ENNReal.toReal b \u2194 a < b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a < ENNReal.toReal b \u2194 \u2191a < b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a < ENNReal.toReal \u2191b \u2194 \u2191a < \u2191b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhle : a \u2264 b + c\nhb : b = \u22a4 \u2192 a = \u22a4\nhc : c = \u22a4 \u2192 a = \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b + ENNReal.toReal c\n[PROOFSTEP]\nrefine le_trans (toReal_mono' hle ?_) toReal_add_le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhle : a \u2264 b + c\nhb : b = \u22a4 \u2192 a = \u22a4\nhc : c = \u22a4 \u2192 a = \u22a4\n\u22a2 b + c = \u22a4 \u2192 a = \u22a4\n[PROOFSTEP]\nsimpa only [add_eq_top, or_imp] using And.intro hb hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\nh : ENNReal.toNNReal a \u2264 ENNReal.toNNReal b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrwa [\u2190 coe_toNNReal ha, \u2190 coe_toNNReal hb, coe_le_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\nh : a < b\n\u22a2 ENNReal.toNNReal a < ENNReal.toNNReal b\n[PROOFSTEP]\nsimpa [\u2190 ENNReal.coe_lt_coe, hb, h.ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\nh : ENNReal.toNNReal a < ENNReal.toNNReal b\n\u22a2 a < b\n[PROOFSTEP]\nrwa [\u2190 coe_toNNReal ha, \u2190 coe_toNNReal hb, coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : a \u2260 \u22a4\nhp : b \u2260 \u22a4\nh : a \u2264 b\n\u22a2 ENNReal.toReal (max a b) = max (ENNReal.toReal a) (ENNReal.toReal b)\n[PROOFSTEP]\nsimp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, max_eq_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhr : a \u2260 \u22a4\nhp : b \u2260 \u22a4\nh : b \u2264 a\n\u22a2 ENNReal.toReal (max a b) = max (ENNReal.toReal a) (ENNReal.toReal b)\n[PROOFSTEP]\nsimp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, max_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhr : a \u2260 \u22a4\nhp : b \u2260 \u22a4\nh : a \u2264 b\n\u22a2 ENNReal.toReal (min a b) = min (ENNReal.toReal a) (ENNReal.toReal b)\n[PROOFSTEP]\nsimp only [h, (ENNReal.toReal_le_toReal hr hp).2 h, min_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\nhr : a \u2260 \u22a4\nhp : b \u2260 \u22a4\nh : b \u2264 a\n\u22a2 ENNReal.toReal (min a b) = min (ENNReal.toReal a) (ENNReal.toReal b)\n[PROOFSTEP]\nsimp only [h, (ENNReal.toReal_le_toReal hp hr).2 h, min_eq_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 0 < ENNReal.toNNReal a \u2194 0 < a \u2227 a < \u22a4\n[PROOFSTEP]\ninduction a using recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 0 < ENNReal.toNNReal \u22a4 \u2194 0 < \u22a4 \u2227 \u22a4 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q x\u271d : \u211d\u22650\n\u22a2 0 < ENNReal.toNNReal \u2191x\u271d \u2194 0 < \u2191x\u271d \u2227 \u2191x\u271d < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nh : p \u2264 q\n\u22a2 ENNReal.ofReal p \u2264 ENNReal.ofReal q\n[PROOFSTEP]\nsimp [ENNReal.ofReal, Real.toNNReal_le_toNNReal h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nh : 0 \u2264 q\n\u22a2 ENNReal.ofReal p \u2264 ENNReal.ofReal q \u2194 p \u2264 q\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, coe_le_coe, Real.toNNReal_le_toNNReal_iff h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhp : 0 \u2264 p\nhq : 0 \u2264 q\n\u22a2 ENNReal.ofReal p = ENNReal.ofReal q \u2194 p = q\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, coe_eq_coe, Real.toNNReal_eq_toNNReal_iff hp hq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nh : 0 < q\n\u22a2 ENNReal.ofReal p < ENNReal.ofReal q \u2194 p < q\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhp : 0 \u2264 p\n\u22a2 ENNReal.ofReal p < ENNReal.ofReal q \u2194 p < q\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff_of_nonneg hp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\n\u22a2 0 < ENNReal.ofReal p \u2194 0 < p\n[PROOFSTEP]\nsimp [ENNReal.ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\n\u22a2 ENNReal.ofReal p = 0 \u2194 p \u2264 0\n[PROOFSTEP]\nsimp [ENNReal.ofReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhq : 0 \u2264 q\n\u22a2 ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q\n[PROOFSTEP]\nobtain h | h := le_total p q\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhq : 0 \u2264 q\nh : p \u2264 q\n\u22a2 ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q\n[PROOFSTEP]\nrw [ofReal_of_nonpos (sub_nonpos_of_le h), tsub_eq_zero_of_le (ofReal_le_ofReal h)]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhq : 0 \u2264 q\nh : q \u2264 p\n\u22a2 ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q\n[PROOFSTEP]\nrefine' ENNReal.eq_sub_of_add_eq ofReal_ne_top _\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhq : 0 \u2264 q\nh : q \u2264 p\n\u22a2 ENNReal.ofReal (p - q) + ENNReal.ofReal q = ENNReal.ofReal p\n[PROOFSTEP]\nrw [\u2190 ofReal_add (sub_nonneg_of_le h) hq, sub_add_cancel]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nb : \u211d\u22650\u221e\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.ofReal a \u2264 b \u2194 a \u2264 ENNReal.toReal b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nb : \u211d\u22650\n\u22a2 ENNReal.ofReal a \u2264 \u2191b \u2194 a \u2264 ENNReal.toReal \u2191b\n[PROOFSTEP]\nsimpa [ENNReal.ofReal, ENNReal.toReal] using Real.toNNReal_le_iff_le_coe\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nb : \u211d\u22650\u221e\nha : 0 \u2264 a\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.ofReal a < b \u2194 a < ENNReal.toReal b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\nha : 0 \u2264 a\nb : \u211d\u22650\n\u22a2 ENNReal.ofReal a < \u2191b \u2194 a < ENNReal.toReal \u2191b\n[PROOFSTEP]\nsimpa [ENNReal.ofReal, ENNReal.toReal] using Real.toNNReal_lt_iff_lt_coe ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\nha : a \u2260 \u22a4\nhb : 0 \u2264 b\n\u22a2 a \u2264 ENNReal.ofReal b \u2194 ENNReal.toReal a \u2264 b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\nhb : 0 \u2264 b\na : \u211d\u22650\n\u22a2 \u2191a \u2264 ENNReal.ofReal b \u2194 ENNReal.toReal \u2191a \u2264 b\n[PROOFSTEP]\nsimpa [ENNReal.ofReal, ENNReal.toReal] using Real.le_toNNReal_iff_coe_le hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\nha : a \u2260 \u22a4\n\u22a2 a < ENNReal.ofReal b \u2194 ENNReal.toReal a < b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\na : \u211d\u22650\n\u22a2 \u2191a < ENNReal.ofReal b \u2194 ENNReal.toReal \u2191a < b\n[PROOFSTEP]\nsimpa [ENNReal.ofReal, ENNReal.toReal] using Real.lt_toNNReal_iff_coe_lt\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhp : 0 \u2264 p\n\u22a2 ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, \u2190 coe_mul, Real.toNNReal_mul hp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\nhq : 0 \u2264 q\n\u22a2 ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q\n[PROOFSTEP]\nrw [mul_comm, ofReal_mul hq, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\nhp : 0 \u2264 p\nn : \u2115\n\u22a2 ENNReal.ofReal (p ^ n) = ENNReal.ofReal p ^ n\n[PROOFSTEP]\nrw [ofReal_eq_coe_nnreal hp, \u2190 coe_pow, \u2190 ofReal_coe_nnreal, NNReal.coe_pow, NNReal.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\nn : \u2115\n\u22a2 ENNReal.ofReal (n \u2022 x) = n \u2022 ENNReal.ofReal x\n[PROOFSTEP]\nsimp only [nsmul_eq_mul, \u2190 ofReal_coe_nat n, \u2190 ofReal_mul n.cast_nonneg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\nhx : 0 < x\n\u22a2 (ENNReal.ofReal x)\u207b\u00b9 = ENNReal.ofReal x\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.ofReal, ENNReal.ofReal, \u2190 @coe_inv (Real.toNNReal x) (by simp [hx]), coe_eq_coe, \u2190 Real.toNNReal_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx : \u211d\nhx : 0 < x\n\u22a2 Real.toNNReal x \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y : \u211d\nhy : 0 < y\n\u22a2 ENNReal.ofReal (x / y) = ENNReal.ofReal x / ENNReal.ofReal y\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, ofReal_mul' (inv_nonneg.2 hy.le), ofReal_inv_of_pos hy]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal (a * \u22a4) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal (\u22a4 * a) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nb : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal (a \u2022 b) = a * ENNReal.toNNReal b\n[PROOFSTEP]\nchange ((a : \u211d\u22650\u221e) * b).toNNReal = a * b.toNNReal\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nb : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal (\u2191a * b) = a * ENNReal.toNNReal b\n[PROOFSTEP]\nsimp only [ENNReal.toNNReal_mul, ENNReal.toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nn : \u2115\n\u22a2 ENNReal.toReal (n \u2022 a) = n \u2022 ENNReal.toReal a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nc : \u211d\na : \u211d\u22650\u221e\nh : 0 \u2264 c\n\u22a2 ENNReal.toReal (ENNReal.ofReal c * a) = c * ENNReal.toReal a\n[PROOFSTEP]\nrw [ENNReal.toReal_mul, ENNReal.toReal_ofReal h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal (a * \u22a4) = 0\n[PROOFSTEP]\nrw [toReal_mul, top_toReal, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal (\u22a4 * a) = 0\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal (a * \u22a4) = 0\n[PROOFSTEP]\nexact toReal_mul_top _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\n\u22a2 ENNReal.toReal a = ENNReal.toReal b \u2194 a = b\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nhb : b \u2260 \u22a4\na : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a = ENNReal.toReal b \u2194 \u2191a = b\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nc d : \u211d\u22650\u221e\nr p q a b : \u211d\u22650\n\u22a2 ENNReal.toReal \u2191a = ENNReal.toReal \u2191b \u2194 \u2191a = \u2191b\n[PROOFSTEP]\nsimp only [coe_eq_coe, NNReal.coe_eq, coe_toReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\ns : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal (r \u2022 s) = r \u2022 ENNReal.toReal s\n[PROOFSTEP]\nrw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\ns : \u211d\u22650\u221e\n\u22a2 \u2191r * ENNReal.toReal s = r \u2022 ENNReal.toReal s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\n\u22a2 p = 0 \u2228 p = \u22a4 \u2228 0 < ENNReal.toReal p\n[PROOFSTEP]\nsimpa only [or_iff_not_imp_left] using toReal_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\n\u22a2 p = 0 \u2227 q = 0 \u2228\n p = 0 \u2227 q = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nrcases eq_or_lt_of_le (bot_le : 0 \u2264 p) with ((rfl : 0 = p) | (hp : 0 < p))\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q\u271d : \u211d\u22650\nq : \u211d\u22650\u221e\nhpq : 0 \u2264 q\n\u22a2 0 = 0 \u2227 q = 0 \u2228\n 0 = 0 \u2227 q = \u22a4 \u2228\n 0 = 0 \u2227 0 < ENNReal.toReal q \u2228\n 0 = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal 0 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal 0 \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal 0 \u2264 ENNReal.toReal q\n[PROOFSTEP]\nsimpa using q.trichotomy\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\n\u22a2 p = 0 \u2227 q = 0 \u2228\n p = 0 \u2227 q = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nrcases eq_or_lt_of_le (le_top : q \u2264 \u221e) with (rfl | hq)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\nhp : 0 < p\nhpq : p \u2264 \u22a4\n\u22a2 p = 0 \u2227 \u22a4 = 0 \u2228\n p = 0 \u2227 \u22a4 = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal \u22a4 \u2228\n p = \u22a4 \u2227 \u22a4 = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 \u22a4 = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal \u22a4 \u2227 ENNReal.toReal p \u2264 ENNReal.toReal \u22a4\n[PROOFSTEP]\nsimpa using p.trichotomy\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n p = 0 \u2227 q = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nrepeat' right\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n p = 0 \u2227 q = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = \u22a4 \u2228\n p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 0 < ENNReal.toReal q \u2228\n p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = \u22a4 \u2227 q = \u22a4 \u2228\n 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h.h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inr.h.h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nhave hq' : 0 < q := lt_of_lt_of_le hp hpq\n[GOAL]\ncase inr.inr.h.h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nhave hp' : p < \u221e := lt_of_le_of_lt hpq hq\n[GOAL]\ncase inr.inr.h.h.h.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\nhp' : p < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q\n[PROOFSTEP]\nsimp [ENNReal.toReal_le_toReal hp'.ne hq.ne, ENNReal.toReal_pos_iff, hpq, hp, hp', hq', hq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 p = \u22a4 \u2228 0 < ENNReal.toReal p \u2227 1 \u2264 ENNReal.toReal p\n[PROOFSTEP]\nsimpa using ENNReal.trichotomy\u2082 (Fact.out : 1 \u2264 p)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal a\u207b\u00b9 = (ENNReal.toNNReal a)\u207b\u00b9\n[PROOFSTEP]\ninduction' a using recTopCoe with a\n[GOAL]\ncase top\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 ENNReal.toNNReal \u22a4\u207b\u00b9 = (ENNReal.toNNReal \u22a4)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase coe\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2191a)\u207b\u00b9 = (ENNReal.toNNReal \u2191a)\u207b\u00b9\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase coe.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u21910)\u207b\u00b9 = (ENNReal.toNNReal \u21910)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase coe.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q a : \u211d\u22650\nha : a \u2260 0\n\u22a2 ENNReal.toNNReal (\u2191a)\u207b\u00b9 = (ENNReal.toNNReal \u2191a)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 coe_inv ha, toNNReal_coe, toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 ENNReal.toNNReal (a / b) = ENNReal.toNNReal a / ENNReal.toNNReal b\n[PROOFSTEP]\nrw [div_eq_mul_inv, toNNReal_mul, toNNReal_inv, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal a\u207b\u00b9 = (ENNReal.toReal a)\u207b\u00b9\n[PROOFSTEP]\nsimp only [ENNReal.toReal, toNNReal_inv, NNReal.coe_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na b : \u211d\u22650\u221e\n\u22a2 ENNReal.toReal (a / b) = ENNReal.toReal a / ENNReal.toReal b\n[PROOFSTEP]\nrw [div_eq_mul_inv, toReal_mul, toReal_inv, div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (i : \u03b1), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 ENNReal.ofReal (\u220f i in s, f i) = \u220f i in s, ENNReal.ofReal (f i)\n[PROOFSTEP]\nsimp_rw [ENNReal.ofReal, \u2190 coe_finset_prod, coe_eq_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (i : \u03b1), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 Real.toNNReal (\u220f i in s, f i) = \u220f a in s, Real.toNNReal (f a)\n[PROOFSTEP]\nexact Real.toNNReal_prod_of_nonneg hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (iInf f) = \u2a05 (i : \u03b9), ENNReal.toNNReal (f i)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\nh\u271d : IsEmpty \u03b9\n\u22a2 ENNReal.toNNReal (iInf f) = \u2a05 (i : \u03b9), ENNReal.toNNReal (f i)\n[PROOFSTEP]\nrw [iInf_of_empty, top_toNNReal, NNReal.iInf_empty]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\nh\u271d : Nonempty \u03b9\n\u22a2 ENNReal.toNNReal (iInf f) = \u2a05 (i : \u03b9), ENNReal.toNNReal (f i)\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\ng : \u03b9 \u2192 \u211d\u22650\u221e\nh\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2a05 (i : \u03b9), \u2191(f i)) = \u2a05 (i : \u03b9), ENNReal.toNNReal ((fun i => \u2191(f i)) i)\n[PROOFSTEP]\nsimp_rw [\u2190 coe_iInf, toNNReal_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhs : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)\n[PROOFSTEP]\nhave hf : \u2200 i, ((\u2191) : s \u2192 \u211d\u22650\u221e) i \u2260 \u221e := fun \u27e8r, rs\u27e9 => hs r rs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhs : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\nhf : \u2200 (i : { x // x \u2208 s }), \u2191i \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (sInf s) = sInf (ENNReal.toNNReal '' s)\n[PROOFSTEP]\nsimpa only [\u2190 sInf_range, \u2190 image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (iSup f) = \u2a06 (i : \u03b9), ENNReal.toNNReal (f i)\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\ng : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2a06 (i : \u03b9), \u2191(f i)) = \u2a06 (i : \u03b9), ENNReal.toNNReal ((fun i => \u2191(f i)) i)\n[PROOFSTEP]\nsimp_rw [toNNReal_coe]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\ng : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u211d\u22650\n\u22a2 ENNReal.toNNReal (\u2a06 (i : \u03b9), \u2191(f i)) = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nby_cases h : BddAbove (range f)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\ng : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u211d\u22650\nh : BddAbove (range f)\n\u22a2 ENNReal.toNNReal (\u2a06 (i : \u03b9), \u2191(f i)) = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nrw [\u2190 coe_iSup h, toNNReal_coe]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\ng : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u211d\u22650\nh : \u00acBddAbove (range f)\n\u22a2 ENNReal.toNNReal (\u2a06 (i : \u03b9), \u2191(f i)) = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nerw [NNReal.iSup_of_not_bddAbove h, (WithTop.iSup_coe_eq_top f).mpr h, top_toNNReal]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhs : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (sSup s) = sSup (ENNReal.toNNReal '' s)\n[PROOFSTEP]\nhave hf : \u2200 i, ((\u2191) : s \u2192 \u211d\u22650\u221e) i \u2260 \u221e := fun \u27e8r, rs\u27e9 => hs r rs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhs : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\nhf : \u2200 (i : { x // x \u2208 s }), \u2191i \u2260 \u22a4\n\u22a2 ENNReal.toNNReal (sSup s) = sSup (ENNReal.toNNReal '' s)\n[PROOFSTEP]\nsimpa only [\u2190 sSup_range, \u2190 image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iSup hf)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\n\u22a2 ENNReal.toReal (iInf f) = \u2a05 (i : \u03b9), ENNReal.toReal (f i)\n[PROOFSTEP]\nsimp only [ENNReal.toReal, toNNReal_iInf hf, NNReal.coe_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhf : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\n\u22a2 ENNReal.toReal (sInf s) = sInf (ENNReal.toReal '' s)\n[PROOFSTEP]\nsimp only [ENNReal.toReal, toNNReal_sInf s hf, NNReal.coe_sInf, Set.image_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b9), f i \u2260 \u22a4\n\u22a2 ENNReal.toReal (iSup f) = \u2a06 (i : \u03b9), ENNReal.toReal (f i)\n[PROOFSTEP]\nsimp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhf : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\n\u22a2 ENNReal.toReal (sSup s) = sSup (ENNReal.toReal '' s)\n[PROOFSTEP]\nsimp only [ENNReal.toReal, toNNReal_sSup s hf, NNReal.coe_sSup, Set.image_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\n\u22a2 a - \u2a05 (i : \u03b9), f i = \u2a06 (i : \u03b9), a - f i\n[PROOFSTEP]\nrefine' eq_of_forall_ge_iff fun c => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\n\u22a2 a - \u2a05 (i : \u03b9), f i \u2264 c \u2194 \u2a06 (i : \u03b9), a - f i \u2264 c\n[PROOFSTEP]\nrw [tsub_le_iff_right, add_comm, iInf_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\n\u22a2 a \u2264 \u2a05 (i : \u03b9), f i + c \u2194 \u2a06 (i : \u03b9), a - f i \u2264 c\n[PROOFSTEP]\nsimp [tsub_le_iff_right, sub_eq_add_neg, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u22a2 sInf s + a = \u2a05 (b : \u211d\u22650\u221e) (_ : b \u2208 s), b + a\n[PROOFSTEP]\nsimp [sInf_eq_iInf, iInf_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\na : \u211d\u22650\u221e\n\u22a2 a + iInf f = \u2a05 (b : \u03b9), a + f b\n[PROOFSTEP]\nrw [add_comm, iInf_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\na : \u211d\u22650\u221e\n\u22a2 \u2a05 (i : \u03b9), f i + a = \u2a05 (b : \u03b9), a + f b\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\nh : \u2200 (i j : \u03b9), \u2203 k, f k + g k \u2264 f i + g j\n\u22a2 \u2a05 (a : \u03b9) (a' : \u03b9), f a + g a' = iInf f + iInf g\n[PROOFSTEP]\nsimp_rw [iInf_add, add_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\n\u22a2 \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\n[PROOFSTEP]\ninduction' s using Finset.cons_induction_on with a s ha ih\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\n\u22a2 \u2a05 (i : \u03b9), \u2211 a in \u2205, f i a = \u2211 a in \u2205, \u2a05 (i : \u03b9), f i a\n[PROOFSTEP]\nsimp only [Finset.sum_empty, ciInf_const]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\n\u22a2 \u2a05 (i : \u03b9), \u2211 a in Finset.cons a s ha, f i a = \u2211 a in Finset.cons a s ha, \u2a05 (i : \u03b9), f i a\n[PROOFSTEP]\nsimp only [Finset.sum_cons, \u2190 ih]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\n\u22a2 \u2a05 (i : \u03b9), f i a + \u2211 a in s, f i a = (\u2a05 (i : \u03b9), f i a) + \u2a05 (i : \u03b9), \u2211 a in s, f i a\n[PROOFSTEP]\nrefine (iInf_add_iInf fun i j => ?_).symm\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\ni j : \u03b9\n\u22a2 \u2203 k, f k a + \u2211 a in s, f k a \u2264 f i a + \u2211 a in s, f j a\n[PROOFSTEP]\nrefine (h (Finset.cons a s ha) i j).imp fun k hk => ?_\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\ni j k : \u03b9\nhk : \u2200 (a_1 : \u03b1), a_1 \u2208 Finset.cons a s ha \u2192 f k a_1 \u2264 f i a_1 \u2227 f k a_1 \u2264 f j a_1\n\u22a2 f k a + \u2211 a in s, f k a \u2264 f i a + \u2211 a in s, f j a\n[PROOFSTEP]\nrw [Finset.forall_mem_cons] at hk \n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf\u271d g : \u03b9 \u2192 \u211d\u22650\u221e\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\ninst\u271d : Nonempty \u03b9\nh : \u2200 (t : Finset \u03b1) (i j : \u03b9), \u2203 k, \u2200 (a : \u03b1), a \u2208 t \u2192 f k a \u2264 f i a \u2227 f k a \u2264 f j a\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u2a05 (i : \u03b9), \u2211 a in s, f i a = \u2211 a in s, \u2a05 (i : \u03b9), f i a\ni j k : \u03b9\nhk : (f k a \u2264 f i a \u2227 f k a \u2264 f j a) \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f k x \u2264 f i x \u2227 f k x \u2264 f j x\n\u22a2 f k a + \u2211 a in s, f k a \u2264 f i a + \u2211 a in s, f j a\n[PROOFSTEP]\nexact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9\u271d : Sort u_3\nf\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e\n\u03b9 : Sort u_4\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\u221e\nx : \u211d\u22650\u221e\nh : x \u2260 \u22a4\n\u22a2 iInf f * x = \u2a05 (i : \u03b9), f i * x\n[PROOFSTEP]\nby_cases h0 : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9\u271d : Sort u_3\nf\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e\n\u03b9 : Sort u_4\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\u221e\nx : \u211d\u22650\u221e\nh : x \u2260 \u22a4\nh0 : x = 0\n\u22a2 iInf f * x = \u2a05 (i : \u03b9), f i * x\n[PROOFSTEP]\nsimp only [h0, mul_zero, iInf_const]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9\u271d : Sort u_3\nf\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e\n\u03b9 : Sort u_4\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\u221e\nx : \u211d\u22650\u221e\nh : x \u2260 \u22a4\nh0 : \u00acx = 0\n\u22a2 iInf f * x = \u2a05 (i : \u03b9), f i * x\n[PROOFSTEP]\nexact iInf_mul_of_ne h0 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9\u271d : Sort u_3\nf\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e\n\u03b9 : Sort u_4\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\u221e\nx : \u211d\u22650\u221e\nh : x \u2260 \u22a4\n\u22a2 x * iInf f = \u2a05 (i : \u03b9), x * f i\n[PROOFSTEP]\nsimpa only [mul_comm] using iInf_mul h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9\u271d : Sort u_3\nf\u271d g : \u03b9\u271d \u2192 \u211d\u22650\u221e\n\u03b9 : Sort u_4\nf : \u03b9 \u2192 \u211d\u22650\u221e\nx : \u211d\u22650\u221e\nh0 : x \u2260 0\nh : x \u2260 \u22a4\n\u22a2 x * iInf f = \u2a05 (i : \u03b9), x * f i\n[PROOFSTEP]\nsimpa only [mul_comm] using iInf_mul_of_ne h0 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\n\u22a2 \u2a06 (x : \u03b9), 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\n\u22a2 OrdConnected (ENNReal.some '' t)\n[PROOFSTEP]\nrefine' \u27e8ball_image_iff.2 fun x hx => ball_image_iff.2 fun y hy z hz => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\u221e\nhz : z \u2208 Icc \u2191x \u2191y\n\u22a2 z \u2208 ENNReal.some '' t\n[PROOFSTEP]\nrcases ENNReal.le_coe_iff.1 hz.2 with \u27e8z, rfl, -\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\nhz : \u2191z \u2208 Icc \u2191x \u2191y\n\u22a2 \u2191z \u2208 ENNReal.some '' t\n[PROOFSTEP]\nexact mem_image_of_mem _ (h.out hx hy \u27e8ENNReal.coe_le_coe.1 hz.1, ENNReal.coe_le_coe.1 hz.2\u27e9)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected s\n\u22a2 OrdConnected (ENNReal.ofReal '' s)\n[PROOFSTEP]\nsimpa only [image_image] using h.image_real_toNNReal.image_coe_nnreal_ennreal\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.ENNReal", "llama_tokens": 64430, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.27262180047939494}} {"text": "[GOAL]\n\u03b1\u271d : Type ?u.229\n\u03b2\u271d : Type ?u.232\n\u03b3\u271d : Type ?u.235\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d \u2192 \u03b3\u271d\na\u271d : Option \u03b1\u271d\nb\u271d : Option \u03b2\u271d\nc : Option \u03b3\u271d\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\n\u22a2 map\u2082 f a b = Seq.seq (f <$> a) fun x => b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1\u271d : Type ?u.229\n\u03b2\u271d : Type ?u.232\n\u03b3\u271d : Type ?u.235\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d \u2192 \u03b3\u271d\na : Option \u03b1\u271d\nb\u271d : Option \u03b2\u271d\nc : Option \u03b3\u271d\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\n\u22a2 map\u2082 f none b = Seq.seq (f <$> none) fun x => b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1\u271d : Type ?u.229\n\u03b2\u271d : Type ?u.232\n\u03b3\u271d : Type ?u.235\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d \u2192 \u03b3\u271d\na : Option \u03b1\u271d\nb\u271d : Option \u03b2\u271d\nc : Option \u03b3\u271d\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) b = Seq.seq (f <$> some val\u271d) fun x => b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\n\u22a2 map\u2082 f a none = none\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f none none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) none = none\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : \u03b2\n\u22a2 map\u2082 f a (some b) = Option.map (fun a => f a b) a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : \u03b2\n\u22a2 map\u2082 f none (some b) = Option.map (fun a => f a b) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : \u03b2\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (some b) = Option.map (fun a => f a b) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc\u271d : Option \u03b3\nc : \u03b3\n\u22a2 c \u2208 map\u2082 f a b \u2194 \u2203 a' b', a' \u2208 a \u2227 b' \u2208 b \u2227 f a' b' = c\n[PROOFSTEP]\nsimp [map\u2082]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u22a2 map\u2082 f a b = none \u2194 a = none \u2228 b = none\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u22a2 map\u2082 f none b = none \u2194 none = none \u2228 b = none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) b = none \u2194 some val\u271d = none \u2228 b = none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u22a2 map\u2082 f none none = none \u2194 none = none \u2228 none = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase none.some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (some val\u271d) = none \u2194 none = none \u2228 some val\u271d = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) none = none \u2194 some val\u271d = none \u2228 none = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (some val\u271d) = none \u2194 some val\u271d\u00b9 = none \u2228 some val\u271d = none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\n\u22a2 map\u2082 f a b = map\u2082 (fun a b => f b a) b a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\n\u22a2 map\u2082 f none b = map\u2082 (fun a b => f b a) b none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) b = map\u2082 (fun a b => f b a) b (some val\u271d)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f none none = map\u2082 (fun a b => f b a) none none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (some val\u271d) = map\u2082 (fun a b => f b a) (some val\u271d) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) none = map\u2082 (fun a b => f b a) none (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (some val\u271d) = map\u2082 (fun a b => f b a) (some val\u271d) (some val\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\n\u22a2 Option.map g (map\u2082 f a b) = map\u2082 (fun a b => g (f a b)) a b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\n\u22a2 Option.map g (map\u2082 f none b) = map\u2082 (fun a b => g (f a b)) none b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 (fun a b => g (f a b)) (some val\u271d) b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\n\u22a2 Option.map g (map\u2082 f none none) = map\u2082 (fun a b => g (f a b)) none none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase none.some\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 (fun a b => g (f a b)) none (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.none\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 (fun a b => g (f a b)) (some val\u271d) none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some.some\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b4\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 (fun a b => g (f a b)) (some val\u271d\u00b9) (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 f (Option.map g a) b = map\u2082 (fun a b => f (g a) b) a b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 f (Option.map g none) b = map\u2082 (fun a b => f (g a) b) none b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b3 \u2192 \u03b2 \u2192 \u03b4\ng : \u03b1 \u2192 \u03b3\nval\u271d : \u03b1\n\u22a2 map\u2082 f (Option.map g (some val\u271d)) b = map\u2082 (fun a b => f (g a) b) (some val\u271d) b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f a (Option.map g b) = map\u2082 (fun a b => f a (g b)) a b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_2\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f a (Option.map g none) = map\u2082 (fun a b => f a (g b)) a none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\nval\u271d : \u03b2\n\u22a2 map\u2082 f a (Option.map g (some val\u271d)) = map\u2082 (fun a b => f a (g b)) a (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nx : Option (\u03b1 \u00d7 \u03b2)\n\u22a2 Option.map (uncurry f) x = map\u2082 f (Option.map Prod.fst x) (Option.map Prod.snd x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u22a2 Option.map (uncurry f) none = map\u2082 f (Option.map Prod.fst none) (Option.map Prod.snd none)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nval\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 Option.map (uncurry f) (some val\u271d) = map\u2082 f (Option.map Prod.fst (some val\u271d)) (Option.map Prod.snd (some val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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 map\u2082 f (map\u2082 g a b) c = map\u2082 f' a (map\u2082 g' b c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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 map\u2082 f (map\u2082 g none b) c = map\u2082 f' none (map\u2082 g' b c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) b) c = map\u2082 f' (some val\u271d) (map\u2082 g' b c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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 map\u2082 f (map\u2082 g none none) c = map\u2082 f' none (map\u2082 g' none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d)) c = map\u2082 f' none (map\u2082 g' (some val\u271d) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) none) c = map\u2082 f' (some val\u271d) (map\u2082 g' none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) c = map\u2082 f' (some val\u271d\u00b9) (map\u2082 g' (some val\u271d) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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 map\u2082 f (map\u2082 g none none) none = map\u2082 f' none (map\u2082 g' none none)\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase none.none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g none none) (some val\u271d) = map\u2082 f' none (map\u2082 g' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase none.some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d)) none = map\u2082 f' none (map\u2082 g' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase none.some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d\u00b9)) (some val\u271d) = map\u2082 f' none (map\u2082 g' (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase some.none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) none) none = map\u2082 f' (some val\u271d) (map\u2082 g' none none)\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase some.none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) none) (some val\u271d) = map\u2082 f' (some val\u271d\u00b9) (map\u2082 g' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase some.some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) none = map\u2082 f' (some val\u271d\u00b9) (map\u2082 g' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\ncase some.some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b5' : Type u_3\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)\nval\u271d\u00b2 : \u03b1\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b2) (some val\u271d\u00b9)) (some val\u271d) = map\u2082 f' (some val\u271d\u00b2) (map\u2082 g' (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [h_assoc]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\n\u22a2 map\u2082 f a b = map\u2082 g b a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\n\u22a2 map\u2082 f none b = map\u2082 g b none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) b = map\u2082 g b (some val\u271d)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\n\u22a2 map\u2082 f none none = map\u2082 g none none\n[PROOFSTEP]\nsimp [h_comm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (some val\u271d) = map\u2082 g (some val\u271d) none\n[PROOFSTEP]\nsimp [h_comm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) none = map\u2082 g none (some val\u271d)\n[PROOFSTEP]\nsimp [h_comm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\ng : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1) (b : \u03b2), f a b = g b a\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (some val\u271d) = map\u2082 g (some val\u271d) (some val\u271d\u00b9)\n[PROOFSTEP]\nsimp [h_comm]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f a (map\u2082 g b c) = map\u2082 g' b (map\u2082 f' a c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f none (map\u2082 g b c) = map\u2082 g' b (map\u2082 f' none c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (map\u2082 g b c) = map\u2082 g' b (map\u2082 f' (some val\u271d) c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f none (map\u2082 g none c) = map\u2082 g' none (map\u2082 f' none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (map\u2082 g (some val\u271d) c) = map\u2082 g' (some val\u271d) (map\u2082 f' none c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (map\u2082 g none c) = map\u2082 g' none (map\u2082 f' (some val\u271d) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (map\u2082 g (some val\u271d) c) = map\u2082 g' (some val\u271d) (map\u2082 f' (some val\u271d\u00b9) c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f none (map\u2082 g none none) = map\u2082 g' none (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase none.none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b3\n\u22a2 map\u2082 f none (map\u2082 g none (some val\u271d)) = map\u2082 g' none (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase none.some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (map\u2082 g (some val\u271d) none) = map\u2082 g' (some val\u271d) (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase none.some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f none (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) = map\u2082 g' (some val\u271d\u00b9) (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (map\u2082 g none none) = map\u2082 g' none (map\u2082 f' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b3\n\u22a2 map\u2082 f (some val\u271d\u00b9) (map\u2082 g none (some val\u271d)) = map\u2082 g' none (map\u2082 f' (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (map\u2082 g (some val\u271d) none) = map\u2082 g' (some val\u271d) (map\u2082 f' (some val\u271d\u00b9) none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_6\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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)\nval\u271d\u00b2 : \u03b1\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f (some val\u271d\u00b2) (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) = map\u2082 g' (some val\u271d\u00b9) (map\u2082 f' (some val\u271d\u00b2) (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f (map\u2082 g a b) c = map\u2082 g' (map\u2082 f' a c) b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f (map\u2082 g none b) c = map\u2082 g' (map\u2082 f' none c) b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) b) c = map\u2082 g' (map\u2082 f' (some val\u271d) c) b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f (map\u2082 g none none) c = map\u2082 g' (map\u2082 f' none c) none\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d)) c = map\u2082 g' (map\u2082 f' none c) (some val\u271d)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) none) c = map\u2082 g' (map\u2082 f' (some val\u271d) c) none\n[PROOFSTEP]\ncases c\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) c = map\u2082 g' (map\u2082 f' (some val\u271d\u00b9) c) (some val\u271d)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase none.none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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 map\u2082 f (map\u2082 g none none) none = map\u2082 g' (map\u2082 f' none none) none\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase none.none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g none none) (some val\u271d) = map\u2082 g' (map\u2082 f' none (some val\u271d)) none\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase none.some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d)) none = map\u2082 g' (map\u2082 f' none none) (some val\u271d)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase none.some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g none (some val\u271d\u00b9)) (some val\u271d) = map\u2082 g' (map\u2082 f' none (some val\u271d)) (some val\u271d\u00b9)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d : \u03b1\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d) none) none = map\u2082 g' (map\u2082 f' (some val\u271d) none) none\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) none) (some val\u271d) = map\u2082 g' (map\u2082 f' (some val\u271d\u00b9) (some val\u271d)) none\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b9) (some val\u271d)) none = map\u2082 g' (map\u2082 f' (some val\u271d\u00b9) none) (some val\u271d)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\u03b4 : Type u_1\n\u03b5 : Type u_2\n\u03b4' : Type u_3\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\nval\u271d\u00b2 : \u03b1\nval\u271d\u00b9 : \u03b2\nval\u271d : \u03b3\n\u22a2 map\u2082 f (map\u2082 g (some val\u271d\u00b2) (some val\u271d\u00b9)) (some val\u271d) = map\u2082 g' (map\u2082 f' (some val\u271d\u00b2) (some val\u271d)) (some val\u271d\u00b9)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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 Option.map g (map\u2082 f a b) = map\u2082 f' (Option.map g\u2081 a) (Option.map g\u2082 b)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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 Option.map g (map\u2082 f none b) = map\u2082 f' (Option.map g\u2081 none) (Option.map g\u2082 b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' (Option.map g\u2081 (some val\u271d)) (Option.map g\u2082 b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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 Option.map g (map\u2082 f none none) = map\u2082 f' (Option.map g\u2081 none) (Option.map g\u2082 none)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' (Option.map g\u2081 none) (Option.map g\u2082 (some val\u271d))\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' (Option.map g\u2081 (some val\u271d)) (Option.map g\u2082 none)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\n\u03b2' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (Option.map g\u2081 (some val\u271d\u00b9)) (Option.map g\u2082 (some val\u271d))\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f a b) = map\u2082 f' (Option.map g' a) b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f none b) = map\u2082 f' (Option.map g' none) b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' (Option.map g' (some val\u271d)) b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f none none) = map\u2082 f' (Option.map g' none) none\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' (Option.map g' none) (some val\u271d)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' (Option.map g' (some val\u271d)) none\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (Option.map g' (some val\u271d\u00b9)) (some val\u271d)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f a b) = map\u2082 f' a (Option.map g' b)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f none b) = map\u2082 f' none (Option.map g' b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' (some val\u271d) (Option.map g' b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f none none) = map\u2082 f' none (Option.map g' none)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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)\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' none (Option.map g' (some val\u271d))\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' (some val\u271d) (Option.map g' none)\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (some val\u271d\u00b9) (Option.map g' (some val\u271d))\n[PROOFSTEP]\nsimp [h_distrib]\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\n\u22a2 map\u2082 f (Option.map g a) b = Option.map g' (map\u2082 f' a b)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\n\u22a2 map\u2082 f (Option.map g none) b = Option.map g' (map\u2082 f' none b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (Option.map g (some val\u271d)) b = Option.map g' (map\u2082 f' (some val\u271d) b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\n\u22a2 map\u2082 f (Option.map g none) none = Option.map g' (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\nval\u271d : \u03b2\n\u22a2 map\u2082 f (Option.map g none) (some val\u271d) = Option.map g' (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (Option.map g (some val\u271d)) none = Option.map g' (map\u2082 f' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_comm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' a b)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (Option.map g (some val\u271d\u00b9)) (some val\u271d) = Option.map g' (map\u2082 f' (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_comm]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\n\u22a2 map\u2082 f a (Option.map g b) = Option.map g' (map\u2082 f' a b)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\n\u22a2 map\u2082 f none (Option.map g b) = Option.map g' (map\u2082 f' none b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (Option.map g b) = Option.map g' (map\u2082 f' (some val\u271d) b)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\n\u22a2 map\u2082 f none (Option.map g none) = Option.map g' (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (Option.map g (some val\u271d)) = Option.map g' (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (Option.map g none) = Option.map g' (map\u2082 f' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b1 \u2192 \u03b2 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_comm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' a b)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (Option.map g (some val\u271d)) = Option.map g' (map\u2082 f' (some val\u271d\u00b9) (some val\u271d))\n[PROOFSTEP]\nsimp [h_right_comm]\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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 Option.map g (map\u2082 f a b) = map\u2082 f' (Option.map g\u2081 b) (Option.map g\u2082 a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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 Option.map g (map\u2082 f none b) = map\u2082 f' (Option.map g\u2081 b) (Option.map g\u2082 none)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' (Option.map g\u2081 b) (Option.map g\u2082 (some val\u271d))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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 Option.map g (map\u2082 f none none) = map\u2082 f' (Option.map g\u2081 none) (Option.map g\u2082 none)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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)\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' (Option.map g\u2081 (some val\u271d)) (Option.map g\u2082 none)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' (Option.map g\u2081 none) (Option.map g\u2082 (some val\u271d))\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_6\n\u03b3 : Type u_4\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\n\u03b1' : Type u_3\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (Option.map g\u2081 (some val\u271d)) (Option.map g\u2082 (some val\u271d\u00b9))\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f a b) = map\u2082 f' (Option.map g' b) a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f none b) = map\u2082 f' (Option.map g' b) none\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' (Option.map g' b) (some val\u271d)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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 Option.map g (map\u2082 f none none) = map\u2082 f' (Option.map g' none) none\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' (Option.map g' (some val\u271d)) none\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' (Option.map g' none) (some val\u271d)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b2' : Type u_2\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\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (Option.map g' (some val\u271d)) (some val\u271d\u00b9)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f a b) = map\u2082 f' b (Option.map g' a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f none b) = map\u2082 f' b (Option.map g' none)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) b) = map\u2082 f' b (Option.map g' (some val\u271d))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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 Option.map g (map\u2082 f none none) = map\u2082 f' none (Option.map g' none)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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)\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f none (some val\u271d)) = map\u2082 f' (some val\u271d) (Option.map g' none)\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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)\nval\u271d : \u03b1\n\u22a2 Option.map g (map\u2082 f (some val\u271d) none) = map\u2082 f' none (Option.map g' (some val\u271d))\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b4 : Type u_1\n\u03b1' : Type u_2\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)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 Option.map g (map\u2082 f (some val\u271d\u00b9) (some val\u271d)) = map\u2082 f' (some val\u271d) (Option.map g' (some val\u271d\u00b9))\n[PROOFSTEP]\nsimp [h_antidistrib]\n[GOAL]\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\n\u22a2 map\u2082 f (Option.map g a) b = Option.map g' (map\u2082 f' b a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\n\u22a2 map\u2082 f (Option.map g none) b = Option.map g' (map\u2082 f' b none)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (Option.map g (some val\u271d)) b = Option.map g' (map\u2082 f' b (some val\u271d))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\n\u22a2 map\u2082 f (Option.map g none) none = Option.map g' (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_left_anticomm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\nval\u271d : \u03b2\n\u22a2 map\u2082 f (Option.map g none) (some val\u271d) = Option.map g' (map\u2082 f' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_left_anticomm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (Option.map g (some val\u271d)) none = Option.map g' (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_left_anticomm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_5\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b1' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1' \u2192 \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b1'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_left_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f (g a) b = g' (f' b a)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (Option.map g (some val\u271d\u00b9)) (some val\u271d) = Option.map g' (map\u2082 f' (some val\u271d) (some val\u271d\u00b9))\n[PROOFSTEP]\nsimp [h_left_anticomm]\n[GOAL]\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\n\u22a2 map\u2082 f a (Option.map g b) = Option.map g' (map\u2082 f' b a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\n\u22a2 map\u2082 f none (Option.map g b) = Option.map g' (map\u2082 f' b none)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : Option \u03b2\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (Option.map g b) = Option.map g' (map\u2082 f' b (some val\u271d))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\n\u22a2 map\u2082 f none (Option.map g none) = Option.map g' (map\u2082 f' none none)\n[PROOFSTEP]\nsimp [h_right_anticomm]\n[GOAL]\ncase none.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\nval\u271d : \u03b2\n\u22a2 map\u2082 f none (Option.map g (some val\u271d)) = Option.map g' (map\u2082 f' (some val\u271d) none)\n[PROOFSTEP]\nsimp [h_right_anticomm]\n[GOAL]\ncase some.none\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\nval\u271d : \u03b1\n\u22a2 map\u2082 f (some val\u271d) (Option.map g none) = Option.map g' (map\u2082 f' none (some val\u271d))\n[PROOFSTEP]\nsimp [h_right_anticomm]\n[GOAL]\ncase some.some\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nc : Option \u03b3\n\u03b2' : Type u_1\n\u03b4 : Type u_2\nf : \u03b1 \u2192 \u03b2' \u2192 \u03b3\ng : \u03b2 \u2192 \u03b2'\nf' : \u03b2 \u2192 \u03b1 \u2192 \u03b4\ng' : \u03b4 \u2192 \u03b3\nh_right_anticomm : \u2200 (a : \u03b1) (b : \u03b2), f a (g b) = g' (f' b a)\nval\u271d\u00b9 : \u03b1\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some val\u271d\u00b9) (Option.map g (some val\u271d)) = Option.map g' (map\u2082 f' (some val\u271d) (some val\u271d\u00b9))\n[PROOFSTEP]\nsimp [h_right_anticomm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.32843\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na : \u03b1\nh : \u2200 (b : \u03b2), f a b = b\no : Option \u03b2\n\u22a2 map\u2082 f (some a) o = o\n[PROOFSTEP]\ncases o\n[GOAL]\ncase none\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.32843\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na : \u03b1\nh : \u2200 (b : \u03b2), f a b = b\n\u22a2 map\u2082 f (some a) none = none\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.32843\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na\u271d : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\na : \u03b1\nh : \u2200 (b : \u03b2), f a b = b\nval\u271d : \u03b2\n\u22a2 map\u2082 f (some a) (some val\u271d) = some val\u271d\n[PROOFSTEP]\nexacts [rfl, congr_arg some (h _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.33033\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb\u271d : Option \u03b2\nc : Option \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\nb : \u03b2\nh : \u2200 (a : \u03b1), f a b = a\no : Option \u03b1\n\u22a2 map\u2082 f o (some b) = o\n[PROOFSTEP]\nsimp [h, map\u2082]\n", "meta": {"mathlib_filename": "Mathlib.Data.Option.NAry", "llama_tokens": 29634, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.27230710087760307}} {"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\u22a2 \u2200 {x y : Set \u2115}, x \u2208 range Ici \u2192 y \u2208 range Ici \u2192 \u2203 z, z \u2208 range Ici \u2227 z \u2286 x \u2229 y\n[PROOFSTEP]\nrintro _ _ \u27e8n, rfl\u27e9 \u27e8m, rfl\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\nn m : \u2115\n\u22a2 \u2203 z, z \u2208 range Ici \u2227 z \u2286 Ici n \u2229 Ici m\n[PROOFSTEP]\nexact \u27e8Ici (max n m), mem_range_self _, Ici_inter_Ici.symm.subset\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\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\n\u22a2 \u2200 {x y : Set \u03b1},\n x \u2208 {t | \u2203 i, p i \u2227 s i = t} \u2192 y \u2208 {t | \u2203 i, p i \u2227 s i = t} \u2192 \u2203 z, z \u2208 {t | \u2203 i, p i \u2227 s i = t} \u2227 z \u2286 x \u2229 y\n[PROOFSTEP]\nrintro _ _ \u27e8i, hi, rfl\u27e9 \u27e8j, hj, rfl\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\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\ni : \u03b9\nhi : p i\nj : \u03b9\nhj : p j\n\u22a2 \u2203 z, z \u2208 {t | \u2203 i, p i \u2227 s i = t} \u2227 z \u2286 s i \u2229 s j\n[PROOFSTEP]\nrcases h.inter hi hj with \u27e8k, hk, hk'\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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\ni : \u03b9\nhi : p i\nj : \u03b9\nhj : p j\nk : \u03b9\nhk : p k\nhk' : s k \u2286 s i \u2229 s j\n\u22a2 \u2203 z, z \u2208 {t | \u2203 i, p i \u2227 s i = t} \u2227 z \u2286 s i \u2229 s j\n[PROOFSTEP]\nexact \u27e8_, \u27e8k, hk, rfl\u27e9, hk'\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\nB : FilterBasis \u03b1\n\u22a2 FilterBasis.filter B = \u2a05 (s : \u2191B.sets), \ud835\udcdf \u2191s\n[PROOFSTEP]\nhave : Directed (\u00b7 \u2265 \u00b7) fun s : B.sets => \ud835\udcdf (s : Set \u03b1) :=\n by\n rintro \u27e8U, U_in\u27e9 \u27e8V, V_in\u27e9\n rcases B.inter_sets U_in V_in with \u27e8W, W_in, W_sub\u27e9\n use\u27e8W, W_in\u27e9\n simp only [ge_iff_le, le_principal_iff, mem_principal, Subtype.coe_mk]\n exact subset_inter_iff.mp W_sub\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\nB : FilterBasis \u03b1\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) fun s => \ud835\udcdf \u2191s\n[PROOFSTEP]\nrintro \u27e8U, U_in\u27e9 \u27e8V, V_in\u27e9\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\u03b9' : Sort u_5\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 B.sets\nV : Set \u03b1\nV_in : V \u2208 B.sets\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := U, property := U_in }) ((fun s => \ud835\udcdf \u2191s) z) \u2227\n (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := V, property := V_in }) ((fun s => \ud835\udcdf \u2191s) z)\n[PROOFSTEP]\nrcases B.inter_sets U_in V_in with \u27e8W, W_in, W_sub\u27e9\n[GOAL]\ncase mk.mk.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\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 B.sets\nV : Set \u03b1\nV_in : V \u2208 B.sets\nW : Set \u03b1\nW_in : W \u2208 B.sets\nW_sub : W \u2286 U \u2229 V\n\u22a2 \u2203 z,\n (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := U, property := U_in }) ((fun s => \ud835\udcdf \u2191s) z) \u2227\n (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := V, property := V_in }) ((fun s => \ud835\udcdf \u2191s) z)\n[PROOFSTEP]\nuse\u27e8W, W_in\u27e9\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\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 B.sets\nV : Set \u03b1\nV_in : V \u2208 B.sets\nW : Set \u03b1\nW_in : W \u2208 B.sets\nW_sub : W \u2286 U \u2229 V\n\u22a2 (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := U, property := U_in })\n ((fun s => \ud835\udcdf \u2191s) { val := W, property := W_in }) \u2227\n (fun x x_1 => x \u2265 x_1) ((fun s => \ud835\udcdf \u2191s) { val := V, property := V_in })\n ((fun s => \ud835\udcdf \u2191s) { val := W, property := W_in })\n[PROOFSTEP]\nsimp only [ge_iff_le, le_principal_iff, mem_principal, Subtype.coe_mk]\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\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 B.sets\nV : Set \u03b1\nV_in : V \u2208 B.sets\nW : Set \u03b1\nW_in : W \u2208 B.sets\nW_sub : W \u2286 U \u2229 V\n\u22a2 W \u2286 U \u2227 W \u2286 V\n[PROOFSTEP]\nexact subset_inter_iff.mp W_sub\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\nB : FilterBasis \u03b1\nthis : Directed (fun x x_1 => x \u2265 x_1) fun s => \ud835\udcdf \u2191s\n\u22a2 FilterBasis.filter B = \u2a05 (s : \u2191B.sets), \ud835\udcdf \u2191s\n[PROOFSTEP]\next U\n[GOAL]\ncase a\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\nB : FilterBasis \u03b1\nthis : Directed (fun x x_1 => x \u2265 x_1) fun s => \ud835\udcdf \u2191s\nU : Set \u03b1\n\u22a2 U \u2208 FilterBasis.filter B \u2194 U \u2208 \u2a05 (s : \u2191B.sets), \ud835\udcdf \u2191s\n[PROOFSTEP]\nsimp [mem_filter_iff, mem_iInf_of_directed 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\nB : FilterBasis \u03b1\n\u22a2 generate B.sets = FilterBasis.filter B\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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nB : FilterBasis \u03b1\n\u22a2 generate B.sets \u2264 FilterBasis.filter B\n[PROOFSTEP]\nintro U U_in\n[GOAL]\ncase a\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\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 FilterBasis.filter B\n\u22a2 U \u2208 generate B.sets\n[PROOFSTEP]\nrcases B.mem_filter_iff.mp U_in with \u27e8V, V_in, h\u27e9\n[GOAL]\ncase a.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\nB : FilterBasis \u03b1\nU : Set \u03b1\nU_in : U \u2208 FilterBasis.filter B\nV : Set \u03b1\nV_in : V \u2208 B\nh : V \u2286 U\n\u22a2 U \u2208 generate B.sets\n[PROOFSTEP]\nexact GenerateSets.superset (GenerateSets.basic V_in) h\n[GOAL]\ncase a\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\nB : FilterBasis \u03b1\n\u22a2 FilterBasis.filter B \u2264 generate B.sets\n[PROOFSTEP]\nrw [le_generate_iff]\n[GOAL]\ncase a\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\nB : FilterBasis \u03b1\n\u22a2 B.sets \u2286 (FilterBasis.filter B).sets\n[PROOFSTEP]\napply mem_filter_of_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\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\nU : Set \u03b1\n\u22a2 U \u2208 IsBasis.filter h \u2194 \u2203 i, p i \u2227 s i \u2286 U\n[PROOFSTEP]\nsimp only [IsBasis.filter, FilterBasis.mem_filter_iff, mem_filterBasis_iff, exists_exists_and_eq_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\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\n\u22a2 IsBasis.filter h = generate {U | \u2203 i, p i \u2227 s i = U}\n[PROOFSTEP]\nerw [h.filterBasis.generate]\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\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\n\u22a2 IsBasis.filter h = FilterBasis.filter (IsBasis.filterBasis h)\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\nU : Set \u03b1\n\u22a2 U \u2208 generate s \u2194 \u2203 i, (Set.Finite i \u2227 i \u2286 s) \u2227 \u22c2\u2080 i \u2286 U\n[PROOFSTEP]\nsimp only [mem_generate_iff, exists_prop, and_assoc, and_left_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\n\u22a2 \u2200 {x y : Set \u03b1},\n x \u2208 sInter '' {t | Set.Finite t \u2227 t \u2286 s} \u2192\n y \u2208 sInter '' {t | Set.Finite t \u2227 t \u2286 s} \u2192 \u2203 z, z \u2208 sInter '' {t | Set.Finite t \u2227 t \u2286 s} \u2227 z \u2286 x \u2229 y\n[PROOFSTEP]\nrintro _ _ \u27e8a, \u27e8fina, suba\u27e9, rfl\u27e9 \u27e8b, \u27e8finb, subb\u27e9, rfl\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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns a : Set (Set \u03b1)\nfina : Set.Finite a\nsuba : a \u2286 s\nb : Set (Set \u03b1)\nfinb : Set.Finite b\nsubb : b \u2286 s\n\u22a2 \u2203 z, z \u2208 sInter '' {t | Set.Finite t \u2227 t \u2286 s} \u2227 z \u2286 \u22c2\u2080 a \u2229 \u22c2\u2080 b\n[PROOFSTEP]\nexact \u27e8\u22c2\u2080 (a \u222a b), mem_image_of_mem _ \u27e8fina.union finb, union_subset suba subb\u27e9, (sInter_union _ _).subset\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p s\n\u22a2 l = l'\n[PROOFSTEP]\next t\n[GOAL]\ncase a\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p s\nt : Set \u03b1\n\u22a2 t \u2208 l \u2194 t \u2208 l'\n[PROOFSTEP]\nrw [hl.mem_iff, hl'.mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : IsBasis p s\nt : Set \u03b1\n\u22a2 t \u2208 IsBasis.filter h \u2194 \u2203 i, p i \u2227 s i \u2286 t\n[PROOFSTEP]\nsimp only [h.mem_filter_iff, exists_prop]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\ni\u271d j\u271d : \u03b9\nhi : p i\u271d\nhj : p j\u271d\n\u22a2 \u2203 k, p k \u2227 s k \u2286 s i\u271d \u2229 s j\u271d\n[PROOFSTEP]\nsimpa only [h.mem_iff] using inter_mem (h.mem_of_mem hi) (h.mem_of_mem hj)\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 IsBasis.filter (_ : IsBasis p s) = l\n[PROOFSTEP]\next U\n[GOAL]\ncase a\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nU : Set \u03b1\n\u22a2 U \u2208 IsBasis.filter (_ : IsBasis p s) \u2194 U \u2208 l\n[PROOFSTEP]\nsimp [h.mem_iff, IsBasis.mem_filter_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 l = generate {U | \u2203 i, p i \u2227 s i = U}\n[PROOFSTEP]\nrw [\u2190 h.isBasis.filter_eq_generate, h.filter_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\n\u22a2 generate s = generate (sInter '' {t | Set.Finite t \u2227 t \u2286 s})\n[PROOFSTEP]\nrw [\u2190 FilterBasis.ofSets_sets, FilterBasis.generate, \u2190 (hasBasis_generate s).filter_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\n\u22a2 IsBasis.filter (_ : IsBasis (fun t => Set.Finite t \u2227 t \u2286 s) fun t => \u22c2\u2080 t) = FilterBasis.filter (FilterBasis.ofSets 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\n\u22a2 FilterBasis.filter (FilterBasis.ofSets s) = generate s\n[PROOFSTEP]\nrw [\u2190 (FilterBasis.ofSets s).generate, FilterBasis.ofSets_sets, \u2190 generate_eq_generate_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\n\u22a2 HasBasis l p' s'\n[PROOFSTEP]\nrefine' \u27e8fun t => \u27e8fun ht => _, fun \u27e8i', hi', ht\u27e9 => mem_of_superset (h' i' hi') ht\u27e9\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht : t \u2208 l\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t\n[PROOFSTEP]\nrcases hl.mem_iff.1 ht with \u27e8i, hi, ht\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t\n[PROOFSTEP]\nrcases h i hi with \u27e8i', hi', hs's\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni'\u271d : \u03b9'\nhl : HasBasis l p s\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 s' i' \u2208 l\nt : Set \u03b1\nht\u271d : t \u2208 l\ni : \u03b9\nhi : p i\nht : s i \u2286 t\ni' : \u03b9'\nhi' : p' i'\nhs's : s' i' \u2286 s i\n\u22a2 \u2203 i, p' i \u2227 s' i \u2286 t\n[PROOFSTEP]\nexact \u27e8i', hi', hs's.trans ht\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, q x) \u2194 \u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 q x\n[PROOFSTEP]\nsimpa using hl.mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2203\u1da0 (x : \u03b1) in l, q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x\n[PROOFSTEP]\nsimp only [Filter.Frequently, hl.eventually_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u00ac\u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 \u00acq x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x\n[PROOFSTEP]\npush_neg\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b9), p i \u2192 Exists fun \u2983x\u2984 => x \u2208 s i \u2227 q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\n\u22a2 (\u2200 {i : \u03b9}, p i \u2192 Set.Nonempty (s i)) \u2194 \u00ac\u2203 i, p i \u2227 s i = \u2205\n[PROOFSTEP]\nsimp only [not_exists, not_and, 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\n\u22a2 (\u2200 {i : Set (Set \u03b1)}, Set.Finite i \u2227 i \u2286 s \u2192 Set.Nonempty (\u22c2\u2080 i)) \u2194\n \u2200 (t : Set (Set \u03b1)), t \u2286 s \u2192 Set.Finite t \u2192 Set.Nonempty (\u22c2\u2080 t)\n[PROOFSTEP]\nsimp only [\u2190 and_imp, and_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\nl\u271d l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nl : Filter \u03b1\nP : Set \u03b1 \u2192 Prop\n\u22a2 HasBasis l (fun s => s \u2208 l \u2227 P s) id \u2194 \u2200 (t : Set \u03b1), t \u2208 l \u2192 \u2203 r, r \u2208 l \u2227 P r \u2227 r \u2286 t\n[PROOFSTEP]\nsimp only [hasBasis_iff, id, and_assoc]\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\nl\u271d l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nl : Filter \u03b1\nP : Set \u03b1 \u2192 Prop\n\u22a2 (\u2200 (t : Set \u03b1), t \u2208 l \u2194 \u2203 i, i \u2208 l \u2227 P i \u2227 i \u2286 t) \u2194 \u2200 (t : Set \u03b1), t \u2208 l \u2192 \u2203 r, r \u2208 l \u2227 P r \u2227 r \u2286 t\n[PROOFSTEP]\nexact forall_congr' fun s => \u27e8fun h => h.1, fun h => \u27e8h, fun \u27e8t, hl, _, hts\u27e9 => mem_of_superset hl hts\u27e9\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nq : \u03b9 \u2192 Prop\nhq : \u2200 (i : \u03b9), p i \u2192 \u2203 j, p j \u2227 q j \u2227 s j \u2286 s i\n\u22a2 HasBasis l (fun i => p i \u2227 q i) s\n[PROOFSTEP]\nrefine' \u27e8fun t => \u27e8fun ht => _, fun \u27e8i, hpi, hti\u27e9 => h.mem_iff.2 \u27e8i, hpi.1, hti\u27e9\u27e9\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nq : \u03b9 \u2192 Prop\nhq : \u2200 (i : \u03b9), p i \u2192 \u2203 j, p j \u2227 q j \u2227 s j \u2286 s i\nt : Set \u03b1\nht : t \u2208 l\n\u22a2 \u2203 i, (p i \u2227 q i) \u2227 s i \u2286 t\n[PROOFSTEP]\nrcases h.mem_iff.1 ht with \u27e8i, hpi, hti\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nq : \u03b9 \u2192 Prop\nhq : \u2200 (i : \u03b9), p i \u2192 \u2203 j, p j \u2227 q j \u2227 s j \u2286 s i\nt : Set \u03b1\nht : t \u2208 l\ni : \u03b9\nhpi : p i\nhti : s i \u2286 t\n\u22a2 \u2203 i, (p i \u2227 q i) \u2227 s i \u2286 t\n[PROOFSTEP]\nrcases hq i hpi with \u27e8j, hpj, hqj, hji\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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nq : \u03b9 \u2192 Prop\nhq : \u2200 (i : \u03b9), p i \u2192 \u2203 j, p j \u2227 q j \u2227 s j \u2286 s i\nt : Set \u03b1\nht : t \u2208 l\ni : \u03b9\nhpi : p i\nhti : s i \u2286 t\nj : \u03b9\nhpj : p j\nhqj : q j\nhji : s j \u2286 s i\n\u22a2 \u2203 i, (p i \u2227 q i) \u2227 s i \u2286 t\n[PROOFSTEP]\nexact \u27e8j, \u27e8hpj, hqj\u27e9, hji.trans hti\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\nl l' : Filter \u03b1\np\u271d : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\np : Set \u03b1 \u2192 Prop\nh : HasBasis l (fun s => s \u2208 l \u2227 p s) id\nV : Set \u03b1\nhV : V \u2208 l\n\u22a2 HasBasis l (fun s => s \u2208 l \u2227 p s \u2227 s \u2286 V) id\n[PROOFSTEP]\nsimpa only [and_assoc] using h.restrict_subset hV\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\n\u22a2 l \u2264 l' \u2194 \u2200 (t : Set \u03b1), t \u2208 l' \u2192 \u2203 i, p i \u2227 s i \u2286 t\n[PROOFSTEP]\nsimp only [le_def, hl.mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 l \u2264 l' \u2194 \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n[PROOFSTEP]\nsimp only [hl'.ge_iff, hl.mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n\u22a2 l = l'\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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n\u22a2 l \u2264 l'\n[PROOFSTEP]\nrw [hl.le_basis_iff hl']\n[GOAL]\ncase a\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n\u22a2 \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n[PROOFSTEP]\nsimpa using h'\n[GOAL]\ncase a\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n\u22a2 l' \u2264 l\n[PROOFSTEP]\nrw [hl'.le_basis_iff hl]\n[GOAL]\ncase a\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nh : \u2200 (i : \u03b9), p i \u2192 \u2203 i', p' i' \u2227 s' i' \u2286 s i\nh' : \u2200 (i' : \u03b9'), p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'\n\u22a2 \u2200 (i' : \u03b9), p i' \u2192 \u2203 i, p' i \u2227 s' i \u2286 s i'\n[PROOFSTEP]\nsimpa 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 l \u2293 l' \u2194 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t\n[PROOFSTEP]\nintro t\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 t \u2208 l \u2293 l' \u2194 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 t \u2208 l \u2293 l' \u2192 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t\n[PROOFSTEP]\nsimp only [mem_inf_iff, hl.mem_iff, hl'.mem_iff]\n[GOAL]\ncase 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 (\u2203 t\u2081, (\u2203 i, p i \u2227 s i \u2286 t\u2081) \u2227 \u2203 t\u2082, (\u2203 i, p' i \u2227 s' i \u2286 t\u2082) \u2227 t = t\u2081 \u2229 t\u2082) \u2192\n \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t\n[PROOFSTEP]\nrintro \u27e8t, \u27e8i, hi, ht\u27e9, t', \u27e8i', hi', ht'\u27e9, 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\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni'\u271d : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\ni : \u03b9\nhi : p i\nht : s i \u2286 t\nt' : Set \u03b1\ni' : \u03b9'\nhi' : p' i'\nht' : s' i' \u2286 t'\n\u22a2 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t \u2229 t'\n[PROOFSTEP]\nexact \u27e8\u27e8i, i'\u27e9, \u27e8hi, hi'\u27e9, inter_subset_inter ht ht'\u27e9\n[GOAL]\ncase 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 (\u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u2229 s' i.snd \u2286 t) \u2192 t \u2208 l \u2293 l'\n[PROOFSTEP]\nrintro \u27e8\u27e8i, i'\u27e9, \u27e8hi, hi'\u27e9, H\u27e9\n[GOAL]\ncase mpr.intro.mk.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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni'\u271d : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\ni : \u03b9\ni' : \u03b9'\nH : s { fst := i, snd := i' }.fst \u2229 s' { fst := i, snd := i' }.snd \u2286 t\nhi : p { fst := i, snd := i' }.fst\nhi' : p' { fst := i, snd := i' }.snd\n\u22a2 t \u2208 l \u2293 l'\n[PROOFSTEP]\nexact mem_inf_of_inter (hl.mem_of_mem hi) (hl'.mem_of_mem hi') 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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\n\u22a2 \u2200 (t : Set \u03b1),\n t \u2208 \u2a05 (i : \u03b9), l i \u2194\n \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 t\n[PROOFSTEP]\nintro t\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i \u2194\n \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i \u2192\n \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 t\n[PROOFSTEP]\nsimp only [mem_iInf', (hl _).mem_iff]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 (\u2203 I,\n Set.Finite I \u2227\n \u2203 V,\n (\u2200 (i : \u03b9), \u2203 i_1, p i i_1 \u2227 s i i_1 \u2286 V i) \u2227\n (\u2200 (i : \u03b9), \u00aci \u2208 I \u2192 V i = univ) \u2227 t = \u22c2 (i : \u03b9) (_ : i \u2208 I), V i \u2227 t = \u22c2 (i : \u03b9), V i) \u2192\n \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 t\n[PROOFSTEP]\nrintro \u27e8I, hI, V, hV, -, rfl, -\u27e9\n[GOAL]\ncase mp.intro.intro.intro.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nI : Set \u03b9\nhI : Set.Finite I\nV : \u03b9 \u2192 Set \u03b1\nhV : \u2200 (i : \u03b9), \u2203 i_1, p i i_1 \u2227 s i i_1 \u2286 V i\n\u22a2 \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 \u22c2 (i : \u03b9) (_ : i \u2208 I), V i\n[PROOFSTEP]\nchoose u hu using hV\n[GOAL]\ncase mp.intro.intro.intro.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nI : Set \u03b9\nhI : Set.Finite I\nV : \u03b9 \u2192 Set \u03b1\nu : (i : \u03b9) \u2192 \u03b9' i\nhu : \u2200 (i : \u03b9), p i (u i) \u2227 s i (u i) \u2286 V i\n\u22a2 \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 \u22c2 (i : \u03b9) (_ : i \u2208 I), V i\n[PROOFSTEP]\nexact \u27e8\u27e8I, u\u27e9, \u27e8hI, fun i _ => (hu i).1\u27e9, iInter\u2082_mono fun i _ => (hu i).2\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 (\u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u03b9), i_1 \u2208 i.fst \u2192 p i_1 (Prod.snd i i_1)) \u2227\n \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i.fst), s i_1 (Prod.snd i i_1) \u2286 t) \u2192\n t \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nrintro \u27e8\u27e8I, f\u27e9, \u27e8hI\u2081, hI\u2082\u27e9, hsub\u27e9\n[GOAL]\ncase mpr.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nI : Set \u03b9\nf : (i : \u03b9) \u2192 \u03b9' i\nhsub : \u22c2 (i : \u03b9) (_ : i \u2208 (I, f).fst), s i (Prod.snd (I, f) i) \u2286 t\nhI\u2081 : Set.Finite (I, f).fst\nhI\u2082 : \u2200 (i : \u03b9), i \u2208 (I, f).fst \u2192 p i (Prod.snd (I, f) i)\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nrefine' mem_of_superset _ hsub\n[GOAL]\ncase mpr.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nI : Set \u03b9\nf : (i : \u03b9) \u2192 \u03b9' i\nhsub : \u22c2 (i : \u03b9) (_ : i \u2208 (I, f).fst), s i (Prod.snd (I, f) i) \u2286 t\nhI\u2081 : Set.Finite (I, f).fst\nhI\u2082 : \u2200 (i : \u03b9), i \u2208 (I, f).fst \u2192 p i (Prod.snd (I, f) i)\n\u22a2 \u22c2 (i : \u03b9) (_ : i \u2208 (I, f).fst), s i (Prod.snd (I, f) i) \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nexact (biInter_mem hI\u2081).mpr fun i hi => mem_iInf_of_mem i <| (hl i).mem_of_mem <| hI\u2082 _ hi\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\n\u22a2 HasBasis (\u2a05 (i : \u03b9), l i) (fun If => Set.Finite If.fst \u2227 \u2200 (i : \u2191If.fst), p (\u2191i) (Sigma.snd If i)) fun If =>\n \u22c2 (i : \u2191If.fst), s (\u2191i) (Sigma.snd If i)\n[PROOFSTEP]\nrefine' \u27e8fun t => \u27e8fun ht => _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nht : t \u2208 \u2a05 (i : \u03b9), l i\n\u22a2 \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227 \u22c2 (i_1 : \u2191i.fst), s (\u2191i_1) (Sigma.snd i i_1) \u2286 t\n[PROOFSTEP]\nrcases(hasBasis_iInf' hl).mem_iff.mp ht with \u27e8\u27e8I, f\u27e9, \u27e8hI, hf\u27e9, hsub\u27e9\n[GOAL]\ncase refine'_1.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nht : t \u2208 \u2a05 (i : \u03b9), l i\nI : Set \u03b9\nf : (i : \u03b9) \u2192 \u03b9' i\nhsub : \u22c2 (i : \u03b9) (_ : i \u2208 (I, f).fst), s i (Prod.snd (I, f) i) \u2286 t\nhI : Set.Finite (I, f).fst\nhf : \u2200 (i : \u03b9), i \u2208 (I, f).fst \u2192 p i (Prod.snd (I, f) i)\n\u22a2 \u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227 \u22c2 (i_1 : \u2191i.fst), s (\u2191i_1) (Sigma.snd i i_1) \u2286 t\n[PROOFSTEP]\nexact \u27e8\u27e8I, fun i => f i\u27e9, \u27e8hI, Subtype.forall.mpr hf\u27e9, trans (iInter_subtype _ _) hsub\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 (\u2203 i,\n (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227\n \u22c2 (i_1 : \u2191i.fst), s (\u2191i_1) (Sigma.snd i i_1) \u2286 t) \u2192\n t \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nrintro \u27e8\u27e8I, f\u27e9, \u27e8hI, hf\u27e9, hsub\u27e9\n[GOAL]\ncase refine'_2.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nI : Set \u03b9\nf : (i : \u2191I) \u2192 \u03b9' \u2191i\nhsub : \u22c2 (i : \u2191{ fst := I, snd := f }.fst), s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t\nhI : Set.Finite { fst := I, snd := f }.fst\nhf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i)\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nrefine' mem_of_superset _ hsub\n[GOAL]\ncase refine'_2.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nI : Set \u03b9\nf : (i : \u2191I) \u2192 \u03b9' \u2191i\nhsub : \u22c2 (i : \u2191{ fst := I, snd := f }.fst), s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t\nhI : Set.Finite { fst := I, snd := f }.fst\nhf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i)\n\u22a2 \u22c2 (i : \u2191{ fst := I, snd := f }.fst), s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\ncases hI.nonempty_fintype\n[GOAL]\ncase refine'_2.intro.mk.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\nI : Set \u03b9\nf : (i : \u2191I) \u2192 \u03b9' \u2191i\nhsub : \u22c2 (i : \u2191{ fst := I, snd := f }.fst), s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t\nhI : Set.Finite { fst := I, snd := f }.fst\nhf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i)\nval\u271d : Fintype \u2191{ fst := I, snd := f }.fst\n\u22a2 \u22c2 (i : \u2191{ fst := I, snd := f }.fst), s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2208 \u2a05 (i : \u03b9), l i\n[PROOFSTEP]\nexact iInter_mem.2 fun i => mem_iInf_of_mem \u2191i <| (hl i).mem_of_mem <| hf _\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\n\u22a2 HasBasis (\u2a05 (i : \u03b9), l i) (fun ii' => p ii'.fst ii'.snd) fun ii' => s ii'.fst ii'.snd\n[PROOFSTEP]\nrefine' \u27e8fun t => _\u27e9\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i \u2194 \u2203 i, p i.fst i.snd \u2227 s i.fst i.snd \u2286 t\n[PROOFSTEP]\nrw [mem_iInf_of_directed h, Sigma.exists]\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\nt : Set \u03b1\n\u22a2 (\u2203 i, t \u2208 l i) \u2194\n \u2203 a b,\n p { fst := a, snd := b }.fst { fst := a, snd := b }.snd \u2227\n s { fst := a, snd := b }.fst { fst := a, snd := b }.snd \u2286 t\n[PROOFSTEP]\nexact exists_congr fun i => (hl i).mem_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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\n\u22a2 HasBasis (\u2a05 (i : \u03b9), l i) (fun ii' => p ii'.fst ii'.snd) fun ii' => s ii'.fst ii'.snd\n[PROOFSTEP]\nrefine' \u27e8fun t => _\u27e9\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9), l i \u2194 \u2203 i, p i.fst i.snd \u2227 s i.fst i.snd \u2286 t\n[PROOFSTEP]\nrw [mem_iInf_of_directed h, Prod.exists]\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ninst\u271d : Nonempty \u03b9\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nh : Directed (fun x x_1 => x \u2265 x_1) l\nt : Set \u03b1\n\u22a2 (\u2203 i, t \u2208 l i) \u2194 \u2203 a b, p (a, b).fst (a, b).snd \u2227 s (a, b).fst (a, b).snd \u2286 t\n[PROOFSTEP]\nexact exists_congr fun i => (hl i).mem_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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\n\u22a2 HasBasis (\u2a05 (i : \u03b9) (_ : i \u2208 dom), l i) (fun ii' => ii'.fst \u2208 dom \u2227 p ii'.fst ii'.snd) fun ii' => s ii'.fst ii'.snd\n[PROOFSTEP]\nrefine' \u27e8fun t => _\u27e9\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 dom), l i \u2194 \u2203 i, (i.fst \u2208 dom \u2227 p i.fst i.snd) \u2227 s i.fst i.snd \u2286 t\n[PROOFSTEP]\nrw [mem_biInf_of_directed h hdom, Sigma.exists]\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\n\u22a2 (\u2203 i, i \u2208 dom \u2227 t \u2208 l i) \u2194\n \u2203 a b,\n ({ fst := a, snd := b }.fst \u2208 dom \u2227 p { fst := a, snd := b }.fst { fst := a, snd := b }.snd) \u2227\n s { fst := a, snd := b }.fst { fst := a, snd := b }.snd \u2286 t\n[PROOFSTEP]\nrefine' exists_congr fun i => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\n\u22a2 i \u2208 dom \u2227 t \u2208 l i \u2192\n \u2203 b,\n ({ fst := i, snd := b }.fst \u2208 dom \u2227 p { fst := i, snd := b }.fst { fst := i, snd := b }.snd) \u2227\n s { fst := i, snd := b }.fst { fst := i, snd := b }.snd \u2286 t\n[PROOFSTEP]\nrintro \u27e8hi, hti\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nhi : i \u2208 dom\nhti : t \u2208 l i\n\u22a2 \u2203 b,\n ({ fst := i, snd := b }.fst \u2208 dom \u2227 p { fst := i, snd := b }.fst { fst := i, snd := b }.snd) \u2227\n s { fst := i, snd := b }.fst { fst := i, snd := b }.snd \u2286 t\n[PROOFSTEP]\nrcases(hl i hi).mem_iff.mp hti with \u27e8b, hb, hbt\u27e9\n[GOAL]\ncase refine'_1.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nhi : i \u2208 dom\nhti : t \u2208 l i\nb : \u03b9' i\nhb : p i b\nhbt : s i b \u2286 t\n\u22a2 \u2203 b,\n ({ fst := i, snd := b }.fst \u2208 dom \u2227 p { fst := i, snd := b }.fst { fst := i, snd := b }.snd) \u2227\n s { fst := i, snd := b }.fst { fst := i, snd := b }.snd \u2286 t\n[PROOFSTEP]\nexact \u27e8b, \u27e8hi, hb\u27e9, hbt\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\n\u22a2 (\u2203 b,\n ({ fst := i, snd := b }.fst \u2208 dom \u2227 p { fst := i, snd := b }.fst { fst := i, snd := b }.snd) \u2227\n s { fst := i, snd := b }.fst { fst := i, snd := b }.snd \u2286 t) \u2192\n i \u2208 dom \u2227 t \u2208 l i\n[PROOFSTEP]\nrintro \u27e8b, \u27e8hi, hb\u27e9, hibt\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\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : \u03b9 \u2192 Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nb : \u03b9' i\nhibt : s { fst := i, snd := b }.fst { fst := i, snd := b }.snd \u2286 t\nhi : { fst := i, snd := b }.fst \u2208 dom\nhb : p { fst := i, snd := b }.fst { fst := i, snd := b }.snd\n\u22a2 i \u2208 dom \u2227 t \u2208 l i\n[PROOFSTEP]\nexact \u27e8hi, (hl i hi).mem_iff.mpr \u27e8b, hb, hibt\u27e9\u27e9\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\n\u22a2 HasBasis (\u2a05 (i : \u03b9) (_ : i \u2208 dom), l i) (fun ii' => ii'.fst \u2208 dom \u2227 p ii'.fst ii'.snd) fun ii' => s ii'.fst ii'.snd\n[PROOFSTEP]\nrefine' \u27e8fun t => _\u27e9\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 dom), l i \u2194 \u2203 i, (i.fst \u2208 dom \u2227 p i.fst i.snd) \u2227 s i.fst i.snd \u2286 t\n[PROOFSTEP]\nrw [mem_biInf_of_directed h hdom, Prod.exists]\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\n\u22a2 (\u2203 i, i \u2208 dom \u2227 t \u2208 l i) \u2194 \u2203 a b, ((a, b).fst \u2208 dom \u2227 p (a, b).fst (a, b).snd) \u2227 s (a, b).fst (a, b).snd \u2286 t\n[PROOFSTEP]\nrefine' exists_congr fun i => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\n\u22a2 i \u2208 dom \u2227 t \u2208 l i \u2192 \u2203 b, ((i, b).fst \u2208 dom \u2227 p (i, b).fst (i, b).snd) \u2227 s (i, b).fst (i, b).snd \u2286 t\n[PROOFSTEP]\nrintro \u27e8hi, hti\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nhi : i \u2208 dom\nhti : t \u2208 l i\n\u22a2 \u2203 b, ((i, b).fst \u2208 dom \u2227 p (i, b).fst (i, b).snd) \u2227 s (i, b).fst (i, b).snd \u2286 t\n[PROOFSTEP]\nrcases(hl i hi).mem_iff.mp hti with \u27e8b, hb, hbt\u27e9\n[GOAL]\ncase refine'_1.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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nhi : i \u2208 dom\nhti : t \u2208 l i\nb : \u03b9'\nhb : p i b\nhbt : s i b \u2286 t\n\u22a2 \u2203 b, ((i, b).fst \u2208 dom \u2227 p (i, b).fst (i, b).snd) \u2227 s (i, b).fst (i, b).snd \u2286 t\n[PROOFSTEP]\nexact \u27e8b, \u27e8hi, hb\u27e9, hbt\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\n\u22a2 (\u2203 b, ((i, b).fst \u2208 dom \u2227 p (i, b).fst (i, b).snd) \u2227 s (i, b).fst (i, b).snd \u2286 t) \u2192 i \u2208 dom \u2227 t \u2208 l i\n[PROOFSTEP]\nrintro \u27e8b, \u27e8hi, hb\u27e9, hibt\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\u03b9\u271d : Sort u_4\n\u03b9'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Type u_6\n\u03b9' : Type u_7\ndom : Set \u03b9\nhdom : Set.Nonempty dom\nl : \u03b9 \u2192 Filter \u03b1\ns : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1\np : \u03b9 \u2192 \u03b9' \u2192 Prop\nhl : \u2200 (i : \u03b9), i \u2208 dom \u2192 HasBasis (l i) (p i) (s i)\nh : DirectedOn (l \u207b\u00b9'o GE.ge) dom\nt : Set \u03b1\ni : \u03b9\nb : \u03b9'\nhibt : s (i, b).fst (i, b).snd \u2286 t\nhi : (i, b).fst \u2208 dom\nhb : p (i, b).fst (i, b).snd\n\u22a2 i \u2208 dom \u2227 t \u2208 l i\n[PROOFSTEP]\nexact \u27e8hi, (hl i hi).mem_iff.mpr \u27e8b, hb, hibt\u27e9\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nt U : Set \u03b1\n\u22a2 U \u2208 \ud835\udcdf t \u2194 \u2203 i, True \u2227 t \u2286 U\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nx : \u03b1\n\u22a2 HasBasis (pure x) (fun x => True) fun x_1 => {x}\n[PROOFSTEP]\nsimp only [\u2190 principal_singleton, hasBasis_principal]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 l \u2294 l' \u2194 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u222a s' i.snd \u2286 t\n[PROOFSTEP]\nintro t\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 t \u2208 l \u2294 l' \u2194 \u2203 i, (p i.fst \u2227 p' i.snd) \u2227 s i.fst \u222a s' i.snd \u2286 t\n[PROOFSTEP]\nsimp_rw [mem_sup, hl.mem_iff, hl'.mem_iff, PProd.exists, union_subset_iff, \u2190 exists_and_right, \u2190 exists_and_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set \u03b1\n\u22a2 (\u2203 x x_1, (p x \u2227 s x \u2286 t) \u2227 p' x_1 \u2227 s' x_1 \u2286 t) \u2194 \u2203 a b, (p a \u2227 p' b) \u2227 s a \u2286 t \u2227 s' b \u2286 t\n[PROOFSTEP]\nsimp only [and_assoc, and_left_comm]\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'\u271d : Sort u_5\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9'\u271d \u2192 Prop\ns' : \u03b9'\u271d \u2192 Set \u03b1\ni' : \u03b9'\u271d\n\u03b9 : Sort u_6\n\u03b9' : \u03b9 \u2192 Type u_7\nl : \u03b9 \u2192 Filter \u03b1\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1\nhl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i)\nt : Set \u03b1\n\u22a2 t \u2208 \u2a06 (i : \u03b9), l i \u2194 \u2203 i, (\u2200 (i_1 : \u03b9), p i_1 (i i_1)) \u2227 \u22c3 (i_1 : \u03b9), s i_1 (i i_1) \u2286 t\n[PROOFSTEP]\nsimp only [hasBasis_iff, (hl _).mem_iff, Classical.skolem, forall_and, iUnion_subset_iff, mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nt u : Set \u03b1\n\u22a2 u \u2208 l \u2294 \ud835\udcdf t \u2194 \u2203 i, p i \u2227 s i \u222a t \u2286 u\n[PROOFSTEP]\nsimp only [(hl.sup' (hasBasis_principal t)).mem_iff, PProd.exists, exists_prop, and_true_iff, Unique.exists_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nx : \u03b1\n\u22a2 HasBasis (l \u2294 pure x) p fun i => s i \u222a {x}\n[PROOFSTEP]\nsimp only [\u2190 principal_singleton, hl.sup_principal]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns'\u271d : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\ns' t : Set \u03b1\n\u22a2 t \u2208 l \u2293 \ud835\udcdf s' \u2194 \u2203 i, p i \u2227 s i \u2229 s' \u2286 t\n[PROOFSTEP]\nsimp only [mem_inf_principal, hl.mem_iff, subset_def, mem_setOf_eq, mem_inter_iff, and_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns'\u271d : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\ns' : Set \u03b1\n\u22a2 HasBasis (\ud835\udcdf s' \u2293 l) p fun i => s' \u2229 s i\n[PROOFSTEP]\nsimpa only [inf_comm, inter_comm] using hl.inf_principal s'\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 (\u2200 {i : PProd \u03b9 \u03b9'}, p i.fst \u2227 p' i.snd \u2192 Set.Nonempty (s i.fst \u2229 s' i.snd)) \u2194\n \u2200 \u2983i : \u03b9\u2984, p i \u2192 \u2200 \u2983i' : \u03b9'\u2984, p' i' \u2192 Set.Nonempty (s i \u2229 s' i')\n[PROOFSTEP]\nsimp [@forall_swap _ \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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 \u00acDisjoint l l' \u2194 \u00ac\u2203 i, p i \u2227 \u2203 i', p' i' \u2227 Disjoint (s i) (s' i')\n[PROOFSTEP]\nsimp only [_root_.disjoint_iff, \u2190 Ne.def, \u2190 neBot_iff, inf_eq_inter, hl.inf_basis_neBot_iff hl', not_exists, not_and,\n bot_eq_empty, \u2190 nonempty_iff_ne_empty]\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_7\ninst\u271d : Finite I\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_6\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nhd : Pairwise (Disjoint on l)\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 Pairwise (Disjoint on fun i => s i (ind i))\n[PROOFSTEP]\nrcases hd.exists_mem_filter_of_disjoint with \u27e8t, htl, hd\u27e9\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_7\ninst\u271d : Finite I\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_6\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nhd\u271d : Pairwise (Disjoint on l)\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\nt : I \u2192 Set \u03b1\nhtl : \u2200 (i : I), t i \u2208 l i\nhd : Pairwise (Disjoint on t)\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 Pairwise (Disjoint on fun i => s i (ind i))\n[PROOFSTEP]\nchoose ind hp ht using fun i => (h i).mem_iff.1 (htl i)\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_7\ninst\u271d : Finite I\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_6\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nhd\u271d : Pairwise (Disjoint on l)\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\nt : I \u2192 Set \u03b1\nhtl : \u2200 (i : I), t i \u2208 l i\nhd : Pairwise (Disjoint on t)\nind : (i : I) \u2192 \u03b9 i\nhp : \u2200 (i : I), p i (ind i)\nht : \u2200 (i : I), s i (ind i) \u2286 t i\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 Pairwise (Disjoint on fun i => s i (ind i))\n[PROOFSTEP]\nexact \u27e8ind, hp, hd.mono fun i j hij => hij.mono (ht _) (ht _)\u27e9\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_6\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_7\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nS : Set I\nhd : PairwiseDisjoint S l\nhS : Set.Finite S\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 PairwiseDisjoint S fun i => s i (ind i)\n[PROOFSTEP]\nrcases hd.exists_mem_filter hS with \u27e8t, htl, hd\u27e9\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_6\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_7\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nS : Set I\nhd\u271d : PairwiseDisjoint S l\nhS : Set.Finite S\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\nt : I \u2192 Set \u03b1\nhtl : \u2200 (i : I), t i \u2208 l i\nhd : PairwiseDisjoint S t\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 PairwiseDisjoint S fun i => s i (ind i)\n[PROOFSTEP]\nchoose ind hp ht using fun i => (h i).mem_iff.1 (htl i)\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\nl\u271d l' : Filter \u03b1\np\u271d : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nI : Type u_6\nl : I \u2192 Filter \u03b1\n\u03b9 : I \u2192 Sort u_7\np : (i : I) \u2192 \u03b9 i \u2192 Prop\ns : (i : I) \u2192 \u03b9 i \u2192 Set \u03b1\nS : Set I\nhd\u271d : PairwiseDisjoint S l\nhS : Set.Finite S\nh : \u2200 (i : I), HasBasis (l i) (p i) (s i)\nt : I \u2192 Set \u03b1\nhtl : \u2200 (i : I), t i \u2208 l i\nhd : PairwiseDisjoint S t\nind : (i : I) \u2192 \u03b9 i\nhp : \u2200 (i : I), p i (ind i)\nht : \u2200 (i : I), s i (ind i) \u2286 t i\n\u22a2 \u2203 ind, (\u2200 (i : I), p i (ind i)) \u2227 PairwiseDisjoint S fun i => s i (ind i)\n[PROOFSTEP]\nexact \u27e8ind, hp, hd.mono ht\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 f \u2194 f \u2293 \ud835\udcdf s\u1d9c = \u22a5\n[PROOFSTEP]\nrefine' not_iff_not.1 ((inf_principal_neBot_iff.trans _).symm.trans neBot_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 (\u2200 (U : Set \u03b1), U \u2208 f \u2192 Set.Nonempty (U \u2229 s\u1d9c)) \u2194 \u00acs \u2208 f\n[PROOFSTEP]\nexact\n \u27e8fun h hs => by simpa [Set.not_nonempty_empty] using h s hs, fun hs t ht =>\n inter_compl_nonempty_iff.2 fun hts => hs <| mem_of_superset ht hts\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : Filter \u03b1\ns : Set \u03b1\nh : \u2200 (U : Set \u03b1), U \u2208 f \u2192 Set.Nonempty (U \u2229 s\u1d9c)\nhs : s \u2208 f\n\u22a2 False\n[PROOFSTEP]\nsimpa [Set.not_nonempty_empty] using h s 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 Disjoint f (\ud835\udcdf s) \u2194 s\u1d9c \u2208 f\n[PROOFSTEP]\nrw [mem_iff_inf_principal_compl, compl_compl, disjoint_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 Disjoint (\ud835\udcdf s) f \u2194 s\u1d9c \u2208 f\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_principal_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns t : Set \u03b1\n\u22a2 Disjoint (\ud835\udcdf s) (\ud835\udcdf t) \u2194 Disjoint s t\n[PROOFSTEP]\nrw [\u2190 subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nx y : \u03b1\n\u22a2 Disjoint (pure x) (pure y) \u2194 x \u2260 y\n[PROOFSTEP]\nsimp only [\u2190 principal_singleton, disjoint_principal_principal, disjoint_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nl\u2081 l\u2082 : Filter \u03b1\n\u22a2 (diagonal \u03b1)\u1d9c \u2208 l\u2081 \u00d7\u02e2 l\u2082 \u2194 Disjoint l\u2081 l\u2082\n[PROOFSTEP]\nsimp only [mem_prod_iff, Filter.disjoint_iff, prod_subset_compl_diagonal_iff_disjoint]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 Disjoint l l' \u2194 \u2203 i, p i \u2227 (s i)\u1d9c \u2208 l'\n[PROOFSTEP]\nsimp only [h.disjoint_iff l'.basis_sets, id, \u2190 disjoint_principal_left,\n (hasBasis_principal _).disjoint_iff l'.basis_sets, true_and, Unique.exists_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf g : Filter \u03b1\n\u22a2 NeBot (f \u2293 g) \u2194 \u2200 {p : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in f, p x) \u2192 \u2203\u1da0 (x : \u03b1) in g, p x\n[PROOFSTEP]\nsimp only [inf_neBot_iff, frequently_iff, and_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf g : Filter \u03b1\n\u22a2 (\u2200 \u2983s : Set \u03b1\u2984, s \u2208 f \u2192 \u2200 \u2983s' : Set \u03b1\u2984, s' \u2208 g \u2192 Set.Nonempty (s \u2229 s')) \u2194\n \u2200 {p : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in f, p x) \u2192 \u2200 {U : Set \u03b1}, U \u2208 g \u2192 \u2203 x, p x \u2227 x \u2208 U\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf g : Filter \u03b1\n\u22a2 NeBot (f \u2293 g) \u2194 \u2200 {p : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in g, p x) \u2192 \u2203\u1da0 (x : \u03b1) in f, p x\n[PROOFSTEP]\nrw [inf_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf g : Filter \u03b1\n\u22a2 NeBot (g \u2293 f) \u2194 \u2200 {p : \u03b1 \u2192 Prop}, (\u2200\u1da0 (x : \u03b1) in g, p x) \u2192 \u2203\u1da0 (x : \u03b1) in f, p x\n[PROOFSTEP]\nexact inf_neBot_iff_frequently_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx\u271d : Set \u03b1\n\u22a2 x\u271d \u2208 l \u2194 \u2203 i, p i \u2227 x\u271d \u2208 \ud835\udcdf (s i)\n[PROOFSTEP]\nsimp only [h.mem_iff, mem_principal, exists_prop]\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l (fun x => True) s\n\u22a2 l = \u2a05 (i : \u03b9), \ud835\udcdf (s i)\n[PROOFSTEP]\nsimpa only [iInf_true] using h.eq_biInf\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2265 x_1) s\ninst\u271d : Nonempty \u03b9\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b9), \ud835\udcdf (s i) \u2194 \u2203 i, True \u2227 s i \u2286 t\n[PROOFSTEP]\nsimpa only [true_and] using mem_iInf_of_directed (h.mono_comp monotone_principal.dual) t\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\nl l' : Filter \u03b1\np : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\n\u03b9 : Type u_6\ns : \u03b9 \u2192 Set \u03b1\n\u22a2 HasBasis (\u2a05 (i : \u03b9), \ud835\udcdf (s i)) (fun t => Set.Finite t) fun t => \u22c2 (i : \u03b9) (_ : i \u2208 t), s i\n[PROOFSTEP]\nrefine' \u27e8fun U => (mem_iInf_finite _).trans _\u27e9\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\nl l' : Filter \u03b1\np : \u03b9\u271d \u2192 Prop\ns\u271d : \u03b9\u271d \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\u271d\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\n\u03b9 : Type u_6\ns : \u03b9 \u2192 Set \u03b1\nU : Set \u03b1\n\u22a2 (\u2203 t, U \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 t), \ud835\udcdf (s i)) \u2194 \u2203 i, Set.Finite i \u2227 \u22c2 (i_1 : \u03b9) (_ : i_1 \u2208 i), s i_1 \u2286 U\n[PROOFSTEP]\nsimp only [iInf_principal_finset, mem_iUnion, mem_principal, exists_prop, exists_finite_iff_finset,\n Finset.set_biInter_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : \u03b2 \u2192 Set \u03b1\nS : Set \u03b2\nh : DirectedOn (s \u207b\u00b9'o fun x x_1 => x \u2265 x_1) S\nne : Set.Nonempty S\nt : Set \u03b1\n\u22a2 t \u2208 \u2a05 (i : \u03b2) (_ : i \u2208 S), \ud835\udcdf (s i) \u2194 \u2203 i, i \u2208 S \u2227 s i \u2286 t\n[PROOFSTEP]\nrefine' mem_biInf_of_directed _ ne\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : \u03b2 \u2192 Set \u03b1\nS : Set \u03b2\nh : DirectedOn (s \u207b\u00b9'o fun x x_1 => x \u2265 x_1) S\nne : Set.Nonempty S\nt : Set \u03b1\n\u22a2 DirectedOn ((fun i => \ud835\udcdf (s i)) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) S\n[PROOFSTEP]\nrw [directedOn_iff_directed, \u2190 directed_comp] at h \u22a2\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : \u03b2 \u2192 Set \u03b1\nS : Set \u03b2\nh : Directed (fun x x_1 => x \u2265 x_1) (s \u2218 Subtype.val)\nne : Set.Nonempty S\nt : Set \u03b1\n\u22a2 Directed (fun x x_1 => x \u2265 x_1) ((fun i => \ud835\udcdf (s i)) \u2218 Subtype.val)\n[PROOFSTEP]\nrefine' h.mono_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : \u03b2 \u2192 Set \u03b1\nS : Set \u03b2\nh : Directed (fun x x_1 => x \u2265 x_1) (s \u2218 Subtype.val)\nne : Set.Nonempty S\nt : Set \u03b1\n\u22a2 \u2200 \u2983x y : Set \u03b1\u2984, x \u2265 y \u2192 \ud835\udcdf x \u2265 \ud835\udcdf y\n[PROOFSTEP]\nexact fun _ _ => principal_mono.2\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : \u03b1 \u2192 \u03b2\nhl : HasBasis l p s\nt : Set \u03b2\n\u22a2 t \u2208 Filter.map f l \u2194 \u2203 i, p i \u2227 f '' s i \u2286 t\n[PROOFSTEP]\nsimp only [mem_map, image_subset_iff, hl.mem_iff, 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : \u03b2 \u2192 \u03b1\nhl : HasBasis l p s\nt : Set \u03b2\n\u22a2 t \u2208 Filter.comap f l \u2194 \u2203 i, p i \u2227 f \u207b\u00b9' s i \u2286 t\n[PROOFSTEP]\nsimp only [mem_comap', hl.mem_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : \u03b2 \u2192 \u03b1\nhl : HasBasis l p s\nt : Set \u03b2\n\u22a2 (\u2203 i, p i \u2227 s i \u2286 {y | \u2200 \u2983x : \u03b2\u2984, f x = y \u2192 x \u2208 t}) \u2194 \u2203 i, p i \u2227 f \u207b\u00b9' s i \u2286 t\n[PROOFSTEP]\nrefine exists_congr (fun i => Iff.rfl.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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : \u03b2 \u2192 \u03b1\nhl : HasBasis l p s\nt : Set \u03b2\ni : \u03b9\n\u22a2 s i \u2286 {y | \u2200 \u2983x : \u03b2\u2984, f x = y \u2192 x \u2208 t} \u2194 f \u207b\u00b9' s i \u2286 t\n[PROOFSTEP]\nexact \u27e8fun h x hx => h hx rfl, fun h y hy x hx => h <| by rwa [mem_preimage, 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt\u271d : Set \u03b1\ni\u271d : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nf : \u03b2 \u2192 \u03b1\nhl : HasBasis l p s\nt : Set \u03b2\ni : \u03b9\nh : f \u207b\u00b9' s i \u2286 t\ny : \u03b1\nhy : y \u2208 s i\nx : \u03b2\nhx : f x = y\n\u22a2 x \u2208 f \u207b\u00b9' s i\n[PROOFSTEP]\nrwa [mem_preimage, 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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx : \u03b1\n\u22a2 (\u2200 (t : Set \u03b1), t \u2208 l \u2192 x \u2208 t) \u2194 \u2200 (i : \u03b9), p i \u2192 x \u2208 s i\n[PROOFSTEP]\nsimp only [h.mem_iff, exists_imp, and_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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx : \u03b1\n\u22a2 (\u2200 (t : Set \u03b1) (x_1 : \u03b9), p x_1 \u2192 s x_1 \u2286 t \u2192 x \u2208 t) \u2194 \u2200 (i : \u03b9), p i \u2192 x \u2208 s i\n[PROOFSTEP]\nexact \u27e8fun h i hi => h (s i) i hi Subset.rfl, fun h t i hi ht => ht (h i hi)\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 \u22c2\u2080 l.sets = \u22c2 (i : \u03b9) (_ : p i), s i\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\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 \u22c2 (i : Set \u03b1) (_ : i \u2208 l.sets), i = \u22c2 (i : \u03b9) (_ : p i), s i\n[PROOFSTEP]\nexact h.biInter_mem monotone_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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhla : HasBasis la pa sa\n\u22a2 Tendsto f la lb \u2194 \u2200 (t : Set \u03b2), t \u2208 lb \u2192 \u2203 i, pa i \u2227 MapsTo f (sa i) t\n[PROOFSTEP]\nsimp only [Tendsto, (hla.map f).le_iff, image_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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhla : HasBasis la pa sa\n\u22a2 (\u2200 (t : Set \u03b2), t \u2208 lb \u2192 \u2203 i, pa i \u2227 sa i \u2286 f \u207b\u00b9' t) \u2194 \u2200 (t : Set \u03b2), t \u2208 lb \u2192 \u2203 i, pa i \u2227 MapsTo f (sa i) t\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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhlb : HasBasis lb pb sb\n\u22a2 Tendsto f la lb \u2194 \u2200 (i : \u03b9'), pb i \u2192 \u2200\u1da0 (x : \u03b1) in la, f x \u2208 sb i\n[PROOFSTEP]\nsimp only [Tendsto, hlb.ge_iff, mem_map', Filter.Eventually]\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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhla : HasBasis la pa sa\nhlb : HasBasis lb pb sb\n\u22a2 Tendsto f la lb \u2194 \u2200 (ib : \u03b9'), pb ib \u2192 \u2203 ia, pa ia \u2227 \u2200 (x : \u03b1), x \u2208 sa ia \u2192 f x \u2208 sb ib\n[PROOFSTEP]\nsimp [hlb.tendsto_right_iff, hla.eventually_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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\n\u22a2 HasBasis (la \u00d7\u02e2 lb) p fun i => sa i \u00d7\u02e2 sb i\n[PROOFSTEP]\nsimp only [hasBasis_iff, (hla.prod_pprod hlb).mem_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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b2)), (\u2203 i, (p i.fst \u2227 p i.snd) \u2227 sa i.fst \u00d7\u02e2 sb i.snd \u2286 t) \u2194 \u2203 i, p i \u2227 sa i \u00d7\u02e2 sb i \u2286 t\n[PROOFSTEP]\nrefine' fun t => \u27e8_, _\u27e9\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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\nt : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 (\u2203 i, (p i.fst \u2227 p i.snd) \u2227 sa i.fst \u00d7\u02e2 sb i.snd \u2286 t) \u2192 \u2203 i, p i \u2227 sa i \u00d7\u02e2 sb i \u2286 t\n[PROOFSTEP]\nrintro \u27e8\u27e8i, j\u27e9, \u27e8hi, hj\u27e9, hsub : sa i \u00d7\u02e2 sb j \u2286 t\u27e9\n[GOAL]\ncase refine'_1.intro.mk.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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\nt : Set (\u03b1 \u00d7 \u03b2)\ni j : \u03b9\nhsub : sa i \u00d7\u02e2 sb j \u2286 t\nhi : p { fst := i, snd := j }.fst\nhj : p { fst := i, snd := j }.snd\n\u22a2 \u2203 i, p i \u2227 sa i \u00d7\u02e2 sb i \u2286 t\n[PROOFSTEP]\nrcases h_dir hi hj with \u27e8k, hk, ki, kj\u27e9\n[GOAL]\ncase refine'_1.intro.mk.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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\nt : Set (\u03b1 \u00d7 \u03b2)\ni j : \u03b9\nhsub : sa i \u00d7\u02e2 sb j \u2286 t\nhi : p { fst := i, snd := j }.fst\nhj : p { fst := i, snd := j }.snd\nk : \u03b9\nhk : p k\nki : sa k \u2286 sa { fst := i, snd := j }.fst\nkj : sb k \u2286 sb { fst := i, snd := j }.snd\n\u22a2 \u2203 i, p i \u2227 sa i \u00d7\u02e2 sb i \u2286 t\n[PROOFSTEP]\nexact \u27e8k, hk, (Set.prod_mono ki kj).trans hsub\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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\nt : Set (\u03b1 \u00d7 \u03b2)\n\u22a2 (\u2203 i, p i \u2227 sa i \u00d7\u02e2 sb i \u2286 t) \u2192 \u2203 i, (p i.fst \u2227 p i.snd) \u2227 sa i.fst \u00d7\u02e2 sb i.snd \u2286 t\n[PROOFSTEP]\nrintro \u27e8i, hi, h\u27e9\n[GOAL]\ncase refine'_2.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\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb\u271d : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\np : \u03b9 \u2192 Prop\nsb : \u03b9 \u2192 Set \u03b2\nhla : HasBasis la p sa\nhlb : HasBasis lb p sb\nh_dir : \u2200 {i j : \u03b9}, p i \u2192 p j \u2192 \u2203 k, p k \u2227 sa k \u2286 sa i \u2227 sb k \u2286 sb j\nt : Set (\u03b1 \u00d7 \u03b2)\ni : \u03b9\nhi : p i\nh : sa i \u00d7\u02e2 sb i \u2286 t\n\u22a2 \u2203 i, (p i.fst \u2227 p i.snd) \u2227 sa i.fst \u00d7\u02e2 sb i.snd \u2286 t\n[PROOFSTEP]\nexact \u27e8\u27e8i, i\u27e9, \u27e8hi, hi\u27e9, 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\u03b9' : Sort u_5\nla : Filter \u03b1\npa : \u03b9 \u2192 Prop\nsa : \u03b9 \u2192 Set \u03b1\nlb : Filter \u03b2\npb : \u03b9' \u2192 Prop\nsb : \u03b9' \u2192 Set \u03b2\nf : \u03b1 \u2192 \u03b2\nhl : HasBasis la pa sa\ni j : \u03b9\nhi : pa i\nhj : pa j\n\u22a2 \u2203 k, pa k \u2227 sa k \u2286 sa i \u2227 sa k \u2286 sa j\n[PROOFSTEP]\nsimpa only [exists_prop, subset_inter_iff] using hl.mem_iff.1 (inter_mem (hl.mem_of_mem hi) (hl.mem_of_mem hj))\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\u03c0 : \u03b1 \u2192 Type u_6\n\u03c0' : \u03b2 \u2192 Type u_7\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\ng : (a : \u03b1) \u2192 \u03c0 a \u2192 \u03c0' (f a)\na : \u03b1\nl : Filter (\u03c0' (f a))\n\u22a2 map (Sigma.mk a) (comap (g a) l) = comap (Sigma.map f g) (map (Sigma.mk (f a)) l)\n[PROOFSTEP]\nrefine' (((basis_sets _).comap _).map _).eq_of_same_basis _\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\u03c0 : \u03b1 \u2192 Type u_6\n\u03c0' : \u03b2 \u2192 Type u_7\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\ng : (a : \u03b1) \u2192 \u03c0 a \u2192 \u03c0' (f a)\na : \u03b1\nl : Filter (\u03c0' (f a))\n\u22a2 HasBasis (comap (Sigma.map f g) (map (Sigma.mk (f a)) l)) (fun s => s \u2208 l) fun i => Sigma.mk a '' (g a \u207b\u00b9' id i)\n[PROOFSTEP]\nconvert ((basis_sets l).map (Sigma.mk (f a))).comap (Sigma.map f g)\n[GOAL]\ncase h.e'_5.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\u03c0 : \u03b1 \u2192 Type u_6\n\u03c0' : \u03b2 \u2192 Type u_7\nf : \u03b1 \u2192 \u03b2\nhf : Function.Injective f\ng : (a : \u03b1) \u2192 \u03c0 a \u2192 \u03c0' (f a)\na : \u03b1\nl : Filter (\u03c0' (f a))\nx\u271d : Set (\u03c0' (f a))\n\u22a2 Sigma.mk a '' (g a \u207b\u00b9' id x\u271d) = Sigma.map f g \u207b\u00b9' (Sigma.mk (f a) '' id x\u271d)\n[PROOFSTEP]\napply image_sigmaMk_preimage_sigmaMap hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2203 t, Antitone t \u2227 \u2a05 (i : \u2115), \ud835\udcdf (s i) = \u2a05 (i : \u2115), \ud835\udcdf (t i)\n[PROOFSTEP]\nuse fun n => \u22c2 m \u2264 n, s m\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 (Antitone fun n => \u22c2 (m : \u2115) (_ : m \u2264 n), s m) \u2227 \u2a05 (i : \u2115), \ud835\udcdf (s i) = \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 Antitone fun n => \u22c2 (m : \u2115) (_ : m \u2264 n), s m\n[PROOFSTEP]\nexact fun i j hij => biInter_mono (Iic_subset_Iic.2 hij) fun n _ => Subset.rfl\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf (s i) = \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf (s i) \u2264 \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m)\n[PROOFSTEP]\nrw [le_iInf_iff]\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m) \u2264 \u2a05 (i : \u2115), \ud835\udcdf (s i)\n[PROOFSTEP]\nrw [le_iInf_iff]\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2200 (i : \u2115), \u2a05 (i : \u2115), \ud835\udcdf (s i) \u2264 \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\n\u22a2 \u2200 (i : \u2115), \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m) \u2264 \ud835\udcdf (s i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\ni : \u2115\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf (s i) \u2264 \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m)\n[PROOFSTEP]\nrw [le_principal_iff]\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\ni : \u2115\n\u22a2 \u22c2 (m : \u2115) (_ : m \u2264 i), s m \u2208 \u2a05 (i : \u2115), \ud835\udcdf (s i)\n[PROOFSTEP]\nrefine' (biInter_mem (finite_le_nat _)).2 fun j _ => _\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\ni j : \u2115\nx\u271d : j \u2208 {i_1 | i_1 \u2264 i}\n\u22a2 s j \u2208 \u2a05 (i : \u2115), \ud835\udcdf (s i)\n[PROOFSTEP]\nexact mem_iInf_of_mem j (mem_principal_self _)\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\ni : \u2115\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf (\u22c2 (m : \u2115) (_ : m \u2264 i), s m) \u2264 \ud835\udcdf (s i)\n[PROOFSTEP]\nrefine iInf_le_of_le i (principal_mono.2 <| iInter\u2082_subset i ?_)\n[GOAL]\ncase h.right.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ns : \u2115 \u2192 Set \u03b1\ni : \u2115\n\u22a2 i \u2264 i\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\n\u03b9' : Sort u_5\ninst\u271d : CompleteLattice \u03b1\nB : Set \u03b9\nBcbl : Set.Countable B\nf : \u03b9 \u2192 \u03b1\ni\u2080 : \u03b9\nh : f i\u2080 = \u22a4\n\u22a2 \u2203 x, \u2a05 (t : \u03b9) (_ : t \u2208 B), f t = \u2a05 (i : \u2115), f (x i)\n[PROOFSTEP]\ncases' B.eq_empty_or_nonempty with hB Bnonempty\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' : Sort u_5\ninst\u271d : CompleteLattice \u03b1\nB : Set \u03b9\nBcbl : Set.Countable B\nf : \u03b9 \u2192 \u03b1\ni\u2080 : \u03b9\nh : f i\u2080 = \u22a4\nhB : B = \u2205\n\u22a2 \u2203 x, \u2a05 (t : \u03b9) (_ : t \u2208 B), f t = \u2a05 (i : \u2115), f (x i)\n[PROOFSTEP]\nrw [hB, iInf_emptyset]\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' : Sort u_5\ninst\u271d : CompleteLattice \u03b1\nB : Set \u03b9\nBcbl : Set.Countable B\nf : \u03b9 \u2192 \u03b1\ni\u2080 : \u03b9\nh : f i\u2080 = \u22a4\nhB : B = \u2205\n\u22a2 \u2203 x, \u22a4 = \u2a05 (i : \u2115), f (x i)\n[PROOFSTEP]\nuse fun _ => i\u2080\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ninst\u271d : CompleteLattice \u03b1\nB : Set \u03b9\nBcbl : Set.Countable B\nf : \u03b9 \u2192 \u03b1\ni\u2080 : \u03b9\nh : f i\u2080 = \u22a4\nhB : B = \u2205\n\u22a2 \u22a4 = \u2a05 (i : \u2115), f i\u2080\n[PROOFSTEP]\nsimp [h]\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' : Sort u_5\ninst\u271d : CompleteLattice \u03b1\nB : Set \u03b9\nBcbl : Set.Countable B\nf : \u03b9 \u2192 \u03b1\ni\u2080 : \u03b9\nh : f i\u2080 = \u22a4\nBnonempty : Set.Nonempty B\n\u22a2 \u2203 x, \u2a05 (t : \u03b9) (_ : t \u2208 B), f t = \u2a05 (i : \u2115), f (x i)\n[PROOFSTEP]\nexact countable_biInf_eq_iInf_seq Bcbl Bnonempty f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ninst\u271d : Preorder \u03b9\nl : Filter \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : HasAntitoneBasis l s\nt : Set \u03b1\n\u22a2 (\u2203 i, True \u2227 s i \u2286 t) \u2194 \u2203 i, s i \u2286 t\n[PROOFSTEP]\nsimp only [exists_prop, true_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' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nobtain \u27e8x', hx'\u27e9 : \u2203 x : \u2115 \u2192 Set \u03b1, f = \u2a05 i, \ud835\udcdf (x i) :=\n by\n rcases h with \u27e8s, hsc, rfl\u27e9\n rw [generate_eq_biInf]\n exact countable_biInf_principal_eq_seq_iInf hsc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\n\u22a2 \u2203 x, f = \u2a05 (i : \u2115), \ud835\udcdf (x i)\n[PROOFSTEP]\nrcases h with \u27e8s, hsc, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\np : \u03b9' \u2192 Prop\ns\u271d : \u03b9' \u2192 Set \u03b1\ns : Set (Set \u03b1)\nhsc : Set.Countable s\nhs : HasBasis (generate s) p s\u271d\n\u22a2 \u2203 x, generate s = \u2a05 (i : \u2115), \ud835\udcdf (x i)\n[PROOFSTEP]\nrw [generate_eq_biInf]\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\np : \u03b9' \u2192 Prop\ns\u271d : \u03b9' \u2192 Set \u03b1\ns : Set (Set \u03b1)\nhsc : Set.Countable s\nhs : HasBasis (generate s) p s\u271d\n\u22a2 \u2203 x, \u2a05 (s_1 : Set \u03b1) (_ : s_1 \u2208 s), \ud835\udcdf s_1 = \u2a05 (i : \u2115), \ud835\udcdf (x i)\n[PROOFSTEP]\nexact countable_biInf_principal_eq_seq_iInf hsc\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nhave : \u2200 i, x' i \u2208 f := fun i => hx'.symm \u25b8 (iInf_le (fun i => \ud835\udcdf (x' i)) i) (mem_principal_self _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nlet x : \u2115 \u2192 { i : \u03b9' // p i } := fun n =>\n Nat.recOn n (hs.index _ <| this 0) fun n xn => hs.index _ <| inter_mem (this <| n + 1) (hs.mem_of_mem xn.2)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nhave x_mono : Antitone fun i => s (x i).1 :=\n antitone_nat_of_succ_le fun i => (hs.set_index_subset _).trans (inter_subset_right _ _)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nhave x_subset : \u2200 i, s (x i).1 \u2286 x' i := by\n rintro (_ | i)\n exacts [hs.set_index_subset _, (hs.set_index_subset _).trans (inter_subset_left _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\n\u22a2 \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\n[PROOFSTEP]\nrintro (_ | i)\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\n\u22a2 s \u2191(x Nat.zero) \u2286 x' Nat.zero\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\ni : \u2115\n\u22a2 s \u2191(x (Nat.succ i)) \u2286 x' (Nat.succ i)\n[PROOFSTEP]\nexacts [hs.set_index_subset _, (hs.set_index_subset _).trans (inter_subset_left _ _)]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\n\u22a2 \u2203 x, (\u2200 (i : \u2115), p (x i)) \u2227 HasAntitoneBasis f fun i => s (x i)\n[PROOFSTEP]\nrefine' \u27e8fun i => (x i).1, fun i => (x i).2, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\n\u22a2 HasAntitoneBasis f fun i => s ((fun i => \u2191(x i)) i)\n[PROOFSTEP]\nhave : (\u2a05 i, \ud835\udcdf (s (x i).1)).HasAntitoneBasis fun i => s (x i).1 :=\n \u27e8hasBasis_iInf_principal (directed_of_sup x_mono), x_mono\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis\u271d : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\nthis : HasAntitoneBasis (\u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))) fun i => s \u2191(x i)\n\u22a2 HasAntitoneBasis f fun i => s ((fun i => \u2191(x i)) i)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis\u271d : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\nthis : HasAntitoneBasis (\u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))) fun i => s \u2191(x i)\n\u22a2 f = \u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))\n[PROOFSTEP]\nexact\n le_antisymm (le_iInf fun i => le_principal_iff.2 <| by cases i <;> apply hs.set_index_mem)\n (hx'.symm \u25b8 le_iInf fun i => le_principal_iff.2 <| this.1.mem_iff.2 \u27e8i, trivial, x_subset i\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' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis\u271d : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\nthis : HasAntitoneBasis (\u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))) fun i => s \u2191(x i)\ni : \u2115\n\u22a2 s \u2191(x i) \u2208 f\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis\u271d : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\nthis : HasAntitoneBasis (\u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))) fun i => s \u2191(x i)\n\u22a2 s \u2191(x Nat.zero) \u2208 f\n[PROOFSTEP]\napply hs.set_index_mem\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b1\nhs : HasBasis f p s\nx' : \u2115 \u2192 Set \u03b1\nhx' : f = \u2a05 (i : \u2115), \ud835\udcdf (x' i)\nthis\u271d : \u2200 (i : \u2115), x' i \u2208 f\nx : \u2115 \u2192 { i // p i } :=\n fun n =>\n Nat.recOn n (index hs (x' 0) (_ : x' 0 \u2208 f)) fun n xn => index hs (x' (n + 1) \u2229 s \u2191xn) (_ : x' (n + 1) \u2229 s \u2191xn \u2208 f)\nx_mono : Antitone fun i => s \u2191(x i)\nx_subset : \u2200 (i : \u2115), s \u2191(x i) \u2286 x' i\nthis : HasAntitoneBasis (\u2a05 (i : \u2115), \ud835\udcdf (s \u2191(x i))) fun i => s \u2191(x i)\nn\u271d : \u2115\n\u22a2 s \u2191(x (Nat.succ n\u271d)) \u2208 f\n[PROOFSTEP]\napply hs.set_index_mem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\ninst\u271d : IsCountablyGenerated f\nx : \u2115 \u2192 Set \u03b1\nhx : HasAntitoneBasis f x\n\u22a2 \u2200 {s : Set \u03b1}, s \u2208 f \u2194 \u2203 i, x i \u2286 s\n[PROOFSTEP]\nsimp [hx.1.mem_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' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\n\u22a2 IsCountablyGenerated (f \u2293 g)\n[PROOFSTEP]\nrcases f.exists_antitone_basis with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\n\u22a2 IsCountablyGenerated (f \u2293 g)\n[PROOFSTEP]\nrcases g.exists_antitone_basis with \u27e8t, ht\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\n\u03b9' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nt : \u2115 \u2192 Set \u03b1\nht : HasAntitoneBasis g t\n\u22a2 IsCountablyGenerated (f \u2293 g)\n[PROOFSTEP]\nexact HasCountableBasis.isCountablyGenerated \u27e8hs.1.inf ht.1, Set.to_countable _\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' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\n\u22a2 IsCountablyGenerated (f \u2294 g)\n[PROOFSTEP]\nrcases f.exists_antitone_basis with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\n\u22a2 IsCountablyGenerated (f \u2294 g)\n[PROOFSTEP]\nrcases g.exists_antitone_basis with \u27e8t, ht\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\n\u03b9' : Sort u_5\nf g : Filter \u03b1\ninst\u271d\u00b9 : IsCountablyGenerated f\ninst\u271d : IsCountablyGenerated g\ns : \u2115 \u2192 Set \u03b1\nhs : HasAntitoneBasis f s\nt : \u2115 \u2192 Set \u03b1\nht : HasAntitoneBasis g t\n\u22a2 IsCountablyGenerated (f \u2294 g)\n[PROOFSTEP]\nexact HasCountableBasis.isCountablyGenerated \u27e8hs.1.sup ht.1, Set.to_countable _\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' : Sort u_5\ninst\u271d : Countable \u03b2\nx : \u03b2 \u2192 Set \u03b1\n\u22a2 IsCountablyGenerated (\u2a05 (i : \u03b2), \ud835\udcdf (x i))\n[PROOFSTEP]\nuse range x, countable_range x\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\ninst\u271d : Countable \u03b2\nx : \u03b2 \u2192 Set \u03b1\n\u22a2 \u2a05 (i : \u03b2), \ud835\udcdf (x i) = generate (range x)\n[PROOFSTEP]\nrw [generate_eq_biInf, iInf_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\nh : \u2203 x, f = \u2a05 (i : \u2115), \ud835\udcdf (x i)\n\u22a2 IsCountablyGenerated f\n[PROOFSTEP]\nrcases h with \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nx : \u2115 \u2192 Set \u03b1\n\u22a2 IsCountablyGenerated (\u2a05 (i : \u2115), \ud835\udcdf (x i))\n[PROOFSTEP]\napply isCountablyGenerated_seq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\nf : Filter \u03b1\n\u22a2 IsCountablyGenerated f \u2194 \u2203 x, HasAntitoneBasis f x\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' : Sort u_5\nf : Filter \u03b1\n\u22a2 IsCountablyGenerated f \u2192 \u2203 x, HasAntitoneBasis f x\n[PROOFSTEP]\nintro h\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' : Sort u_5\nf : Filter \u03b1\nh : IsCountablyGenerated f\n\u22a2 \u2203 x, HasAntitoneBasis f x\n[PROOFSTEP]\nexact f.exists_antitone_basis\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' : Sort u_5\nf : Filter \u03b1\n\u22a2 (\u2203 x, HasAntitoneBasis f x) \u2192 IsCountablyGenerated f\n[PROOFSTEP]\nrintro \u27e8x, h\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\n\u03b9' : Sort u_5\nf : Filter \u03b1\nx : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f x\n\u22a2 IsCountablyGenerated f\n[PROOFSTEP]\nrw [h.1.eq_iInf]\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' : Sort u_5\nf : Filter \u03b1\nx : \u2115 \u2192 Set \u03b1\nh : HasAntitoneBasis f x\n\u22a2 IsCountablyGenerated (\u2a05 (i : \u2115), \ud835\udcdf (x i))\n[PROOFSTEP]\nexact isCountablyGenerated_seq x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\na : \u03b1\n\u22a2 IsCountablyGenerated (pure a)\n[PROOFSTEP]\nrw [\u2190 principal_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Sort u_5\na : \u03b1\n\u22a2 IsCountablyGenerated (\ud835\udcdf {a})\n[PROOFSTEP]\nexact isCountablyGenerated_principal _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f i)\n\u22a2 IsCountablyGenerated (\u2a05 (i : \u03b9), f i)\n[PROOFSTEP]\nchoose s hs using fun i => exists_antitone_basis (f i)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f i) (s i)\n\u22a2 IsCountablyGenerated (\u2a05 (i : \u03b9), f i)\n[PROOFSTEP]\nrw [\u2190 PLift.down_surjective.iInf_comp]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f i) (s i)\n\u22a2 IsCountablyGenerated (\u2a05 (x : PLift \u03b9), f x.down)\n[PROOFSTEP]\nrefine' HasCountableBasis.isCountablyGenerated \u27e8hasBasis_iInf fun n => (hs _).1, _\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f i) (s i)\n\u22a2 Set.Countable {If | Set.Finite If.fst \u2227 (\u2191If.fst \u2192 True)}\n[PROOFSTEP]\nrefine' (countable_range <| Sigma.map ((\u2191) : Finset (PLift \u03b9) \u2192 Set (PLift \u03b9)) fun _ => id).mono _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f i) (s i)\n\u22a2 {If | Set.Finite If.fst \u2227 (\u2191If.fst \u2192 True)} \u2286 range (Sigma.map Finset.toSet fun x => id)\n[PROOFSTEP]\nrintro \u27e8I, f\u27e9 \u27e8hI, -\u27e9\n[GOAL]\ncase mk.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f\u271d i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f\u271d i) (s i)\nI : Set (PLift \u03b9)\nf : \u2191I \u2192 \u2115\nhI : Set.Finite { fst := I, snd := f }.fst\n\u22a2 { fst := I, snd := f } \u2208 range (Sigma.map Finset.toSet fun x => id)\n[PROOFSTEP]\nlift I to Finset (PLift \u03b9) using hI\n[GOAL]\ncase mk.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Sort u_5\n\u03b9 : Sort u\n\u03b1 : Type v\ninst\u271d\u00b9 : Countable \u03b9\nf\u271d : \u03b9 \u2192 Filter \u03b1\ninst\u271d : \u2200 (i : \u03b9), IsCountablyGenerated (f\u271d i)\ns : \u03b9 \u2192 \u2115 \u2192 Set \u03b1\nhs : \u2200 (i : \u03b9), HasAntitoneBasis (f\u271d i) (s i)\nI : Finset (PLift \u03b9)\nf : \u2191\u2191I \u2192 \u2115\n\u22a2 { fst := \u2191I, snd := f } \u2208 range (Sigma.map Finset.toSet fun x => id)\n[PROOFSTEP]\nexact \u27e8\u27e8I, f\u27e9, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Bases", "llama_tokens": 52216, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5621764862150634, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.2723070937713914}} {"text": "[GOAL]\nA : Mon_ (Type u)\nx y z : A.X\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\nconvert congr_fun A.mul_assoc ((x, y), z)\n[GOAL]\nA : Mon_ (Type u)\nx : A.X\n\u22a2 1 * x = x\n[PROOFSTEP]\nconvert congr_fun A.one_mul (PUnit.unit, x)\n[GOAL]\nA : Mon_ (Type u)\nx : A.X\n\u22a2 x * 1 = x\n[PROOFSTEP]\nconvert congr_fun A.mul_one (x, PUnit.unit)\n[GOAL]\nA : MonCat\n\u22a2 (MonoidalCategory.tensorHom (fun x => 1) (\ud835\udfd9 \u2191A) \u226b fun p => p.fst * p.snd) = (MonoidalCategory.leftUnitor \u2191A).hom\n[PROOFSTEP]\next \u27e8_, _\u27e9\n[GOAL]\ncase h.mk\nA : MonCat\nfst\u271d : MonoidalCategory.tensorUnit (Type u)\nsnd\u271d : \u2191A\n\u22a2 (MonoidalCategory.tensorHom (fun x => 1) (\ud835\udfd9 \u2191A) \u226b fun p => p.fst * p.snd) (fst\u271d, snd\u271d) =\n (MonoidalCategory.leftUnitor \u2191A).hom (fst\u271d, snd\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.mk\nA : MonCat\nfst\u271d : MonoidalCategory.tensorUnit (Type u)\nsnd\u271d : \u2191A\n\u22a2 1 * snd\u271d = snd\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nA : MonCat\n\u22a2 ((MonoidalCategory.tensorHom (\ud835\udfd9 \u2191A) fun x => 1) \u226b fun p => p.fst * p.snd) = (MonoidalCategory.rightUnitor \u2191A).hom\n[PROOFSTEP]\next \u27e8_, _\u27e9\n[GOAL]\ncase h.mk\nA : MonCat\nfst\u271d : \u2191A\nsnd\u271d : MonoidalCategory.tensorUnit (Type u)\n\u22a2 ((MonoidalCategory.tensorHom (\ud835\udfd9 \u2191A) fun x => 1) \u226b fun p => p.fst * p.snd) (fst\u271d, snd\u271d) =\n (MonoidalCategory.rightUnitor \u2191A).hom (fst\u271d, snd\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.mk\nA : MonCat\nfst\u271d : \u2191A\nsnd\u271d : MonoidalCategory.tensorUnit (Type u)\n\u22a2 fst\u271d * 1 = fst\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nA : MonCat\n\u22a2 (MonoidalCategory.tensorHom (fun p => p.fst * p.snd) (\ud835\udfd9 \u2191A) \u226b fun p => p.fst * p.snd) =\n (MonoidalCategory.associator \u2191A \u2191A \u2191A).hom \u226b\n (MonoidalCategory.tensorHom (\ud835\udfd9 \u2191A) fun p => p.fst * p.snd) \u226b fun p => p.fst * p.snd\n[PROOFSTEP]\next \u27e8\u27e8x, y\u27e9, z\u27e9\n[GOAL]\ncase h.mk.mk\nA : MonCat\nz x y : \u2191A\n\u22a2 (MonoidalCategory.tensorHom (fun p => p.fst * p.snd) (\ud835\udfd9 \u2191A) \u226b fun p => p.fst * p.snd) ((x, y), z) =\n ((MonoidalCategory.associator \u2191A \u2191A \u2191A).hom \u226b\n (MonoidalCategory.tensorHom (\ud835\udfd9 \u2191A) fun p => p.fst * p.snd) \u226b fun p => p.fst * p.snd)\n ((x, y), z)\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\n\u22a2 \u2200 {X Y : Mon_ (Type u)} (f : X \u27f6 Y),\n (\ud835\udfed (Mon_ (Type u))).map f \u226b\n ((fun A =>\n Iso.mk (Mon_.Hom.mk (\ud835\udfd9 ((\ud835\udfed (Mon_ (Type u))).obj A).X)) (Mon_.Hom.mk (\ud835\udfd9 ((functor \u22d9 inverse).obj A).X)))\n Y).hom =\n ((fun A => Iso.mk (Mon_.Hom.mk (\ud835\udfd9 ((\ud835\udfed (Mon_ (Type u))).obj A).X)) (Mon_.Hom.mk (\ud835\udfd9 ((functor \u22d9 inverse).obj A).X)))\n X).hom \u226b\n (functor \u22d9 inverse).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 \u2200 {X Y : MonCat} (f : X \u27f6 Y),\n (inverse \u22d9 functor).map f \u226b\n ((fun A =>\n Iso.mk\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((inverse \u22d9 functor).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) }\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((\ud835\udfed MonCat).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })\n Y).hom =\n ((fun A =>\n Iso.mk\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((inverse \u22d9 functor).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) }\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((\ud835\udfed MonCat).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })\n X).hom \u226b\n (\ud835\udfed MonCat).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 \u2200 {X Y : Mon_ (Type u)} (f : X \u27f6 Y),\n (functor \u22d9 forget MonCat).map f \u226b ((fun A => Iso.refl ((functor \u22d9 forget MonCat).obj A)) Y).hom =\n ((fun A => Iso.refl ((functor \u22d9 forget MonCat).obj A)) X).hom \u226b (Mon_.forget (Type u)).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nA : CommMon_ (Type u)\nsrc\u271d : Monoid A.X := monMonoid A.toMon_\nx y : A.X\n\u22a2 x * y = y * x\n[PROOFSTEP]\nconvert congr_fun A.mul_comm (y, x)\n[GOAL]\nA : CommMonCat\nsrc\u271d : Mon_ (Type u) := MonTypeEquivalenceMon.inverse.obj ((forget\u2082 CommMonCat MonCat).obj A)\n\u22a2 (\u03b2_ (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).X (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).X).hom \u226b\n (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).mul =\n (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).mul\n[PROOFSTEP]\next \u27e8x : A, y : A\u27e9\n[GOAL]\ncase h.mk\nA : CommMonCat\nsrc\u271d : Mon_ (Type u) := MonTypeEquivalenceMon.inverse.obj ((forget\u2082 CommMonCat MonCat).obj A)\nx y : \u2191A\n\u22a2 ((\u03b2_ (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).X (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).X).hom \u226b\n (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul).mul)\n (x, y) =\n Mon_.mul (Mon_.mk src\u271d.X src\u271d.one src\u271d.mul) (x, y)\n[PROOFSTEP]\nexact CommMonoid.mul_comm y x\n[GOAL]\n\u22a2 \u2200 {X Y : CommMon_ (Type u)} (f : X \u27f6 Y),\n (\ud835\udfed (CommMon_ (Type u))).map f \u226b\n ((fun A =>\n Iso.mk (Mon_.Hom.mk (\ud835\udfd9 ((\ud835\udfed (CommMon_ (Type u))).obj A).X))\n (Mon_.Hom.mk\n (\ud835\udfd9 ((CommMonTypeEquivalenceCommMon.functor \u22d9 CommMonTypeEquivalenceCommMon.inverse).obj A).X)))\n Y).hom =\n ((fun A =>\n Iso.mk (Mon_.Hom.mk (\ud835\udfd9 ((\ud835\udfed (CommMon_ (Type u))).obj A).X))\n (Mon_.Hom.mk\n (\ud835\udfd9 ((CommMonTypeEquivalenceCommMon.functor \u22d9 CommMonTypeEquivalenceCommMon.inverse).obj A).X)))\n X).hom \u226b\n (CommMonTypeEquivalenceCommMon.functor \u22d9 CommMonTypeEquivalenceCommMon.inverse).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 \u2200 {X Y : CommMonCat} (f : X \u27f6 Y),\n (CommMonTypeEquivalenceCommMon.inverse \u22d9 CommMonTypeEquivalenceCommMon.functor).map f \u226b\n ((fun A =>\n Iso.mk\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200\n (x y :\n \u2191((CommMonTypeEquivalenceCommMon.inverse \u22d9 CommMonTypeEquivalenceCommMon.functor).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) }\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((\ud835\udfed CommMonCat).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })\n Y).hom =\n ((fun A =>\n Iso.mk\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200\n (x y :\n \u2191((CommMonTypeEquivalenceCommMon.inverse \u22d9 CommMonTypeEquivalenceCommMon.functor).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) }\n { toOneHom := { toFun := id, map_one' := (_ : id 1 = id 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191((\ud835\udfed CommMonCat).obj A)),\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y) =\n OneHom.toFun { toFun := id, map_one' := (_ : id 1 = id 1) } (x * y)) })\n X).hom \u226b\n (\ud835\udfed CommMonCat).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 \u2200 {X Y : CommMon_ (Type u)} (f : X \u27f6 Y),\n (CommMonTypeEquivalenceCommMon.functor \u22d9 forget\u2082 CommMonCat MonCat).map f \u226b\n ((fun A => Iso.refl ((CommMonTypeEquivalenceCommMon.functor \u22d9 forget\u2082 CommMonCat MonCat).obj A)) Y).hom =\n ((fun A => Iso.refl ((CommMonTypeEquivalenceCommMon.functor \u22d9 forget\u2082 CommMonCat MonCat).obj A)) X).hom \u226b\n (CommMon_.forget\u2082Mon_ (Type u) \u22d9 MonTypeEquivalenceMon.functor).map f\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Internal.Types", "llama_tokens": 3685, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.411110869232168, "lm_q1q2_score": 0.27212368885534755}} {"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : a \u2208 dedup l\n\u22a2 \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter (fun x x_1 => x \u2260 x_1) l \u2192 a \u2260 b\n[PROOFSTEP]\nsimpa only [forall_mem_ne, not_not] using h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nh : \u00aca \u2208 dedup l\n\u22a2 \u2200 (b : \u03b1), b \u2208 pwFilter (fun x x_1 => x \u2260 x_1) l \u2192 a \u2260 b\n[PROOFSTEP]\nsimpa only [forall_mem_ne] using h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 a \u2208 dedup l \u2194 a \u2208 l\n[PROOFSTEP]\nhave := not_congr (@forall_mem_pwFilter \u03b1 (\u00b7 \u2260 \u00b7) _ ?_ a l)\n[GOAL]\ncase refine_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nthis : (\u00ac\u2200 (b : \u03b1), b \u2208 pwFilter (fun x x_1 => x \u2260 x_1) l \u2192 a \u2260 b) \u2194 \u00ac\u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b\n\u22a2 a \u2208 dedup l \u2194 a \u2208 l\ncase refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 \u2200 {x y z : \u03b1}, (fun x x_1 => x \u2260 x_1) x z \u2192 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z\n[PROOFSTEP]\nsimpa only [dedup, forall_mem_ne, not_not] using this\n[GOAL]\ncase refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 \u2200 {x y z : \u03b1}, (fun x x_1 => x \u2260 x_1) x z \u2192 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z\n[PROOFSTEP]\nintros x y z xz\n[GOAL]\ncase refine_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nx y z : \u03b1\nxz : x \u2260 z\n\u22a2 (fun x x_1 => x \u2260 x_1) x y \u2228 (fun x x_1 => x \u2260 x_1) y z\n[PROOFSTEP]\nexact not_and_or.1 <| mt (fun h \u21a6 h.1.trans h.2) xz\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 headI (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then headI (dedup (tail (a :: l))) else headI (a :: l)\n[PROOFSTEP]\nby_cases ha : a \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nha : a \u2208 l\n\u22a2 headI (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then headI (dedup (tail (a :: l))) else headI (a :: l)\n[PROOFSTEP]\nsimp [ha, List.dedup_cons_of_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nha : \u00aca \u2208 l\n\u22a2 headI (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then headI (dedup (tail (a :: l))) else headI (a :: l)\n[PROOFSTEP]\nsimp [ha, List.dedup_cons_of_mem]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 tail (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then tail (dedup (tail (a :: l))) else dedup (tail (a :: l))\n[PROOFSTEP]\nby_cases ha : a \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nha : a \u2208 l\n\u22a2 tail (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then tail (dedup (tail (a :: l))) else dedup (tail (a :: l))\n[PROOFSTEP]\nsimp [ha, List.dedup_cons_of_mem]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Inhabited \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nha : \u00aca \u2208 l\n\u22a2 tail (dedup (a :: l)) = if headI (a :: l) \u2208 tail (a :: l) then tail (dedup (tail (a :: l))) else dedup (tail (a :: l))\n[PROOFSTEP]\nsimp [ha, List.dedup_cons_of_mem]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\n\u22a2 dedup l = a :: l' \u2194 a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : dedup l = a :: l'\n\u22a2 a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\n[PROOFSTEP]\nrefine' \u27e8mem_dedup.1 (h.symm \u25b8 mem_cons_self _ _), fun ha => _, by rw [h, tail_cons]\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : dedup l = a :: l'\n\u22a2 tail (dedup l) = l'\n[PROOFSTEP]\nrw [h, tail_cons]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : dedup l = a :: l'\nha : a \u2208 l'\n\u22a2 False\n[PROOFSTEP]\nhave : count a l.dedup \u2264 1 := nodup_iff_count_le_one.1 (nodup_dedup l) a\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : dedup l = a :: l'\nha : a \u2208 l'\nthis : count a (dedup l) \u2264 1\n\u22a2 False\n[PROOFSTEP]\nrw [h, count_cons_self, add_le_iff_nonpos_left] at this \n[GOAL]\ncase refine'_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : dedup l = a :: l'\nha : a \u2208 l'\nthis : count a l' \u2264 0\n\u22a2 False\n[PROOFSTEP]\nexact not_le_of_lt (count_pos.2 ha) this\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\n\u22a2 dedup l = a :: l'\n[PROOFSTEP]\nhave := @List.cons_head!_tail \u03b1 \u27e8a\u27e9 _ (ne_nil_of_mem (mem_dedup.2 h.1))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\nthis : head! (dedup l) :: tail (dedup l) = dedup l\n\u22a2 dedup l = a :: l'\n[PROOFSTEP]\nhave hal : a \u2208 l.dedup := mem_dedup.2 h.1\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\nthis : head! (dedup l) :: tail (dedup l) = dedup l\nhal : a \u2208 dedup l\n\u22a2 dedup l = a :: l'\n[PROOFSTEP]\nrw [\u2190 this, mem_cons, or_iff_not_imp_right] at hal \n[GOAL]\ncase refine'_2\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\nl' : List \u03b1\nh : a \u2208 l \u2227 \u00aca \u2208 l' \u2227 tail (dedup l) = l'\nthis : head! (dedup l) :: tail (dedup l) = dedup l\nhal : \u00aca \u2208 tail (dedup l) \u2192 a = head! (dedup l)\n\u22a2 dedup l = a :: l'\n[PROOFSTEP]\nexact this \u25b8 h.2.2.symm \u25b8 cons_eq_cons.2 \u27e8(hal (h.2.2.symm \u25b8 h.2.1)).symm, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 dedup l = [] \u2194 l = []\n[PROOFSTEP]\ninduction' l with a l hl\n[GOAL]\ncase nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\n\u22a2 dedup [] = [] \u2194 [] = []\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nhl : dedup l = [] \u2194 l = []\n\u22a2 dedup (a :: l) = [] \u2194 a :: l = []\n[PROOFSTEP]\nby_cases h : a \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nhl : dedup l = [] \u2194 l = []\nh : a \u2208 l\n\u22a2 dedup (a :: l) = [] \u2194 a :: l = []\n[PROOFSTEP]\nsimp only [List.dedup_cons_of_mem h, hl, List.ne_nil_of_mem h]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nhl : dedup l = [] \u2194 l = []\nh : \u00aca \u2208 l\n\u22a2 dedup (a :: l) = [] \u2194 a :: l = []\n[PROOFSTEP]\nsimp only [List.dedup_cons_of_not_mem h, List.cons_ne_nil]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with a l\u2081 IH\n[GOAL]\ncase nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\n\u22a2 dedup ([] ++ l\u2082) = [] \u222a dedup l\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nIH : dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\n\u22a2 dedup (a :: l\u2081 ++ l\u2082) = a :: l\u2081 \u222a dedup l\u2082\n[PROOFSTEP]\nsimp only [cons_union] at *\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nIH : dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\n\u22a2 dedup (a :: l\u2081 ++ l\u2082) = List.insert a (l\u2081 \u222a dedup l\u2082)\n[PROOFSTEP]\nrw [\u2190 IH, cons_append]\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nIH : dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\n\u22a2 dedup (a :: (l\u2081 ++ l\u2082)) = List.insert a (dedup (l\u2081 ++ l\u2082))\n[PROOFSTEP]\nby_cases h : a \u2208 dedup (l\u2081 ++ l\u2082)\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nIH : dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\nh : a \u2208 dedup (l\u2081 ++ l\u2082)\n\u22a2 dedup (a :: (l\u2081 ++ l\u2082)) = List.insert a (dedup (l\u2081 ++ l\u2082))\n[PROOFSTEP]\nrw [dedup_cons_of_mem' h, insert_of_mem h]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nIH : dedup (l\u2081 ++ l\u2082) = l\u2081 \u222a dedup l\u2082\nh : \u00aca \u2208 dedup (l\u2081 ++ l\u2082)\n\u22a2 dedup (a :: (l\u2081 ++ l\u2082)) = List.insert a (dedup (l\u2081 ++ l\u2082))\n[PROOFSTEP]\nrw [dedup_cons_of_not_mem' h, insert_of_not_mem h]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nn : \u2115\nx\u271d : n + 2 \u2260 0\n\u22a2 dedup (replicate (n + 2) x) = [x]\n[PROOFSTEP]\nrw [replicate_succ, dedup_cons_of_mem (mem_replicate.2 \u27e8n.succ_ne_zero, rfl\u27e9), replicate_dedup n.succ_ne_zero]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\na : \u03b1\n\u22a2 count a (dedup l) = if a \u2208 l then 1 else 0\n[PROOFSTEP]\nsimp_rw [count_eq_of_nodup <| nodup_dedup l, mem_dedup]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\n\u22a2 sum (map (fun x => count x l) (filter p (dedup l))) = countp p l\n[PROOFSTEP]\ninduction' l with a as h\n[GOAL]\ncase nil\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\n\u22a2 sum (map (fun x => count x []) (filter p (dedup []))) = countp p []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n\u22a2 sum (map (fun x => count x (a :: as)) (filter p (dedup (a :: as)))) = countp p (a :: as)\n[PROOFSTEP]\nsimp_rw [List.countp_cons, List.count_cons', List.sum_map_add]\n[GOAL]\ncase cons\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n\u22a2 sum (map (fun i => count i as) (filter p (dedup (a :: as)))) +\n sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) =\n countp p as + if p a = true then 1 else 0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase cons.e_a\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n\u22a2 sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = countp p as\n[PROOFSTEP]\nrefine' _root_.trans _ h\n[GOAL]\ncase cons.e_a\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n\u22a2 sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))\n[PROOFSTEP]\nby_cases ha : a \u2208 as\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : a \u2208 as\n\u22a2 sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))\n[PROOFSTEP]\nsimp [dedup_cons_of_mem ha]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : \u00aca \u2208 as\n\u22a2 sum (map (fun i => count i as) (filter p (dedup (a :: as)))) = sum (map (fun x => count x as) (filter p (dedup as)))\n[PROOFSTEP]\nsimp only [dedup_cons_of_not_mem ha, List.filter]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : \u00aca \u2208 as\n\u22a2 sum\n (map (fun i => count i as)\n (match p a with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))\n[PROOFSTEP]\nmatch p a with\n| true => simp only [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]\n| false => simp only\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : \u00aca \u2208 as\n\u22a2 sum\n (map (fun i => count i as)\n (match true with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))\n[PROOFSTEP]\nsimp only [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nha : \u00aca \u2208 as\n\u22a2 sum\n (map (fun i => count i as)\n (match false with\n | true => a :: filter p (dedup as)\n | false => filter p (dedup as))) =\n sum (map (fun i => count i as) (filter p (dedup as)))\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase cons.e_a\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\n\u22a2 sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0\n[PROOFSTEP]\nby_cases hp : p a\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n\u22a2 sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0\n[PROOFSTEP]\nrefine' _root_.trans (sum_map_eq_nsmul_single a _ fun _ h _ => by simp [h]) _\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh\u271d : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\nx\u271d\u00b9 : \u03b1\nh : x\u271d\u00b9 \u2260 a\nx\u271d : x\u271d\u00b9 \u2208 filter p (dedup (a :: as))\n\u22a2 (if x\u271d\u00b9 = a then 1 else 0) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : p a = true\n\u22a2 (count a (filter p (dedup (a :: as))) \u2022 if a = a then 1 else 0) = if p a = true then 1 else 0\n[PROOFSTEP]\nsimp [hp, count_dedup]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\n\u22a2 sum (map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))) = if p a = true then 1 else 0\n[PROOFSTEP]\nrefine' _root_.trans (List.sum_eq_zero fun n hn => _) (by simp [hp])\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\n\u22a2 0 = if p a = true then 1 else 0\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\nn : \u2115\nhn : n \u2208 map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\n\u22a2 n = 0\n[PROOFSTEP]\nobtain \u27e8a', ha'\u27e9 := List.mem_map.1 hn\n[GOAL]\ncase neg.intro\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\nn : \u2115\nhn : n \u2208 map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : \u03b1\nha' : a' \u2208 filter p (dedup (a :: as)) \u2227 (if a' = a then 1 else 0) = n\n\u22a2 n = 0\n[PROOFSTEP]\nsplit_ifs at ha' with ha\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\nn : \u2115\nhn : n \u2208 map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : \u03b1\nha : a' = a\nha' : a' \u2208 filter p (dedup (a :: as)) \u2227 1 = n\n\u22a2 n = 0\n[PROOFSTEP]\nsimp only [ha, mem_filter, mem_dedup, find?, mem_cons, true_or, hp, and_false, false_and] at ha' \n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\np : \u03b1 \u2192 Bool\na : \u03b1\nas : List \u03b1\nh : sum (map (fun x => count x as) (filter p (dedup as))) = countp p as\nhp : \u00acp a = true\nn : \u2115\nhn : n \u2208 map (fun i => if i = a then 1 else 0) (filter p (dedup (a :: as)))\na' : \u03b1\nha : \u00aca' = a\nha' : a' \u2208 filter p (dedup (a :: as)) \u2227 0 = n\n\u22a2 n = 0\n[PROOFSTEP]\nexact ha'.2.symm\n[GOAL]\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 sum (map (fun x => count x l) (dedup l)) = length l\n[PROOFSTEP]\nsimpa using sum_map_count_dedup_filter_eq_countp (fun _ => True) l\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Dedup", "llama_tokens": 7103, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.2719598375416356}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\n\u22a2 \u03b1 \u2243 Additive \u03b1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\n\u22a2 \u03b1 \u2243 Multiplicative \u03b1\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.TypeTags", "llama_tokens": 86, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.4339814648038985, "lm_q1q2_score": 0.27172628758267486}} {"text": "[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nh : degree f \u2260 0\n\u22a2 span {f} \u2260 \u22a4\n[PROOFSTEP]\nsimp_rw [Ne.def, span_singleton_eq_top, Polynomial.isUnit_iff, not_exists, not_and]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nh : degree f \u2260 0\n\u22a2 \u2200 (x : R), IsUnit x \u2192 \u00ac\u2191C x = f\n[PROOFSTEP]\nrintro x hx rfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nx : R\nhx : IsUnit x\nh : degree (\u2191C x) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact h (degree_C hx.ne_zero)\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : DistribSMul S R\ninst\u271d : IsScalarTower S R R\na : S\nx : R\n\u22a2 a \u2022 \u2191(of f) x = \u2191(of f) (a \u2022 x)\n[PROOFSTEP]\nrw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf g : R[X]\n\u22a2 f \u2223 g - 0 \u2194 f \u2223 g\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf : R[X]\n\u22a2 f \u2223 -f + 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\nx : R\n\u22a2 \u2191(aeval (root f)) (\u2191C x) = \u2191(mk f) (\u2191C x)\n[PROOFSTEP]\nrw [aeval_C]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\nx : R\n\u22a2 \u2191(algebraMap R (AdjoinRoot f)) x = \u2191(mk f) (\u2191C x)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p\u271d p q : R[X]\nihp : \u2191(aeval (root f)) p = \u2191(mk f) p\nihq : \u2191(aeval (root f)) q = \u2191(mk f) q\n\u22a2 \u2191(aeval (root f)) (p + q) = \u2191(mk f) (p + q)\n[PROOFSTEP]\nrw [AlgHom.map_add, RingHom.map_add, ihp, ihq]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\nn : \u2115\nx : R\nx\u271d : \u2191(aeval (root f)) (\u2191C x * X ^ n) = \u2191(mk f) (\u2191C x * X ^ n)\n\u22a2 \u2191(aeval (root f)) (\u2191C x * X ^ (n + 1)) = \u2191(mk f) (\u2191C x * X ^ (n + 1))\n[PROOFSTEP]\nrw [AlgHom.map_mul, aeval_C, AlgHom.map_pow, aeval_X, RingHom.map_mul, mk_C, RingHom.map_pow, mk_X]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\nn : \u2115\nx : R\nx\u271d : \u2191(aeval (root f)) (\u2191C x * X ^ n) = \u2191(mk f) (\u2191C x * X ^ n)\n\u22a2 \u2191(algebraMap R (AdjoinRoot f)) x * root f ^ (n + 1) = \u2191(of f) x * root f ^ (n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf : R[X]\n\u22a2 Algebra.adjoin R {root f} = \u22a4\n[PROOFSTEP]\nrefine Algebra.eq_top_iff.2 fun x => ?_\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf : R[X]\nx : AdjoinRoot f\n\u22a2 x \u2208 Algebra.adjoin R {root f}\n[PROOFSTEP]\ninduction x using AdjoinRoot.induction_on with\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm \u25b8 \u27e8p, aeval_eq p\u27e9\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf : R[X]\nx : AdjoinRoot f\n\u22a2 x \u2208 Algebra.adjoin R {root f}\n[PROOFSTEP]\ninduction x using AdjoinRoot.induction_on with\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm \u25b8 \u27e8p, aeval_eq p\u27e9\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\n\u22a2 \u2191(mk f) p \u2208 Algebra.adjoin R {root f}\n[PROOFSTEP]\n\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm \u25b8 \u27e8p, aeval_eq p\u27e9\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf p : R[X]\n\u22a2 \u2191(mk f) p \u2208 Algebra.adjoin R {root f}\n[PROOFSTEP]\nexact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm \u25b8 \u27e8p, aeval_eq p\u27e9\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf\u271d f : R[X]\n\u22a2 eval\u2082 (of f) (root f) f = 0\n[PROOFSTEP]\nrw [\u2190 algebraMap_eq, \u2190 aeval_def, aeval_eq, mk_self]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nf\u271d f : R[X]\n\u22a2 IsRoot (Polynomial.map (of f) f) (root f)\n[PROOFSTEP]\nrw [IsRoot, eval_map, eval\u2082_root]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\n\u22a2 Function.Injective \u2191(of f)\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\n\u22a2 \u2200 (a : R), \u2191(of f) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro p hp\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\np : R\nhp : \u2191(of f) p = 0\n\u22a2 p = 0\n[PROOFSTEP]\nrw [AdjoinRoot.of, RingHom.comp_apply, AdjoinRoot.mk_eq_zero] at hp \n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\np : R\nhp : f \u2223 \u2191C p\n\u22a2 p = 0\n[PROOFSTEP]\nby_cases h : f = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\np : R\nhp : f \u2223 \u2191C p\nh : f = 0\n\u22a2 p = 0\n[PROOFSTEP]\nexact C_eq_zero.mp (eq_zero_of_zero_dvd (by rwa [h] at hp ))\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\np : R\nhp : f \u2223 \u2191C p\nh : f = 0\n\u22a2 0 \u2223 \u2191C p\n[PROOFSTEP]\nrwa [h] at hp \n[GOAL]\ncase neg\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\nhf : degree f \u2260 0\np : R\nhp : f \u2223 \u2191C p\nh : \u00acf = 0\n\u22a2 p = 0\n[PROOFSTEP]\ncontrapose! hf with h_contra\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\np : R\nhp : f \u2223 \u2191C p\nh : \u00acf = 0\nh_contra : p \u2260 0\n\u22a2 degree f = 0\n[PROOFSTEP]\nrw [\u2190 degree_C h_contra]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\np : R\nhp : f \u2223 \u2191C p\nh : \u00acf = 0\nh_contra : p \u2260 0\n\u22a2 degree f = degree (\u2191C p)\n[PROOFSTEP]\napply le_antisymm (degree_le_of_dvd hp (by rwa [Ne.def, C_eq_zero])) _\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\np : R\nhp : f \u2223 \u2191C p\nh : \u00acf = 0\nh_contra : p \u2260 0\n\u22a2 \u2191C p \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, C_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : IsDomain R\np : R\nhp : f \u2223 \u2191C p\nh : \u00acf = 0\nh_contra : p \u2260 0\n\u22a2 degree (\u2191C p) \u2264 degree f\n[PROOFSTEP]\nrwa [degree_C h_contra, zero_le_degree_iff]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\nx : S\nh : eval\u2082 i x f = 0\n\u22a2 AdjoinRoot f \u2192+* S\n[PROOFSTEP]\napply Ideal.Quotient.lift _ (eval\u2082RingHom i x)\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\nx : S\nh : eval\u2082 i x f = 0\n\u22a2 \u2200 (a : R[X]), a \u2208 span {f} \u2192 \u2191(eval\u2082RingHom i x) a = 0\n[PROOFSTEP]\nintro g H\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\nx : S\nh : eval\u2082 i x f = 0\ng : R[X]\nH : g \u2208 span {f}\n\u22a2 \u2191(eval\u2082RingHom i x) g = 0\n[PROOFSTEP]\nrcases mem_span_singleton.1 H with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\nx : S\nh : eval\u2082 i x f = 0\ng : R[X]\nH : g \u2208 span {f}\ny : R[X]\nhy : g = f * y\n\u22a2 \u2191(eval\u2082RingHom i x) g = 0\n[PROOFSTEP]\nrw [hy, RingHom.map_mul, coe_eval\u2082RingHom, h, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\n\u22a2 \u2191(lift i a h) (root f) = a\n[PROOFSTEP]\nrw [root, lift_mk, eval\u2082_X]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\nf : R[X]\ninst\u271d : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\nx : R\n\u22a2 \u2191(lift i a h) (\u2191(of f) x) = \u2191i x\n[PROOFSTEP]\nrw [\u2190 mk_C x, lift_mk, eval\u2082_C]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\n\u22a2 \u2191(aeval (\u2191\u03d5 (root f))) f = 0\n[PROOFSTEP]\nhave h : \u03d5.toRingHom.comp (of f) = algebraMap R S := RingHom.ext_iff.mpr \u03d5.commutes\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh\u271d : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\nh : RingHom.comp (\u2191\u03d5) (of f) = algebraMap R S\n\u22a2 \u2191(aeval (\u2191\u03d5 (root f))) f = 0\n[PROOFSTEP]\nrw [aeval_def, \u2190 h, \u2190 RingHom.map_zero \u03d5.toRingHom, \u2190 eval\u2082_root f, hom_eval\u2082]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh\u271d : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\nh : RingHom.comp (\u2191\u03d5) (of f) = algebraMap R S\n\u22a2 eval\u2082 (RingHom.comp (\u2191\u03d5) (of f)) (\u2191\u03d5 (root f)) f = eval\u2082 (RingHom.comp (\u2191\u03d5) (of f)) (\u2191\u2191\u03d5 (root f)) f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f\u271d = 0\ninst\u271d : Algebra R S\nf : R[X]\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\n\u22a2 liftHom f (\u2191\u03d5 (root f)) (_ : \u2191(aeval (\u2191\u03d5 (root f))) f = 0) = \u03d5\n[PROOFSTEP]\nsuffices \u03d5.equalizer (liftHom f (\u03d5 (root f)) (aeval_algHom_eq_zero f \u03d5)) = \u22a4 by\n exact (AlgHom.ext fun x => (SetLike.ext_iff.mp this x).mpr Algebra.mem_top).symm\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f\u271d = 0\ninst\u271d : Algebra R S\nf : R[X]\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\nthis : AlgHom.equalizer \u03d5 (liftHom f (\u2191\u03d5 (root f)) (_ : \u2191(aeval (\u2191\u03d5 (root f))) f = 0)) = \u22a4\n\u22a2 liftHom f (\u2191\u03d5 (root f)) (_ : \u2191(aeval (\u2191\u03d5 (root f))) f = 0) = \u03d5\n[PROOFSTEP]\nexact (AlgHom.ext fun x => (SetLike.ext_iff.mp this x).mpr Algebra.mem_top).symm\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f\u271d = 0\ninst\u271d : Algebra R S\nf : R[X]\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\n\u22a2 AlgHom.equalizer \u03d5 (liftHom f (\u2191\u03d5 (root f)) (_ : \u2191(aeval (\u2191\u03d5 (root f))) f = 0)) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 adjoinRoot_eq_top, Algebra.adjoin_le_iff, Set.singleton_subset_iff]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f\u271d = 0\ninst\u271d : Algebra R S\nf : R[X]\n\u03d5 : AdjoinRoot f \u2192\u2090[R] S\n\u22a2 root f \u2208 \u2191(AlgHom.equalizer \u03d5 (liftHom f (\u2191\u03d5 (root f)) (_ : \u2191(aeval (\u2191\u03d5 (root f))) f = 0)))\n[PROOFSTEP]\nexact (@lift_root _ _ _ _ _ _ _ (aeval_algHom_eq_zero f \u03d5)).symm\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\nhfx : \u2191(aeval a) f = 0\nr : R\n\u22a2 \u2191(of (\u2191C r * X - 1)) r * root (\u2191C r * X - 1) = 1\n[PROOFSTEP]\nconvert sub_eq_zero.1 ((eval\u2082_sub _).symm.trans <| eval\u2082_root <| C r * X - 1)\n[GOAL]\ncase h.e'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\nhfx : \u2191(aeval a) f = 0\nr : R\n\u22a2 \u2191(of (\u2191C r * X - 1)) r * root (\u2191C r * X - 1) = eval\u2082 (of (\u2191C r * X - 1)) (root (\u2191C r * X - 1)) (\u2191C r * X)\n[PROOFSTEP]\nsimp only [eval\u2082_mul, eval\u2082_C, eval\u2082_X, eval\u2082_one]\n[GOAL]\ncase h.e'_3\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\nf : R[X]\ninst\u271d\u00b9 : CommRing S\ni : R \u2192+* S\na : S\nh : eval\u2082 i a f = 0\ninst\u271d : Algebra R S\nhfx : \u2191(aeval a) f = 0\nr : R\n\u22a2 1 = eval\u2082 (of (\u2191C r * X - 1)) (root (\u2191C r * X - 1)) 1\n[PROOFSTEP]\nsimp only [eval\u2082_mul, eval\u2082_C, eval\u2082_X, eval\u2082_one]\n[GOAL]\nR : Type u\nS\u271d : Type v\nK : Type w\ninst\u271d\u2074 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b3 : CommRing S\u271d\ni : R \u2192+* S\u271d\na : S\u271d\nh : eval\u2082 i a f\u271d = 0\ninst\u271d\u00b2 : Algebra R S\u271d\nhfx : \u2191(aeval a) f\u271d = 0\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nr : R\nf g : AdjoinRoot (\u2191C r * X - 1) \u2192\u2090[R] S\n\u22a2 \u2191(algebraMap R S) r * \u2191f (root (\u2191C r * X - 1)) = 1\n[PROOFSTEP]\nrw [\u2190 f.commutes, \u2190 f.map_mul, algebraMap_eq, root_isInv, map_one]\n[GOAL]\nR : Type u\nS\u271d : Type v\nK : Type w\ninst\u271d\u2074 : CommRing R\nf\u271d : R[X]\ninst\u271d\u00b3 : CommRing S\u271d\ni : R \u2192+* S\u271d\na : S\u271d\nh : eval\u2082 i a f\u271d = 0\ninst\u271d\u00b2 : Algebra R S\u271d\nhfx : \u2191(aeval a) f\u271d = 0\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nr : R\nf g : AdjoinRoot (\u2191C r * X - 1) \u2192\u2090[R] S\n\u22a2 \u2191(algebraMap R S) r * \u2191g (root (\u2191C r * X - 1)) = 1\n[PROOFSTEP]\nrw [\u2190 g.commutes, \u2190 g.map_mul, algebraMap_eq, root_isInv, map_one]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : Field K\nf : K[X]\ninst\u271d : Fact (Irreducible f)\nsrc\u271d : GroupWithZero (K[X] \u29f8 span {f}) := Quotient.groupWithZero (span {f})\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nletI : GroupWithZero (AdjoinRoot f) :=\n Ideal.Quotient.groupWithZero\n _\n -- porting note: was\n -- `rw [Rat.cast_mk' (K := \u211a), _root_.map_mul, _root_.map_intCast, map_inv\u2080, map_natCast]`\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : Field K\nf : K[X]\ninst\u271d : Fact (Irreducible f)\nsrc\u271d : GroupWithZero (K[X] \u29f8 span {f}) := Quotient.groupWithZero (span {f})\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\nthis : GroupWithZero (AdjoinRoot f) := Quotient.groupWithZero (span {f})\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nconvert_to ((Rat.mk' a b h1 h2 : K) : AdjoinRoot f) = ((\u2191a * (\u2191b)\u207b\u00b9 : K) : AdjoinRoot f)\n[GOAL]\ncase h.e'_3\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : Field K\nf : K[X]\ninst\u271d : Fact (Irreducible f)\nsrc\u271d : GroupWithZero (K[X] \u29f8 span {f}) := Quotient.groupWithZero (span {f})\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\nthis : GroupWithZero (AdjoinRoot f) := Quotient.groupWithZero (span {f})\n\u22a2 \u2191a * (\u2191b)\u207b\u00b9 = \u2191(of f) (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [_root_.map_mul, map_intCast, map_inv\u2080, map_natCast]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : Field K\nf : K[X]\ninst\u271d : Fact (Irreducible f)\nsrc\u271d : GroupWithZero (K[X] \u29f8 span {f}) := Quotient.groupWithZero (span {f})\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\nthis : GroupWithZero (AdjoinRoot f) := Quotient.groupWithZero (span {f})\n\u22a2 \u2191(of f) \u2191(Rat.mk' a b) = \u2191(of f) (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [Rat.cast_mk', _root_.map_mul, map_intCast, map_inv\u2080, map_natCast]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : Field K\nf : K[X]\ninst\u271d : Fact (Irreducible f)\nsrc\u271d : GroupWithZero (K[X] \u29f8 span {f}) := Quotient.groupWithZero (span {f})\na : \u211a\nx : AdjoinRoot f\np : K[X]\n\u22a2 (fun y => a \u2022 y = \u2191(of f) \u2191a * y) (\u2191(mk f) p)\n[PROOFSTEP]\nsimp only [smul_mk, of, RingHom.comp_apply, \u2190 (mk f).map_mul, Polynomial.rat_smul_eq_C_mul]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\n\u22a2 Function.LeftInverse \u2191(mk g) \u2191(modByMonicHom hg)\n[PROOFSTEP]\nintro f\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nf : AdjoinRoot g\n\u22a2 \u2191(mk g) (\u2191(modByMonicHom hg) f) = f\n[PROOFSTEP]\ninduction f using AdjoinRoot.induction_on\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 \u2191(mk g) (\u2191(modByMonicHom hg) (\u2191(mk g) p\u271d)) = \u2191(mk g) p\u271d\n[PROOFSTEP]\nrw [modByMonicHom_mk hg, mk_eq_mk, modByMonic_eq_sub_mul_div _ hg, sub_sub_cancel_left, dvd_neg]\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 g \u2223 g * (p\u271d /\u2098 g)\n[PROOFSTEP]\napply dvd_mul_right\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nf\u2081 f\u2082 : AdjoinRoot g\ni : Fin (natDegree g)\n\u22a2 (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) i =\n ((fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 + (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) i\n[PROOFSTEP]\nsimp only [(modByMonicHom hg).map_add, coeff_add, Pi.add_apply]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nf\u2081 : R\nf\u2082 : AdjoinRoot g\ni : Fin (natDegree g)\n\u22a2 AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 + (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) i =\n (\u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082)\n i\n[PROOFSTEP]\nsimp only [(modByMonicHom hg).map_smul, coeff_smul, Pi.smul_apply, RingHom.id_apply]\n -- porting note: another proof that I converted to tactic mode\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\n\u22a2 Function.LeftInverse (fun c => \u2191(mk g) (\u2211 i : Fin (natDegree g), \u2191(monomial \u2191i) (c i)))\n {\n toAddHom :=\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (f\u2081 : R) (f\u2082 : AdjoinRoot g),\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) =\n \u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nf : AdjoinRoot g\n\u22a2 (fun c => \u2191(mk g) (\u2211 i : Fin (natDegree g), \u2191(monomial \u2191i) (c i)))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (f\u2081 : R) (f\u2082 : AdjoinRoot g),\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) =\n \u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082) }.toAddHom\n f) =\n f\n[PROOFSTEP]\ninduction f using AdjoinRoot.induction_on\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 (fun c => \u2191(mk g) (\u2211 i : Fin (natDegree g), \u2191(monomial \u2191i) (c i)))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (f\u2081 : R) (f\u2082 : AdjoinRoot g),\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) =\n \u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082) }.toAddHom\n (\u2191(mk g) p\u271d)) =\n \u2191(mk g) p\u271d\n[PROOFSTEP]\nsimp only [modByMonicHom_mk, sum_modByMonic_coeff hg degree_le_natDegree]\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 \u2191(mk g) (p\u271d %\u2098 g) = \u2191(mk g) p\u271d\n[PROOFSTEP]\nrefine (mk_eq_mk.mpr ?_).symm\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 g \u2223 p\u271d - p\u271d %\u2098 g\n[PROOFSTEP]\nrw [modByMonic_eq_sub_mul_div _ hg, sub_sub_cancel]\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\np\u271d : R[X]\n\u22a2 g \u2223 g * (p\u271d /\u2098 g)\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (f\u2081 : R) (f\u2082 : AdjoinRoot g),\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) =\n \u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082) }.toAddHom\n ((fun c => \u2191(mk g) (\u2211 i : Fin (natDegree g), \u2191(monomial \u2191i) (c i))) x) i =\n x i\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u271d : Nontrivial R\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) },\n map_smul' :=\n (_ :\n \u2200 (f\u2081 : R) (f\u2082 : AdjoinRoot g),\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n (f\u2081 \u2022 f\u2082) =\n \u2191(RingHom.id R) f\u2081 \u2022\n AddHom.toFun\n { toFun := fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i,\n map_add' :=\n (_ :\n \u2200 (f\u2081 f\u2082 : AdjoinRoot g),\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) (f\u2081 + f\u2082) =\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2081 +\n (fun f i => coeff (\u2191(modByMonicHom hg) f) \u2191i) f\u2082) }\n f\u2082) }.toAddHom\n ((fun c => \u2191(mk g) (\u2211 i : Fin (natDegree g), \u2191(monomial \u2191i) (c i))) x) i =\n x i\n[PROOFSTEP]\nsimp only [modByMonicHom_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u271d : Nontrivial R\n\u22a2 coeff ((\u2211 x_1 : Fin (natDegree g), \u2191(monomial \u2191x_1) (x x_1)) %\u2098 g) \u2191i = x i\n[PROOFSTEP]\nrw [(modByMonic_eq_self_iff hg).mpr, finset_sum_coeff]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u271d : Nontrivial R\n\u22a2 \u2211 b : Fin (natDegree g), coeff (\u2191(monomial \u2191b) (x b)) \u2191i = x i\n[PROOFSTEP]\nsimp_rw [coeff_monomial, Fin.val_eq_val, Finset.sum_ite_eq', if_pos (Finset.mem_univ _)]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u271d : Nontrivial R\n\u22a2 degree (\u2211 x_1 : Fin (natDegree g), \u2191(monomial \u2191x_1) (x x_1)) < degree g\n[PROOFSTEP]\nsimp_rw [\u2190 C_mul_X_pow_eq_monomial]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\nx : Fin (natDegree g) \u2192 R\ni : Fin (natDegree g)\n\u271d : Nontrivial R\n\u22a2 degree (\u2211 x_1 : Fin (natDegree g), \u2191C (x x_1) * X ^ \u2191x_1) < degree g\n[PROOFSTEP]\nexact (degree_eq_natDegree <| hg.ne_zero).symm \u25b8 degree_sum_fin_lt _\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\n\u22a2 \u2191(powerBasisAux' hg) i = root g ^ \u2191i\n[PROOFSTEP]\nsimp only [powerBasisAux', Basis.coe_ofEquivFun, LinearEquiv.coe_symm_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\n\u22a2 \u2191(mk g) (\u2211 x : Fin (natDegree g), \u2191(monomial \u2191x) (Function.update 0 i 1 x)) = root g ^ \u2191i\n[PROOFSTEP]\nrw [Finset.sum_eq_single i]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\n\u22a2 \u2191(mk g) (\u2191(monomial \u2191i) (Function.update 0 i 1 i)) = root g ^ \u2191i\n[PROOFSTEP]\nrw [Function.update_same, monomial_one_right_eq_X_pow, (mk g).map_pow, mk_X]\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\n\u22a2 \u2200 (b : Fin (natDegree g)), b \u2208 Finset.univ \u2192 b \u2260 i \u2192 \u2191(monomial \u2191b) (Function.update 0 i 1 b) = 0\n[PROOFSTEP]\nintro j _ hj\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni j : Fin (natDegree g)\na\u271d : j \u2208 Finset.univ\nhj : j \u2260 i\n\u22a2 \u2191(monomial \u2191j) (Function.update 0 i 1 j) = 0\n[PROOFSTEP]\nrw [\u2190 monomial_zero_right _]\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni j : Fin (natDegree g)\na\u271d : j \u2208 Finset.univ\nhj : j \u2260 i\n\u22a2 \u2191(monomial \u2191j) (Function.update 0 i 1 j) = \u2191(monomial ?m.1619036) 0\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni j : Fin (natDegree g)\na\u271d : j \u2208 Finset.univ\nhj : j \u2260 i\n\u22a2 \u2115\n[PROOFSTEP]\nconvert\n congr_arg _\n (Function.update_noteq hj _ _)\n -- Fix `DecidableEq` mismatch\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\n\u22a2 \u00aci \u2208 Finset.univ \u2192 \u2191(monomial \u2191i) (Function.update 0 i 1 i) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\na\u271d : \u00aci \u2208 Finset.univ\n\u22a2 \u2191(monomial \u2191i) (Function.update 0 i 1 i) = 0\n[PROOFSTEP]\nhave := Finset.mem_univ i\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\ng : R[X]\nhg : Monic g\ni : Fin (natDegree g)\na\u271d : \u00aci \u2208 Finset.univ\nthis : i \u2208 Finset.univ\n\u22a2 \u2191(monomial \u2191i) (Function.update 0 i 1 i) = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\n\u22a2 minpoly K (root f) = f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n[PROOFSTEP]\nhave f'_monic : Monic _ := monic_mul_leadingCoeff_inv hf\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\n\u22a2 minpoly K (root f) = f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n[PROOFSTEP]\nrefine' (minpoly.unique K _ f'_monic _ _).symm\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\n\u22a2 \u2191(aeval (root f)) (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) = 0\n[PROOFSTEP]\nrw [AlgHom.map_mul, aeval_eq, mk_self, zero_mul]\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\n\u22a2 \u2200 (q : K[X]), Monic q \u2192 \u2191(aeval (root f)) q = 0 \u2192 degree (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) \u2264 degree q\n[PROOFSTEP]\nintro q q_monic q_aeval\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\n\u22a2 degree (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) \u2264 degree q\n[PROOFSTEP]\nhave commutes : (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval).comp (mk q) = mk f :=\n by\n ext\n \u00b7 simp only [RingHom.comp_apply, mk_C, lift_of]\n rfl\n \u00b7 simp only [RingHom.comp_apply, mk_X, lift_root]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\n\u22a2 RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.a\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\nx\u271d : K\n\u22a2 \u2191(RingHom.comp (RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q)) C) x\u271d =\n \u2191(RingHom.comp (mk f) C) x\u271d\n[PROOFSTEP]\nsimp only [RingHom.comp_apply, mk_C, lift_of]\n[GOAL]\ncase h\u2081.a\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\nx\u271d : K\n\u22a2 \u2191(algebraMap K (AdjoinRoot f)) x\u271d = \u2191(of f) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\n\u22a2 \u2191(RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q)) X = \u2191(mk f) X\n[PROOFSTEP]\nsimp only [RingHom.comp_apply, mk_X, lift_root]\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 degree (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) \u2264 degree q\n[PROOFSTEP]\nrw [degree_eq_natDegree f'_monic.ne_zero, degree_eq_natDegree q_monic.ne_zero, Nat.cast_withBot, Nat.cast_withBot,\n -- porting note: addedWithBot.coe_le_coe, natDegree_mul hf, natDegree_C, add_zero]\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 natDegree f \u2264 natDegree q\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\napply natDegree_le_of_dvd\n[GOAL]\ncase refine'_2.h1\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 f \u2223 q\n[PROOFSTEP]\nhave : mk f q = 0 := by rw [\u2190 commutes, RingHom.comp_apply, mk_self, RingHom.map_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 \u2191(mk f) q = 0\n[PROOFSTEP]\nrw [\u2190 commutes, RingHom.comp_apply, mk_self, RingHom.map_zero]\n[GOAL]\ncase refine'_2.h1\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\nthis : \u2191(mk f) q = 0\n\u22a2 f \u2223 q\n[PROOFSTEP]\nexact mk_eq_zero.1 this\n[GOAL]\ncase refine'_2.h2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 q \u2260 0\n[PROOFSTEP]\nexact q_monic.ne_zero\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf'_monic : Monic (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9)\nq : K[X]\nq_monic : Monic q\nq_aeval : \u2191(aeval (root f)) q = 0\ncommutes : RingHom.comp (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval) (mk q) = mk f\n\u22a2 \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, C_eq_zero, inv_eq_zero, leadingCoeff_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\n\u22a2 Basis (Fin (natDegree f)) K (AdjoinRoot f)\n[PROOFSTEP]\nlet f' := f * C f.leadingCoeff\u207b\u00b9\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n\u22a2 Basis (Fin (natDegree f)) K (AdjoinRoot f)\n[PROOFSTEP]\nhave deg_f' : f'.natDegree = f.natDegree :=\n by\n rw [natDegree_mul hf, natDegree_C, add_zero]\n \u00b7 rwa [Ne.def, C_eq_zero, inv_eq_zero, leadingCoeff_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n\u22a2 natDegree f' = natDegree f\n[PROOFSTEP]\nrw [natDegree_mul hf, natDegree_C, add_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n\u22a2 \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, C_eq_zero, inv_eq_zero, leadingCoeff_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\n\u22a2 Basis (Fin (natDegree f)) K (AdjoinRoot f)\n[PROOFSTEP]\nhave minpoly_eq : minpoly K (root f) = f' := minpoly_root hf\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\n\u22a2 Basis (Fin (natDegree f)) K (AdjoinRoot f)\n[PROOFSTEP]\napply @Basis.mk _ _ _ fun i : Fin f.natDegree => root f ^ i.val\n[GOAL]\ncase hli\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\n\u22a2 LinearIndependent K fun i => root f ^ \u2191i\n[PROOFSTEP]\nrw [\u2190 deg_f', \u2190 minpoly_eq]\n[GOAL]\ncase hli\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\n\u22a2 LinearIndependent K fun i => root f ^ \u2191i\n[PROOFSTEP]\nexact linearIndependent_pow (root f)\n[GOAL]\ncase hsp\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\n\u22a2 \u22a4 \u2264 Submodule.span K (Set.range fun i => root f ^ \u2191i)\n[PROOFSTEP]\nrintro y -\n[GOAL]\ncase hsp\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\n\u22a2 y \u2208 Submodule.span K (Set.range fun i => root f ^ \u2191i)\n[PROOFSTEP]\nrw [\u2190 deg_f', \u2190 minpoly_eq]\n[GOAL]\ncase hsp\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\n\u22a2 y \u2208 Submodule.span K (Set.range fun i => root f ^ \u2191i)\n[PROOFSTEP]\napply (isIntegral_root hf).mem_span_pow\n[GOAL]\ncase hsp\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\n\u22a2 \u2203 f_1, y = \u2191(aeval (root f)) f_1\n[PROOFSTEP]\nobtain \u27e8g\u27e9 := y\n[GOAL]\ncase hsp.mk\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng\u271d : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\ng : K[X]\n\u22a2 \u2203 f_1, Quot.mk Setoid.r g = \u2191(aeval (root f)) f_1\n[PROOFSTEP]\nuse g\n[GOAL]\ncase h\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng\u271d : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\ng : K[X]\n\u22a2 Quot.mk Setoid.r g = \u2191(aeval (root f)) g\n[PROOFSTEP]\nrw [aeval_eq]\n[GOAL]\ncase h\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng\u271d : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\nf' : K[X] := f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\ndeg_f' : natDegree f' = natDegree f\nminpoly_eq : minpoly K (root f) = f'\ny : AdjoinRoot f\ng : K[X]\n\u22a2 Quot.mk Setoid.r g = \u2191(mk f) g\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\n\u22a2 \u2200 (i : Fin (natDegree f)), \u2191(powerBasisAux hf) i = root f ^ \u2191i\n[PROOFSTEP]\nsimp [powerBasisAux]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : f \u2260 0\n\u22a2 minpoly K (powerBasis hf).gen = f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9\n[PROOFSTEP]\nrw [powerBasis_gen, minpoly_root hf]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b9 : CommRing R\ng : R[X]\ninst\u271d : Field K\nf : K[X]\nhf : Monic f\nhf' : optParam (f \u2260 0) (_ : f \u2260 0)\n\u22a2 minpoly K (powerBasis hf').gen = f\n[PROOFSTEP]\nrw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_one, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 \u2191(aeval { val := x, property := (_ : x \u2208 adjoin R {x}) }) (minpoly R x) = 0\n[PROOFSTEP]\nsimp [\u2190 Subalgebra.coe_eq_zero, aeval_subalgebra_coe]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\na : AdjoinRoot (minpoly R x)\n\u22a2 \u2191(aeval { val := x, property := (_ : x \u2208 adjoin R {x}) }) (minpoly R x) = 0\n[PROOFSTEP]\nsimp [\u2190 Subalgebra.coe_eq_zero, aeval_subalgebra_coe]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 \u2191(toAdjoin R x) (\u2191(mk (minpoly R x)) X) = { val := x, property := (_ : x \u2208 adjoin R {x}) }\n[PROOFSTEP]\nsimp [toAdjoin]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 Function.Surjective \u2191(toAdjoin R x)\n[PROOFSTEP]\nrw [\u2190 range_top_iff_surjective, _root_.eq_top_iff, \u2190 adjoin_adjoin_coe_preimage]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 adjoin R (Subtype.val \u207b\u00b9' {x}) \u2264 AlgHom.range (toAdjoin R x)\n[PROOFSTEP]\nrefine' adjoin_le _\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 Subtype.val \u207b\u00b9' {x} \u2286 \u2191(AlgHom.range (toAdjoin R x))\n[PROOFSTEP]\nsimp only [AlgHom.coe_range, Set.mem_range]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 Subtype.val \u207b\u00b9' {x} \u2286 Set.range \u2191(toAdjoin R x)\n[PROOFSTEP]\nrintro \u27e8y\u2081, y\u2082\u27e9 h\n[GOAL]\ncase mk\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx y\u2081 : S\ny\u2082 : y\u2081 \u2208 \u2191(adjoin R {x})\nh : { val := y\u2081, property := y\u2082 } \u2208 Subtype.val \u207b\u00b9' {x}\n\u22a2 { val := y\u2081, property := y\u2082 } \u2208 Set.range \u2191(toAdjoin R x)\n[PROOFSTEP]\nrefine' \u27e8mk (minpoly R x) X, by simpa [toAdjoin] using h.symm\u27e9\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx y\u2081 : S\ny\u2082 : y\u2081 \u2208 \u2191(adjoin R {x})\nh : { val := y\u2081, property := y\u2082 } \u2208 Subtype.val \u207b\u00b9' {x}\n\u22a2 \u2191(toAdjoin R x) (\u2191(mk (minpoly R x)) X) = { val := y\u2081, property := y\u2082 }\n[PROOFSTEP]\nsimpa [toAdjoin] using h.symm\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\ng : R[X]\npb : PowerBasis R S\nh\u2081 : \u2191(aeval (root g)) (minpoly R pb.gen) = 0\nh\u2082 : \u2191(aeval pb.gen) g = 0\nsrc\u271d : AdjoinRoot g \u2192\u2090[R] S := liftHom g pb.gen h\u2082\nx : AdjoinRoot g\n\u22a2 \u2191(PowerBasis.lift pb (root g) h\u2081) (\u2191(liftHom g pb.gen h\u2082) x) = x\n[PROOFSTEP]\ninduction x using AdjoinRoot.induction_on\n[GOAL]\ncase ih\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\ng : R[X]\npb : PowerBasis R S\nh\u2081 : \u2191(aeval (root g)) (minpoly R pb.gen) = 0\nh\u2082 : \u2191(aeval pb.gen) g = 0\nsrc\u271d : AdjoinRoot g \u2192\u2090[R] S := liftHom g pb.gen h\u2082\np\u271d : R[X]\n\u22a2 \u2191(PowerBasis.lift pb (root g) h\u2081) (\u2191(liftHom g pb.gen h\u2082) (\u2191(mk g) p\u271d)) = \u2191(mk g) p\u271d\n[PROOFSTEP]\nrw [liftHom_mk, pb.lift_aeval, aeval_eq]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\ng : R[X]\npb : PowerBasis R S\nh\u2081 : \u2191(aeval (root g)) (minpoly R pb.gen) = 0\nh\u2082 : \u2191(aeval pb.gen) g = 0\nsrc\u271d : AdjoinRoot g \u2192\u2090[R] S := liftHom g pb.gen h\u2082\nx : S\n\u22a2 \u2191(liftHom g pb.gen h\u2082) (\u2191(PowerBasis.lift pb (root g) h\u2081) x) = x\n[PROOFSTEP]\nnontriviality S\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\ng : R[X]\npb : PowerBasis R S\nh\u2081 : \u2191(aeval (root g)) (minpoly R pb.gen) = 0\nh\u2082 : \u2191(aeval pb.gen) g = 0\nsrc\u271d : AdjoinRoot g \u2192\u2090[R] S := liftHom g pb.gen h\u2082\nx : S\n\u271d : Nontrivial S\n\u22a2 \u2191(liftHom g pb.gen h\u2082) (\u2191(PowerBasis.lift pb (root g) h\u2081) x) = x\n[PROOFSTEP]\nobtain \u27e8f, _hf, rfl\u27e9 := pb.exists_eq_aeval x\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\ng : R[X]\npb : PowerBasis R S\nh\u2081 : \u2191(aeval (root g)) (minpoly R pb.gen) = 0\nh\u2082 : \u2191(aeval pb.gen) g = 0\nsrc\u271d : AdjoinRoot g \u2192\u2090[R] S := liftHom g pb.gen h\u2082\n\u271d : Nontrivial S\nf : R[X]\n_hf : natDegree f < pb.dim\n\u22a2 \u2191(liftHom g pb.gen h\u2082) (\u2191(PowerBasis.lift pb (root g) h\u2081) (\u2191(aeval pb.gen) f)) = \u2191(aeval pb.gen) f\n[PROOFSTEP]\nrw [pb.lift_aeval, aeval_eq, liftHom_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\nL : Type u_1\nF : Type u_2\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\npb : PowerBasis F K\nf : F[X]\nhf : f \u2260 0\nx : L\n\u22a2 x \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F (powerBasis hf).gen)) \u2194\n \u2191(Equiv.refl L) x \u2208 roots (Polynomial.map (algebraMap F L) f)\n[PROOFSTEP]\nrw [powerBasis_gen, minpoly_root hf, Polynomial.map_mul, roots_mul, Polynomial.map_C, roots_C, add_zero,\n Equiv.refl_apply]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\nL : Type u_1\nF : Type u_2\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\npb : PowerBasis F K\nf : F[X]\nhf : f \u2260 0\nx : L\n\u22a2 Polynomial.map (algebraMap F L) f * Polynomial.map (algebraMap F L) (\u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) \u2260 0\n[PROOFSTEP]\nrw [\u2190 Polynomial.map_mul]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\nL : Type u_1\nF : Type u_2\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\npb : PowerBasis F K\nf : F[X]\nhf : f \u2260 0\nx : L\n\u22a2 Polynomial.map (algebraMap F L) (f * \u2191C (Polynomial.leadingCoeff f)\u207b\u00b9) \u2260 0\n[PROOFSTEP]\nexact map_monic_ne_zero (monic_mul_leadingCoeff_inv hf)\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf : R[X]\n\u22a2 Ideal.map (of f) I = Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I)\n[PROOFSTEP]\nrw [of, AdjoinRoot.mk, Ideal.map_map]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf : R[X]\nx : AdjoinRoot f\n\u22a2 \u2191(RingEquiv.symm (quotMapOfEquivQuotMapCMapSpanMk I f))\n (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I))) x) =\n \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) x\n[PROOFSTEP]\nrw [quotMapOfEquivQuotMapCMapSpanMk, Ideal.quotEquivOfEq_symm]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf : R[X]\nx : AdjoinRoot f\n\u22a2 \u2191(quotEquivOfEq (_ : Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I) = Ideal.map (of f) I))\n (\u2191(Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (Ideal.map C I))) x) =\n \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) x\n[PROOFSTEP]\nexact Ideal.quotEquivOfEq_mk _ _\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf : R[X]\n\u22a2 span {\u2191(Ideal.Quotient.mk (Ideal.map C I)) f} =\n Ideal.map (\u2191(polynomialQuotientEquivQuotientPolynomial I)) (span {Polynomial.map (Ideal.Quotient.mk I) f})\n[PROOFSTEP]\nrw [map_span, Set.image_singleton, RingEquiv.coe_toRingHom, polynomialQuotientEquivQuotientPolynomial_map_mk I f]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf p : R[X]\n\u22a2 \u2191(quotQuotEquivComm I f)\n (\u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) p)) =\n \u2191(Ideal.Quotient.mk (span {\u2191(Ideal.Quotient.mk (Ideal.map C I)) f})) (\u2191(Ideal.Quotient.mk (Ideal.map C I)) p)\n[PROOFSTEP]\nsimp only [Polynomial.quotQuotEquivComm, quotientEquiv_mk, polynomialQuotientEquivQuotientPolynomial_map_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf p : R[X]\n\u22a2 \u2191(RingEquiv.symm (quotQuotEquivComm I f))\n (\u2191(Ideal.Quotient.mk (span {\u2191(Ideal.Quotient.mk (Ideal.map C I)) f})) (\u2191(Ideal.Quotient.mk (Ideal.map C I)) p)) =\n \u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) p)\n[PROOFSTEP]\nsimp only [Polynomial.quotQuotEquivComm, quotientEquiv_symm_mk, polynomialQuotientEquivQuotientPolynomial_symm_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf : R[X]\n\u22a2 Ideal.map (Ideal.Quotient.mk (Ideal.map C I)) (span {f}) = span {\u2191(Ideal.Quotient.mk (Ideal.map C I)) f}\n[PROOFSTEP]\nrw [map_span, Set.image_singleton]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI : Ideal R\nf p : R[X]\n\u22a2 \u2191(RingEquiv.symm (quotAdjoinRootEquivQuotPolynomialQuot I f))\n (\u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) p)) =\n \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) (\u2191(mk f) p)\n[PROOFSTEP]\nrw [quotAdjoinRootEquivQuotPolynomialQuot, RingEquiv.symm_trans_apply, RingEquiv.symm_trans_apply,\n RingEquiv.symm_trans_apply, RingEquiv.symm_symm, Polynomial.quotQuotEquivComm_mk, Ideal.quotEquivOfEq_symm,\n Ideal.quotEquivOfEq_mk, \u2190 RingHom.comp_apply, \u2190 DoubleQuot.quotQuotMk,\n quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk, quotMapOfEquivQuotMapCMapSpanMk_symm_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI\u271d : Ideal R\nf\u271d f : R[X]\nI : Ideal R\nx : R\n\u22a2 \u2191(quotAdjoinRootEquivQuotPolynomialQuot I f) (\u2191(algebraMap R (AdjoinRoot f \u29f8 Ideal.map (of f) I)) x) =\n \u2191(algebraMap R ((R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) f})) x\n[PROOFSTEP]\nhave :\n algebraMap R (AdjoinRoot f \u29f8 Ideal.map (of f) I) x =\n Ideal.Quotient.mk (Ideal.map (AdjoinRoot.of f) I) ((mk f) (C x)) :=\n rfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI\u271d : Ideal R\nf\u271d f : R[X]\nI : Ideal R\nx : R\nthis :\n \u2191(algebraMap R (AdjoinRoot f \u29f8 Ideal.map (of f) I)) x = \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) (\u2191(mk f) (\u2191C x))\n\u22a2 \u2191(quotAdjoinRootEquivQuotPolynomialQuot I f) (\u2191(algebraMap R (AdjoinRoot f \u29f8 Ideal.map (of f) I)) x) =\n \u2191(algebraMap R ((R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) f})) x\n[PROOFSTEP]\nrw [this, quotAdjoinRootEquivQuotPolynomialQuot_mk_of, map_C]\n -- Porting note: the following `rfl` was not needed\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI\u271d : Ideal R\nf\u271d f : R[X]\nI : Ideal R\nx : R\nthis :\n \u2191(algebraMap R (AdjoinRoot f \u29f8 Ideal.map (of f) I)) x = \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) (\u2191(mk f) (\u2191C x))\n\u22a2 \u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (\u2191C (\u2191(Ideal.Quotient.mk I) x)) =\n \u2191(algebraMap R ((R \u29f8 I)[X] \u29f8 span {Polynomial.map (Ideal.Quotient.mk I) f})) x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI\u271d : Ideal R\nf\u271d f g : R[X]\nI : Ideal R\n\u22a2 \u2191(quotEquivQuotMap f I) (\u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) (\u2191(mk f) g)) =\n \u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)\n[PROOFSTEP]\nrw [AdjoinRoot.quotEquivQuotMap_apply, AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_mk_of]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d : CommRing R\nI\u271d : Ideal R\nf\u271d f g : R[X]\nI : Ideal R\n\u22a2 \u2191(AlgEquiv.symm (quotEquivQuotMap f I))\n (\u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) g)) =\n \u2191(Ideal.Quotient.mk (Ideal.map (of f) I)) (\u2191(mk f) g)\n[PROOFSTEP]\nrw [AdjoinRoot.quotEquivQuotMap_symm_apply, AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n\u22a2 \u2191(aeval (root (minpoly R pb.gen))) (minpoly R pb.gen) = 0\n[PROOFSTEP]\nrw [AdjoinRoot.aeval_eq, AdjoinRoot.mk_self]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n\u22a2 Ideal.map (of (minpoly R pb.gen)) I =\n Ideal.map\n (\u2191(toRingEquiv\n (AlgEquiv.symm\n (equiv' (minpoly R pb.gen) pb (_ : \u2191(aeval (root (minpoly R pb.gen))) (minpoly R pb.gen) = 0)\n (_ : \u2191(aeval pb.gen) (minpoly R pb.gen) = 0)))))\n (Ideal.map (algebraMap R S) I)\n[PROOFSTEP]\nrw [Ideal.map_map, AlgEquiv.toRingEquiv_eq_coe, \u2190 AlgEquiv.coe_ringHom_commutes, \u2190 AdjoinRoot.algebraMap_eq,\n AlgHom.comp_algebraMap]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n\u22a2 \u2191(quotientEquiv (Ideal.map (algebraMap R S) I) (Ideal.map (of (minpoly R pb.gen)) I)\n (toRingEquiv\n (AlgEquiv.symm\n (equiv' (minpoly R pb.gen) pb (_ : \u2191(aeval (root (minpoly R pb.gen))) (minpoly R pb.gen) = 0)\n (_ : \u2191(aeval pb.gen) (minpoly R pb.gen) = 0))))\n (_ :\n Ideal.map (of (minpoly R pb.gen)) I =\n Ideal.map\n (\u2191(toRingEquiv\n (AlgEquiv.symm\n (equiv' (minpoly R pb.gen) pb (_ : \u2191(aeval (root (minpoly R pb.gen))) (minpoly R pb.gen) = 0)\n (_ : \u2191(aeval pb.gen) (minpoly R pb.gen) = 0)))))\n (Ideal.map (algebraMap R S) I)))\n (\u2191(algebraMap R (S \u29f8 Ideal.map (algebraMap R S) I)) x) =\n \u2191(algebraMap R (AdjoinRoot (minpoly R pb.gen) \u29f8 Ideal.map (of (minpoly R pb.gen)) I)) x\n[PROOFSTEP]\nrw [\u2190 Ideal.Quotient.mk_algebraMap, Ideal.quotientEquiv_apply, RingHom.toFun_eq_coe, Ideal.quotientMap_mk,\n AlgEquiv.toRingEquiv_eq_coe, RingEquiv.coe_toRingHom, AlgEquiv.coe_ringEquiv, AlgEquiv.commutes,\n Quotient.mk_algebraMap]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\nx : R\n\u22a2 \u2191(Ideal.Quotient.mk (Ideal.map (of (minpoly R pb.gen)) I)) (\u2191(algebraMap R (AdjoinRoot (minpoly R pb.gen))) x) =\n \u2191(algebraMap R (AdjoinRoot (minpoly R pb.gen) \u29f8 Ideal.map (of (minpoly R pb.gen)) I)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\ng : R[X]\n\u22a2 \u2191(quotientEquivQuotientMinpolyMap pb I) (\u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (\u2191(aeval pb.gen) g)) =\n \u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R pb.gen)}))\n (Polynomial.map (Ideal.Quotient.mk I) g)\n[PROOFSTEP]\nrw [PowerBasis.quotientEquivQuotientMinpolyMap, AlgEquiv.trans_apply, AlgEquiv.ofRingEquiv_apply, quotientEquiv_mk,\n AlgEquiv.coe_ringEquiv', AdjoinRoot.equiv'_symm_apply, PowerBasis.lift_aeval, AdjoinRoot.aeval_eq,\n AdjoinRoot.quotEquivQuotMap_apply_mk]\n[GOAL]\nR : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\npb : PowerBasis R S\nI : Ideal R\ng : R[X]\n\u22a2 \u2191(AlgEquiv.symm (quotientEquivQuotientMinpolyMap pb I))\n (\u2191(Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R pb.gen)}))\n (Polynomial.map (Ideal.Quotient.mk I) g)) =\n \u2191(Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (\u2191(aeval pb.gen) g)\n[PROOFSTEP]\nsimp only [quotientEquivQuotientMinpolyMap, toRingEquiv_eq_coe, symm_trans_apply, quotEquivQuotMap_symm_apply_mk,\n ofRingEquiv_symm_apply, quotientEquiv_symm_mk, toRingEquiv_symm, RingEquiv.symm_symm, AdjoinRoot.equiv'_apply,\n coe_ringEquiv, liftHom_mk, symm_toRingEquiv]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.AdjoinRoot", "llama_tokens": 27404, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.27158017776866955}} {"text": "[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : L \u2192 R\nm : M\n\u22a2 m \u2208 preWeightSpace M \u03c7 \u2194 \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nsimp [preWeightSpace]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 M\nx : L\n\u22a2 \u2203 k, preWeightSpace M 0 \u2264 LinearMap.ker (\u2191(toEndomorphism R L M) x ^ k)\n[PROOFSTEP]\nuse(toEndomorphism R L M x).maximalGeneralizedEigenspaceIndex 0\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 M\nx : L\n\u22a2 preWeightSpace M 0 \u2264\n LinearMap.ker\n (\u2191(toEndomorphism R L M) x ^ Module.End.maximalGeneralizedEigenspaceIndex (\u2191(toEndomorphism R L M) x) 0)\n[PROOFSTEP]\nsimp only [\u2190 Module.End.generalizedEigenspace_zero, preWeightSpace, Pi.zero_apply, iInf_le, \u2190\n (toEndomorphism R L M x).maximalGeneralizedEigenspace_eq]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\n\u22a2 LinearMap.range (LinearMap.comp (\u2191g) (TensorProduct.mapIncl (preWeightSpace M\u2081 \u03c7\u2081) (preWeightSpace M\u2082 \u03c7\u2082))) \u2264\n preWeightSpace M\u2083 (\u03c7\u2081 + \u03c7\u2082)\n[PROOFSTEP]\nintro m\u2083\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nm\u2083 : M\u2083\n\u22a2 m\u2083 \u2208 LinearMap.range (LinearMap.comp (\u2191g) (TensorProduct.mapIncl (preWeightSpace M\u2081 \u03c7\u2081) (preWeightSpace M\u2082 \u03c7\u2082))) \u2192\n m\u2083 \u2208 preWeightSpace M\u2083 (\u03c7\u2081 + \u03c7\u2082)\n[PROOFSTEP]\nsimp only [LieModuleHom.coe_toLinearMap, Pi.add_apply, Function.comp_apply, mem_preWeightSpace, LinearMap.coe_comp,\n TensorProduct.mapIncl, exists_imp, LinearMap.mem_range]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nm\u2083 : M\u2083\n\u22a2 \u2200 (x : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }),\n \u2191g (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082))) x) =\n m\u2083 \u2192\n \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k) m\u2083 = 0\n[PROOFSTEP]\nrintro t rfl x\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t)) =\n 0\n[PROOFSTEP]\nlet F : Module.End R M\u2083 :=\n toEndomorphism R L M\u2083 x -\n (\u03c7\u2081 x + \u03c7\u2082 x) \u2022\n \u21911\n -- The goal is linear in `t` so use induction to reduce to the case that `t` is a pure tensor.\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t)) =\n 0\n[PROOFSTEP]\nrefine t.induction_on ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n 0)) =\n 0\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ 0)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n 0)) =\n 0\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, LieModuleHom.map_zero]\n[GOAL]\ncase refine_2\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2200 (x_1 : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 }) (y : { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }),\n \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (x_1 \u2297\u209c[R] y))) =\n 0\ncase refine_3\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2200 (x_1 y : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }),\n (\u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n x_1)) =\n 0) \u2192\n (\u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n y)) =\n 0) \u2192\n \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (x_1 + y))) =\n 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_3\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2200 (x_1 y : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }),\n (\u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n x_1)) =\n 0) \u2192\n (\u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n y)) =\n 0) \u2192\n \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081))\n (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (x_1 + y))) =\n 0\n[PROOFSTEP]\nrintro t\u2081 t\u2082 \u27e8k\u2081, hk\u2081\u27e9 \u27e8k\u2082, hk\u2082\u27e9\n[GOAL]\ncase refine_3.intro.intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nt\u2081 t\u2082 : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nk\u2081 : \u2115\nhk\u2081 :\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k\u2081)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t\u2081)) =\n 0\nk\u2082 : \u2115\nhk\u2082 :\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k\u2082)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t\u2082)) =\n 0\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (t\u2081 + t\u2082))) =\n 0\n[PROOFSTEP]\nuse max k\u2081 k\u2082\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nt\u2081 t\u2082 : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nk\u2081 : \u2115\nhk\u2081 :\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k\u2081)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t\u2081)) =\n 0\nk\u2082 : \u2115\nhk\u2082 :\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k\u2082)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n t\u2082)) =\n 0\n\u22a2 \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ max k\u2081 k\u2082)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (t\u2081 + t\u2082))) =\n 0\n[PROOFSTEP]\nsimp only [LieModuleHom.map_add, LinearMap.map_add, LinearMap.pow_map_zero_of_le (le_max_left k\u2081 k\u2082) hk\u2081,\n LinearMap.pow_map_zero_of_le (le_max_right k\u2081 k\u2082) hk\u2082, add_zero]\n -- Now the main argument: pure tensors.\n[GOAL]\ncase refine_2\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\n\u22a2 \u2200 (x_1 : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 }) (y : { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }),\n \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n (x_1 \u2297\u209c[R] y))) =\n 0\n[PROOFSTEP]\nrintro \u27e8m\u2081, hm\u2081\u27e9\n \u27e8m\u2082, hm\u2082\u27e9\n -- change \u2203 k, (F ^ k) ((g : M\u2081 \u2297[R] M\u2082 \u2192\u2097[R] M\u2083) (m\u2081 \u2297\u209c m\u2082)) = 0\n -- Eliminate `g` from the picture.\n[GOAL]\ncase refine_2.mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nlet f\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := (toEndomorphism R L M\u2081 x - \u03c7\u2081 x \u2022 \u21911).rTensor M\u2082\n[GOAL]\ncase refine_2.mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nlet f\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := (toEndomorphism R L M\u2082 x - \u03c7\u2082 x \u2022 \u21911).lTensor M\u2081\n[GOAL]\ncase refine_2.mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nhave h_comm_square : F \u2218\u2097 \u2191g = (g : M\u2081 \u2297[R] M\u2082 \u2192\u2097[R] M\u2083).comp (f\u2081 + f\u2082) :=\n by\n ext m\u2081 m\u2082;\n simp only [\u2190 g.map_lie x (m\u2081 \u2297\u209c m\u2082), add_smul, sub_tmul, tmul_sub, smul_tmul, lie_tmul_right, tmul_smul,\n toEndomorphism_apply_apply, LieModuleHom.map_smul, LinearMap.one_apply, LieModuleHom.coe_toLinearMap,\n LinearMap.smul_apply, Function.comp_apply, LinearMap.coe_comp, LinearMap.rTensor_tmul, LieModuleHom.map_add,\n LinearMap.add_apply, LieModuleHom.map_sub, LinearMap.sub_apply, LinearMap.lTensor_tmul,\n AlgebraTensorModule.curry_apply, curry_apply, LinearMap.toFun_eq_coe, LinearMap.coe_restrictScalars]\n abel\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\n\u22a2 LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\n[PROOFSTEP]\next m\u2081 m\u2082\n[GOAL]\ncase a.h.h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081\u271d : M\u2081\nhm\u2081 : m\u2081\u271d \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082\u271d : M\u2082\nhm\u2082 : m\u2082\u271d \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nm\u2081 : M\u2081\nm\u2082 : M\u2082\n\u22a2 \u2191(\u2191(AlgebraTensorModule.curry (LinearMap.comp F \u2191g)) m\u2081) m\u2082 =\n \u2191(\u2191(AlgebraTensorModule.curry (LinearMap.comp (\u2191g) (f\u2081 + f\u2082))) m\u2081) m\u2082\n[PROOFSTEP]\nsimp only [\u2190 g.map_lie x (m\u2081 \u2297\u209c m\u2082), add_smul, sub_tmul, tmul_sub, smul_tmul, lie_tmul_right, tmul_smul,\n toEndomorphism_apply_apply, LieModuleHom.map_smul, LinearMap.one_apply, LieModuleHom.coe_toLinearMap,\n LinearMap.smul_apply, Function.comp_apply, LinearMap.coe_comp, LinearMap.rTensor_tmul, LieModuleHom.map_add,\n LinearMap.add_apply, LieModuleHom.map_sub, LinearMap.sub_apply, LinearMap.lTensor_tmul,\n AlgebraTensorModule.curry_apply, curry_apply, LinearMap.toFun_eq_coe, LinearMap.coe_restrictScalars]\n[GOAL]\ncase a.h.h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081\u271d : M\u2081\nhm\u2081 : m\u2081\u271d \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082\u271d : M\u2082\nhm\u2082 : m\u2082\u271d \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nm\u2081 : M\u2081\nm\u2082 : M\u2082\n\u22a2 \u2191g (\u2045x, m\u2081\u2046 \u2297\u209c[R] m\u2082) + \u2191g (m\u2081 \u2297\u209c[R] \u2045x, m\u2082\u2046) - (\u03c7\u2081 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082) + \u03c7\u2082 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082)) =\n \u2191g (\u2045x, m\u2081\u2046 \u2297\u209c[R] m\u2082) - \u03c7\u2081 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082) + (\u2191g (m\u2081 \u2297\u209c[R] \u2045x, m\u2082\u2046) - \u03c7\u2082 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082))\n[PROOFSTEP]\nabel\n[GOAL]\ncase a.h.h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081\u271d : M\u2081\nhm\u2081 : m\u2081\u271d \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082\u271d : M\u2082\nhm\u2082 : m\u2082\u271d \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nm\u2081 : M\u2081\nm\u2082 : M\u2082\n\u22a2 \u2191g (\u2045x, m\u2081\u2046 \u2297\u209c[R] m\u2082) + \u2191g (m\u2081 \u2297\u209c[R] \u2045x, m\u2082\u2046) - (\u03c7\u2081 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082) + \u03c7\u2082 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082)) =\n \u2191g (\u2045x, m\u2081\u2046 \u2297\u209c[R] m\u2082) - \u03c7\u2081 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082) + (\u2191g (m\u2081 \u2297\u209c[R] \u2045x, m\u2082\u2046) - \u03c7\u2082 x \u2022 \u2191g (m\u2081 \u2297\u209c[R] m\u2082))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine_2.mk.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nrsuffices \u27e8k, hk\u27e9 : \u2203 k : \u2115, ((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c m\u2082) = 0\n[GOAL]\ncase refine_2.mk.mk.intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nk : \u2115\nhk : \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2203 k,\n \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nk : \u2115\nhk : \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2191((\u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1) ^ k)\n (\u2191g\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace M\u2081 \u03c7\u2081)) (Submodule.subtype (preWeightSpace M\u2082 \u03c7\u2082)))\n ({ val := m\u2081, property := hm\u2081 } \u2297\u209c[R] { val := m\u2082, property := hm\u2082 }))) =\n 0\n[PROOFSTEP]\nchange (F ^ k) (g.toLinearMap (m\u2081 \u2297\u209c[R] m\u2082)) = 0\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nk : \u2115\nhk : \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2191(F ^ k) (\u2191\u2191g (m\u2081 \u2297\u209c[R] m\u2082)) = 0\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_apply, LinearMap.commute_pow_left_of_commute h_comm_square, LinearMap.comp_apply, hk,\n LinearMap.map_zero]\n -- Unpack the information we have about `m\u2081`, `m\u2082`.\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nhm\u2081 : m\u2081 \u2208 preWeightSpace M\u2081 \u03c7\u2081\nm\u2082 : M\u2082\nhm\u2082 : m\u2082 \u2208 preWeightSpace M\u2082 \u03c7\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nsimp only [mem_preWeightSpace] at hm\u2081 hm\u2082 \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nobtain \u27e8k\u2081, hk\u2081\u27e9 := hm\u2081 x\n[GOAL]\ncase intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nobtain \u27e8k\u2082, hk\u2082\u27e9 := hm\u2082 x\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nhave hf\u2081 : (f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c m\u2082) = 0 := by simp only [hk\u2081, zero_tmul, LinearMap.rTensor_tmul, LinearMap.rTensor_pow]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\n\u22a2 \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nsimp only [hk\u2081, zero_tmul, LinearMap.rTensor_tmul, LinearMap.rTensor_pow]\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nhave hf\u2082 : (f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c m\u2082) = 0 := by\n simp only [hk\u2082, tmul_zero, LinearMap.lTensor_tmul, LinearMap.lTensor_pow]\n -- It's now just an application of the binomial theorem.\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nsimp only [hk\u2082, tmul_zero, LinearMap.lTensor_tmul, LinearMap.lTensor_pow]\n -- It's now just an application of the binomial theorem.\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2203 k, \u2191((f\u2081 + f\u2082) ^ k) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nuse k\u2081 + k\u2082 - 1\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 \u2191((f\u2081 + f\u2082) ^ (k\u2081 + k\u2082 - 1)) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nhave hf_comm : Commute f\u2081 f\u2082 := by\n ext m\u2081 m\u2082\n simp only [LinearMap.mul_apply, LinearMap.rTensor_tmul, LinearMap.lTensor_tmul, AlgebraTensorModule.curry_apply,\n LinearMap.toFun_eq_coe, LinearMap.lTensor_tmul, curry_apply, LinearMap.coe_restrictScalars]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 Commute f\u2081 f\u2082\n[PROOFSTEP]\next m\u2081 m\u2082\n[GOAL]\ncase a.h.h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081\u271d : M\u2081\nm\u2082\u271d : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081\u271d = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082\u271d = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081\u271d = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082\u271d = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081\u271d \u2297\u209c[R] m\u2082\u271d) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081\u271d \u2297\u209c[R] m\u2082\u271d) = 0\nm\u2081 : M\u2081\nm\u2082 : M\u2082\n\u22a2 \u2191(\u2191(AlgebraTensorModule.curry (f\u2081 * f\u2082)) m\u2081) m\u2082 = \u2191(\u2191(AlgebraTensorModule.curry (f\u2082 * f\u2081)) m\u2081) m\u2082\n[PROOFSTEP]\nsimp only [LinearMap.mul_apply, LinearMap.rTensor_tmul, LinearMap.lTensor_tmul, AlgebraTensorModule.curry_apply,\n LinearMap.toFun_eq_coe, LinearMap.lTensor_tmul, curry_apply, LinearMap.coe_restrictScalars]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\n\u22a2 \u2191((f\u2081 + f\u2082) ^ (k\u2081 + k\u2082 - 1)) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nrw [hf_comm.add_pow']\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\n\u22a2 \u2191(\u2211 m in Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1), Nat.choose (k\u2081 + k\u2082 - 1) m.fst \u2022 (f\u2081 ^ m.fst * f\u2082 ^ m.snd))\n (m\u2081 \u2297\u209c[R] m\u2082) =\n 0\n[PROOFSTEP]\nsimp only [TensorProduct.mapIncl, Submodule.subtype_apply, Finset.sum_apply, Submodule.coe_mk, LinearMap.coeFn_sum,\n TensorProduct.map_tmul, LinearMap.smul_apply]\n -- The required sum is zero because each individual term is zero.\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\n\u22a2 \u2211 x_1 in Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1),\n Nat.choose (k\u2081 + k\u2082 - 1) x_1.fst \u2022\n \u2191(LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ x_1.fst *\n LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ x_1.snd)\n (m\u2081 \u2297\u209c[R] m\u2082) =\n 0\n[PROOFSTEP]\napply Finset.sum_eq_zero\n[GOAL]\ncase h.h\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\n\u22a2 \u2200 (x_1 : \u2115 \u00d7 \u2115),\n x_1 \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1) \u2192\n Nat.choose (k\u2081 + k\u2082 - 1) x_1.fst \u2022\n \u2191(LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ x_1.fst *\n LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ x_1.snd)\n (m\u2081 \u2297\u209c[R] m\u2082) =\n 0\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase h.h.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\n\u22a2 Nat.choose (k\u2081 + k\u2082 - 1) (i, j).fst \u2022\n \u2191(LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ (i, j).fst *\n LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ (i, j).snd)\n (m\u2081 \u2297\u209c[R] m\u2082) =\n 0\n[PROOFSTEP]\nsuffices (f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c m\u2082) = 0 by rw [this]; apply smul_zero\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\nthis : \u2191(f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 Nat.choose (k\u2081 + k\u2082 - 1) (i, j).fst \u2022\n \u2191(LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ (i, j).fst *\n LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ (i, j).snd)\n (m\u2081 \u2297\u209c[R] m\u2082) =\n 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\nthis : \u2191(f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n\u22a2 Nat.choose (k\u2081 + k\u2082 - 1) (i, j).fst \u2022 0 = 0\n[PROOFSTEP]\napply smul_zero\n[GOAL]\ncase h.h.mk\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\n\u22a2 \u2191(f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\ncases' Nat.le_or_le_of_add_eq_add_pred (Finset.Nat.mem_antidiagonal.mp hij) with hi hj\n[GOAL]\ncase h.h.mk.inl\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\nhi : k\u2081 \u2264 (i, j).fst\n\u22a2 \u2191(f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nrw [(hf_comm.pow_pow i j).eq, LinearMap.mul_apply, LinearMap.pow_map_zero_of_le hi hf\u2081, LinearMap.map_zero]\n[GOAL]\ncase h.h.mk.inr\nR : Type u\nL : Type v\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : LieRing L\ninst\u271d\u00b9\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u00b9\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b9\u2075 : AddCommGroup M\ninst\u271d\u00b9\u2074 : Module R M\ninst\u271d\u00b9\u00b3 : LieRingModule L M\ninst\u271d\u00b9\u00b2 : LieModule R L M\nM\u2081 : Type w\u2081\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R M\u2081\ninst\u271d\u2079 : LieRingModule L M\u2081\ninst\u271d\u2078 : LieModule R L M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : Module R M\u2082\ninst\u271d\u2075 : LieRingModule L M\u2082\ninst\u271d\u2074 : LieModule R L M\u2082\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : Module R M\u2083\ninst\u271d\u00b9 : LieRingModule L M\u2083\ninst\u271d : LieModule R L M\u2083\ng : M\u2081 \u2297[R] M\u2082 \u2192\u2097\u2045R,L\u2046 M\u2083\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nt : { x // x \u2208 preWeightSpace M\u2081 \u03c7\u2081 } \u2297[R] { x // x \u2208 preWeightSpace M\u2082 \u03c7\u2082 }\nx : L\nF : Module.End R M\u2083 := \u2191(toEndomorphism R L M\u2083) x - (\u03c7\u2081 x + \u03c7\u2082 x) \u2022 1\nm\u2081 : M\u2081\nm\u2082 : M\u2082\nf\u2081 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.rTensor M\u2082 (\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1)\nf\u2082 : Module.End R (M\u2081 \u2297[R] M\u2082) := LinearMap.lTensor M\u2081 (\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1)\nh_comm_square : LinearMap.comp F \u2191g = LinearMap.comp (\u2191g) (f\u2081 + f\u2082)\nhm\u2081 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k) m\u2081 = 0\nhm\u2082 : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k) m\u2082 = 0\nk\u2081 : \u2115\nhk\u2081 : \u2191((\u2191(toEndomorphism R L M\u2081) x - \u03c7\u2081 x \u2022 1) ^ k\u2081) m\u2081 = 0\nk\u2082 : \u2115\nhk\u2082 : \u2191((\u2191(toEndomorphism R L M\u2082) x - \u03c7\u2082 x \u2022 1) ^ k\u2082) m\u2082 = 0\nhf\u2081 : \u2191(f\u2081 ^ k\u2081) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf\u2082 : \u2191(f\u2082 ^ k\u2082) (m\u2081 \u2297\u209c[R] m\u2082) = 0\nhf_comm : Commute f\u2081 f\u2082\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (k\u2081 + k\u2082 - 1)\nhj : k\u2082 \u2264 (i, j).snd\n\u22a2 \u2191(f\u2081 ^ i * f\u2082 ^ j) (m\u2081 \u2297\u209c[R] m\u2082) = 0\n[PROOFSTEP]\nrw [LinearMap.mul_apply, LinearMap.pow_map_zero_of_le hj hf\u2082, LinearMap.map_zero]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nx : L\nm : M\nhx : x \u2208 preWeightSpace L \u03c7\u2081\nhm : m \u2208 preWeightSpace M \u03c7\u2082\n\u22a2 \u2045x, m\u2046 \u2208 preWeightSpace M (\u03c7\u2081 + \u03c7\u2082)\n[PROOFSTEP]\napply LieModule.weight_vector_multiplication L L M M (toModuleHom R L M) \u03c7\u2081 \u03c7\u2082\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nx : L\nm : M\nhx : x \u2208 preWeightSpace L \u03c7\u2081\nhm : m \u2208 preWeightSpace M \u03c7\u2082\n\u22a2 \u2045x, m\u2046 \u2208\n LinearMap.range\n (LinearMap.comp (\u2191(toModuleHom R L M)) (TensorProduct.mapIncl (preWeightSpace L \u03c7\u2081) (preWeightSpace M \u03c7\u2082)))\n[PROOFSTEP]\nsimp only [LieModuleHom.coe_toLinearMap, Function.comp_apply, LinearMap.coe_comp, TensorProduct.mapIncl,\n LinearMap.mem_range]\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nx : L\nm : M\nhx : x \u2208 preWeightSpace L \u03c7\u2081\nhm : m \u2208 preWeightSpace M \u03c7\u2082\n\u22a2 \u2203 y,\n \u2191(toModuleHom R L M)\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace L \u03c7\u2081)) (Submodule.subtype (preWeightSpace M \u03c7\u2082))) y) =\n \u2045x, m\u2046\n[PROOFSTEP]\nuse\u27e8x, hx\u27e9 \u2297\u209c \u27e8m, hm\u27e9\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : L \u2192 R\nx : L\nm : M\nhx : x \u2208 preWeightSpace L \u03c7\u2081\nhm : m \u2208 preWeightSpace M \u03c7\u2082\n\u22a2 \u2191(toModuleHom R L M)\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace L \u03c7\u2081)) (Submodule.subtype (preWeightSpace M \u03c7\u2082)))\n ({ val := x, property := hx } \u2297\u209c[R] { val := m, property := hm })) =\n \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [Submodule.subtype_apply, toModuleHom_apply, TensorProduct.map_tmul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x\u271d, m\u271d\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x\u271d, m\u271d\u2046 \u2208 (preWeightSpace M \u03c7).toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [\u2190 zero_add \u03c7]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x\u271d, m\u271d\u2046 \u2208 (preWeightSpace M (0 + \u03c7)).toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrefine lie_mem_preWeightSpace_of_mem_preWeightSpace ?_ hm\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 x\u271d \u2208 preWeightSpace L 0\n[PROOFSTEP]\nsuffices preWeightSpace L (0 : L \u2192 R) = \u22a4 by simp only [this, Submodule.mem_top]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nthis : preWeightSpace L 0 = \u22a4\n\u22a2 x\u271d \u2208 preWeightSpace L 0\n[PROOFSTEP]\nsimp only [this, Submodule.mem_top]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nsrc\u271d : Submodule R M := preWeightSpace M \u03c7\nx\u271d : L\nm\u271d : M\nhm :\n m\u271d \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 preWeightSpace L 0 = \u22a4\n[PROOFSTEP]\nexact LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 weightSpace M 0 = \u22a4\n[PROOFSTEP]\nrw [\u2190 LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.top_coeSubmodule]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 \u2191(weightSpace M 0) = \u22a4\n[PROOFSTEP]\nexact iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L M\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\n\u22a2 \u2191(weightSpace M (\u03c7 \u2218 \u2191(LieSubalgebra.incl \u22a4))) = \u2191(weightSpace M \u03c7)\n[PROOFSTEP]\next m\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\n\u22a2 m \u2208 \u2191(weightSpace M (\u03c7 \u2218 \u2191(LieSubalgebra.incl \u22a4))) \u2194 m \u2208 \u2191(weightSpace M \u03c7)\n[PROOFSTEP]\nsimp only [weightSpace, LieSubmodule.coe_toSubmodule_mk, LieSubalgebra.coe_bracket_of_module, Function.comp_apply,\n mem_preWeightSpace]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\n\u22a2 (\u2200 (x : { x // x \u2208 \u22a4 }),\n \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0) \u2194\n \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\n\u22a2 (\u2200 (x : { x // x \u2208 \u22a4 }),\n \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0) \u2192\n \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase h.mpr\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\n\u22a2 (\u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0) \u2192\n \u2200 (x : { x // x \u2208 \u22a4 }),\n \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase h.mp\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh :\n \u2200 (x : { x // x \u2208 \u22a4 }), \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\nx : L\n\u22a2 \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := h \u27e8x, Set.mem_univ x\u27e9\n[GOAL]\ncase h.mp.intro\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh :\n \u2200 (x : { x // x \u2208 \u22a4 }), \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\nx : L\nk : \u2115\nhk :\n \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) { val := x, property := (_ : x \u2208 Set.univ) } -\n \u03c7 (\u2191(LieSubalgebra.incl \u22a4) { val := x, property := (_ : x \u2208 Set.univ) }) \u2022 1) ^\n k)\n m =\n 0\n\u22a2 \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh :\n \u2200 (x : { x // x \u2208 \u22a4 }), \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\nx : L\nk : \u2115\nhk :\n \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) { val := x, property := (_ : x \u2208 Set.univ) } -\n \u03c7 (\u2191(LieSubalgebra.incl \u22a4) { val := x, property := (_ : x \u2208 Set.univ) }) \u2022 1) ^\n k)\n m =\n 0\n\u22a2 \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nexact hk\n[GOAL]\ncase h.mpr\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\nx : { x // x \u2208 \u22a4 }\n\u22a2 \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := h x\n[GOAL]\ncase h.mpr.intro\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\nx : { x // x \u2208 \u22a4 }\nk : \u2115\nhk : \u2191((\u2191(toEndomorphism R L M) \u2191x - \u03c7 \u2191x \u2022 1) ^ k) m = 0\n\u22a2 \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\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 : LieAlgebra.IsNilpotent R L\n\u03c7 : L \u2192 R\nm : M\nh : \u2200 (x : L), \u2203 k, \u2191((\u2191(toEndomorphism R L M) x - \u03c7 x \u2022 1) ^ k) m = 0\nx : { x // x \u2208 \u22a4 }\nk : \u2115\nhk : \u2191((\u2191(toEndomorphism R L M) \u2191x - \u03c7 \u2191x \u2022 1) ^ k) m = 0\n\u22a2 \u2191((\u2191(toEndomorphism R { x // x \u2208 \u22a4 } M) x - \u03c7 (\u2191(LieSubalgebra.incl \u22a4) x) \u2022 1) ^ k) m = 0\n[PROOFSTEP]\nexact hk\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 weightSpace M 0 = \u22a4\n[PROOFSTEP]\nhave h\u2080 : (0 : L \u2192 R) \u2218 (\u22a4 : LieSubalgebra R L).incl = 0 := by ext; rfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 0 \u2218 \u2191(LieSubalgebra.incl \u22a4) = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\nx\u271d : { x // x \u2208 \u22a4 }\n\u22a2 (0 \u2218 \u2191(LieSubalgebra.incl \u22a4)) x\u271d = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\nh\u2080 : 0 \u2218 \u2191(LieSubalgebra.incl \u22a4) = 0\n\u22a2 weightSpace M 0 = \u22a4\n[PROOFSTEP]\nrw [\u2190 LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.top_coeSubmodule, \u2190 h\u2080, coe_weightSpace_of_top, \u2190\n iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L M]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\nh\u2080 : 0 \u2218 \u2191(LieSubalgebra.incl \u22a4) = 0\n\u22a2 \u2191(weightSpace M 0) = \u2a05 (x : L), Module.End.maximalGeneralizedEigenspace (\u2191(toEndomorphism R L M) x) 0\n[PROOFSTEP]\nrfl\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\nH : LieSubalgebra R L\ninst\u271d\u2077 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : LieRingModule L M\ninst\u271d\u00b3 : LieModule R L M\ninst\u271d\u00b2 : Nontrivial M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 IsWeight \u22a4 M 0\n[PROOFSTEP]\nrw [IsWeight, LieHom.coe_zero, zero_weightSpace_eq_top_of_nilpotent]\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\nH : LieSubalgebra R L\ninst\u271d\u2077 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : LieRingModule L M\ninst\u271d\u00b3 : LieModule R L M\ninst\u271d\u00b2 : Nontrivial M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNilpotent R L M\n\u22a2 \u22a4 \u2260 \u22a5\n[PROOFSTEP]\nexact top_ne_bot\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNoetherian R M\nx : L\n\u22a2 _root_.IsNilpotent (\u2191(toEndomorphism R L { x // x \u2208 \u2191(weightSpace M 0) }) x)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := exists_preWeightSpace_zero_le_ker_of_isNoetherian R M x\n[GOAL]\ncase intro\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNoetherian R M\nx : L\nk : \u2115\nhk : preWeightSpace M 0 \u2264 LinearMap.ker (\u2191(toEndomorphism R L M) x ^ k)\n\u22a2 _root_.IsNilpotent (\u2191(toEndomorphism R L { x // x \u2208 \u2191(weightSpace M 0) }) x)\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNoetherian R M\nx : L\nk : \u2115\nhk : preWeightSpace M 0 \u2264 LinearMap.ker (\u2191(toEndomorphism R L M) x ^ k)\n\u22a2 \u2191(toEndomorphism R L { x // x \u2208 \u2191(weightSpace M 0) }) x ^ k = 0\n[PROOFSTEP]\next \u27e8m, hm\u27e9\n[GOAL]\ncase h.h.mk.a\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNoetherian R M\nx : L\nk : \u2115\nhk : preWeightSpace M 0 \u2264 LinearMap.ker (\u2191(toEndomorphism R L M) x ^ k)\nm : M\nhm : m \u2208 \u2191(weightSpace M 0)\n\u22a2 \u2191(\u2191(\u2191(toEndomorphism R L { x // x \u2208 \u2191(weightSpace M 0) }) x ^ k) { val := m, property := hm }) =\n \u2191(\u21910 { val := m, property := hm })\n[PROOFSTEP]\nrw [LinearMap.zero_apply, LieSubmodule.coe_zero, Submodule.coe_eq_zero, \u2190\n LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism, LinearMap.pow_restrict, \u2190 SetLike.coe_eq_coe,\n LinearMap.restrict_apply, Submodule.coe_mk, Submodule.coe_zero]\n[GOAL]\ncase h.h.mk.a\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2076 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieAlgebra.IsNilpotent R L\ninst\u271d : IsNoetherian R M\nx : L\nk : \u2115\nhk : preWeightSpace M 0 \u2264 LinearMap.ker (\u2191(toEndomorphism R L M) x ^ k)\nm : M\nhm : m \u2208 \u2191(weightSpace M 0)\n\u22a2 \u2191(\u2191(toEndomorphism R L M) x ^ k) \u2191{ val := m, property := hm } = 0\n[PROOFSTEP]\nexact hk hm\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\n\u22a2 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) = weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\n\u22a2 x \u2208 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) \u2194 x \u2208 weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\nlet f : H \u2192 Module.End R L := fun y => toEndomorphism R H L y - \u03c7 y \u2022 \u21911\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\n\u22a2 x \u2208 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) \u2194 x \u2208 weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\nlet g : H \u2192 Module.End R H := fun y => toEndomorphism R H H y - \u03c7 y \u2022 \u21911\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\n\u22a2 x \u2208 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) \u2194 x \u2208 weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\nsuffices\n (\u2200 y : H, \u2203 k : \u2115, (f y ^ k).comp (H.incl : H \u2192\u2097[R] L) x = 0) \u2194\n \u2200 y : H, \u2203 k : \u2115, (H.incl : H \u2192\u2097[R] L).comp (g y ^ k) x = 0\n by\n simp only [LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, Function.comp_apply, LinearMap.coe_comp,\n Submodule.coe_eq_zero] at this \n simp only [mem_weightSpace, mem_preWeightSpace, LieSubalgebra.coe_incl', LieSubmodule.mem_comap, this]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\nthis :\n (\u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (f y ^ k) \u2191(LieSubalgebra.incl H)) x = 0) \u2194\n \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g y ^ k)) x = 0\n\u22a2 x \u2208 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) \u2194 x \u2208 weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\nsimp only [LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, Function.comp_apply, LinearMap.coe_comp,\n Submodule.coe_eq_zero] at this \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\nthis :\n (\u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1) ^ k) \u2191x = 0) \u2194\n \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191((\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1) ^ k) x = 0\n\u22a2 x \u2208 LieSubmodule.comap (LieSubalgebra.incl' H) (rootSpace H \u03c7) \u2194 x \u2208 weightSpace { x // x \u2208 H } \u03c7\n[PROOFSTEP]\nsimp only [mem_weightSpace, mem_preWeightSpace, LieSubalgebra.coe_incl', LieSubmodule.mem_comap, this]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\n\u22a2 (\u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (f y ^ k) \u2191(LieSubalgebra.incl H)) x = 0) \u2194\n \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g y ^ k)) x = 0\n[PROOFSTEP]\nhave hfg : \u2200 y : H, (f y).comp (H.incl : H \u2192\u2097[R] L) = (H.incl : H \u2192\u2097[R] L).comp (g y) :=\n by\n rintro \u27e8y, hy\u27e9; ext \u27e8z, _\u27e9\n simp only [Submodule.coe_sub, toEndomorphism_apply_apply, LieHom.coe_toLinearMap, LinearMap.one_apply,\n LieSubalgebra.coe_incl, LieSubalgebra.coe_bracket_of_module, LieSubalgebra.coe_bracket, LinearMap.smul_apply,\n Function.comp_apply, Submodule.coe_smul_of_tower, LinearMap.coe_comp, LinearMap.sub_apply]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\n\u22a2 \u2200 (y : { x // x \u2208 H }), LinearMap.comp (f y) \u2191(LieSubalgebra.incl H) = LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g y)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\ny : L\nhy : y \u2208 H\n\u22a2 LinearMap.comp (f { val := y, property := hy }) \u2191(LieSubalgebra.incl H) =\n LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g { val := y, property := hy })\n[PROOFSTEP]\next \u27e8z, _\u27e9\n[GOAL]\ncase mk.h.mk\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\ny : L\nhy : y \u2208 H\nz : L\nproperty\u271d : z \u2208 H\n\u22a2 \u2191(LinearMap.comp (f { val := y, property := hy }) \u2191(LieSubalgebra.incl H)) { val := z, property := property\u271d } =\n \u2191(LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g { val := y, property := hy })) { val := z, property := property\u271d }\n[PROOFSTEP]\nsimp only [Submodule.coe_sub, toEndomorphism_apply_apply, LieHom.coe_toLinearMap, LinearMap.one_apply,\n LieSubalgebra.coe_incl, LieSubalgebra.coe_bracket_of_module, LieSubalgebra.coe_bracket, LinearMap.smul_apply,\n Function.comp_apply, Submodule.coe_smul_of_tower, LinearMap.coe_comp, LinearMap.sub_apply]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nx : { x // x \u2208 H }\nf : { x // x \u2208 H } \u2192 Module.End R L := fun y => \u2191(toEndomorphism R { x // x \u2208 H } L) y - \u03c7 y \u2022 1\ng : { x // x \u2208 H } \u2192 Module.End R { x // x \u2208 H } :=\n fun y => \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y - \u03c7 y \u2022 1\nhfg :\n \u2200 (y : { x // x \u2208 H }), LinearMap.comp (f y) \u2191(LieSubalgebra.incl H) = LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g y)\n\u22a2 (\u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (f y ^ k) \u2191(LieSubalgebra.incl H)) x = 0) \u2194\n \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(LinearMap.comp (\u2191(LieSubalgebra.incl H)) (g y ^ k)) x = 0\n[PROOFSTEP]\nsimp_rw [LinearMap.commute_pow_left_of_commute (hfg _)]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : { x // x \u2208 H } \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\n\u22a2 \u2045x, m\u2046 \u2208 weightSpace M (\u03c7\u2081 + \u03c7\u2082)\n[PROOFSTEP]\napply LieModule.weight_vector_multiplication H L M M ((toModuleHom R L M).restrictLie H) \u03c7\u2081 \u03c7\u2082\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : { x // x \u2208 H } \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\n\u22a2 \u2045x, m\u2046 \u2208\n LinearMap.range\n (LinearMap.comp (\u2191(LieModuleHom.restrictLie (toModuleHom R L M) H))\n (TensorProduct.mapIncl (preWeightSpace L \u03c7\u2081) (preWeightSpace M \u03c7\u2082)))\n[PROOFSTEP]\nsimp only [LieModuleHom.coe_toLinearMap, Function.comp_apply, LinearMap.coe_comp, TensorProduct.mapIncl,\n LinearMap.mem_range]\n[GOAL]\ncase a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : { x // x \u2208 H } \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\n\u22a2 \u2203 y,\n \u2191(LieModuleHom.restrictLie (toModuleHom R L M) H)\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace L \u03c7\u2081)) (Submodule.subtype (preWeightSpace M \u03c7\u2082))) y) =\n \u2045x, m\u2046\n[PROOFSTEP]\nuse\u27e8x, hx\u27e9 \u2297\u209c \u27e8m, hm\u27e9\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : { x // x \u2208 H } \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\n\u22a2 \u2191(LieModuleHom.restrictLie (toModuleHom R L M) H)\n (\u2191(TensorProduct.map (Submodule.subtype (preWeightSpace L \u03c7\u2081)) (Submodule.subtype (preWeightSpace M \u03c7\u2082)))\n ({ val := x, property := hx } \u2297\u209c[R] { val := m, property := hm })) =\n \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [Submodule.subtype_apply, toModuleHom_apply, Submodule.coe_mk, LieModuleHom.coe_restrictLie,\n TensorProduct.map_tmul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nm n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n\n[PROOFSTEP]\nsimp only [LieSubmodule.coe_add, lie_add]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nm n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 { val := \u2045\u2191x, \u2191m\u2046 + \u2045\u2191x, \u2191n\u2046, property := (_ : (fun x => x \u2208 \u2191(weightSpace M \u03c7\u2083)) (\u2045\u2191x, \u2191m\u2046 + \u2045\u2191x, \u2191n\u2046)) } =\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) } +\n { val := \u2045\u2191x, \u2191n\u2046, property := (_ : \u2045\u2191x, \u2191n\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n m\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 { val := \u2045\u2191x, \u2191(t \u2022 m)\u2046, property := (_ : \u2045\u2191x, \u2191(t \u2022 m)\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) } =\n \u2191(RingHom.id R) t \u2022 { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }\n[PROOFSTEP]\nconv_lhs =>\n congr\n rw [LieSubmodule.coe_smul, lie_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n| { val := \u2045\u2191x, \u2191(t \u2022 m)\u2046, property := (_ : \u2045\u2191x, \u2191(t \u2022 m)\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }\n[PROOFSTEP]\n congr\n rw [LieSubmodule.coe_smul, lie_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n| { val := \u2045\u2191x, \u2191(t \u2022 m)\u2046, property := (_ : \u2045\u2191x, \u2191(t \u2022 m)\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }\n[PROOFSTEP]\n congr\n rw [LieSubmodule.coe_smul, lie_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n| { val := \u2045\u2191x, \u2191(t \u2022 m)\u2046, property := (_ : \u2045\u2191x, \u2191(t \u2022 m)\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase val\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nt : R\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n| \u2045\u2191x, \u2191(t \u2022 m)\u2046\n[PROOFSTEP]\nrw [LieSubmodule.coe_smul, lie_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx y : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\n\u22a2 (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n m) })\n (x + y) =\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n x +\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n y\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx y : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 \u2191(\u2191((fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n (x + y))\n m) =\n \u2191(\u2191((fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n x +\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n y)\n m)\n[PROOFSTEP]\nsimp only [AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, add_lie, LinearMap.coe_mk, AddHom.coe_mk,\n LinearMap.add_apply, AddSubmonoid.mk_add_mk]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nt : R\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\n\u22a2 AddHom.toFun\n {\n toFun := fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) },\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }),\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n (x + y) =\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n x +\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n y) }\n (t \u2022 x) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) },\n map_add' :=\n (_ :\n \u2200 (x y : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }),\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n (x + y) =\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n x +\n (fun x =>\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191x, \u2191m\u2046,\n property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n y) }\n x\n[PROOFSTEP]\nsimp only [RingHom.id_apply]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nt : R\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\n\u22a2 {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t_1 : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t_1 \u2022 m) =\n \u2191(RingHom.id R) t_1 \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) } =\n t \u2022\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) }\n m) }\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nt : R\nx : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 \u2191(\u2191{\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) },\n map_smul' :=\n (_ :\n \u2200 (t_1 : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t_1 \u2022 m) =\n \u2191(RingHom.id R) t_1 \u2022\n AddHom.toFun\n {\n toFun := fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046, property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046,\n property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n m +\n (fun m =>\n { val := \u2045\u2191(t \u2022 x), \u2191m\u2046,\n property := (_ : \u2045\u2191(t \u2022 x), \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) }\n m) =\n \u2191(\u2191(t \u2022\n {\n toAddHom :=\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) n) },\n map_smul' :=\n (_ :\n \u2200 (t : R) (m : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n (t \u2022 m) =\n \u2191(RingHom.id R) t \u2022\n AddHom.toFun\n { toFun := fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) },\n map_add' :=\n (_ :\n \u2200 (m n : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }),\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n (m + n) =\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) }) m +\n (fun m => { val := \u2045\u2191x, \u2191m\u2046, property := (_ : \u2045\u2191x, \u2191m\u2046 \u2208 \u2191(weightSpace M \u03c7\u2083)) })\n n) }\n m) })\n m)\n[PROOFSTEP]\nsimp only [SetLike.val_smul, smul_lie, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.smul_apply, SetLike.mk_smul_mk]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 H }\ny : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\n\u22a2 AddHom.toFun (rootSpaceWeightSpaceProductAux R L H M h\u03c7).toAddHom \u2045x, y\u2046 =\n \u2045x, AddHom.toFun (rootSpaceWeightSpaceProductAux R L H M h\u03c7).toAddHom y\u2046\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 H }\ny : { x // x \u2208 \u2191(rootSpace H \u03c7\u2081) }\nm : { x // x \u2208 \u2191(weightSpace M \u03c7\u2082) }\n\u22a2 \u2191(\u2191(AddHom.toFun (rootSpaceWeightSpaceProductAux R L H M h\u03c7).toAddHom \u2045x, y\u2046) m) =\n \u2191(\u2191\u2045x, AddHom.toFun (rootSpaceWeightSpaceProductAux R L H M h\u03c7).toAddHom y\u2046 m)\n[PROOFSTEP]\nsimp only [rootSpaceWeightSpaceProductAux, LieSubmodule.coe_bracket, LieSubalgebra.coe_bracket_of_module, lie_lie,\n LinearMap.coe_mk, AddHom.coe_mk, Subtype.coe_mk, LieHom.lie_apply, LieSubmodule.coe_sub]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 rootSpace H \u03c7\u2081 }\nm : { x // x \u2208 weightSpace M \u03c7\u2082 }\n\u22a2 \u2191(\u2191(rootSpaceWeightSpaceProduct R L H M \u03c7\u2081 \u03c7\u2082 \u03c7\u2083 h\u03c7) (x \u2297\u209c[R] m)) = \u2045\u2191x, \u2191m\u2046\n[PROOFSTEP]\nsimp only [rootSpaceWeightSpaceProduct, rootSpaceWeightSpaceProductAux, coe_liftLie_eq_lift_coe, AddHom.toFun_eq_coe,\n LinearMap.coe_toAddHom, lift_apply, LinearMap.coe_mk, AddHom.coe_mk, Submodule.coe_mk]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 \u03c7\u2083 : { x // x \u2208 H } \u2192 R\nh\u03c7 : \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2083\nx : { x // x \u2208 rootSpace H \u03c7\u2081 }\ny : { x // x \u2208 rootSpace H \u03c7\u2082 }\n\u22a2 \u2191(\u2191(rootSpaceProduct R L H \u03c7\u2081 \u03c7\u2082 \u03c7\u2083 h\u03c7) (x \u2297\u209c[R] y)) = \u2045\u2191x, \u2191y\u2046\n[PROOFSTEP]\nsimp only [rootSpaceProduct_def, coe_rootSpaceWeightSpaceProduct_tmul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nsrc\u271d : Submodule R L := \u2191(rootSpace H 0)\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]\nlet xy : rootSpace H 0 \u2297[R] rootSpace H 0 := \u27e8x, hx\u27e9 \u2297\u209c \u27e8y, hy\u27e9\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nsrc\u271d : Submodule R L := \u2191(rootSpace H 0)\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\nxy : { x // x \u2208 \u2191(rootSpace H 0) } \u2297[R] { x // x \u2208 \u2191(rootSpace H 0) } :=\n { val := x, property := hx } \u2297\u209c[R] { val := y, property := hy }\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 (rootSpaceProduct R L H 0 0 0 (add_zero 0) xy : L) \u2208 rootSpace H 0 by\n rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nsrc\u271d : Submodule R L := \u2191(rootSpace H 0)\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\nxy : { x // x \u2208 \u2191(rootSpace H 0) } \u2297[R] { x // x \u2208 \u2191(rootSpace H 0) } :=\n { val := x, property := hx } \u2297\u209c[R] { val := y, property := hy }\nthis : \u2191(\u2191(rootSpaceProduct R L H 0 0 0 (_ : 0 + 0 = 0)) xy) \u2208 rootSpace H 0\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]\nrwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nsrc\u271d : Submodule R L := \u2191(rootSpace H 0)\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\nxy : { x // x \u2208 \u2191(rootSpace H 0) } \u2297[R] { x // x \u2208 \u2191(rootSpace H 0) } :=\n { val := x, property := hx } \u2297\u209c[R] { val := y, property := hy }\n\u22a2 \u2191(\u2191(rootSpaceProduct R L H 0 0 0 (_ : 0 + 0 = 0)) xy) \u2208 rootSpace H 0\n[PROOFSTEP]\nexact (rootSpaceProduct R L H 0 0 0 (add_zero 0) xy).property\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\n\u22a2 x \u2208 zeroRootSubalgebra R L H \u2194 \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nrw [zeroRootSubalgebra]\n -- Porting note: added the following `change` otherwise the `simp` fails\n -- See https://github.com/leanprover-community/mathlib4/issues/5026\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\n\u22a2 (x \u2208\n let src := \u2191(rootSpace H 0);\n {\n toSubmodule :=\n { toAddSubmonoid := src.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208 (\u2191(rootSpace H 0)).toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208 (\u2191(rootSpace H 0)).toAddSubmonoid.toAddSubsemigroup.carrier) },\n lie_mem' :=\n (_ :\n \u2200 {x y : L}\n {hx :\n x \u2208\n { toAddSubmonoid := (\u2191(rootSpace H 0)).toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208 (\u2191(rootSpace H 0)).toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n (\u2191(rootSpace H\n 0)).toAddSubmonoid.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier}\n {hy :\n y \u2208\n { toAddSubmonoid := (\u2191(rootSpace H 0)).toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : L},\n x \u2208 (\u2191(rootSpace H 0)).toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n c \u2022 x \u2208\n (\u2191(rootSpace H\n 0)).toAddSubmonoid.toAddSubsemigroup.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier},\n \u2045x, y\u2046 \u2208 rootSpace H 0) }) \u2194\n \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nchange x \u2208 rootSpace H 0 \u2194 _\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\n\u22a2 x \u2208 rootSpace H 0 \u2194 \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nsimp only [mem_weightSpace, mem_preWeightSpace, Pi.zero_apply, zero_smul, sub_zero]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 LieSubalgebra.toLieSubmodule H \u2264 rootSpace H 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 LieSubalgebra.toLieSubmodule H\n\u22a2 x \u2208 rootSpace H 0\n[PROOFSTEP]\nsimp only [LieSubalgebra.mem_toLieSubmodule] at hx \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\n\u22a2 x \u2208 rootSpace H 0\n[PROOFSTEP]\nsimp only [mem_weightSpace, mem_preWeightSpace, Pi.zero_apply, sub_zero, zero_smul]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\n\u22a2 \u2200 (x_1 : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) x_1 ^ k) x = 0\n[PROOFSTEP]\nintro y\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := (inferInstance : IsNilpotent R H)\n[GOAL]\ncase mk.intro\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\n\u22a2 \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nlet f : Module.End R H := toEndomorphism R H H y\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\n\u22a2 \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nlet g : Module.End R L := toEndomorphism R H L y\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\n\u22a2 \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nhave hfg : g.comp (H : Submodule R L).subtype = (H : Submodule R L).subtype.comp f :=\n by\n ext z\n simp only [toEndomorphism_apply_apply, Submodule.subtype_apply, LieSubalgebra.coe_bracket_of_module,\n LieSubalgebra.coe_bracket, Function.comp_apply, LinearMap.coe_comp]\n rfl\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\n\u22a2 LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\n[PROOFSTEP]\next z\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nz : { x // x \u2208 H.toSubmodule }\n\u22a2 \u2191(LinearMap.comp g (Submodule.subtype H.toSubmodule)) z = \u2191(LinearMap.comp (Submodule.subtype H.toSubmodule) f) z\n[PROOFSTEP]\nsimp only [toEndomorphism_apply_apply, Submodule.subtype_apply, LieSubalgebra.coe_bracket_of_module,\n LieSubalgebra.coe_bracket, Function.comp_apply, LinearMap.coe_comp]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nz : { x // x \u2208 H.toSubmodule }\n\u22a2 \u2045\u2191y, \u2191z\u2046 = \u2191(\u2191(\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y) z)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\n\u22a2 \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nchange (g ^ k).comp (H : Submodule R L).subtype \u27e8x, hx\u27e9 = 0\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\n\u22a2 \u2191(LinearMap.comp (g ^ k) (Submodule.subtype H.toSubmodule)) { val := x, property := hx } = 0\n[PROOFSTEP]\nrw [LinearMap.commute_pow_left_of_commute hfg k]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\n\u22a2 \u2191(LinearMap.comp (Submodule.subtype H.toSubmodule) (f ^ k)) { val := x, property := hx } = 0\n[PROOFSTEP]\nhave h := iterate_toEndomorphism_mem_lowerCentralSeries R H H y \u27e8x, hx\u27e9 k\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\nh :\n (\u2191(\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y))^[k] { val := x, property := hx } \u2208\n lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k\n\u22a2 \u2191(LinearMap.comp (Submodule.subtype H.toSubmodule) (f ^ k)) { val := x, property := hx } = 0\n[PROOFSTEP]\nrw [hk, LieSubmodule.mem_bot] at h \n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\nh : (\u2191(\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y))^[k] { val := x, property := hx } = 0\n\u22a2 \u2191(LinearMap.comp (Submodule.subtype H.toSubmodule) (f ^ k)) { val := x, property := hx } = 0\n[PROOFSTEP]\nsimp only [Submodule.subtype_apply, Function.comp_apply, LinearMap.pow_apply, LinearMap.coe_comp, Submodule.coe_eq_zero]\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 H\ny : { x // x \u2208 H }\nk : \u2115\nhk : lowerCentralSeries R { x // x \u2208 H } { x // x \u2208 H } k = \u22a5\nf : Module.End R { x // x \u2208 H } := \u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y\ng : Module.End R L := \u2191(toEndomorphism R { x // x \u2208 H } L) y\nhfg : LinearMap.comp g (Submodule.subtype H.toSubmodule) = LinearMap.comp (Submodule.subtype H.toSubmodule) f\nh : (\u2191(\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y))^[k] { val := x, property := hx } = 0\n\u22a2 (\u2191(\u2191(toEndomorphism R { x // x \u2208 H } { x // x \u2208 H }) y))^[k] { val := x, property := hx } = 0\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 H \u2264 zeroRootSubalgebra R L H\n[PROOFSTEP]\nrw [\u2190 LieSubalgebra.coe_submodule_le_coe_submodule, \u2190 H.coe_toLieSubmodule, coe_zeroRootSubalgebra,\n LieSubmodule.coeSubmodule_le_coeSubmodule]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 LieSubalgebra.toLieSubmodule H \u2264 rootSpace H 0\n[PROOFSTEP]\nexact toLieSubmodule_le_rootSpace_zero R L H\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 LieSubalgebra.normalizer (zeroRootSubalgebra R L H) = zeroRootSubalgebra R L H\n[PROOFSTEP]\nrefine' le_antisymm _ (LieSubalgebra.le_normalizer _)\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 LieSubalgebra.normalizer (zeroRootSubalgebra R L H) \u2264 zeroRootSubalgebra R L H\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : x \u2208 LieSubalgebra.normalizer (zeroRootSubalgebra R L H)\n\u22a2 x \u2208 zeroRootSubalgebra R L H\n[PROOFSTEP]\nrw [LieSubalgebra.mem_normalizer_iff] at hx \n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 (y : L), y \u2208 zeroRootSubalgebra R L H \u2192 \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 x \u2208 zeroRootSubalgebra R L H\n[PROOFSTEP]\nrw [mem_zeroRootSubalgebra]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 (y : L), y \u2208 zeroRootSubalgebra R L H \u2192 \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2200 (y : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y ^ k) x = 0\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx : L\nhx : \u2200 (y : L), y \u2208 zeroRootSubalgebra R L H \u2192 \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\ny : L\nhy : y \u2208 H\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) x = 0\n[PROOFSTEP]\nspecialize hx y (le_zeroRootSubalgebra R L H hy)\n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2045x, y\u2046 \u2208 zeroRootSubalgebra R L H\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) x = 0\n[PROOFSTEP]\nrw [mem_zeroRootSubalgebra] at hx \n[GOAL]\ncase mk\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y_1 ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) x = 0\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hx \u27e8y, hy\u27e9\n[GOAL]\ncase mk.intro\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) \u2045x, y\u2046 = 0\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) x = 0\n[PROOFSTEP]\nrw [\u2190 lie_skew, LinearMap.map_neg, neg_eq_zero] at hk \n[GOAL]\ncase mk.intro\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) \u2045y, x\u2046 = 0\n\u22a2 \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) x = 0\n[PROOFSTEP]\nuse k + 1\n[GOAL]\ncase h\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nhy : y \u2208 H\nhx : \u2200 (y_1 : { x // x \u2208 H }), \u2203 k, \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) y_1 ^ k) \u2045x, y\u2046 = 0\nk : \u2115\nhk : \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ k) \u2045y, x\u2046 = 0\n\u22a2 \u2191(\u2191(toEndomorphism R { x // x \u2208 H } L) { val := y, property := hy } ^ (k + 1)) x = 0\n[PROOFSTEP]\nrw [LinearMap.iterate_succ, LinearMap.coe_comp, Function.comp_apply, toEndomorphism_apply_apply,\n LieSubalgebra.coe_bracket_of_module, Submodule.coe_mk, hk]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nh : zeroRootSubalgebra R L H = H\n\u22a2 LieSubalgebra.normalizer H = H\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nh : zeroRootSubalgebra R L H = H\n\u22a2 LieSubalgebra.normalizer (zeroRootSubalgebra R L H) = zeroRootSubalgebra R L H\n[PROOFSTEP]\nexact zeroRootSubalgebra_normalizer_eq_self R L H\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\n\u22a2 zeroRootSubalgebra R L H = H\n[PROOFSTEP]\nrefine' le_antisymm _ (le_zeroRootSubalgebra R L H)\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\n\u22a2 zeroRootSubalgebra R L H \u2264 H\n[PROOFSTEP]\nsuffices rootSpace H 0 \u2264 H.toLieSubmodule by exact fun x hx => this hx\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\nthis : rootSpace H 0 \u2264 LieSubalgebra.toLieSubmodule H\n\u22a2 zeroRootSubalgebra R L H \u2264 H\n[PROOFSTEP]\nexact fun x hx => this hx\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\n\u22a2 rootSpace H 0 \u2264 LieSubalgebra.toLieSubmodule H\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := (rootSpace H 0).isNilpotent_iff_exists_self_le_ucs.mp (by infer_instance)\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\n\u22a2 LieModule.IsNilpotent R { x // x \u2208 H } { x // x \u2208 \u2191(rootSpace H 0) }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase intro\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\nk : \u2115\nhk : rootSpace H 0 \u2264 LieSubmodule.ucs k \u22a5\n\u22a2 rootSpace H 0 \u2264 LieSubalgebra.toLieSubmodule H\n[PROOFSTEP]\nexact hk.trans (LieSubmodule.ucs_le_of_normalizer_eq_self (by simp) k)\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : LieRing L\ninst\u271d\u2077 : LieAlgebra R L\nH\u271d : LieSubalgebra R L\ninst\u271d\u2076 : IsNilpotent R { x // x \u2208 H\u271d }\nM : Type w\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nH : LieSubalgebra R L\ninst\u271d\u00b9 : LieSubalgebra.IsCartanSubalgebra H\ninst\u271d : IsNoetherian R L\nk : \u2115\nhk : rootSpace H 0 \u2264 LieSubmodule.ucs k \u22a5\n\u22a2 LieSubmodule.normalizer (LieSubalgebra.toLieSubmodule H) = LieSubalgebra.toLieSubmodule H\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : IsNilpotent R { x // x \u2208 H }\nM : Type w\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 LieSubalgebra.IsCartanSubalgebra H \u2192 zeroRootSubalgebra R L H = H\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2075 : IsNilpotent R { x // x \u2208 H }\nM : Type w\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\na\u271d : LieSubalgebra.IsCartanSubalgebra H\n\u22a2 zeroRootSubalgebra R L H = H\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, m\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nhave hx : (x : L) \u2208 rootSpace H 0 := by rw [\u2190 LieSubmodule.mem_coeSubmodule, \u2190 coe_zeroRootSubalgebra]; exact x.prop\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2191x \u2208 rootSpace H 0\n[PROOFSTEP]\nrw [\u2190 LieSubmodule.mem_coeSubmodule, \u2190 coe_zeroRootSubalgebra]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2191x \u2208 (zeroRootSubalgebra R L H).toSubmodule\n[PROOFSTEP]\nexact x.prop\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhx : \u2191x \u2208 rootSpace H 0\n\u22a2 \u2045x, m\u2046 \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhx : \u2191x \u2208 rootSpace H 0\n\u22a2 \u2045x, m\u2046 \u2208 (\u2191(weightSpace M \u03c7)).toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [\u2190 zero_add \u03c7]\n[GOAL]\nR : Type u\nL : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : LieAlgebra.IsNilpotent R { x // x \u2208 H }\nM : Type w\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7 : { x // x \u2208 H } \u2192 R\nsrc\u271d : Submodule R M := \u2191(weightSpace M \u03c7)\nx : { x // x \u2208 zeroRootSubalgebra R L H }\nm : M\nhm :\n m \u2208\n { toAddSubmonoid := src\u271d.toAddSubmonoid,\n smul_mem' :=\n (_ :\n \u2200 (c : R) {x : M}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhx : \u2191x \u2208 rootSpace H 0\n\u22a2 \u2045x, m\u2046 \u2208 (\u2191(weightSpace M (0 + \u03c7))).toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact lie_mem_weightSpace_of_mem_weightSpace hx hm\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Weights", "llama_tokens": 91295, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.4148988457967688, "lm_q1q2_score": 0.2702463759381016}} {"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 : Fintype \u03b1\na : Option \u03b1\n\u22a2 a \u2208 \u2191insertNone univ\n[PROOFSTEP]\nsimp\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 : Fintype \u03b1\n\u22a2 \u00acnone \u2208 map Embedding.some univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Trunc (P (ULift (Fin zero)))\n[PROOFSTEP]\nhave : card PEmpty = card (ULift (Fin 0)) := by simp only [card_fin, card_pempty, card_ulift]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card PEmpty = card (ULift (Fin 0))\n[PROOFSTEP]\nsimp only [card_fin, card_pempty, card_ulift]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : card PEmpty = card (ULift (Fin 0))\n\u22a2 Trunc (P (ULift (Fin zero)))\n[PROOFSTEP]\napply Trunc.bind (truncEquivOfCardEq this)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : card PEmpty = card (ULift (Fin 0))\n\u22a2 PEmpty \u2243 ULift (Fin 0) \u2192 Trunc (P (ULift (Fin zero)))\n[PROOFSTEP]\nintro e\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : card PEmpty = card (ULift (Fin 0))\ne : PEmpty \u2243 ULift (Fin 0)\n\u22a2 Trunc (P (ULift (Fin zero)))\n[PROOFSTEP]\napply Trunc.mk\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : card PEmpty = card (ULift (Fin 0))\ne : PEmpty \u2243 ULift (Fin 0)\n\u22a2 P (ULift (Fin zero))\n[PROOFSTEP]\nrefine' of_equiv e h_empty\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 Trunc (P (ULift (Fin (succ n))))\n[PROOFSTEP]\nhave : card (Option (ULift (Fin n))) = card (ULift (Fin n.succ)) := by simp only [card_fin, card_option, card_ulift]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u22a2 card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\n[PROOFSTEP]\nsimp only [card_fin, card_option, card_ulift]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nthis : card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\n\u22a2 Trunc (P (ULift (Fin (succ n))))\n[PROOFSTEP]\napply Trunc.bind (truncEquivOfCardEq this)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nthis : card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\n\u22a2 Option (ULift (Fin n)) \u2243 ULift (Fin (succ n)) \u2192 Trunc (P (ULift (Fin (succ n))))\n[PROOFSTEP]\nintro e\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nthis : card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\ne : Option (ULift (Fin n)) \u2243 ULift (Fin (succ n))\n\u22a2 Trunc (P (ULift (Fin (succ n))))\n[PROOFSTEP]\napply Trunc.map _ (ind n)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nthis : card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\ne : Option (ULift (Fin n)) \u2243 ULift (Fin (succ n))\n\u22a2 P (ULift (Fin n)) \u2192 P (ULift (Fin (succ n)))\n[PROOFSTEP]\nintro ih\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u2115\nthis : card (Option (ULift (Fin n))) = card (ULift (Fin (succ n)))\ne : Option (ULift (Fin n)) \u2243 ULift (Fin (succ n))\nih : P (ULift (Fin n))\n\u22a2 P (ULift (Fin (succ n)))\n[PROOFSTEP]\nrefine' of_equiv e (h_option ih)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Trunc (P \u03b1)\n[PROOFSTEP]\nsuffices \u2200 n : \u2115, Trunc (P (ULift <| Fin n))\n by\n apply Trunc.bind (this (Fintype.card \u03b1))\n intro h\n apply Trunc.map _ (Fintype.truncEquivFin \u03b1)\n intro e\n exact of_equiv (Equiv.ulift.trans e.symm) h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\n\u22a2 Trunc (P \u03b1)\n[PROOFSTEP]\napply Trunc.bind (this (Fintype.card \u03b1))\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\n\u22a2 P (ULift (Fin (card \u03b1))) \u2192 Trunc (P \u03b1)\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\nh : P (ULift (Fin (card \u03b1)))\n\u22a2 Trunc (P \u03b1)\n[PROOFSTEP]\napply Trunc.map _ (Fintype.truncEquivFin \u03b1)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\nh : P (ULift (Fin (card \u03b1)))\n\u22a2 \u03b1 \u2243 Fin (card \u03b1) \u2192 P \u03b1\n[PROOFSTEP]\nintro e\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\nh : P (ULift (Fin (card \u03b1)))\ne : \u03b1 \u2243 Fin (card \u03b1)\n\u22a2 P \u03b1\n[PROOFSTEP]\nexact of_equiv (Equiv.ulift.trans e.symm) h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Sort v\nof_equiv : {\u03b1 \u03b2 : Type u} \u2192 \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : {\u03b1 : Type u} \u2192 [inst : Fintype \u03b1] \u2192 [inst : DecidableEq \u03b1] \u2192 P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (n : \u2115) \u2192 Trunc (P (ULift (Fin n)))\n[PROOFSTEP]\napply ind\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : (\u03b1 : Type u) \u2192 [inst : Fintype \u03b1] \u2192 Prop\nof_equiv : \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2), P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\nh_fintype : Fintype \u03b1\n\u22a2 P \u03b1\n[PROOFSTEP]\nobtain \u27e8p\u27e9 :=\n let f_empty := fun i => by convert h_empty\n let h_option :\n \u2200 {\u03b1 : Type u} [Fintype \u03b1] [DecidableEq \u03b1], (\u2200 (h : Fintype \u03b1), P \u03b1) \u2192 \u2200 (h : Fintype (Option \u03b1)), P (Option \u03b1) :=\n by\n rintro \u03b1 h\u03b1 - P\u03b1 h\u03b1'\n convert h_option \u03b1 (P\u03b1 _)\n @truncRecEmptyOption (fun \u03b1 => \u2200 h, @P \u03b1 h) (@fun \u03b1 \u03b2 e h\u03b1 h\u03b2 => @of_equiv \u03b1 \u03b2 h\u03b2 e (h\u03b1 _)) f_empty h_option \u03b1 _\n (Classical.decEq \u03b1)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : (\u03b1 : Type u) \u2192 [inst : Fintype \u03b1] \u2192 Prop\nof_equiv : \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2), P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\nh_fintype : Fintype \u03b1\ni : Fintype PEmpty\n\u22a2 P PEmpty\n[PROOFSTEP]\nconvert h_empty\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : (\u03b1 : Type u) \u2192 [inst : Fintype \u03b1] \u2192 Prop\nof_equiv : \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2), P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\nh_fintype : Fintype \u03b1\nf_empty : \u2200 (i : Fintype PEmpty), P PEmpty := fun i => ?m.7851 i\n\u22a2 \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1] [inst : DecidableEq \u03b1],\n (\u2200 (h : Fintype \u03b1), P \u03b1) \u2192 \u2200 (h : Fintype (Option \u03b1)), P (Option \u03b1)\n[PROOFSTEP]\nrintro \u03b1 h\u03b1 - P\u03b1 h\u03b1'\n[GOAL]\n\u03b1\u271d\u00b9 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : (\u03b1 : Type u) \u2192 [inst : Fintype \u03b1] \u2192 Prop\nof_equiv : \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2), P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1\u271d : Type u\nh_fintype : Fintype \u03b1\u271d\nf_empty : \u2200 (i : Fintype PEmpty), P PEmpty := fun i => ?m.7851 i\n\u03b1 : Type u\nh\u03b1 : Fintype \u03b1\nP\u03b1 : \u2200 (h : Fintype \u03b1), P \u03b1\nh\u03b1' : Fintype (Option \u03b1)\n\u22a2 P (Option \u03b1)\n[PROOFSTEP]\nconvert h_option \u03b1 (P\u03b1 _)\n[GOAL]\ncase mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : (\u03b1 : Type u) \u2192 [inst : Fintype \u03b1] \u2192 Prop\nof_equiv : \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2] (e : \u03b1 \u2243 \u03b2), P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\nh_fintype : Fintype \u03b1\nx\u271d : Trunc (\u2200 (h : Fintype \u03b1), P \u03b1)\np : \u2200 (h : Fintype \u03b1), P \u03b1\n\u22a2 P \u03b1\n[PROOFSTEP]\nexact\n p\n _\n -- \u00b7\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Prop\nof_equiv : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d : Finite \u03b1\n\u22a2 P \u03b1\n[PROOFSTEP]\ncases nonempty_fintype \u03b1\n[GOAL]\ncase intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Prop\nof_equiv : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 P \u03b1\n[PROOFSTEP]\nrefine' Fintype.induction_empty_option _ _ _ \u03b1\n[GOAL]\ncase intro.refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Prop\nof_equiv : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 \u2200 (\u03b1 \u03b2 : Type u) [inst : Fintype \u03b2], \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\ncase intro.refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Prop\nof_equiv : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 P PEmpty\ncase intro.refine'_3\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nP : Type u \u2192 Prop\nof_equiv : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2\nh_empty : P PEmpty\nh_option : \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n\u03b1 : Type u\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 \u2200 (\u03b1 : Type u) [inst : Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)\n[PROOFSTEP]\nexacts [fun \u03b1 \u03b2 _ => of_equiv, h_empty, @h_option]\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Option", "llama_tokens": 6142, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.26990240396285026}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_4, u_1} C\nD : Type u_2\ninst\u271d\u00b9 : Category.{?u.745, u_2} D\nE : Type u_3\ninst\u271d : Category.{?u.752, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nU\u271d : C\nS\u271d : Sieve ((\ud835\udfed C).obj U\u271d)\nh : S\u271d \u2208 GrothendieckTopology.sieves J ((\ud835\udfed C).obj U\u271d)\n\u22a2 Sieve.functorPullback (\ud835\udfed C) S\u271d \u2208 GrothendieckTopology.sieves J U\u271d\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\n\u22a2 X \u27f6 \u2131.val.obj Y.right\n[PROOFSTEP]\nletI hom_sh := whiskerRight ((Ran.adjunction A G.op).counit.app \u2131.val) (coyoneda.obj (op X))\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\nhom_sh : (ran G.op \u22d9 (whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj \u2131.val \u22d9 coyoneda.obj (op X) \u27f6\n (\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X) :=\n whiskerRight (NatTrans.app (Ran.adjunction A G.op).counit \u2131.val) (coyoneda.obj (op X))\n\u22a2 X \u27f6 \u2131.val.obj Y.right\n[PROOFSTEP]\nhaveI S' := K.pullback_stable Y.hom.unop hS\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\nhom_sh : (ran G.op \u22d9 (whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj \u2131.val \u22d9 coyoneda.obj (op X) \u27f6\n (\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X) :=\n whiskerRight (NatTrans.app (Ran.adjunction A G.op).counit \u2131.val) (coyoneda.obj (op X))\nS' : Sieve.pullback Y.hom.unop S \u2208 GrothendieckTopology.sieves K (G.op.obj Y.right).unop\n\u22a2 X \u27f6 \u2131.val.obj Y.right\n[PROOFSTEP]\nhaveI hs' := ((hx.pullback Y.3.unop).functorPullback G).compPresheafMap hom_sh\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\nhom_sh : (ran G.op \u22d9 (whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj \u2131.val \u22d9 coyoneda.obj (op X) \u27f6\n (\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X) :=\n whiskerRight (NatTrans.app (Ran.adjunction A G.op).counit \u2131.val) (coyoneda.obj (op X))\nS' : Sieve.pullback Y.hom.unop S \u2208 GrothendieckTopology.sieves K (G.op.obj Y.right).unop\nhs' : Compatible (compPresheafMap hom_sh (FamilyOfElements.functorPullback G (FamilyOfElements.pullback Y.hom.unop x)))\n\u22a2 X \u27f6 \u2131.val.obj Y.right\n[PROOFSTEP]\nexact (\u2131.2 X _ (hu.cover_lift S')).amalgamate _ hs'\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\ny : ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op Y.right.unop)\nH : IsAmalgamation (pulledbackFamily \u2131 S x Y) y\n\u22a2 y = getSection hu \u2131 hS hx Y\n[PROOFSTEP]\napply IsSheafFor.isSeparatedFor _ (pulledbackFamily \u2131 S x Y)\n[GOAL]\ncase a\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\ny : ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op Y.right.unop)\nH : IsAmalgamation (pulledbackFamily \u2131 S x Y) y\n\u22a2 IsAmalgamation (pulledbackFamily \u2131 S x Y) y\n[PROOFSTEP]\nexact H\n[GOAL]\ncase a\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\ny : ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op Y.right.unop)\nH : IsAmalgamation (pulledbackFamily \u2131 S x Y) y\n\u22a2 IsAmalgamation (pulledbackFamily \u2131 S x Y) (getSection hu \u2131 hS hx Y)\n[PROOFSTEP]\napply getSection_isAmalgamation\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY : StructuredArrow (op U) G.op\ny : ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op Y.right.unop)\nH : IsAmalgamation (pulledbackFamily \u2131 S x Y) y\n\u22a2 IsSheafFor ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X))\n (Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows)\n[PROOFSTEP]\nexact \u2131.2 X _ (hu.cover_lift (K.pullback_stable Y.hom.unop hS))\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\n\u22a2 getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right = getSection hu \u2131 hS hx Z\n[PROOFSTEP]\napply getSection_is_unique\n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\n\u22a2 IsAmalgamation (pulledbackFamily \u2131 S x Z) (getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right)\n[PROOFSTEP]\nintro V' fV' hV'\n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right) =\n pulledbackFamily \u2131 S x Z fV' hV'\n[PROOFSTEP]\nhave eq : Z.hom = Y.hom \u226b (G.map f.right.unop).op := by\n convert f.w\n erw [Category.id_comp]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\n\u22a2 Z.hom = Y.hom \u226b (G.map f.right.unop).op\n[PROOFSTEP]\nconvert f.w\n[GOAL]\ncase h.e'_2.h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\ne_1\u271d :\n ((Functor.fromPUnit (op U)).obj Z.left \u27f6 G.op.obj Z.right) =\n ((Functor.fromPUnit (op U)).obj Y.left \u27f6 G.op.obj Z.right)\n\u22a2 Z.hom = (Functor.fromPUnit (op U)).map f.left \u226b Z.hom\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right) =\n pulledbackFamily \u2131 S x Z fV' hV'\n[PROOFSTEP]\nrw [eq] at hV' \n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right) =\n pulledbackFamily \u2131 S x Z fV' hV'\u271d\n[PROOFSTEP]\nconvert getSection_isAmalgamation hu \u2131 hS hx Y (fV' \u226b f.right.unop) _ using 1\n[GOAL]\ncase h.e'_2\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (getSection hu \u2131 hS hx Y \u226b \u2131.val.map f.right) =\n ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map (fV' \u226b f.right.unop).op (getSection hu \u2131 hS hx Y)\n[PROOFSTEP]\naesop_cat\n -- porting note: the below proof was mildly rewritten because `simp` changed behaviour\n -- slightly (a rewrite which seemed to work in Lean 3, didn't work in Lean 4 because of\n -- motive is not type correct issues)\n[GOAL]\ncase h.e'_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 pulledbackFamily \u2131 S x Z fV' hV'\u271d = pulledbackFamily \u2131 S x Y (fV' \u226b f.right.unop) ?H\n[PROOFSTEP]\nrw [pulledbackFamily_apply, pulledbackFamily_apply]\n[GOAL]\ncase h.e'_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 x (G.map fV' \u226b Z.hom.unop) hV'\u271d \u226b NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit \u2131.val) (op V') =\n x (G.map (fV' \u226b f.right.unop) \u226b Y.hom.unop) ?H \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit \u2131.val) (op V')\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.e'_3.e_a.e_f\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 G.map fV' \u226b Z.hom.unop = G.map (fV' \u226b f.right.unop) \u226b Y.hom.unop\n[PROOFSTEP]\nsimp [eq]\n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 Presieve.functorPullback G (Sieve.pullback Y.hom.unop S).arrows (fV' \u226b f.right.unop)\n[PROOFSTEP]\nchange S (G.map _ \u226b Y.hom.unop)\n[GOAL]\ncase H\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nY Z : StructuredArrow (op U) G.op\nf : Y \u27f6 Z\nV' : C\nfV' : V' \u27f6 Z.right.unop\nhV'\u271d : Presieve.functorPullback G (Sieve.pullback Z.hom.unop S).arrows fV'\nhV' : Presieve.functorPullback G (Sieve.pullback (Y.hom \u226b (G.map f.right.unop).op).unop S).arrows fV'\neq : Z.hom = Y.hom \u226b (G.map f.right.unop).op\n\u22a2 S.arrows (G.map (fV' \u226b f.right.unop) \u226b Y.hom.unop)\n[PROOFSTEP]\nsimpa only [Functor.map_comp, Category.assoc] using hV'\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n NatTrans.app (gluedLimitCone hu \u2131 hS hx).\u03c0 ((StructuredArrow.map f.op).obj W)\n[PROOFSTEP]\ndsimp only [gluedLimitCone_\u03c0_app]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W = getSection hu \u2131 hS hx ((StructuredArrow.map f.op).obj W)\n[PROOFSTEP]\napply getSection_is_unique hu \u2131 hS hx ((StructuredArrow.map f.op).obj W)\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\n\u22a2 IsAmalgamation (pulledbackFamily \u2131 S x ((StructuredArrow.map f.op).obj W))\n (y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W)\n[PROOFSTEP]\nintro V' fV' hV'\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W) =\n pulledbackFamily \u2131 S x ((StructuredArrow.map f.op).obj W) fV' hV'\n[PROOFSTEP]\ndsimp only [Ran.adjunction, Ran.equiv, pulledbackFamily_apply]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W) =\n x (G.map fV' \u226b ((StructuredArrow.map f.op).obj W).hom.unop) hV' \u226b\n NatTrans.app\n (NatTrans.app\n (Adjunction.adjunctionOfEquivRight\n (fun F G_1 =>\n {\n toFun := fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op G_1 (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))),\n invFun := fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op G_1 x) (Ran.cone x f),\n left_inv :=\n (_ :\n \u2200 (x : F \u27f6 Ran.loc G.op G_1),\n (fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op G_1 x) (Ran.cone x f))\n ((fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op G_1 (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj F \u27f6 G_1),\n (fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op G_1 (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n ((fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op G_1 x) (Ran.cone x f)) x) =\n x) }.symm)\n (_ :\n \u2200 (X' X : D\u1d52\u1d56 \u2964 A) (Y : C\u1d52\u1d56 \u2964 A) (f : X' \u27f6 X) (g : ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj X \u27f6 Y),\n \u2191((fun F G_1 => (Ran.equiv G.op G_1 F).symm) X' Y) (((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).map f \u226b g) =\n f \u226b \u2191((fun F G_1 => (Ran.equiv G.op G_1 F).symm) X Y) g)).counit\n \u2131.val)\n (op V')\n[PROOFSTEP]\nerw [Adjunction.adjunctionOfEquivRight_counit_app]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W) =\n x (G.map fV' \u226b ((StructuredArrow.map f.op).obj W).hom.unop) hV' \u226b\n NatTrans.app\n (\u2191{\n toFun := fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))),\n invFun := fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f),\n left_inv :=\n (_ :\n \u2200 (x : Ran.loc G.op \u2131.val \u27f6 Ran.loc G.op \u2131.val),\n (fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f))\n ((fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj (Ran.loc G.op \u2131.val) \u27f6 \u2131.val),\n (fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n ((fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f)) x) =\n x) }.symm.symm\n (\ud835\udfd9 (Ran.loc G.op \u2131.val)))\n (op V')\n[PROOFSTEP]\nhave :\n y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op =\n x (G.map fV' \u226b W.hom.unop \u226b f) (by simpa only using hV') :=\n by\n convert H (show S ((G.map fV' \u226b W.hom.unop) \u226b f) by simpa only [Category.assoc] using hV') using 2\n simp only [Category.assoc]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 S.arrows (G.map fV' \u226b W.hom.unop \u226b f)\n[PROOFSTEP]\nsimpa only using hV'\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\n[PROOFSTEP]\nconvert H (show S ((G.map fV' \u226b W.hom.unop) \u226b f) by simpa only [Category.assoc] using hV') using 2\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 S.arrows ((G.map fV' \u226b W.hom.unop) \u226b f)\n[PROOFSTEP]\nsimpa only [Category.assoc] using hV'\n[GOAL]\ncase h.e'_3.h.e'_2\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\n\u22a2 G.map fV' \u226b W.hom.unop \u226b f = (G.map fV' \u226b W.hom.unop) \u226b f\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\n\u22a2 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV'.op (y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W) =\n x (G.map fV' \u226b ((StructuredArrow.map f.op).obj W).hom.unop) hV' \u226b\n NatTrans.app\n (\u2191{\n toFun := fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))),\n invFun := fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f),\n left_inv :=\n (_ :\n \u2200 (x : Ran.loc G.op \u2131.val \u27f6 Ran.loc G.op \u2131.val),\n (fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f))\n ((fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj (Ran.loc G.op \u2131.val) \u27f6 \u2131.val),\n (fun f =>\n NatTrans.mk fun x =>\n NatTrans.app f (G.op.obj x) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj x)) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj x))))\n ((fun f => NatTrans.mk fun x => limit.lift (Ran.diagram G.op \u2131.val x) (Ran.cone x f)) x) =\n x) }.symm.symm\n (\ud835\udfd9 (Ran.loc G.op \u2131.val)))\n (op V')\n[PROOFSTEP]\nsimp only [Quiver.Hom.unop_op, Equiv.symm_symm, StructuredArrow.map_obj_hom, unop_comp, Equiv.coe_fn_mk,\n Functor.comp_map, coyoneda_obj_map, Category.assoc, \u2190 this, op_comp, ran_obj_map, NatTrans.id_app]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W \u226b ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).map fV'.op =\n y \u226b\n limit.pre (Ran.diagram G.op \u2131.val (op V)) (StructuredArrow.map (W.hom.unop.op \u226b (G.map fV').op)) \u226b\n \ud835\udfd9 ((Ran.loc G.op \u2131.val).obj (G.op.obj (op V'))) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (G.op.obj (op V'))) (StructuredArrow.mk (\ud835\udfd9 (G.op.obj (op V'))))\n[PROOFSTEP]\nerw [Category.id_comp, limit.pre_\u03c0]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W \u226b ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).map fV'.op =\n y \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (op V))\n ((StructuredArrow.map (W.hom.unop.op \u226b (G.map fV').op)).obj (StructuredArrow.mk (\ud835\udfd9 (G.op.obj (op V')))))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\n\u22a2 limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W \u226b ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).map fV'.op =\n limit.\u03c0 (Ran.diagram G.op \u2131.val (op V))\n ((StructuredArrow.map (W.hom.unop.op \u226b (G.map fV').op)).obj (StructuredArrow.mk (\ud835\udfd9 (G.op.obj (op V')))))\n[PROOFSTEP]\nconvert limit.w (Ran.diagram G.op \u2131.val (op V)) (StructuredArrow.homMk' W fV'.op)\n[GOAL]\ncase h.e'_3.h.h.e'_7\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\ne_1\u271d :\n (((ran G.op).obj \u2131.val).obj (op V) \u27f6 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).obj (op V')) =\n (limit (Ran.diagram G.op \u2131.val (op V)) \u27f6\n (Ran.diagram G.op \u2131.val (op V)).obj (StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)))\n\u22a2 (StructuredArrow.map (W.hom.unop.op \u226b (G.map fV').op)).obj (StructuredArrow.mk (\ud835\udfd9 (G.op.obj (op V')))) =\n StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)\n[PROOFSTEP]\nrw [StructuredArrow.map_mk]\n[GOAL]\ncase h.e'_3.h.h.e'_7\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\ne_1\u271d :\n (((ran G.op).obj \u2131.val).obj (op V) \u27f6 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).obj (op V')) =\n (limit (Ran.diagram G.op \u2131.val (op V)) \u27f6\n (Ran.diagram G.op \u2131.val (op V)).obj (StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)))\n\u22a2 StructuredArrow.mk ((W.hom.unop.op \u226b (G.map fV').op) \u226b \ud835\udfd9 (G.op.obj (op V'))) =\n StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)\n[PROOFSTEP]\nerw [Category.comp_id]\n[GOAL]\ncase h.e'_3.h.h.e'_7\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nf : V \u27f6 U\ny : X \u27f6 ((ran G.op).obj \u2131.val).obj (op V)\nW : StructuredArrow (op V) G.op\nH : \u2200 {V' : C} {fV : G.obj V' \u27f6 V} (hV : S.arrows (fV \u226b f)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b f) hV\nV' : C\nfV' : V' \u27f6 ((StructuredArrow.map f.op).obj W).right.unop\nhV' : Presieve.functorPullback G (Sieve.pullback ((StructuredArrow.map f.op).obj W).hom.unop S).arrows fV'\nthis : y \u226b ((ran G.op).obj \u2131.val).map (G.map fV' \u226b W.hom.unop).op = x (G.map fV' \u226b W.hom.unop \u226b f) hV'\ne_1\u271d :\n (((ran G.op).obj \u2131.val).obj (op V) \u27f6 ((\ud835\udfed (C\u1d52\u1d56 \u2964 A)).obj \u2131.val).obj (op V')) =\n (limit (Ran.diagram G.op \u2131.val (op V)) \u27f6\n (Ran.diagram G.op \u2131.val (op V)).obj (StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)))\n\u22a2 StructuredArrow.mk (W.hom.unop.op \u226b (G.map fV').op) = StructuredArrow.mk (W.hom \u226b G.op.map fV'.op)\n[PROOFSTEP]\nsimp only [Quiver.Hom.unop_op, Functor.op_map, Quiver.Hom.op_unop]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 IsAmalgamation x (gluedSection hu \u2131 hS hx)\n[PROOFSTEP]\nintro V fV hV\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) = x fV hV\n[PROOFSTEP]\nrefine limit.hom_ext (\u03bb (W : StructuredArrow (op V) G.op) => ?_)\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W\n[PROOFSTEP]\nsimp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 limit.lift (StructuredArrow.map fV.op \u22d9 Ran.diagram G.op \u2131.val (op U))\n (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W\n[PROOFSTEP]\nerw [limit.lift_\u03c0]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 NatTrans.app (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0 W =\n x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W\n[PROOFSTEP]\nsymm\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n NatTrans.app (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0 W\n[PROOFSTEP]\nconvert helper hu \u2131 hS hx _ (x fV hV) _ _ using 1\n[GOAL]\ncase convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 \u2200 {V' : C} {fV_1 : G.obj V' \u27f6 V} (hV_1 : S.arrows (fV_1 \u226b fV)),\n x fV hV \u226b ((ran G.op).obj \u2131.val).map fV_1.op = x (fV_1 \u226b fV) hV_1\n[PROOFSTEP]\nintro V' fV' hV'\n[GOAL]\ncase convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\n\u22a2 x fV hV \u226b ((ran G.op).obj \u2131.val).map fV'.op = x (fV' \u226b fV) hV'\n[PROOFSTEP]\nconvert hx fV' (\ud835\udfd9 _) hV hV' (by rw [Category.id_comp])\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\n\u22a2 fV' \u226b fV = \ud835\udfd9 (G.obj V') \u226b fV' \u226b fV\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\ncase h.e'_3.h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\ne_1\u271d :\n (X \u27f6 ((ran G.op).obj \u2131.val).obj (op (G.obj V'))) = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op (G.obj V'))\n\u22a2 x (fV' \u226b fV) hV' = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map (\ud835\udfd9 (G.obj V')).op (x (fV' \u226b fV) hV')\n[PROOFSTEP]\nsimp only [op_id, FunctorToTypes.map_id_apply]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n\u22a2 y = gluedSection hu \u2131 hS hx\n[PROOFSTEP]\nunfold gluedSection limit.lift\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\n\u22a2 y = IsLimit.lift (limit.isLimit (Ran.diagram G.op \u2131.val (op U))) (gluedLimitCone hu \u2131 hS hx)\n[PROOFSTEP]\nrefine limit.hom_ext (\u03bb (W : StructuredArrow (op U) G.op) => ?_)\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op U)) W =\n IsLimit.lift (limit.isLimit (Ran.diagram G.op \u2131.val (op U))) (gluedLimitCone hu \u2131 hS hx) \u226b\n limit.\u03c0 (Ran.diagram G.op \u2131.val (op U)) W\n[PROOFSTEP]\nerw [limit.lift_\u03c0]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\n\u22a2 y \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op U)) W = NatTrans.app (gluedLimitCone hu \u2131 hS hx).\u03c0 W\n[PROOFSTEP]\nconvert helper hu \u2131 hS hx (\ud835\udfd9 _) y W _\n[GOAL]\ncase h.e'_3.h.h.e'_8\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\ne_1\u271d : ((op X).unop \u27f6 (Ran.diagram G.op \u2131.val (op U)).obj W) = (X \u27f6 (Ran.diagram G.op \u2131.val (op U)).obj W)\n\u22a2 W = (StructuredArrow.map (\ud835\udfd9 U).op).obj W\n[PROOFSTEP]\nsimp only [op_id, StructuredArrow.map_id]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\n\u22a2 \u2200 {V' : C} {fV : G.obj V' \u27f6 U} (hV : S.arrows (fV \u226b \ud835\udfd9 U)), y \u226b ((ran G.op).obj \u2131.val).map fV.op = x (fV \u226b \ud835\udfd9 U) hV\n[PROOFSTEP]\nintro V' fV' hV'\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\nV' : C\nfV' : G.obj V' \u27f6 U\nhV' : S.arrows (fV' \u226b \ud835\udfd9 U)\n\u22a2 y \u226b ((ran G.op).obj \u2131.val).map fV'.op = x (fV' \u226b \ud835\udfd9 U) hV'\n[PROOFSTEP]\nconvert hy fV' (by simpa only [Category.comp_id] using hV')\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\nV' : C\nfV' : G.obj V' \u27f6 U\nhV' : S.arrows (fV' \u226b \ud835\udfd9 U)\n\u22a2 S.arrows fV'\n[PROOFSTEP]\nsimpa only [Category.comp_id] using hV'\n[GOAL]\ncase h.e'_3.h.h.e'_2\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\ny : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\nhy : IsAmalgamation x y\nW : StructuredArrow (op U) G.op\nV' : C\nfV' : G.obj V' \u27f6 U\nhV' : S.arrows (fV' \u226b \ud835\udfd9 U)\ne_1\u271d :\n (X \u27f6 ((ran G.op).obj \u2131.val).obj (op (G.obj V'))) = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op (G.obj V'))\n\u22a2 fV' \u226b \ud835\udfd9 U = fV'\n[PROOFSTEP]\nerw [Category.comp_id]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\n\u22a2 Presheaf.IsSheaf K ((ran G.op).obj \u2131.val)\n[PROOFSTEP]\nintro X U S hS x hx\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 (fun t => IsAmalgamation x t) ?w \u2227\n \u2200 (y : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)), (fun t => IsAmalgamation x t) y \u2192 y = ?w\ncase w\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\nswap\n[GOAL]\ncase w\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)\n[PROOFSTEP]\napply RanIsSheafOfCoverLifting.gluedSection hG \u2131 hS hx\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 (fun t => IsAmalgamation x t) (RanIsSheafOfCoverLifting.gluedSection hG \u2131 hS hx) \u2227\n \u2200 (y : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y \u2192 y = RanIsSheafOfCoverLifting.gluedSection hG \u2131 hS hx\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 (fun t => IsAmalgamation x t) (RanIsSheafOfCoverLifting.gluedSection hG \u2131 hS hx)\n[PROOFSTEP]\napply RanIsSheafOfCoverLifting.gluedSection_isAmalgamation\n[GOAL]\ncase h.right\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhG : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 \u2200 (y : ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op U)),\n (fun t => IsAmalgamation x t) y \u2192 y = RanIsSheafOfCoverLifting.gluedSection hG \u2131 hS hx\n[PROOFSTEP]\napply RanIsSheafOfCoverLifting.gluedSection_is_unique\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : (pullback A Hc Hp).obj X \u27f6 Y\n\u22a2 (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f) =\n f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : (pullback A Hc Hp).obj X \u27f6 Y\n\u22a2 ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f)).val =\n f.val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : (pullback A Hc Hp).obj X \u27f6 Y\n\u22a2 \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm\n (\u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val) =\n f.val\n[PROOFSTEP]\nrw [Equiv.symm_apply_apply]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : X \u27f6 (copullback A Hl).obj Y\n\u22a2 (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val }) f) =\n f\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : X \u27f6 (copullback A Hl).obj Y\n\u22a2 ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val }) f)).val =\n f.val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX : Sheaf K A\nY : Sheaf J A\nf : X \u27f6 (copullback A Hl).obj Y\n\u22a2 \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val)\n (\u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val) =\n f.val\n[PROOFSTEP]\nrw [Equiv.apply_symm_apply]\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\n\u22a2 \u2200 {X : Sheaf K A} {Y : Sheaf J A} {f : (pullback A Hc Hp).obj X \u27f6 Y},\n \u2191((fun X Y =>\n { toFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val },\n invFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val },\n left_inv :=\n (_ :\n \u2200 (f : (pullback A Hc Hp).obj X \u27f6 Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 (copullback A Hl).obj Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n f) =\n f) })\n X Y)\n f =\n NatTrans.app (NatTrans.mk fun X => { val := NatTrans.app (Ran.adjunction A G.op).unit X.val }) X \u226b\n (copullback A Hl).map f\n[PROOFSTEP]\nrefine Sheaf.Hom.ext _ _ ?_\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX\u271d : Sheaf K A\nY\u271d : Sheaf J A\nf\u271d : (pullback A Hc Hp).obj X\u271d \u27f6 Y\u271d\n\u22a2 (\u2191((fun X Y =>\n { toFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val },\n invFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val },\n left_inv :=\n (_ :\n \u2200 (f : (pullback A Hc Hp).obj X \u27f6 Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 (copullback A Hl).obj Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n f) =\n f) })\n X\u271d Y\u271d)\n f\u271d).val =\n (NatTrans.app (NatTrans.mk fun X => { val := NatTrans.app (Ran.adjunction A G.op).unit X.val }) X\u271d \u226b\n (copullback A Hl).map f\u271d).val\n[PROOFSTEP]\napply (Ran.adjunction A G.op).homEquiv_unit\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\n\u22a2 \u2200 {X : Sheaf K A} {Y : Sheaf J A} {g : X \u27f6 (copullback A Hl).obj Y},\n \u2191((fun X Y =>\n { toFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val },\n invFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val },\n left_inv :=\n (_ :\n \u2200 (f : (pullback A Hc Hp).obj X \u27f6 Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 (copullback A Hl).obj Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n f) =\n f) })\n X Y).symm\n g =\n (pullback A Hc Hp).map g \u226b\n NatTrans.app (NatTrans.mk fun X => { val := NatTrans.app (Ran.adjunction A G.op).counit X.val }) Y\n[PROOFSTEP]\nrefine Sheaf.Hom.ext _ _ ?_\n[GOAL]\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nX\u271d : Sheaf K A\nY\u271d : Sheaf J A\ng\u271d : X\u271d \u27f6 (copullback A Hl).obj Y\u271d\n\u22a2 (\u2191((fun X Y =>\n { toFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val },\n invFun := fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val },\n left_inv :=\n (_ :\n \u2200 (f : (pullback A Hc Hp).obj X \u27f6 Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val }) f) =\n f),\n right_inv :=\n (_ :\n \u2200 (f : X \u27f6 (copullback A Hl).obj Y),\n (fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val) f.val })\n ((fun f => { val := \u2191(Adjunction.homEquiv (Ran.adjunction A G.op) X.val Y.val).symm f.val })\n f) =\n f) })\n X\u271d Y\u271d).symm\n g\u271d).val =\n ((pullback A Hc Hp).map g\u271d \u226b\n NatTrans.app (NatTrans.mk fun X => { val := NatTrans.app (Ran.adjunction A G.op).counit X.val }) Y\u271d).val\n[PROOFSTEP]\napply (Ran.adjunction A G.op).homEquiv_counit\n[GOAL]\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (NatTrans.app (pullbackSheafificationCompatibility A Hp Hl Hc).hom F).val =\n whiskerLeft G.op (GrothendieckTopology.toSheafify K F)\n[PROOFSTEP]\ndsimp [pullbackSheafificationCompatibility, Adjunction.leftAdjointUniq]\n[GOAL]\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (coyoneda.preimage\n (NatTrans.app\n (Adjunction.leftAdjointsCoyonedaEquiv\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc))\n (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom\n (op F))).unop.val =\n whiskerLeft G.op (GrothendieckTopology.toSheafify K F)\n[PROOFSTEP]\napply Quiver.Hom.op_inj\n[GOAL]\ncase a\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (coyoneda.preimage\n (NatTrans.app\n (Adjunction.leftAdjointsCoyonedaEquiv\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc))\n (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom\n (op F))).unop.val).op =\n (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op\n[PROOFSTEP]\napply coyoneda.map_injective\n[GOAL]\ncase a.a\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 coyoneda.map\n (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (coyoneda.preimage\n (NatTrans.app\n (Adjunction.leftAdjointsCoyonedaEquiv\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc))\n (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom\n (op F))).unop.val).op =\n coyoneda.map (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op\n[PROOFSTEP]\next E : 2\n[GOAL]\ncase a.a.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\n\u22a2 NatTrans.app\n (coyoneda.map\n (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (coyoneda.preimage\n (NatTrans.app\n (Adjunction.leftAdjointsCoyonedaEquiv\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc))\n (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A))).hom\n (op F))).unop.val).op)\n E =\n NatTrans.app (coyoneda.map (whiskerLeft G.op (GrothendieckTopology.toSheafify K F)).op) E\n[PROOFSTEP]\ndsimp [Functor.preimage, Full.preimage, coyoneda, Adjunction.leftAdjointsCoyonedaEquiv]\n[GOAL]\ncase a.a.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\n\u22a2 (fun g =>\n (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (\u2191(Adjunction.homEquiv (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A)) F\n (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F))).symm\n (\u2191(Adjunction.homEquiv\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc)) F\n (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)))\n (\ud835\udfd9 (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F))))).val) \u226b\n g) =\n fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b g\n[PROOFSTEP]\nerw [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]\n[GOAL]\ncase a.a.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\n\u22a2 (fun g =>\n (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n (((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op \u22d9 presheafToSheaf J A).map\n (NatTrans.app\n (Adjunction.comp (sheafificationAdjunction K A) (pullbackCopullbackAdjunction A Hp Hl Hc)).unit F \u226b\n (sheafToPresheaf J A \u22d9 ran G.op).map (\ud835\udfd9 (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)))) \u226b\n NatTrans.app (Adjunction.comp (Ran.adjunction A G.op) (sheafificationAdjunction J A)).counit\n (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F))).val) \u226b\n g) =\n fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b g\n[PROOFSTEP]\ndsimp [Adjunction.comp]\n[GOAL]\ncase a.a.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\n\u22a2 (fun g =>\n (GrothendieckTopology.toSheafify J (G.op \u22d9 F) \u226b\n GrothendieckTopology.sheafifyMap J\n ((whiskerLeft G.op (NatTrans.app (sheafificationAdjunction K A).unit F) \u226b\n whiskerLeft G.op (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) \u226b\n \ud835\udfd9 (G.op \u22d9 ((copullback A Hl).obj (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F))).val)) \u226b\n whiskerLeft G.op ((ran G.op).map (\ud835\udfd9 (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val))) \u226b\n \ud835\udfd9\n (GrothendieckTopology.sheafify J\n (G.op \u22d9 (ran G.op).obj (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val)) \u226b\n GrothendieckTopology.sheafifyMap J\n (NatTrans.app (Ran.adjunction A G.op).counit\n (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val) \u226b\n GrothendieckTopology.sheafifyLift J (\ud835\udfd9 (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val)\n (_ : Presheaf.IsSheaf J (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val)) \u226b\n g) =\n fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b g\n[PROOFSTEP]\nsimp only [sheafificationAdjunction_unit_app, Category.comp_id, Functor.map_id, whiskerLeft_id',\n GrothendieckTopology.sheafifyMap_comp, GrothendieckTopology.sheafifyMap_sheafifyLift, Category.id_comp,\n Category.assoc, GrothendieckTopology.toSheafify_sheafifyLift]\n[GOAL]\ncase a.a.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\n\u22a2 (fun g =>\n whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b\n whiskerLeft G.op (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) \u226b\n NatTrans.app (Ran.adjunction A G.op).counit (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u226b g) =\n fun g => whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b g\n[PROOFSTEP]\next t s : 3\n[GOAL]\ncase a.a.w.h.h.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\n\u22a2 NatTrans.app\n (whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b\n whiskerLeft G.op (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) \u226b\n NatTrans.app (Ran.adjunction A G.op).counit (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u226b t)\n s =\n NatTrans.app (whiskerLeft G.op (GrothendieckTopology.toSheafify K F) \u226b t) s\n[PROOFSTEP]\ndsimp [pullbackSheaf]\n[GOAL]\ncase a.a.w.h.h.w.h\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\n\u22a2 NatTrans.app (GrothendieckTopology.toSheafify K F) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s \u226b\n NatTrans.app t s =\n NatTrans.app (GrothendieckTopology.toSheafify K F) (op (G.obj s.unop)) \u226b NatTrans.app t s\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase a.a.w.h.h.w.h.e_a\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s \u226b\n NatTrans.app t s =\n NatTrans.app t s\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase a.a.w.h.h.w.h.e_a\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\n\u22a2 (NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s) \u226b\n NatTrans.app t s =\n NatTrans.app t s\n[PROOFSTEP]\nconvert Category.id_comp (obj := A) _\n[GOAL]\ncase h.e'_2.h.h.e'_6\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s =\n \ud835\udfd9 ((GrothendieckTopology.sheafify K F).obj (op (G.obj s.unop)))\n[PROOFSTEP]\nhave := (Ran.adjunction A G.op).left_triangle\n[GOAL]\ncase h.e'_2.h.h.e'_6\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\nthis :\n whiskerRight (Ran.adjunction A G.op).unit ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op) \u226b\n whiskerLeft ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op) (Ran.adjunction A G.op).counit =\n \ud835\udfd9 (\ud835\udfed (D\u1d52\u1d56 \u2964 A) \u22d9 (whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op)\n\u22a2 NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s =\n \ud835\udfd9 ((GrothendieckTopology.sheafify K F).obj (op (G.obj s.unop)))\n[PROOFSTEP]\napply_fun (fun e => (e.app (K.sheafify F)).app s) at this \n[GOAL]\ncase h.e'_2.h.h.e'_6\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\nE : C\u1d52\u1d56 \u2964 A\nt : (pullbackSheaf Hc Hp ((presheafToSheaf K A).obj F)).val \u27f6 E\ns : C\u1d52\u1d56\nthis :\n NatTrans.app\n (NatTrans.app\n (whiskerRight (Ran.adjunction A G.op).unit ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op) \u226b\n whiskerLeft ((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op) (Ran.adjunction A G.op).counit)\n (GrothendieckTopology.sheafify K F))\n s =\n NatTrans.app\n (NatTrans.app (\ud835\udfd9 (\ud835\udfed (D\u1d52\u1d56 \u2964 A) \u22d9 (whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op)) (GrothendieckTopology.sheafify K F)) s\n\u22a2 NatTrans.app (NatTrans.app (Ran.adjunction A G.op).unit (GrothendieckTopology.sheafify K F)) (op (G.obj s.unop)) \u226b\n NatTrans.app (NatTrans.app (Ran.adjunction A G.op).counit (G.op \u22d9 GrothendieckTopology.sheafify K F)) s =\n \ud835\udfd9 ((GrothendieckTopology.sheafify K F).obj (op (G.obj s.unop)))\n[PROOFSTEP]\nexact this\n[GOAL]\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 (NatTrans.app (pullbackSheafificationCompatibility A Hp Hl Hc).hom F).val =\n GrothendieckTopology.sheafifyLift J (whiskerLeft G.op (GrothendieckTopology.toSheafify K F))\n (_ : Presheaf.IsSheaf J ((presheafToSheaf K A \u22d9 pullback A Hc Hp).obj F).val)\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC D : Type u\ninst\u271d\u00b9\u2070 : Category.{v, u} C\ninst\u271d\u2079 : Category.{v, u} D\nA : Type w\ninst\u271d\u2078 : Category.{max u v, w} A\ninst\u271d\u2077 : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst\u271d\u2076 : ConcreteCategory A\ninst\u271d\u2075 : PreservesLimits (forget A)\ninst\u271d\u2074 : ReflectsIsomorphisms (forget A)\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget A)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 A\ninst\u271d\u00b9 : (X : D) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 (forget A)\ninst\u271d : \u2200 (X : D), HasColimitsOfShape (GrothendieckTopology.Cover K X)\u1d52\u1d56 A\nG : C \u2964 D\nHp : CoverPreserving J K G\nHl : CoverLifting J K G\nHc : CompatiblePreserving K G\nF : D\u1d52\u1d56 \u2964 A\n\u22a2 GrothendieckTopology.toSheafify J (((whiskeringLeft C\u1d52\u1d56 D\u1d52\u1d56 A).obj G.op).obj F) \u226b\n (NatTrans.app (pullbackSheafificationCompatibility A Hp Hl Hc).hom F).val =\n whiskerLeft G.op (GrothendieckTopology.toSheafify K F)\n[PROOFSTEP]\napply toSheafify_pullbackSheafificationCompatibility\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.CoverLifting", "llama_tokens": 43759, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.269741456353371}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\n\u22a2 HasSum (fun n => 0 ^ n \u2022 a n) (a 0)\n[PROOFSTEP]\nconvert hasSum_single (\u03b1 := E) 0 fun b h => _\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\n\u22a2 a 0 = 0 ^ 0 \u2022 a 0\n[PROOFSTEP]\nfirst\n| simp [Nat.pos_of_ne_zero h]\n| simp\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\n\u22a2 a 0 = 0 ^ 0 \u2022 a 0\n[PROOFSTEP]\nsimp [Nat.pos_of_ne_zero h]\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\n\u22a2 a 0 = 0 ^ 0 \u2022 a 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\nb : \u2115\nh : b \u2260 0\n\u22a2 0 ^ b \u2022 a b = 0\n[PROOFSTEP]\nfirst\n| simp [Nat.pos_of_ne_zero h]\n| simp\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na\u271d a : \u2115 \u2192 E\nb : \u2115\nh : b \u2260 0\n\u22a2 0 ^ b \u2022 a b = 0\n[PROOFSTEP]\nsimp [Nat.pos_of_ne_zero h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\n\u22a2 \u2203 t, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t\n[PROOFSTEP]\nobtain rfl | hn := n.eq_zero_or_pos\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < 0 \u2192 a k = 0\n\u22a2 \u2203 t, z ^ 0 \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + 0)) t\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\n\u22a2 \u2203 t, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t\n[PROOFSTEP]\nby_cases h : z = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : z = 0\n\u22a2 \u2203 t, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t\n[PROOFSTEP]\nhave : s = 0 := hs.unique (by simpa [ha 0 hn, h] using hasSum_at_zero a)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : z = 0\n\u22a2 HasSum (fun m => z ^ m \u2022 a m) 0\n[PROOFSTEP]\nsimpa [ha 0 hn, h] using hasSum_at_zero a\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : z = 0\nthis : s = 0\n\u22a2 \u2203 t, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t\n[PROOFSTEP]\nexact \u27e8a n, by simp [h, hn, this], by simpa [h] using hasSum_at_zero fun m => a (m + n)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : z = 0\nthis : s = 0\n\u22a2 z ^ n \u2022 a n = s\n[PROOFSTEP]\nsimp [h, hn, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : z = 0\nthis : s = 0\n\u22a2 HasSum (fun m => z ^ m \u2022 a (m + n)) (a n)\n[PROOFSTEP]\nsimpa [h] using hasSum_at_zero fun m => a (m + n)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\n\u22a2 \u2203 t, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t\n[PROOFSTEP]\nrefine \u27e8(z ^ n)\u207b\u00b9 \u2022 s, by field_simp [smul_smul], ?_\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\n\u22a2 z ^ n \u2022 (z ^ n)\u207b\u00b9 \u2022 s = s\n[PROOFSTEP]\nfield_simp [smul_smul]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\n\u22a2 HasSum (fun m => z ^ m \u2022 a (m + n)) ((z ^ n)\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nhave h1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0 :=\n Finset.sum_eq_zero fun k hk => by simp [ha k (Finset.mem_range.mp hk)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nk : \u2115\nhk : k \u2208 Finset.range n\n\u22a2 z ^ k \u2022 a k = 0\n[PROOFSTEP]\nsimp [ha k (Finset.mem_range.mp hk)]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nh1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0\n\u22a2 HasSum (fun m => z ^ m \u2022 a (m + n)) ((z ^ n)\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nhave h2 : HasSum (fun m => z ^ (m + n) \u2022 a (m + n)) s := by simpa [h1] using (hasSum_nat_add_iff' n).mpr hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nh1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0\n\u22a2 HasSum (fun m => z ^ (m + n) \u2022 a (m + n)) s\n[PROOFSTEP]\nsimpa [h1] using (hasSum_nat_add_iff' n).mpr hs\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nh1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0\nh2 : HasSum (fun m => z ^ (m + n) \u2022 a (m + n)) s\n\u22a2 HasSum (fun m => z ^ m \u2022 a (m + n)) ((z ^ n)\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nconvert h2.const_smul (z\u207b\u00b9 ^ n) using 1\n[GOAL]\ncase h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nh1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0\nh2 : HasSum (fun m => z ^ (m + n) \u2022 a (m + n)) s\n\u22a2 (fun m => z ^ m \u2022 a (m + n)) = fun i => z\u207b\u00b9 ^ n \u2022 z ^ (i + n) \u2022 a (i + n)\n[PROOFSTEP]\nfield_simp [pow_add, smul_smul]\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\na : \u2115 \u2192 E\nhs : HasSum (fun m => z ^ m \u2022 a m) s\nha : \u2200 (k : \u2115), k < n \u2192 a k = 0\nhn : n > 0\nh : \u00acz = 0\nh1 : \u2211 i in Finset.range n, z ^ i \u2022 a i = 0\nh2 : HasSum (fun m => z ^ (m + n) \u2022 a (m + n)) s\n\u22a2 (z ^ n)\u207b\u00b9 \u2022 s = z\u207b\u00b9 ^ n \u2022 s\n[PROOFSTEP]\nsimp only [inv_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 HasFPowerSeriesAt (dslope f z\u2080) (fslope p) z\u2080\n[PROOFSTEP]\nhave hpd : deriv f z\u2080 = p.coeff 1 := hp.deriv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhpd : deriv f z\u2080 = coeff p 1\n\u22a2 HasFPowerSeriesAt (dslope f z\u2080) (fslope p) z\u2080\n[PROOFSTEP]\nhave hp0 : p.coeff 0 = f z\u2080 := hp.coeff_zero 1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\n\u22a2 HasFPowerSeriesAt (dslope f z\u2080) (fslope p) z\u2080\n[PROOFSTEP]\nsimp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\n\u22a2 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + z))\n[PROOFSTEP]\nrefine hp.mono fun x hx => ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\n\u22a2 HasSum (fun n => x ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + x))\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : x = 0\n\u22a2 HasSum (fun n => x ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + x))\n[PROOFSTEP]\nconvert hasSum_single (\u03b1 := E) 0 _\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : x = 0\n\u22a2 dslope f z\u2080 (z\u2080 + x) = x ^ 0 \u2022 coeff p (0 + 1)\n[PROOFSTEP]\nintros\n[GOAL]\ncase pos.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : x = 0\n\u22a2 \u2200 (b' : \u2115), b' \u2260 0 \u2192 x ^ b' \u2022 coeff p (b' + 1) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : x = 0\n\u22a2 dslope f z\u2080 (z\u2080 + x) = x ^ 0 \u2022 coeff p (0 + 1)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos.convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : x = 0\nb'\u271d : \u2115\nx\u271d : b'\u271d \u2260 0\n\u22a2 x ^ b'\u271d \u2022 coeff p (b'\u271d + 1) = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : \u00acx = 0\n\u22a2 HasSum (fun n => x ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + x))\n[PROOFSTEP]\nhave hxx : \u2200 n : \u2115, x\u207b\u00b9 * x ^ (n + 1) = x ^ n := fun n => by field_simp [h, _root_.pow_succ']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn\u271d : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : \u00acx = 0\nn : \u2115\n\u22a2 x\u207b\u00b9 * x ^ (n + 1) = x ^ n\n[PROOFSTEP]\nfield_simp [h, _root_.pow_succ']\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : \u00acx = 0\nhxx : \u2200 (n : \u2115), x\u207b\u00b9 * x ^ (n + 1) = x ^ n\n\u22a2 HasSum (fun n => x ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + x))\n[PROOFSTEP]\nsuffices HasSum (fun n => x\u207b\u00b9 \u2022 x ^ (n + 1) \u2022 p.coeff (n + 1)) (x\u207b\u00b9 \u2022 (f (z\u2080 + x) - f z\u2080)) by\n simpa [dslope, slope, h, smul_smul, hxx] using this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : \u00acx = 0\nhxx : \u2200 (n : \u2115), x\u207b\u00b9 * x ^ (n + 1) = x ^ n\nthis : HasSum (fun n => x\u207b\u00b9 \u2022 x ^ (n + 1) \u2022 coeff p (n + 1)) (x\u207b\u00b9 \u2022 (f (z\u2080 + x) - f z\u2080))\n\u22a2 HasSum (fun n => x ^ n \u2022 coeff p (n + 1)) (dslope f z\u2080 (z\u2080 + x))\n[PROOFSTEP]\nsimpa [dslope, slope, h, smul_smul, hxx] using this\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhpd : deriv f z\u2080 = coeff p 1\nhp0 : coeff p 0 = f z\u2080\nhp : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd 0, HasSum (fun n => z ^ n \u2022 coeff p n) (f (z\u2080 + z))\nx : \ud835\udd5c\nhx : HasSum (fun n => x ^ n \u2022 coeff p n) (f (z\u2080 + x))\nh : \u00acx = 0\nhxx : \u2200 (n : \u2115), x\u207b\u00b9 * x ^ (n + 1) = x ^ n\n\u22a2 HasSum (fun n => x\u207b\u00b9 \u2022 x ^ (n + 1) \u2022 coeff p (n + 1)) (x\u207b\u00b9 \u2022 (f (z\u2080 + x) - f z\u2080))\n[PROOFSTEP]\nsimpa [hp0] using ((hasSum_nat_add_iff' 1).mpr hx).const_smul x\u207b\u00b9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn\u271d : \u2115\nz z\u2080 : \ud835\udd5c\nn : \u2115\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 HasFPowerSeriesAt ((swap dslope z\u2080)^[n] f) (fslope^[n] p) z\u2080\n[PROOFSTEP]\ninduction' n with n ih generalizing f p\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf\u271d g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp\u271d : HasFPowerSeriesAt f\u271d p\u271d z\u2080\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 HasFPowerSeriesAt ((swap dslope z\u2080)^[zero] f) (fslope^[zero] p) z\u2080\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf\u271d g : \ud835\udd5c \u2192 E\nn\u271d : \u2115\nz z\u2080 : \ud835\udd5c\nhp\u271d : HasFPowerSeriesAt f\u271d p\u271d z\u2080\nn : \u2115\nih :\n \u2200 {p : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E} {f : \ud835\udd5c \u2192 E},\n HasFPowerSeriesAt f p z\u2080 \u2192 HasFPowerSeriesAt ((swap dslope z\u2080)^[n] f) (fslope^[n] p) z\u2080\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 HasFPowerSeriesAt ((swap dslope z\u2080)^[succ n] f) (fslope^[succ n] p) z\u2080\n[PROOFSTEP]\nsimpa using ih (has_fpower_series_dslope_fslope hp)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\n\u22a2 (swap dslope z\u2080)^[order p] f z\u2080 \u2260 0\n[PROOFSTEP]\nrw [\u2190 coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\n\u22a2 \u2191(fslope^[order p] p 0) 1 \u2260 0\n[PROOFSTEP]\nsimpa [coeff_eq_zero] using apply_order_ne_zero h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = (z - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f z\n[PROOFSTEP]\nhave hq := hasFPowerSeriesAt_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\n\u22a2 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = (z - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f z\n[PROOFSTEP]\nfilter_upwards [hq, hasFPowerSeriesAt_iff'.mp hp] with x hx1 hx2\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f x)\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\n\u22a2 f x = (x - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f x\n[PROOFSTEP]\nhave : \u2200 k < p.order, p.coeff k = 0 := fun k hk => by simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f x)\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\nk : \u2115\nhk : k < order p\n\u22a2 coeff p k = 0\n[PROOFSTEP]\nsimpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f x)\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\nthis : \u2200 (k : \u2115), k < order p \u2192 coeff p k = 0\n\u22a2 f x = (x - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f x\n[PROOFSTEP]\nobtain \u27e8s, hs1, hs2\u27e9 := HasSum.exists_hasSum_smul_of_apply_eq_zero hx2 this\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns\u271d : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f x)\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\nthis : \u2200 (k : \u2115), k < order p \u2192 coeff p k = 0\ns : E\nhs1 : (x - z\u2080) ^ order p \u2022 s = f x\nhs2 : HasSum (fun m => (x - z\u2080) ^ m \u2022 coeff p (m + order p)) s\n\u22a2 f x = (x - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f x\n[PROOFSTEP]\nconvert hs1.symm\n[GOAL]\ncase h.e'_3.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns\u271d : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f x)\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\nthis : \u2200 (k : \u2115), k < order p \u2192 coeff p k = 0\ns : E\nhs1 : (x - z\u2080) ^ order p \u2022 s = f x\nhs2 : HasSum (fun m => (x - z\u2080) ^ m \u2022 coeff p (m + order p)) s\n\u22a2 (swap dslope z\u2080)^[order p] f x = s\n[PROOFSTEP]\nsimp only [coeff_iterate_fslope] at hx1 \n[GOAL]\ncase h.e'_3.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns\u271d : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nhq : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, HasSum (fun n => (z - z\u2080) ^ n \u2022 coeff (fslope^[order p] p) n) ((swap dslope z\u2080)^[order p] f z)\nx : \ud835\udd5c\nhx2 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p n) (f x)\nthis : \u2200 (k : \u2115), k < order p \u2192 coeff p k = 0\ns : E\nhs1 : (x - z\u2080) ^ order p \u2022 s = f x\nhs2 : HasSum (fun m => (x - z\u2080) ^ m \u2022 coeff p (m + order p)) s\nhx1 : HasSum (fun n => (x - z\u2080) ^ n \u2022 coeff p (n + order p)) ((swap dslope z\u2080)^[order p] f x)\n\u22a2 (swap dslope z\u2080)^[order p] f x = s\n[PROOFSTEP]\nexact hx1.unique hs2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\n\u22a2 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 0\n[PROOFSTEP]\nrw [eventually_nhdsWithin_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\n\u22a2 \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd z\u2080, x \u2208 {z\u2080}\u1d9c \u2192 f x \u2260 0\n[PROOFSTEP]\nhave h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).continuousAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\nh2 : ContinuousAt ((swap dslope z\u2080)^[order p] f) z\u2080\n\u22a2 \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd z\u2080, x \u2208 {z\u2080}\u1d9c \u2192 f x \u2260 0\n[PROOFSTEP]\nhave h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\nh2 : ContinuousAt ((swap dslope z\u2080)^[order p] f) z\u2080\nh3 : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, (swap dslope z\u2080)^[order p] f z \u2260 0\n\u22a2 \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd z\u2080, x \u2208 {z\u2080}\u1d9c \u2192 f x \u2260 0\n[PROOFSTEP]\nfilter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz\u271d z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p \u2260 0\nh2 : ContinuousAt ((swap dslope z\u2080)^[order p] f) z\u2080\nh3 : \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, (swap dslope z\u2080)^[order p] f z \u2260 0\nz : \ud835\udd5c\ne1 : f z = (z - z\u2080) ^ order p \u2022 (swap dslope z\u2080)^[order p] f z\ne2 : (swap dslope z\u2080)^[order p] f z \u2260 0\ne3 : z \u2208 {z\u2080}\u1d9c\n\u22a2 f z \u2260 0\n[PROOFSTEP]\nsimpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhp : HasFPowerSeriesAt f p z\u2080\nh : p = 0\n\u22a2 HasFPowerSeriesAt (fun z => f z) 0 z\u2080\n[PROOFSTEP]\nrwa [h] at hp \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhf : AnalyticAt \ud835\udd5c f z\u2080\n\u22a2 (\u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 0\n[PROOFSTEP]\nrcases hf with \u27e8p, hp\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nhp : HasFPowerSeriesAt f p z\u2080\n\u22a2 (\u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 0\n[PROOFSTEP]\nby_cases h : p = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nhp : HasFPowerSeriesAt f p z\u2080\nh : p = 0\n\u22a2 (\u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 0\n[PROOFSTEP]\nexact Or.inl (HasFPowerSeriesAt.eventually_eq_zero (by rwa [h] at hp ))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nhp : HasFPowerSeriesAt f p z\u2080\nh : p = 0\n\u22a2 HasFPowerSeriesAt (fun z => f z) 0 z\u2080\n[PROOFSTEP]\nrwa [h] at hp \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np\u271d q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nhp : HasFPowerSeriesAt f p z\u2080\nh : \u00acp = 0\n\u22a2 (\u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 0\n[PROOFSTEP]\nexact Or.inr (hp.locally_ne_zero h)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhf : AnalyticAt \ud835\udd5c f z\u2080\nhg : AnalyticAt \ud835\udd5c g z\u2080\n\u22a2 (\u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = g z) \u2228 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z \u2260 g z\n[PROOFSTEP]\nsimpa [sub_eq_zero] using (hf.sub hg).eventually_eq_zero_or_eventually_ne_zero\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nhf : AnalyticAt \ud835\udd5c f z\u2080\nhg : AnalyticAt \ud835\udd5c g z\u2080\n\u22a2 (\u2203\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z = g z) \u2194 \u2200\u1da0 (z : \ud835\udd5c) in \ud835\udcdd z\u2080, f z = g z\n[PROOFSTEP]\nsimpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nU : Set \ud835\udd5c\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nh\u2080 : z\u2080 \u2208 U\nhfg : \u2203\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z = g z\n\u22a2 EqOn f g U\n[PROOFSTEP]\nhave hfg' : \u2203\u1da0 z in \ud835\udcdd[\u2260] z\u2080, (f - g) z = 0 := hfg.mono fun z h => by rw [Pi.sub_apply, h, sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz\u271d z\u2080 : \ud835\udd5c\nU : Set \ud835\udd5c\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nh\u2080 : z\u2080 \u2208 U\nhfg : \u2203\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z = g z\nz : \ud835\udd5c\nh : f z = g z\n\u22a2 (f - g) z = 0\n[PROOFSTEP]\nrw [Pi.sub_apply, h, sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np q : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf g : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\nU : Set \ud835\udd5c\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nh\u2080 : z\u2080 \u2208 U\nhfg : \u2203\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, f z = g z\nhfg' : \u2203\u1da0 (z : \ud835\udd5c) in \ud835\udcdd[{z\u2080}\u1d9c] z\u2080, (f - g) z = 0\n\u22a2 EqOn f g U\n[PROOFSTEP]\nsimpa [sub_eq_zero] using fun z hz => (hf.sub hg).eqOn_zero_of_preconnected_of_frequently_eq_zero hU h\u2080 hfg' hz\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Analytic.IsolatedZeros", "llama_tokens": 16117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.26935992552575533}} {"text": "[GOAL]\nJ : Type v\ninst\u271d : Category.{v', v} J\nF : J \u2964 Discrete PUnit\nc s : Cone F\n\u22a2 s.pt = c.pt\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type v\ninst\u271d : Category.{v', v} J\nF : J \u2964 Discrete PUnit\nc s : Cocone F\n\u22a2 c.pt = s.pt\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Unit", "llama_tokens": 119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.26931112502820426}} {"text": "[GOAL]\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nh : (forget\u2082 FinBoolAlgCat FinPartOrd).map f = (forget\u2082 FinBoolAlgCat FinPartOrd).map g\n\u22a2 f = g\n[PROOFSTEP]\ndsimp at *\n[GOAL]\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nh : \u2191f = \u2191g\n\u22a2 f = g\n[PROOFSTEP]\napply FunLike.coe_injective\n[GOAL]\ncase a\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nh : \u2191f = \u2191g\n\u22a2 (fun f => \u2191f) f = (fun f => \u2191f) g\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nh : \u2191f = \u2191g\n\u22a2 \u2191f = \u2191g\n[PROOFSTEP]\next x\n[GOAL]\ncase a.h\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nh : \u2191f = \u2191g\nx : \u2191X.toBoolAlgCat\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\napply_fun (fun f => f x) at h \n[GOAL]\ncase a.h\nX Y : FinBoolAlgCat\nf g : X \u27f6 Y\nx : \u2191X.toBoolAlgCat\nh : \u2191\u2191f x = \u2191\u2191g x\n\u22a2 \u2191f x = \u2191g x\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 \u03b2 : FinBoolAlgCat\ne : \u2191\u03b1.toBoolAlgCat \u2243o \u2191\u03b2.toBoolAlgCat\n\u22a2 ((let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBoolAlgCat),\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 (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\n \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.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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 : FinBoolAlgCat\ne : \u2191\u03b1.toBoolAlgCat \u2243o \u2191\u03b2.toBoolAlgCat\nx\u271d : (forget FinBoolAlgCat).obj \u03b1\n\u22a2 \u2191((let src :=\n { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBoolAlgCat),\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 (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\n \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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 : FinBoolAlgCat\ne : \u2191\u03b1.toBoolAlgCat \u2243o \u2191\u03b2.toBoolAlgCat\n\u22a2 ((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\n \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.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBoolAlgCat),\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 : FinBoolAlgCat\ne : \u2191\u03b1.toBoolAlgCat \u2243o \u2191\u03b2.toBoolAlgCat\nx\u271d : (forget FinBoolAlgCat).obj \u03b2\n\u22a2 \u2191((let src :=\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\n \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191(OrderIso.symm e),\n map_sup' :=\n (_ :\n \u2200 (a b : \u2191\u03b2.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat),\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.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \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.toBoolAlgCat),\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n (a \u2294 b) =\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n a \u2294\n SupHom.toFun\n {\n toSupHom :=\n { toFun := \u2191e,\n map_sup' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n map_inf' := (_ : \u2200 (a b : \u2191\u03b1.toBoolAlgCat), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) }.toSupHom\n b) },\n map_inf' :=\n (_ :\n \u2200 (a b : \u2191\u03b1.toBoolAlgCat),\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.FinBoolAlgCat", "llama_tokens": 7968, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.39233683016710835, "lm_q1q2_score": 0.26912322979508657}} {"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nhave := I.locally_compact\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis : LocallyCompactSpace H\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nhave := ChartedSpace.locallyCompact H M\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d : LocallyCompactSpace H\nthis : LocallyCompactSpace M\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nhave := I.secondCountableTopology\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b9 : LocallyCompactSpace H\nthis\u271d : LocallyCompactSpace M\nthis : SecondCountableTopology H\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nhave := ChartedSpace.secondCountable_of_sigma_compact H M\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b2 : LocallyCompactSpace H\nthis\u271d\u00b9 : LocallyCompactSpace M\nthis\u271d : SecondCountableTopology H\nthis : SecondCountableTopology M\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nhave := ManifoldWithCorners.metrizableSpace I M\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\nlet _ : MetricSpace M := TopologicalSpace.metrizableSpaceMetric M\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = 0\n[PROOFSTEP]\napply ae_eq_zero_of_forall_set_integral_isCompact_eq_zero' hf (fun s hs \u21a6 Eq.symm ?_)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4, 0 < \u03b4 \u2227 IsCompact (cthickening \u03b4 s) := hs.exists_isCompact_cthickening\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8u, -, u_pos, u_lim\u27e9 : \u2203 u, StrictAnti u \u2227 (\u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4) \u2227 Tendsto u atTop (\ud835\udcdd 0) :=\n exists_seq_strictAnti_tendsto' \u03b4pos\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nlet v : \u2115 \u2192 Set M := fun n \u21a6 thickening (u n) s\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8K, K_compact, vK\u27e9 : \u2203 K, IsCompact K \u2227 \u2200 n, v n \u2286 K :=\n \u27e8_, h\u03b4, fun n \u21a6 thickening_subset_cthickening_of_le (u_pos n).2.le _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 n, \u2203 (g : M \u2192 \u211d), support g = v n \u2227 Smooth I \ud835\udcd8(\u211d) g \u2227 Set.range g \u2286 Set.Icc 0 1 \u2227 \u2200 x \u2208 s, g x = 1 :=\n by\n intro n\n rcases exists_msmooth_support_eq_eq_one_iff I isOpen_thickening hs.isClosed\n (self_subset_thickening (u_pos n).1 s) with\n \u27e8g, g_smooth, g_range, g_supp, hg\u27e9\n exact \u27e8g, g_supp, g_smooth, g_range, fun x hx \u21a6 (hg x).1 hx\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\n\u22a2 \u2200 (n : \u2115), \u2203 g, support g = v n \u2227 Smooth I \ud835\udcd8(\u211d, \u211d) g \u2227 range g \u2286 Icc 0 1 \u2227 \u2200 (x : M), x \u2208 s \u2192 g x = 1\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\nn : \u2115\n\u22a2 \u2203 g, support g = v n \u2227 Smooth I \ud835\udcd8(\u211d, \u211d) g \u2227 range g \u2286 Icc 0 1 \u2227 \u2200 (x : M), x \u2208 s \u2192 g x = 1\n[PROOFSTEP]\nrcases exists_msmooth_support_eq_eq_one_iff I isOpen_thickening hs.isClosed (self_subset_thickening (u_pos n).1 s) with\n \u27e8g, g_smooth, g_range, g_supp, hg\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\nn : \u2115\ng : M \u2192 \u211d\ng_smooth : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_range : range g \u2286 Icc 0 1\ng_supp : support g = thickening (u n) s\nhg : \u2200 (x : M), x \u2208 s \u2194 g x = 1\n\u22a2 \u2203 g, support g = v n \u2227 Smooth I \ud835\udcd8(\u211d, \u211d) g \u2227 range g \u2286 Icc 0 1 \u2227 \u2200 (x : M), x \u2208 s \u2192 g x = 1\n[PROOFSTEP]\nexact \u27e8g, g_supp, g_smooth, g_range, fun x hx \u21a6 (hg x).1 hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\nthis : \u2200 (n : \u2115), \u2203 g, support g = v n \u2227 Smooth I \ud835\udcd8(\u211d, \u211d) g \u2227 range g \u2286 Icc 0 1 \u2227 \u2200 (x : M), x \u2208 s \u2192 g x = 1\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nchoose g g_supp g_diff g_range hg using this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nhave L : Tendsto (fun n \u21a6 \u222b x, g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b x in s, f x \u2202\u03bc)) :=\n by\n rw [\u2190 integral_indicator hs.measurableSet]\n let bound : M \u2192 \u211d := K.indicator (fun x \u21a6 \u2016f x\u2016)\n have A : \u2200 n, AEStronglyMeasurable (fun x \u21a6 g n x \u2022 f x) \u03bc := fun n \u21a6\n (g_diff n).continuous.aestronglyMeasurable.smul hf.aestronglyMeasurable\n have B : Integrable bound \u03bc := by\n rw [integrable_indicator_iff K_compact.measurableSet]\n exact (hf.integrableOn_isCompact K_compact).norm\n have C : \u2200 n, \u2200\u1d50 x \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x := by\n intro n\n apply eventually_of_forall (fun x \u21a6 ?_)\n rw [norm_smul]\n refine le_indicator_apply (fun _ \u21a6 ?_) (fun hxK \u21a6 ?_)\n \u00b7 have : \u2016g n x\u2016 \u2264 1 := by\n have := g_range n (mem_range_self (f := g n) x)\n rw [Real.norm_of_nonneg this.1]\n exact this.2\n exact mul_le_of_le_one_left (norm_nonneg _) this\n \u00b7 have : g n x = 0 := by rw [\u2190 nmem_support, g_supp]; contrapose! hxK; exact vK n hxK\n simp [this]\n have D : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (s.indicator f x)) :=\n by\n apply eventually_of_forall (fun x \u21a6 ?_)\n by_cases hxs : x \u2208 s\n \u00b7 have : \u2200 n, g n x = 1 := fun n \u21a6 hg n x hxs\n simp [this, indicator_of_mem hxs f]\n \u00b7 simp_rw [indicator_of_not_mem hxs f]\n apply tendsto_const_nhds.congr'\n suffices H : \u2200\u1da0 n in atTop, g n x = 0\n \u00b7 filter_upwards [H] with n hn using by simp [hn]\n obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5, 0 < \u03b5 \u2227 x \u2209 thickening \u03b5 s :=\n by\n rw [\u2190 hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs \n simpa using hxs\n filter_upwards [(tendsto_order.1 u_lim).2 _ \u03b5pos] with n hn\n rw [\u2190 nmem_support, g_supp]\n contrapose! h\u03b5\n exact thickening_mono hn.le s h\u03b5\n exact tendsto_integral_of_dominated_convergence bound A B C D\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\n[PROOFSTEP]\nrw [\u2190 integral_indicator hs.measurableSet]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nlet bound : M \u2192 \u211d := K.indicator (fun x \u21a6 \u2016f x\u2016)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nhave A : \u2200 n, AEStronglyMeasurable (fun x \u21a6 g n x \u2022 f x) \u03bc := fun n \u21a6\n (g_diff n).continuous.aestronglyMeasurable.smul hf.aestronglyMeasurable\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nhave B : Integrable bound \u03bc := by\n rw [integrable_indicator_iff K_compact.measurableSet]\n exact (hf.integrableOn_isCompact K_compact).norm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\n\u22a2 Integrable bound\n[PROOFSTEP]\nrw [integrable_indicator_iff K_compact.measurableSet]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\n\u22a2 IntegrableOn (fun x => \u2016f x\u2016) K\n[PROOFSTEP]\nexact (hf.integrableOn_isCompact K_compact).norm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nhave C : \u2200 n, \u2200\u1d50 x \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x := by\n intro n\n apply eventually_of_forall (fun x \u21a6 ?_)\n rw [norm_smul]\n refine le_indicator_apply (fun _ \u21a6 ?_) (fun hxK \u21a6 ?_)\n \u00b7 have : \u2016g n x\u2016 \u2264 1 := by\n have := g_range n (mem_range_self (f := g n) x)\n rw [Real.norm_of_nonneg this.1]\n exact this.2\n exact mul_le_of_le_one_left (norm_nonneg _) this\n \u00b7 have : g n x = 0 := by rw [\u2190 nmem_support, g_supp]; contrapose! hxK; exact vK n hxK\n simp [this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\n[PROOFSTEP]\napply eventually_of_forall (fun x \u21a6 ?_)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\n\u22a2 \u2016g n x \u2022 f x\u2016 \u2264 bound x\n[PROOFSTEP]\nrw [norm_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\n\u22a2 \u2016g n x\u2016 * \u2016f x\u2016 \u2264 bound x\n[PROOFSTEP]\nrefine le_indicator_apply (fun _ \u21a6 ?_) (fun hxK \u21a6 ?_)\n[GOAL]\ncase refine_1\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d\u00b9 : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nx\u271d : x \u2208 K\n\u22a2 \u2016g n x\u2016 * \u2016f x\u2016 \u2264 \u2016f x\u2016\n[PROOFSTEP]\nhave : \u2016g n x\u2016 \u2264 1 := by\n have := g_range n (mem_range_self (f := g n) x)\n rw [Real.norm_of_nonneg this.1]\n exact this.2\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d\u00b9 : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nx\u271d : x \u2208 K\n\u22a2 \u2016g n x\u2016 \u2264 1\n[PROOFSTEP]\nhave := g_range n (mem_range_self (f := g n) x)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d\u00b9 : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nx\u271d : x \u2208 K\nthis : g n x \u2208 Icc 0 1\n\u22a2 \u2016g n x\u2016 \u2264 1\n[PROOFSTEP]\nrw [Real.norm_of_nonneg this.1]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d\u00b9 : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nx\u271d : x \u2208 K\nthis : g n x \u2208 Icc 0 1\n\u22a2 g n x \u2264 1\n[PROOFSTEP]\nexact this.2\n[GOAL]\ncase refine_1\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d\u00b9 : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nx\u271d : x \u2208 K\nthis : \u2016g n x\u2016 \u2264 1\n\u22a2 \u2016g n x\u2016 * \u2016f x\u2016 \u2264 \u2016f x\u2016\n[PROOFSTEP]\nexact mul_le_of_le_one_left (norm_nonneg _) this\n[GOAL]\ncase refine_2\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nhxK : \u00acx \u2208 K\n\u22a2 \u2016g n x\u2016 * \u2016f x\u2016 \u2264 0\n[PROOFSTEP]\nhave : g n x = 0 := by rw [\u2190 nmem_support, g_supp]; contrapose! hxK; exact vK n hxK\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nhxK : \u00acx \u2208 K\n\u22a2 g n x = 0\n[PROOFSTEP]\nrw [\u2190 nmem_support, g_supp]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nhxK : \u00acx \u2208 K\n\u22a2 \u00acx \u2208 v n\n[PROOFSTEP]\ncontrapose! hxK\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nhxK : x \u2208 (fun n => thickening (u n) s) n\n\u22a2 x \u2208 K\n[PROOFSTEP]\nexact vK n hxK\n[GOAL]\ncase refine_2\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nn : \u2115\nx : M\nhxK : \u00acx \u2208 K\nthis : g n x = 0\n\u22a2 \u2016g n x\u2016 * \u2016f x\u2016 \u2264 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nhave D : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (s.indicator f x)) :=\n by\n apply eventually_of_forall (fun x \u21a6 ?_)\n by_cases hxs : x \u2208 s\n \u00b7 have : \u2200 n, g n x = 1 := fun n \u21a6 hg n x hxs\n simp [this, indicator_of_mem hxs f]\n \u00b7 simp_rw [indicator_of_not_mem hxs f]\n apply tendsto_const_nhds.congr'\n suffices H : \u2200\u1da0 n in atTop, g n x = 0\n \u00b7 filter_upwards [H] with n hn using by simp [hn]\n obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5, 0 < \u03b5 \u2227 x \u2209 thickening \u03b5 s :=\n by\n rw [\u2190 hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs \n simpa using hxs\n filter_upwards [(tendsto_order.1 u_lim).2 _ \u03b5pos] with n hn\n rw [\u2190 nmem_support, g_supp]\n contrapose! h\u03b5\n exact thickening_mono hn.le s h\u03b5\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n[PROOFSTEP]\napply eventually_of_forall (fun x \u21a6 ?_)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\n\u22a2 Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n[PROOFSTEP]\nby_cases hxs : x \u2208 s\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : x \u2208 s\n\u22a2 Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n[PROOFSTEP]\nhave : \u2200 n, g n x = 1 := fun n \u21a6 hg n x hxs\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : x \u2208 s\nthis : \u2200 (n : \u2115), g n x = 1\n\u22a2 Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n[PROOFSTEP]\nsimp [this, indicator_of_mem hxs f]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u22a2 Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n[PROOFSTEP]\nsimp_rw [indicator_of_not_mem hxs f]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u22a2 Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_const_nhds.congr'\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u22a2 (fun x => 0) =\u1da0[atTop] fun n => g n x \u2022 f x\n[PROOFSTEP]\nsuffices H : \u2200\u1da0 n in atTop, g n x = 0\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH\u271d : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\u271d\nI : ModelWithCorners \u211d E H\u271d\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H\u271d M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\u271d\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\u271d\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\nH : \u2200\u1da0 (n : \u2115) in atTop, g n x = 0\n\u22a2 (fun x => 0) =\u1da0[atTop] fun n => g n x \u2022 f x\n[PROOFSTEP]\nfilter_upwards [H] with n hn using by simp [hn]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH\u271d : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\u271d\nI : ModelWithCorners \u211d E H\u271d\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H\u271d M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\u271d\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\u271d\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\nH : \u2200\u1da0 (n : \u2115) in atTop, g n x = 0\nn : \u2115\nhn : g n x = 0\n\u22a2 0 = g n x \u2022 f x\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g n x = 0\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5, 0 < \u03b5 \u2227 x \u2209 thickening \u03b5 s :=\n by\n rw [\u2190 hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs \n simpa using hxs\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u00acx \u2208 thickening \u03b5 s\n[PROOFSTEP]\nrw [\u2190 hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), thickening \u03b4 s\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u00acx \u2208 thickening \u03b5 s\n[PROOFSTEP]\nsimpa using hxs\n[GOAL]\ncase H.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u00acx \u2208 thickening \u03b5 s\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, g n x = 0\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 u_lim).2 _ \u03b5pos] with n hn\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u00acx \u2208 thickening \u03b5 s\nn : \u2115\nhn : u n < \u03b5\n\u22a2 g n x = 0\n[PROOFSTEP]\nrw [\u2190 nmem_support, g_supp]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u00acx \u2208 thickening \u03b5 s\nn : \u2115\nhn : u n < \u03b5\n\u22a2 \u00acx \u2208 v n\n[PROOFSTEP]\ncontrapose! h\u03b5\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nx : M\nhxs : \u00acx \u2208 s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n < \u03b5\nh\u03b5 : x \u2208 (fun n => thickening (u n) s) n\n\u22a2 x \u2208 thickening \u03b5 s\n[PROOFSTEP]\nexact thickening_mono hn.le s h\u03b5\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nbound : M \u2192 \u211d := indicator K fun x => \u2016f x\u2016\nA : \u2200 (n : \u2115), AEStronglyMeasurable (fun x => g n x \u2022 f x) \u03bc\nB : Integrable bound\nC : \u2200 (n : \u2115), \u2200\u1d50 (x : M) \u2202\u03bc, \u2016g n x \u2022 f x\u2016 \u2264 bound x\nD : \u2200\u1d50 (x : M) \u2202\u03bc, Tendsto (fun n => g n x \u2022 f x) atTop (\ud835\udcdd (indicator s f x))\n\u22a2 Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M), indicator s (fun x => f x) x \u2202\u03bc))\n[PROOFSTEP]\nexact tendsto_integral_of_dominated_convergence bound A B C D\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nL : Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 n, \u222b x, g n x \u2022 f x \u2202\u03bc = 0 := by\n refine' fun n \u21a6 h _ (g_diff n) _\n apply HasCompactSupport.of_support_subset_isCompact K_compact\n simpa [g_supp] using vK n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nL : Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\n\u22a2 \u2200 (n : \u2115), \u222b (x : M), g n x \u2022 f x \u2202\u03bc = 0\n[PROOFSTEP]\nrefine' fun n \u21a6 h _ (g_diff n) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nL : Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\nn : \u2115\n\u22a2 HasCompactSupport fun x => g n x\n[PROOFSTEP]\napply HasCompactSupport.of_support_subset_isCompact K_compact\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u00b3 : LocallyCompactSpace H\nthis\u271d\u00b2 : LocallyCompactSpace M\nthis\u271d\u00b9 : SecondCountableTopology H\nthis\u271d : SecondCountableTopology M\nthis : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nL : Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\nn : \u2115\n\u22a2 (support fun x => g n x) \u2286 K\n[PROOFSTEP]\nsimpa [g_supp] using vK n\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = 0\nthis\u271d\u2074 : LocallyCompactSpace H\nthis\u271d\u00b3 : LocallyCompactSpace M\nthis\u271d\u00b2 : SecondCountableTopology H\nthis\u271d\u00b9 : SecondCountableTopology M\nthis\u271d : MetrizableSpace M\nx\u271d : MetricSpace M := metrizableSpaceMetric M\ns : Set M\nhs : IsCompact s\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : IsCompact (cthickening \u03b4 s)\nu : \u2115 \u2192 \u211d\nu_pos : \u2200 (n : \u2115), u n \u2208 Ioo 0 \u03b4\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nv : \u2115 \u2192 Set M := fun n => thickening (u n) s\nK : Set M\nK_compact : IsCompact K\nvK : \u2200 (n : \u2115), v n \u2286 K\ng : \u2115 \u2192 M \u2192 \u211d\ng_supp : \u2200 (n : \u2115), support (g n) = v n\ng_diff : \u2200 (n : \u2115), Smooth I \ud835\udcd8(\u211d, \u211d) (g n)\ng_range : \u2200 (n : \u2115), range (g n) \u2286 Icc 0 1\nhg : \u2200 (n : \u2115) (x : M), x \u2208 s \u2192 g n x = 1\nL : Tendsto (fun n => \u222b (x : M), g n x \u2022 f x \u2202\u03bc) atTop (\ud835\udcdd (\u222b (x : M) in s, f x \u2202\u03bc))\nthis : \u2200 (n : \u2115), \u222b (x : M), g n x \u2022 f x \u2202\u03bc = 0\n\u22a2 0 = \u222b (x : M) in s, f x \u2202\u03bc\n[PROOFSTEP]\nsimpa [this] using L\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = f' x\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, (f - f') x = 0 :=\n by\n apply ae_eq_zero_of_integral_smooth_smul_eq_zero I (hf.sub hf')\n intro g g_diff g_supp\n simp only [Pi.sub_apply, smul_sub]\n rw [integral_sub, sub_eq_zero]\n \u00b7 exact h g g_diff g_supp\n \u00b7 exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n \u00b7 exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, (f - f') x = 0\n[PROOFSTEP]\napply ae_eq_zero_of_integral_smooth_smul_eq_zero I (hf.sub hf')\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\n\u22a2 \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 (f - f') x \u2202\u03bc = 0\n[PROOFSTEP]\nintro g g_diff g_supp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\ng : M \u2192 \u211d\ng_diff : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_supp : HasCompactSupport g\n\u22a2 \u222b (x : M), g x \u2022 (f - f') x \u2202\u03bc = 0\n[PROOFSTEP]\nsimp only [Pi.sub_apply, smul_sub]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\ng : M \u2192 \u211d\ng_diff : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_supp : HasCompactSupport g\n\u22a2 \u222b (x : M), g x \u2022 f x - g x \u2022 f' x \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub, sub_eq_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\ng : M \u2192 \u211d\ng_diff : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_supp : HasCompactSupport g\n\u22a2 \u222b (a : M), g a \u2022 f a \u2202\u03bc = \u222b (a : M), g a \u2022 f' a \u2202\u03bc\n[PROOFSTEP]\nexact h g g_diff g_supp\n[GOAL]\ncase hf\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\ng : M \u2192 \u211d\ng_diff : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_supp : HasCompactSupport g\n\u22a2 Integrable fun x => g x \u2022 f x\n[PROOFSTEP]\nexact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n[GOAL]\ncase hg\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\ng : M \u2192 \u211d\ng_diff : Smooth I \ud835\udcd8(\u211d, \u211d) g\ng_supp : HasCompactSupport g\n\u22a2 Integrable fun x => g x \u2022 f' x\n[PROOFSTEP]\nexact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\nthis : \u2200\u1d50 (x : M) \u2202\u03bc, (f - f') x = 0\n\u22a2 \u2200\u1d50 (x : M) \u2202\u03bc, f x = f' x\n[PROOFSTEP]\nfilter_upwards [this] with x hx\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : SmoothManifoldWithCorners I M\ninst\u271d\u00b3 : MeasurableSpace M\ninst\u271d\u00b2 : BorelSpace M\ninst\u271d\u00b9 : SigmaCompactSpace M\ninst\u271d : T2Space M\nf f' : M \u2192 F\n\u03bc : Measure M\nhf : LocallyIntegrable f\nhf' : LocallyIntegrable f'\nh : \u2200 (g : M \u2192 \u211d), Smooth I \ud835\udcd8(\u211d, \u211d) g \u2192 HasCompactSupport g \u2192 \u222b (x : M), g x \u2022 f x \u2202\u03bc = \u222b (x : M), g x \u2022 f' x \u2202\u03bc\nthis : \u2200\u1d50 (x : M) \u2202\u03bc, (f - f') x = 0\nx : M\nhx : (f - f') x = 0\n\u22a2 f x = f' x\n[PROOFSTEP]\nsimpa [sub_eq_zero] using hx\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff", "llama_tokens": 48597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.26906946907552276}} {"text": "[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\ns' s : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nh : \u00acs \u2264 s'\n\u22a2 \u00ac\u2203 d, s' = s + d\n[PROOFSTEP]\nrintro \u27e8d, rfl\u27e9\n[GOAL]\ncase intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\ns : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\nh : \u00acs \u2264 s + d\n\u22a2 False\n[PROOFSTEP]\nexact h le_self_add\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(monomial i) r \u2223 \u2191(monomial j) s \u2194 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(monomial i) r \u2223 \u2191(monomial j) s \u2192 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2191(monomial j) s = \u2191(monomial i) r * x\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nrw [MvPolynomial.ext_iff] at hx \n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nhave hj := hx j\n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhj : coeff j (\u2191(monomial j) s) = coeff j (\u2191(monomial i) r * x)\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nhave hi := hx i\n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhj : coeff j (\u2191(monomial j) s) = coeff j (\u2191(monomial i) r * x)\nhi : coeff i (\u2191(monomial j) s) = coeff i (\u2191(monomial i) r * x)\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nclassical\nsimp_rw [coeff_monomial, if_pos] at hj hi \nsimp_rw [coeff_monomial_mul'] at hi hj \nsplit_ifs at hi hj with hi hi\n\u00b7 exact \u27e8Or.inr hi, _, hj\u27e9\n\u00b7\n exact\n \u27e8Or.inl hj, hj.symm \u25b8 dvd_zero _\u27e9\n -- Porting note: two goals remain at this point in Lean 4\n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhj : coeff j (\u2191(monomial j) s) = coeff j (\u2191(monomial i) r * x)\nhi : coeff i (\u2191(monomial j) s) = coeff i (\u2191(monomial i) r * x)\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nsimp_rw [coeff_monomial, if_pos] at hj hi \n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi : (if j = i then s else 0) = coeff i (\u2191(monomial i) r * x)\nhj : s = coeff j (\u2191(monomial i) r * x)\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nsimp_rw [coeff_monomial_mul'] at hi hj \n[GOAL]\ncase mp.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi : (if j = i then s else 0) = if i \u2264 i then r * coeff (i - i) x else 0\nhj : s = if i \u2264 j then r * coeff (j - i) x else 0\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nsplit_ifs at hi hj with hi hi\n[GOAL]\ncase pos\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi\u271d\u00b9 : (if j = i then s else 0) = if i \u2264 i then r * coeff (i - i) x else 0\nhi\u271d : j = i\nhi : i \u2264 j\nhj : s = r * coeff (j - i) x\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nexact \u27e8Or.inr hi, _, hj\u27e9\n[GOAL]\ncase neg\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi\u271d\u00b9 : (if j = i then s else 0) = if i \u2264 i then r * coeff (i - i) x else 0\nhi\u271d : j = i\nhi : \u00aci \u2264 j\nhj : s = 0\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nexact\n \u27e8Or.inl hj, hj.symm \u25b8 dvd_zero _\u27e9\n -- Porting note: two goals remain at this point in Lean 4\n[GOAL]\ncase pos\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi\u271d : (if j = i then s else 0) = if i \u2264 i then r * coeff (i - i) x else 0\nhi : \u00acj = i\nh\u271d : i \u2264 j\nhj : s = r * coeff (j - i) x\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nsimp_all only [or_true, dvd_mul_right]\n[GOAL]\ncase neg\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\nx : MvPolynomial \u03c3 R\nhx : \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m (\u2191(monomial j) s) = coeff m (\u2191(monomial i) r * x)\nhi\u271d : (if j = i then s else 0) = if i \u2264 i then r * coeff (i - i) x else 0\nhi : \u00acj = i\nh\u271d : \u00aci \u2264 j\nhj : s = 0\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s\n[PROOFSTEP]\nsimp_all only [ite_self, le_refl, ite_true, dvd_mul_right]\n[GOAL]\ncase mpr\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr s : R\ni j : \u03c3 \u2192\u2080 \u2115\n\u22a2 (s = 0 \u2228 i \u2264 j) \u2227 r \u2223 s \u2192 \u2191(monomial i) r \u2223 \u2191(monomial j) s\n[PROOFSTEP]\nrintro \u27e8h | hij, d, rfl\u27e9\n[GOAL]\ncase mpr.intro.inl.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr : R\ni j : \u03c3 \u2192\u2080 \u2115\nd : R\nh : r * d = 0\n\u22a2 \u2191(monomial i) r \u2223 \u2191(monomial j) (r * d)\n[PROOFSTEP]\nsimp_rw [h, monomial_zero, dvd_zero]\n[GOAL]\ncase mpr.intro.inr.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr : R\ni j : \u03c3 \u2192\u2080 \u2115\nhij : i \u2264 j\nd : R\n\u22a2 \u2191(monomial i) r \u2223 \u2191(monomial j) (r * d)\n[PROOFSTEP]\nrefine' \u27e8monomial (j - i) d, _\u27e9\n[GOAL]\ncase mpr.intro.inr.intro\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\nr : R\ni j : \u03c3 \u2192\u2080 \u2115\nhij : i \u2264 j\nd : R\n\u22a2 \u2191(monomial j) (r * d) = \u2191(monomial i) r * \u2191(monomial (j - i)) d\n[PROOFSTEP]\nrw [monomial_mul, add_tsub_cancel_of_le hij]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\ni j : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(monomial i) 1 \u2223 \u2191(monomial j) 1 \u2194 i \u2264 j\n[PROOFSTEP]\nrw [monomial_dvd_monomial]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\ni j : \u03c3 \u2192\u2080 \u2115\n\u22a2 (1 = 0 \u2228 i \u2264 j) \u2227 1 \u2223 1 \u2194 i \u2264 j\n[PROOFSTEP]\nsimp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\ni j : \u03c3\n\u22a2 X i \u2223 X j \u2194 i = j\n[PROOFSTEP]\nrefine' monomial_one_dvd_monomial_one.trans _\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\ni j : \u03c3\n\u22a2 Finsupp.single i 1 \u2264 Finsupp.single j 1 \u2194 i = j\n[PROOFSTEP]\nsimp_rw [Finsupp.single_le_iff, Nat.one_le_iff_ne_zero, Finsupp.single_apply_ne_zero, and_true_iff]\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\ni : \u03c3\nj : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 X i \u2223 \u2191(monomial j) r \u2194 r = 0 \u2228 \u2191j i \u2260 0\n[PROOFSTEP]\nrefine' monomial_dvd_monomial.trans _\n[GOAL]\n\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : CommSemiring R\ni : \u03c3\nj : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (r = 0 \u2228 Finsupp.single i 1 \u2264 j) \u2227 1 \u2223 r \u2194 r = 0 \u2228 \u2191j i \u2260 0\n[PROOFSTEP]\nsimp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]\n", "meta": {"mathlib_filename": "Mathlib.Data.MvPolynomial.Division", "llama_tokens": 3715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.2686331518300428}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082\u271d a\u2082 : \u03b1\nh : HasSum f a\u2081\n\u22a2 HasSum (fun i => a\u2082 * f i) (a\u2082 * a\u2081)\n[PROOFSTEP]\nsimpa only using h.map (AddMonoidHom.mulLeft a\u2082) (continuous_const.mul continuous_id)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082\u271d a\u2082 : \u03b1\nhf : HasSum f a\u2081\n\u22a2 HasSum (fun i => f i * a\u2082) (a\u2081 * a\u2082)\n[PROOFSTEP]\nsimpa only using hf.map (AddMonoidHom.mulRight a\u2082) (continuous_id.mul continuous_const)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : HasSum f a\nb : \u03b1\n\u22a2 HasSum (fun i => f i / b) (a / b)\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, h.mul_right b\u207b\u00b9]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a\u2082 \u2260 0\nH : HasSum (fun i => a\u2082 * f i) (a\u2082 * a\u2081)\n\u22a2 HasSum f a\u2081\n[PROOFSTEP]\nsimpa only [inv_mul_cancel_left\u2080 h] using H.mul_left a\u2082\u207b\u00b9\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a\u2082 \u2260 0\nH : HasSum (fun i => f i * a\u2082) (a\u2081 * a\u2082)\n\u22a2 HasSum f a\u2081\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right\u2080 h] using H.mul_right a\u2082\u207b\u00b9\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a\u2082 \u2260 0\n\u22a2 HasSum (fun i => f i / a\u2082) (a\u2081 / a\u2082) \u2194 HasSum f a\u2081\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hasSum_mul_right_iff (inv_ne_zero h)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a \u2260 0\nH : Summable fun i => a * f i\n\u22a2 Summable f\n[PROOFSTEP]\nsimpa only [inv_mul_cancel_left\u2080 h] using H.mul_left a\u207b\u00b9\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a \u2260 0\nH : Summable fun i => f i * a\n\u22a2 Summable f\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right\u2080 h] using H.mul_right a\u207b\u00b9\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a \u2260 0\n\u22a2 (Summable fun i => f i / a) \u2194 Summable f\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using summable_mul_right_iff (inv_ne_zero h)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : DivisionSemiring \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\nha : a = 0\n\u22a2 \u2211' (x : \u03b9), a * f x = a * \u2211' (x : \u03b9), f x\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : DivisionSemiring \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\nha : \u00aca = 0\n\u22a2 \u2211' (x : \u03b9), a * f x = a * \u2211' (x : \u03b9), f x\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt (summable_mul_left_iff ha).mp hf), mul_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : DivisionSemiring \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\nha : a = 0\n\u22a2 \u2211' (x : \u03b9), f x * a = (\u2211' (x : \u03b9), f x) * a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : DivisionSemiring \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\nhf : \u00acSummable f\nha : \u00aca = 0\n\u22a2 \u2211' (x : \u03b9), f x * a = (\u2211' (x : \u03b9), f x) * a\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt (summable_mul_right_iff ha).mp hf), zero_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : DivisionSemiring \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d : T2Space \u03b1\n\u22a2 \u2211' (x : \u03b9), f x / a = (\u2211' (x : \u03b9), f x) / a\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using tsum_mul_right\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f x.fst * g x.snd\n\u22a2 Summable fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\nrw [summable_mul_prod_iff_summable_mul_sigma_antidiagonal] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n\u22a2 Summable fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\nconv => congr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n| Summable fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n| Summable fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n| Summable fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n| fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\nx\u271d : \u2115\n| \u2211 kl in Nat.antidiagonal x\u271d, f kl.fst * g kl.snd\n[PROOFSTEP]\nrw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f (\u2191x.snd).fst * g (\u2191x.snd).snd\n\u22a2 Summable fun x => \u2211' (b : \u2191\u2191(Nat.antidiagonal x)), f (\u2191b).fst * g (\u2191b).snd\n[PROOFSTEP]\nexact h.sigma' fun n => (hasSum_fintype _).summable\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\nconv_rhs => congr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n| \u2211' (n : \u2115), \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n| \u2211' (n : \u2115), \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr; ext; rw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n| \u2211' (n : \u2115), \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n| fun n => \u2211 kl in Nat.antidiagonal n, f kl.fst * g kl.snd\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\nx\u271d : \u2115\n| \u2211 kl in Nat.antidiagonal x\u271d, f kl.fst * g kl.snd\n[PROOFSTEP]\nrw [\u2190 Finset.sum_finset_coe, \u2190 tsum_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (x : \u2115) (b : \u2191\u2191(Nat.antidiagonal x)), f (\u2191b).fst * g (\u2191b).snd\n[PROOFSTEP]\nrw [tsum_mul_tsum hf hg hfg, \u2190 Nat.sigmaAntidiagonalEquivProd.tsum_eq (_ : \u2115 \u00d7 \u2115 \u2192 \u03b1)]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n\u22a2 \u2211' (c : (n : \u2115) \u00d7 { x // x \u2208 Nat.antidiagonal n }),\n f (\u2191Nat.sigmaAntidiagonalEquivProd c).fst * g (\u2191Nat.sigmaAntidiagonalEquivProd c).snd =\n \u2211' (x : \u2115) (b : \u2191\u2191(Nat.antidiagonal x)), f (\u2191b).fst * g (\u2191b).snd\n[PROOFSTEP]\nexact tsum_sigma' (fun n => (hasSum_fintype _).summable) (summable_mul_prod_iff_summable_mul_sigma_antidiagonal.mp hfg)\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f x.fst * g x.snd\n\u22a2 Summable fun n => \u2211 k in range (n + 1), f k * g (n - k)\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nh : Summable fun x => f x.fst * g x.snd\n\u22a2 Summable fun n => \u2211 ij in Nat.antidiagonal n, f ij.fst * g ij.snd\n[PROOFSTEP]\nexact summable_sum_mul_antidiagonal_of_summable_mul h\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 k in range (n + 1), f k * g (n - k)\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]\n[GOAL]\n\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.fst * g x.snd\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 ij in Nat.antidiagonal n, f ij.fst * g ij.snd\n[PROOFSTEP]\nexact tsum_mul_tsum_eq_tsum_sum_antidiagonal hf hg hfg\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.InfiniteSum.Ring", "llama_tokens": 6161, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.26853923292627646}} {"text": "[GOAL]\n\u03b1 : Type ?u.58\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03c3 : Type u\nx\u271d : Nonempty (Cardinal.{u} \u21aa \u03c3)\nf : Cardinal.{u} \u2192 \u03c3\nhf : Injective f\ng : \u03c3 \u2192 Cardinal.{u} := Function.invFun f\nx : \u03c3\nhx : g x = 2 ^ sum g\nthis : g x \u2264 sum g\n\u22a2 g x > sum g\n[PROOFSTEP]\nrw [hx]\n[GOAL]\n\u03b1 : Type ?u.58\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03c3 : Type u\nx\u271d : Nonempty (Cardinal.{u} \u21aa \u03c3)\nf : Cardinal.{u} \u2192 \u03c3\nhf : Injective f\ng : \u03c3 \u2192 Cardinal.{u} := Function.invFun f\nx : \u03c3\nhx : g x = 2 ^ sum g\nthis : g x \u2264 sum g\n\u22a2 2 ^ sum g > sum g\n[PROOFSTEP]\nexact cantor _\n[GOAL]\n\u03b1 : Type ?u.20118\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : WellOrder\n\u22a2 { \u03b1 := o.\u03b1, r := o.r, wo := (_ : IsWellOrder o.\u03b1 o.r) } = o\n[PROOFSTEP]\ncases o\n[GOAL]\ncase mk\n\u03b1 : Type ?u.20118\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1\u271d : Type u_3\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 { \u03b1 := { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.\u03b1, r := { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r,\n wo := (_ : IsWellOrder { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.\u03b1 { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r) } =\n { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.24026\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nw : WellOrder\n\u22a2 Quotient.mk isEquivalent w = type w.r\n[PROOFSTEP]\ncases w\n[GOAL]\ncase mk\n\u03b1 : Type ?u.24026\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1\u271d : Type u_3\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d } = type { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo : IsWellOrder \u03b1 r\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1, r := r, wo := wo } = type r\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.24255\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 type (Quotient.out o).r = o\n[PROOFSTEP]\nrw [Ordinal.type, WellOrder.eta, Quotient.out_eq]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 type r \u2260 0 \u2194 Nonempty \u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.28990\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 IsEmpty (Quotient.out o).\u03b1 \u2194 o = 0\n[PROOFSTEP]\nrw [\u2190 @type_eq_zero_iff_isEmpty o.out.\u03b1 (\u00b7 < \u00b7), type_lt]\n[GOAL]\n\u03b1 : Type ?u.29806\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 Nonempty (Quotient.out o).\u03b1 \u2194 o \u2260 0\n[PROOFSTEP]\nrw [\u2190 @type_ne_zero_iff_nonempty o.out.\u03b1 (\u00b7 < \u00b7), type_lt]\n[GOAL]\n\u03b1\u271d : Type ?u.31118\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nf : \u03b2 \u2243 \u03b1\n\u22a2 (type fun x y => r (\u2191f x) (\u2191f y)) = type r\n[PROOFSTEP]\nconvert (RelIso.preimage f r).ordinal_type_eq\n[GOAL]\n\u03b1\u271d : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69219}\nh : \u03b1 \u2264 \u03b2\n\u22a2 (fun x x_1 => x < x_1) \u227ci fun x x_1 => x < x_1\n[PROOFSTEP]\nchange \u03b1.out.r \u227ci \u03b2.out.r\n[GOAL]\n\u03b1\u271d : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69728}\nh : \u03b1 \u2264 \u03b2\n\u22a2 (Quotient.out \u03b1).r \u227ci (Quotient.out \u03b2).r\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq \u03b1, \u2190 Quotient.out_eq \u03b2] at h \n[GOAL]\n\u03b1\u271d : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69728}\nh : Quotient.mk isEquivalent (Quotient.out \u03b1) \u2264 Quotient.mk isEquivalent (Quotient.out \u03b2)\n\u22a2 (Quotient.out \u03b1).r \u227ci (Quotient.out \u03b2).r\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1\u271d : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69728}\n\u22a2 Quotient.mk isEquivalent (Quotient.out \u03b1) \u2264 Quotient.mk isEquivalent (Quotient.out \u03b2) \u2192\n (Quotient.out \u03b1).r \u227ci (Quotient.out \u03b2).r\n[PROOFSTEP]\ncases Quotient.out \u03b1\n[GOAL]\ncase mk\n\u03b1\u271d\u00b9 : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d\u00b9 \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69884}\n\u03b1\u271d : Type ?u.69884\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d } \u2264 Quotient.mk isEquivalent (Quotient.out \u03b2) \u2192\n { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r \u227ci (Quotient.out \u03b2).r\n[PROOFSTEP]\ncases Quotient.out \u03b2\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b2 : Type ?u.68609\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d\u00b2 \u2192 \u03b1\u271d\u00b2 \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.69942}\n\u03b1\u271d\u00b9 : Type ?u.69942\nr\u271d\u00b9 : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d\u00b9 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b1\u271d\u00b9 r\u271d\u00b9\n\u03b1\u271d : Type ?u.69942\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d\u00b9, r := r\u271d\u00b9, wo := wo\u271d\u00b9 } \u2264\n Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d } \u2192\n { \u03b1 := \u03b1\u271d\u00b9, r := r\u271d\u00b9, wo := wo\u271d\u00b9 }.r \u227ci { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r\n[PROOFSTEP]\nexact Classical.choice\n[GOAL]\n\u03b1\u271d : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.70918}\nh : \u03b1 < \u03b2\n\u22a2 (fun x x_1 => x < x_1) \u227ai fun x x_1 => x < x_1\n[PROOFSTEP]\nchange \u03b1.out.r \u227ai \u03b2.out.r\n[GOAL]\n\u03b1\u271d : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.71427}\nh : \u03b1 < \u03b2\n\u22a2 (Quotient.out \u03b1).r \u227ai (Quotient.out \u03b2).r\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq \u03b1, \u2190 Quotient.out_eq \u03b2] at h \n[GOAL]\n\u03b1\u271d : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.71427}\nh : Quotient.mk isEquivalent (Quotient.out \u03b1) < Quotient.mk isEquivalent (Quotient.out \u03b2)\n\u22a2 (Quotient.out \u03b1).r \u227ai (Quotient.out \u03b2).r\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1\u271d : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.71427}\n\u22a2 Quotient.mk isEquivalent (Quotient.out \u03b1) < Quotient.mk isEquivalent (Quotient.out \u03b2) \u2192\n (Quotient.out \u03b1).r \u227ai (Quotient.out \u03b2).r\n[PROOFSTEP]\ncases Quotient.out \u03b1\n[GOAL]\ncase mk\n\u03b1\u271d\u00b9 : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d\u00b9 \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.71563}\n\u03b1\u271d : Type ?u.71563\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d } < Quotient.mk isEquivalent (Quotient.out \u03b2) \u2192\n { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r \u227ai (Quotient.out \u03b2).r\n[PROOFSTEP]\ncases Quotient.out \u03b2\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b2 : Type ?u.70308\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d\u00b2 \u2192 \u03b1\u271d\u00b2 \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Ordinal.{?u.71621}\n\u03b1\u271d\u00b9 : Type ?u.71621\nr\u271d\u00b9 : \u03b1\u271d\u00b9 \u2192 \u03b1\u271d\u00b9 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b1\u271d\u00b9 r\u271d\u00b9\n\u03b1\u271d : Type ?u.71621\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\nwo\u271d : IsWellOrder \u03b1\u271d r\u271d\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d\u00b9, r := r\u271d\u00b9, wo := wo\u271d\u00b9 } <\n Quotient.mk isEquivalent { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d } \u2192\n { \u03b1 := \u03b1\u271d\u00b9, r := r\u271d\u00b9, wo := wo\u271d\u00b9 }.r \u227ai { \u03b1 := \u03b1\u271d, r := r\u271d, wo := wo\u271d }.r\n[PROOFSTEP]\nexact Classical.choice\n[GOAL]\n\u03b1 : Type ?u.72085\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\ni : (Quotient.out o).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) i < o\n[PROOFSTEP]\nsimp_rw [\u2190 type_lt o]\n[GOAL]\n\u03b1 : Type ?u.72085\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\ni : (Quotient.out o).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) i < type fun x x_1 => x < x_1\n[PROOFSTEP]\napply typein_lt_type\n[GOAL]\n\u03b1\u271d : Type ?u.73586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ai s\nx\u271d : \u2191{b | s b f.top}\na : \u03b2\nh : a \u2208 {b | s b f.top}\n\u22a2 \u2203 a_1,\n \u2191(RelEmbedding.codRestrict {b | s b f.top}\n { toRelEmbedding := f.toRelEmbedding,\n init' :=\n (_ :\n \u2200 (x : \u03b1) (x_1 : \u03b2),\n s x_1 (\u2191f.toRelEmbedding x) \u2192 \u2203 a', \u2191f.toRelEmbedding a' = x_1) }.toRelEmbedding\n (_ : \u2200 (a : \u03b1), s (\u2191f.toRelEmbedding a) f.top))\n a_1 =\n { val := a, property := h }\n[PROOFSTEP]\nrcases f.down.1 h with \u27e8b, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.73586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ai s\nx\u271d : \u2191{b | s b f.top}\nb : \u03b1\nh : \u2191f.toRelEmbedding b \u2208 {b | s b f.top}\n\u22a2 \u2203 a,\n \u2191(RelEmbedding.codRestrict {b | s b f.top}\n { toRelEmbedding := f.toRelEmbedding,\n init' :=\n (_ :\n \u2200 (x : \u03b1) (x_1 : \u03b2),\n s x_1 (\u2191f.toRelEmbedding x) \u2192 \u2203 a', \u2191f.toRelEmbedding a' = x_1) }.toRelEmbedding\n (_ : \u2200 (a : \u03b1), s (\u2191f.toRelEmbedding a) f.top))\n a =\n { val := \u2191f.toRelEmbedding b, property := h }\n[PROOFSTEP]\nexact \u27e8b, rfl\u27e9\n[GOAL]\n\u03b1\u271d : Type ?u.74324\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ci s\na : \u03b1\nx\u271d : \u2191{b | r b a}\nx : \u03b1\nh : x \u2208 {b | r b a}\n\u22a2 \u2191(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding) { val := x, property := h } \u2208\n {b | s b (\u2191f a)}\n[PROOFSTEP]\nrw [RelEmbedding.trans_apply]\n[GOAL]\n\u03b1\u271d : Type ?u.74324\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ci s\na : \u03b1\nx\u271d : \u2191{b | r b a}\nx : \u03b1\nh : x \u2208 {b | r b a}\n\u22a2 \u2191f.toRelEmbedding (\u2191(Subrel.relEmbedding r {b | r b a}) { val := x, property := h }) \u2208 {b | s b (\u2191f a)}\n[PROOFSTEP]\nexact f.toRelEmbedding.map_rel_iff.2 h\n[GOAL]\n\u03b1\u271d : Type ?u.74324\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ci s\na : \u03b1\nx\u271d : \u2191{b | s b (\u2191f a)}\ny : \u03b2\nh : y \u2208 {b | s b (\u2191f a)}\n\u22a2 \u2203 a_1,\n \u2191(RelEmbedding.codRestrict {b | s b (\u2191f a)}\n (RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding)\n (_ :\n \u2200 (x : \u2191{b | r b a}),\n \u2191(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding) x \u2208 {b | s b (\u2191f a)}))\n a_1 =\n { val := y, property := h }\n[PROOFSTEP]\nrcases f.init h with \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.74324\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u227ci s\na\u271d : \u03b1\nx\u271d : \u2191{b | s b (\u2191f a\u271d)}\na : \u03b1\nh : \u2191f a \u2208 {b | s b (\u2191f a\u271d)}\n\u22a2 \u2203 a_1,\n \u2191(RelEmbedding.codRestrict {b | s b (\u2191f a\u271d)}\n (RelEmbedding.trans (Subrel.relEmbedding r {b | r b a\u271d}) f.toRelEmbedding)\n (_ :\n \u2200 (x : \u2191{b | r b a\u271d}),\n \u2191(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a\u271d}) f.toRelEmbedding) x \u2208 {b | s b (\u2191f a\u271d)}))\n a_1 =\n { val := \u2191f a, property := h }\n[PROOFSTEP]\nexact \u27e8\u27e8a, f.toRelEmbedding.map_rel_iff.1 h\u27e9, Subtype.eq <| RelEmbedding.trans_apply _ _ _\u27e9\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\n\u22a2 r a b\n[PROOFSTEP]\nhave : f.top.1 = a := by\n let f' := PrincipalSeg.ofElement r a\n let g' := f.trans (PrincipalSeg.ofElement r b)\n have : g'.top = f'.top := by rw [Subsingleton.elim f' g']\n exact this\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\n\u22a2 \u2191f.top = a\n[PROOFSTEP]\nlet f' := PrincipalSeg.ofElement r a\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nf' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.ofElement r a\n\u22a2 \u2191f.top = a\n[PROOFSTEP]\nlet g' := f.trans (PrincipalSeg.ofElement r b)\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nf' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.ofElement r a\ng' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.trans f (PrincipalSeg.ofElement r b)\n\u22a2 \u2191f.top = a\n[PROOFSTEP]\nhave : g'.top = f'.top := by rw [Subsingleton.elim f' g']\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nf' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.ofElement r a\ng' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.trans f (PrincipalSeg.ofElement r b)\n\u22a2 g'.top = f'.top\n[PROOFSTEP]\nrw [Subsingleton.elim f' g']\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nf' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.ofElement r a\ng' : Subrel r {b | r b a} \u227ai r := PrincipalSeg.trans f (PrincipalSeg.ofElement r b)\nthis : g'.top = f'.top\n\u22a2 \u2191f.top = a\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nthis : \u2191f.top = a\n\u22a2 r a b\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\na b : \u03b1\nx\u271d : typein r a < typein r b\nf : Subrel r {b | r b a} \u227ai Subrel r {b_1 | r b_1 b}\nthis : \u2191f.top = a\n\u22a2 r (\u2191f.top) b\n[PROOFSTEP]\nexact f.top.2\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\no\u2081 o\u2082 : Ordinal.{u_3}\nh\u2081 : o\u2081 < type r\nh\u2082 : o\u2082 < type r\n\u22a2 r (enum r o\u2081 h\u2081) (enum r o\u2082 h\u2082) \u2194 o\u2081 < o\u2082\n[PROOFSTEP]\nrw [\u2190 typein_lt_typein r, typein_enum, typein_enum]\n[GOAL]\n\u03b1\u271d : Type ?u.98270\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\n\u22a2 \u2200 (hr : o < type r) (hs : o < type s), \u2191f (enum r o hr) = enum s o hs\n[PROOFSTEP]\nrefine' inductionOn o _\n[GOAL]\n\u03b1\u271d : Type ?u.98270\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\n\u22a2 \u2200 (\u03b1_1 : Type u) (r_1 : \u03b1_1 \u2192 \u03b1_1 \u2192 Prop) [inst : IsWellOrder \u03b1_1 r_1] (hr : type r_1 < type r)\n (hs : type r_1 < type s), \u2191f (enum r (type r_1) hr) = enum s (type r_1) hs\n[PROOFSTEP]\nrintro \u03b3 t wo \u27e8g\u27e9 \u27e8h\u27e9\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type ?u.98270\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo : IsWellOrder \u03b3 t\ng : t \u227ai r\nh : t \u227ai s\n\u22a2 \u2191f (enum r (type t) (_ : Nonempty (t \u227ai r))) = enum s (type t) (_ : Nonempty (t \u227ai s))\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type ?u.98270\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo : IsWellOrder \u03b3 t\ng : t \u227ai r\nh : t \u227ai s\n\u22a2 \u2191f (enum r (type t) (_ : Nonempty (t \u227ai r))) = enum s (type t) (_ : Nonempty (t \u227ai s))\n[PROOFSTEP]\nrw [enum_type g, enum_type (PrincipalSeg.ltEquiv g f)]\n[GOAL]\ncase intro.intro\n\u03b1\u271d : Type ?u.98270\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo : IsWellOrder \u03b3 t\ng : t \u227ai r\nh : t \u227ai s\n\u22a2 \u2191f g.top = (PrincipalSeg.ltEquiv g f).top\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type ?u.98738\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\nhr : o < type r\n\u22a2 o < type s\n[PROOFSTEP]\nconvert hr using 1\n[GOAL]\ncase h.e'_4\n\u03b1\u271d : Type ?u.98738\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\nhr : o < type r\n\u22a2 type s = type r\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase h.e'_4.a\n\u03b1\u271d : Type ?u.98738\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 \u03b2 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nf : r \u2243r s\no : Ordinal.{u}\nhr : o < type r\n\u22a2 { \u03b1 := \u03b2, r := s, wo := inst\u271d } \u2248 { \u03b1 := \u03b1, r := r, wo := inst\u271d\u00b9 }\n[PROOFSTEP]\nexact \u27e8f.symm\u27e9\n[GOAL]\n\u03b1\u271d : Type ?u.99459\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo : IsWellOrder \u03b1 r\na x : \u03b1\nx\u271d : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y x \u2192 Acc (fun x x_1 => x < x_1) (typein r y)\no : Ordinal.{u_3}\nh : o < typein r x\n\u22a2 Acc (fun x x_1 => x < x_1) o\n[PROOFSTEP]\nrcases typein_surj r (lt_trans h (typein_lt_type r _)) with \u27e8b, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.99459\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo : IsWellOrder \u03b1 r\na x : \u03b1\nx\u271d : \u2200 (y : \u03b1), r y x \u2192 Acc r y\nIH : \u2200 (y : \u03b1), r y x \u2192 Acc (fun x x_1 => x < x_1) (typein r y)\nb : \u03b1\nh : typein r b < typein r x\n\u22a2 Acc (fun x x_1 => x < x_1) (typein r b)\n[PROOFSTEP]\nexact IH _ ((typein_lt_typein r).1 h)\n[GOAL]\n\u03b1\u271d : Type ?u.101731\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx\u271d : IsWellOrder \u03b1 r\nh : card (type r) = 0\n\u22a2 type r = 0\n[PROOFSTEP]\nhaveI := Cardinal.mk_eq_zero_iff.1 h\n[GOAL]\n\u03b1\u271d : Type ?u.101731\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx\u271d : IsWellOrder \u03b1 r\nh : card (type r) = 0\nthis : IsEmpty { \u03b1 := \u03b1, r := r, wo := x\u271d }.\u03b1\n\u22a2 type r = 0\n[PROOFSTEP]\napply type_eq_zero_of_empty\n[GOAL]\n\u03b1 : Type ?u.101731\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\ne : o = 0\n\u22a2 card o = 0\n[PROOFSTEP]\nsimp only [e, card_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 type (ULift.down \u207b\u00b9'o r) = lift (type r)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 (type fun x y => r x.down y.down) = lift (type r)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type ?u.108304\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\n\u22a2 lift (type r) < lift (type s) \u2194 Nonempty (r \u227ai s)\n[PROOFSTEP]\nhaveI := @RelEmbedding.isWellOrder _ _ (@Equiv.ulift.{max v w} \u03b1 \u207b\u00b9'o r) r (RelIso.preimage Equiv.ulift.{max v w} r) _\n[GOAL]\n\u03b1\u271d : Type ?u.108304\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nthis : IsWellOrder (ULift \u03b1) (\u2191Equiv.ulift \u207b\u00b9'o r)\n\u22a2 lift (type r) < lift (type s) \u2194 Nonempty (r \u227ai s)\n[PROOFSTEP]\nhaveI := @RelEmbedding.isWellOrder _ _ (@Equiv.ulift.{max u w} \u03b2 \u207b\u00b9'o s) s (RelIso.preimage Equiv.ulift.{max u w} s) _\n[GOAL]\n\u03b1\u271d : Type ?u.108304\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nthis\u271d : IsWellOrder (ULift \u03b1) (\u2191Equiv.ulift \u207b\u00b9'o r)\nthis : IsWellOrder (ULift \u03b2) (\u2191Equiv.ulift \u207b\u00b9'o s)\n\u22a2 lift (type r) < lift (type s) \u2194 Nonempty (r \u227ai s)\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8f\u27e9 =>\n \u27e8(f.equivLT (RelIso.preimage Equiv.ulift r).symm).ltLe (InitialSeg.ofIso (RelIso.preimage Equiv.ulift s))\u27e9,\n fun \u27e8f\u27e9 =>\n \u27e8(f.equivLT (RelIso.preimage Equiv.ulift r)).ltLe (InitialSeg.ofIso (RelIso.preimage Equiv.ulift s).symm)\u27e9\u27e9\n[GOAL]\n\u03b1\u271d : Type ?u.110330\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{v}\n\u03b1 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1 r\n\u03b2 : Type v\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 lift (type r) \u2264 lift (type s) \u2194 type r \u2264 type s\n[PROOFSTEP]\nrw [\u2190 lift_umax]\n[GOAL]\n\u03b1\u271d : Type ?u.110330\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{v}\n\u03b1 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1 r\n\u03b2 : Type v\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 lift (type r) \u2264 lift (type s) \u2194 type r \u2264 type s\n[PROOFSTEP]\nexact lift_type_le.{_, _, u}\n[GOAL]\n\u03b1 : Type ?u.110723\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{v}\n\u22a2 lift a = lift b \u2194 a = b\n[PROOFSTEP]\nsimp only [le_antisymm_iff, lift_le]\n[GOAL]\n\u03b1 : Type ?u.110994\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{v}\n\u22a2 lift a < lift b \u2194 a < b\n[PROOFSTEP]\nsimp only [lt_iff_le_not_le, lift_le]\n[GOAL]\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\ne' : Cardinal.lift #\u03b1 = card (type s)\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nrw [card_type, \u2190 Cardinal.lift_id'.{max u v, u} #\u03b2, \u2190 Cardinal.lift_umax.{u, v}, lift_mk_eq.{u, max u v, max u v}] at e' \n[GOAL]\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\ne' : Nonempty (\u03b1 \u2243 \u03b2)\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\ncases' e' with f\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nf : \u03b1 \u2243 \u03b2\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nhave g := RelIso.preimage f s\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nf : \u03b1 \u2243 \u03b2\ng : \u2191f \u207b\u00b9'o s \u2243r s\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nhaveI := (g : f \u207b\u00b9'o s \u21aar s).isWellOrder\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nf : \u03b1 \u2243 \u03b2\ng : \u2191f \u207b\u00b9'o s \u2243r s\nthis : IsWellOrder \u03b1 (\u2191f \u207b\u00b9'o s)\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nhave := lift_type_eq.{u, max u v, max u v}.2 \u27e8g\u27e9\n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nf : \u03b1 \u2243 \u03b2\ng : \u2191f \u207b\u00b9'o s \u2243r s\nthis\u271d : IsWellOrder \u03b1 (\u2191f \u207b\u00b9'o s)\nthis : lift (type (\u2191f \u207b\u00b9'o s)) = lift (type s)\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nrw [lift_id, lift_umax.{u, v}] at this \n[GOAL]\ncase intro\n\u03b1\u271d : Type ?u.112020\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Cardinal.{u}\nb : Ordinal.{max u v}\nh : card b \u2264 Cardinal.lift a\nc : Cardinal.{u}\ne : Cardinal.lift c = card b\n\u03b1 : Type u\n\u03b2 : Type (max u v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nf : \u03b1 \u2243 \u03b2\ng : \u2191f \u207b\u00b9'o s \u2243r s\nthis\u271d : IsWellOrder \u03b1 (\u2191f \u207b\u00b9'o s)\nthis : lift (type (\u2191f \u207b\u00b9'o s)) = type s\n\u22a2 \u2203 a', lift a' = type s\n[PROOFSTEP]\nexact \u27e8_, this\u27e9\n[GOAL]\n\u03b1 : Type ?u.113147\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal.{u}\nb : Ordinal.{max u v}\nh : b \u2264 lift a\n\u22a2 card b \u2264 Cardinal.lift (card a)\n[PROOFSTEP]\nrw [lift_card]\n[GOAL]\n\u03b1 : Type ?u.113147\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal.{u}\nb : Ordinal.{max u v}\nh : b \u2264 lift a\n\u22a2 card b \u2264 card (lift a)\n[PROOFSTEP]\nexact card_le_card h\n[GOAL]\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\n\u22a2 \u2200 {a b : (\u03b1 \u2295 \u03b2) \u2295 \u03b3},\n Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) a) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) b) \u2194 Sum.Lex (Sum.Lex r s) t a b\n[PROOFSTEP]\nintros a b\n[GOAL]\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na b : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) a) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) b) \u2194 Sum.Lex (Sum.Lex r s) t a b\n[PROOFSTEP]\nrcases a with (\u27e8a | a\u27e9 | a)\n[GOAL]\ncase inl.inl\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\na : \u03b1\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) b) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inl a)) b\n[PROOFSTEP]\nrcases b with (\u27e8b | b\u27e9 | b)\n[GOAL]\ncase inl.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\na : \u03b2\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) b) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inr a)) b\n[PROOFSTEP]\nrcases b with (\u27e8b | b\u27e9 | b)\n[GOAL]\ncase inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\nb : (\u03b1 \u2295 \u03b2) \u2295 \u03b3\na : \u03b3\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr a)) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) b) \u2194 Sum.Lex (Sum.Lex r s) t (Sum.inr a) b\n[PROOFSTEP]\nrcases b with (\u27e8b | b\u27e9 | b)\n[GOAL]\ncase inl.inl.inl.inl\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na b : \u03b1\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inl a)) (Sum.inl (Sum.inl b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inl.inl.inl.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b1\nb : \u03b2\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inl a)) (Sum.inl (Sum.inr b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inl.inl.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b1\nb : \u03b3\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr b)) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inl a)) (Sum.inr b)\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inl.inr.inl.inl\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b2\nb : \u03b1\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inr a)) (Sum.inl (Sum.inl b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inl.inr.inl.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na b : \u03b2\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inr a)) (Sum.inl (Sum.inr b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inl.inr.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b2\nb : \u03b3\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr a))) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr b)) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inl (Sum.inr a)) (Sum.inr b)\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inr.inl.inl\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b3\nb : \u03b1\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr a)) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inl b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inr a) (Sum.inl (Sum.inl b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inr.inl.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na : \u03b3\nb : \u03b2\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr a)) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inl (Sum.inr b))) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inr a) (Sum.inl (Sum.inr b))\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\ncase inr.inr\n\u03b1\u271d : Type ?u.132524\n\u03b2\u271d : Type u_1\n\u03b3\u271d : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3\u271d \u2192 \u03b3\u271d \u2192 Prop\no\u2081 o\u2082 o\u2083 : Ordinal.{u}\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : WellOrder\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo\u271d\u00b2 : IsWellOrder \u03b1 r\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nwo\u271d\u00b9 : IsWellOrder \u03b2 s\n\u03b3 : Type u\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nwo\u271d : IsWellOrder \u03b3 t\na b : \u03b3\n\u22a2 Sum.Lex r (Sum.Lex s t) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr a)) (\u2191(sumAssoc \u03b1 \u03b2 \u03b3) (Sum.inr b)) \u2194\n Sum.Lex (Sum.Lex r s) t (Sum.inr a) (Sum.inr b)\n[PROOFSTEP]\nsimp only [sumAssoc_apply_inl_inl, sumAssoc_apply_inl_inr, sumAssoc_apply_inr, Sum.lex_inl_inl, Sum.lex_inr_inr,\n Sum.Lex.sep, Sum.lex_inr_inl]\n[GOAL]\n\u03b1 : Type ?u.137403\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn : \u2115\n\u22a2 card \u2191n = \u2191n\n[PROOFSTEP]\ninduction n <;> [simp; simp only [card_add, card_one, Nat.cast_succ, *]]\n[GOAL]\n\u03b1 : Type ?u.137403\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn : \u2115\n\u22a2 card \u2191n = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type ?u.137403\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 card \u2191Nat.zero = \u2191Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type ?u.137403\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn\u271d : \u2115\nn_ih\u271d : card \u2191n\u271d = \u2191n\u271d\n\u22a2 card \u2191(Nat.succ n\u271d) = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp only [card_add, card_one, Nat.cast_succ, *]\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc a b : Ordinal.{u}\nh : a \u2264 b\n\u22a2 c + a \u2264 c + b\n[PROOFSTEP]\nrevert h c\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u22a2 \u2200 (c : Ordinal.{u}), a \u2264 b \u2192 c + a \u2264 c + b\n[PROOFSTEP]\nrefine inductionOn a (fun \u03b1\u2081 r\u2081 _ \u21a6 ?_)\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2081 r\u2081\n\u22a2 \u2200 (c : Ordinal.{u}), type r\u2081 \u2264 b \u2192 c + type r\u2081 \u2264 c + b\n[PROOFSTEP]\nrefine inductionOn b (fun \u03b1\u2082 r\u2082 _ \u21a6 ?_)\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2082 r\u2082\n\u22a2 \u2200 (c : Ordinal.{u}), type r\u2081 \u2264 type r\u2082 \u2192 c + type r\u2081 \u2264 c + type r\u2082\n[PROOFSTEP]\nrintro c \u27e8\u27e8\u27e8f, fo\u27e9, fi\u27e9\u27e9\n[GOAL]\ncase intro.mk.mk\n\u03b1 : Type ?u.138586\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u22a2 c + type r\u2081 \u2264 c + type r\u2082\n[PROOFSTEP]\nrefine inductionOn c (fun \u03b2 s _ \u21a6 ?_)\n[GOAL]\ncase intro.mk.mk\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 type s + type r\u2081 \u2264 type s + type r\u2082\n[PROOFSTEP]\nhave := (Embedding.refl \u03b2).sumMap f\n[GOAL]\ncase intro.mk.mk\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\n\u22a2 type s + type r\u2081 \u2264 type s + type r\u2082\n[PROOFSTEP]\nrefine \u27e8\u27e8\u27e8(Embedding.refl.{u + 1} _).sumMap f, ?_\u27e9, ?_\u27e9\u27e9\n[GOAL]\ncase intro.mk.mk.refine_1\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\n\u22a2 \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b\n[PROOFSTEP]\nintros a b\n[GOAL]\ncase intro.mk.mk.refine_1\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na b : \u03b2 \u2295 \u03b1\u2081\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b\n[PROOFSTEP]\nmatch a, b with\n| Sum.inl a, Sum.inl b => exact Sum.lex_inl_inl.trans Sum.lex_inl_inl.symm\n| Sum.inl a, Sum.inr b => apply iff_of_true <;> apply Sum.Lex.sep\n| Sum.inr a, Sum.inl b => apply iff_of_false <;> exact Sum.lex_inr_inl\n| Sum.inr a, Sum.inr b => exact Sum.lex_inr_inr.trans <| fo.trans Sum.lex_inr_inr.symm\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na b : \u03b2\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl b)) \u2194\n Sum.Lex s r\u2081 (Sum.inl a) (Sum.inl b)\n[PROOFSTEP]\nexact Sum.lex_inl_inl.trans Sum.lex_inl_inl.symm\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b2\nb : \u03b1\u2081\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr b)) \u2194\n Sum.Lex s r\u2081 (Sum.inl a) (Sum.inr b)\n[PROOFSTEP]\napply iff_of_true\n[GOAL]\ncase ha\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b2\nb : \u03b1\u2081\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr b))\n[PROOFSTEP]\napply Sum.Lex.sep\n[GOAL]\ncase hb\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b2\nb : \u03b1\u2081\n\u22a2 Sum.Lex s r\u2081 (Sum.inl a) (Sum.inr b)\n[PROOFSTEP]\napply Sum.Lex.sep\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b1\u2081\nb : \u03b2\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl b)) \u2194\n Sum.Lex s r\u2081 (Sum.inr a) (Sum.inl b)\n[PROOFSTEP]\napply iff_of_false\n[GOAL]\ncase ha\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b1\u2081\nb : \u03b2\n\u22a2 \u00acSum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inl b))\n[PROOFSTEP]\nexact Sum.lex_inr_inl\n[GOAL]\ncase hb\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na : \u03b1\u2081\nb : \u03b2\n\u22a2 \u00acSum.Lex s r\u2081 (Sum.inr a) (Sum.inl b)\n[PROOFSTEP]\nexact Sum.lex_inr_inl\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d b\u271d : \u03b2 \u2295 \u03b1\u2081\na b : \u03b1\u2081\n\u22a2 Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr a))\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) (Sum.inr b)) \u2194\n Sum.Lex s r\u2081 (Sum.inr a) (Sum.inr b)\n[PROOFSTEP]\nexact Sum.lex_inr_inr.trans <| fo.trans Sum.lex_inr_inr.symm\n[GOAL]\ncase intro.mk.mk.refine_2\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\n\u22a2 \u2200 (a : \u03b2 \u2295 \u03b1\u2081) (b : \u03b2 \u2295 \u03b1\u2082),\n Sum.Lex s r\u2082 b\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a) \u2192\n \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n b\n[PROOFSTEP]\nintros a b H\n[GOAL]\ncase intro.mk.mk.refine_2\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na : \u03b2 \u2295 \u03b1\u2081\nb : \u03b2 \u2295 \u03b1\u2082\nH :\n Sum.Lex s r\u2082 b\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a)\n\u22a2 \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n b\n[PROOFSTEP]\nmatch a, b, H with\n| _, Sum.inl b, _ => exact \u27e8Sum.inl b, rfl\u27e9\n| Sum.inl a, Sum.inr b, H => exact (Sum.lex_inr_inl H).elim\n| Sum.inr a, Sum.inr b, H =>\n let \u27e8w, h\u27e9 := fi _ _ (Sum.lex_inr_inr.1 H)\n exact \u27e8Sum.inr w, congr_arg Sum.inr h\u27e9\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u2074 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b3 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na : \u03b2 \u2295 \u03b1\u2081\nb\u271d : \u03b2 \u2295 \u03b1\u2082\nH :\n Sum.Lex s r\u2082 b\u271d\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a)\nx\u271d\u00b9 : \u03b2 \u2295 \u03b1\u2081\nb : \u03b2\nx\u271d :\n Sum.Lex s r\u2082 (Sum.inl b)\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n x\u271d\u00b9)\n\u22a2 \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n Sum.inl b\n[PROOFSTEP]\nexact \u27e8Sum.inl b, rfl\u27e9\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d : \u03b2 \u2295 \u03b1\u2081\nb\u271d : \u03b2 \u2295 \u03b1\u2082\nH\u271d :\n Sum.Lex s r\u2082 b\u271d\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a\u271d)\na : \u03b2\nb : \u03b1\u2082\nH :\n Sum.Lex s r\u2082 (Sum.inr b)\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n (Sum.inl a))\n\u22a2 \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n Sum.inr b\n[PROOFSTEP]\nexact (Sum.lex_inr_inl H).elim\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d : \u03b2 \u2295 \u03b1\u2081\nb\u271d : \u03b2 \u2295 \u03b1\u2082\nH\u271d :\n Sum.Lex s r\u2082 b\u271d\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a\u271d)\na : \u03b1\u2081\nb : \u03b1\u2082\nH :\n Sum.Lex s r\u2082 (Sum.inr b)\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n (Sum.inr a))\n\u22a2 \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n Sum.inr b\n[PROOFSTEP]\nlet \u27e8w, h\u27e9 := fi _ _ (Sum.lex_inr_inr.1 H)\n[GOAL]\n\u03b1 : Type ?u.138586\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nthis : \u03b2 \u2295 \u03b1\u2081 \u21aa \u03b2 \u2295 \u03b1\u2082\na\u271d : \u03b2 \u2295 \u03b1\u2081\nb\u271d : \u03b2 \u2295 \u03b1\u2082\nH\u271d :\n Sum.Lex s r\u2082 b\u271d\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a\u271d)\na : \u03b1\u2081\nb : \u03b1\u2082\nH :\n Sum.Lex s r\u2082 (Sum.inr b)\n (\u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a) (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n (Sum.inr a))\nw : \u03b1\u2081\nh : \u2191{ toEmbedding := f, map_rel_iff' := fo } w = b\n\u22a2 \u2203 a',\n \u2191{ toEmbedding := Embedding.sumMap (Embedding.refl \u03b2) f,\n map_rel_iff' :=\n (_ :\n \u2200 {a b : \u03b2 \u2295 \u03b1\u2081},\n Sum.Lex s r\u2082 (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) a)\n (\u2191(Embedding.sumMap (Embedding.refl \u03b2) f) b) \u2194\n Sum.Lex s r\u2081 a b) }\n a' =\n Sum.inr b\n[PROOFSTEP]\nexact \u27e8Sum.inr w, congr_arg Sum.inr h\u27e9\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc a b : Ordinal.{u}\nh : a \u2264 b\n\u22a2 Function.swap (fun x x_1 => x + x_1) c a \u2264 Function.swap (fun x x_1 => x + x_1) c b\n[PROOFSTEP]\nrevert h c\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u22a2 \u2200 (c : Ordinal.{u}), a \u2264 b \u2192 Function.swap (fun x x_1 => x + x_1) c a \u2264 Function.swap (fun x x_1 => x + x_1) c b\n[PROOFSTEP]\nrefine inductionOn a (fun \u03b1\u2081 r\u2081 _ \u21a6 ?_)\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2081 r\u2081\n\u22a2 \u2200 (c : Ordinal.{u}),\n type r\u2081 \u2264 b \u2192 Function.swap (fun x x_1 => x + x_1) c (type r\u2081) \u2264 Function.swap (fun x x_1 => x + x_1) c b\n[PROOFSTEP]\nrefine inductionOn b (fun \u03b1\u2082 r\u2082 _ \u21a6 ?_)\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2082 r\u2082\n\u22a2 \u2200 (c : Ordinal.{u}),\n type r\u2081 \u2264 type r\u2082 \u2192\n Function.swap (fun x x_1 => x + x_1) c (type r\u2081) \u2264 Function.swap (fun x x_1 => x + x_1) c (type r\u2082)\n[PROOFSTEP]\nrintro c \u27e8\u27e8\u27e8f, fo\u27e9, fi\u27e9\u27e9\n[GOAL]\ncase intro.mk.mk\n\u03b1 : Type ?u.140748\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u22a2 Function.swap (fun x x_1 => x + x_1) c (type r\u2081) \u2264 Function.swap (fun x x_1 => x + x_1) c (type r\u2082)\n[PROOFSTEP]\nrefine inductionOn c (fun \u03b2 s _ \u21a6 ?_)\n[GOAL]\ncase intro.mk.mk\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 Function.swap (fun x x_1 => x + x_1) (type s) (type r\u2081) \u2264 Function.swap (fun x x_1 => x + x_1) (type s) (type r\u2082)\n[PROOFSTEP]\nexact\n @RelEmbedding.ordinal_type_le _ _ (Sum.Lex r\u2081 s) (Sum.Lex r\u2082 s) _ _\n \u27e8f.sumMap (Embedding.refl _), by\n intro a b\n constructor <;> intro H\n \u00b7 cases' a with a a <;> cases' b with b b <;> cases H <;> constructor <;> [rwa [\u2190 fo]; assumption]\n \u00b7 cases H <;> constructor <;> [rwa [fo]; assumption]\u27e9\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 \u2200 {a b : \u03b1\u2081 \u2295 \u03b2},\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b) \u2194\n Sum.Lex r\u2081 s a b\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b) \u2194\n Sum.Lex r\u2081 s a b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b) \u2192\n Sum.Lex r\u2081 s a b\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\n\u22a2 Sum.Lex r\u2081 s a b \u2192\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\nH : Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n\u22a2 Sum.Lex r\u2081 s a b\n[PROOFSTEP]\ncases' a with a a <;> cases' b with b b <;> cases H <;> constructor <;> [rwa [\u2190 fo]; assumption]\n[GOAL]\ncase mp\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\nH : Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n\u22a2 Sum.Lex r\u2081 s a b\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase mp.inl\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nb : \u03b1\u2081 \u2295 \u03b2\na : \u03b1\u2081\nH : Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a)) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n\u22a2 Sum.Lex r\u2081 s (Sum.inl a) b\n[PROOFSTEP]\ncases' b with b b\n[GOAL]\ncase mp.inr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nb : \u03b1\u2081 \u2295 \u03b2\na : \u03b2\nH : Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr a)) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n\u22a2 Sum.Lex r\u2081 s (Sum.inr a) b\n[PROOFSTEP]\ncases' b with b b\n[GOAL]\ncase mp.inl.inl\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081\nH :\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl b))\n\u22a2 Sum.Lex r\u2081 s (Sum.inl a) (Sum.inl b)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase mp.inl.inr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na : \u03b1\u2081\nb : \u03b2\nH :\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr b))\n\u22a2 Sum.Lex r\u2081 s (Sum.inl a) (Sum.inr b)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase mp.inr.inl\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na : \u03b2\nb : \u03b1\u2081\nH :\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr a))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl b))\n\u22a2 Sum.Lex r\u2081 s (Sum.inr a) (Sum.inl b)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase mp.inr.inr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b2\nH :\n Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr a))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr b))\n\u22a2 Sum.Lex r\u2081 s (Sum.inr a) (Sum.inr b)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase mp.inl.inl.inl\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081\nh\u271d : r\u2082 (\u2191f a) (\u2191f b)\n\u22a2 Sum.Lex r\u2081 s (Sum.inl a) (Sum.inl b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.inl.inr.sep\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na : \u03b1\u2081\nb : \u03b2\n\u22a2 Sum.Lex r\u2081 s (Sum.inl a) (Sum.inr b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.inr.inr.inr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b2\nh\u271d : s (\u2191(Embedding.refl \u03b2) a) (\u2191(Embedding.refl \u03b2) b)\n\u22a2 Sum.Lex r\u2081 s (Sum.inr a) (Sum.inr b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.inl.inl.inl.h\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081\nh\u271d : r\u2082 (\u2191f a) (\u2191f b)\n\u22a2 r\u2081 a b\n[PROOFSTEP]\nrwa [\u2190 fo]\n[GOAL]\ncase mp.inr.inr.inr.h\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b2\nh\u271d : s (\u2191(Embedding.refl \u03b2) a) (\u2191(Embedding.refl \u03b2) b)\n\u22a2 s a b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\nH : Sum.Lex r\u2081 s a b\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n[PROOFSTEP]\ncases H <;> constructor <;> [rwa [fo]; assumption]\n[GOAL]\ncase mpr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na b : \u03b1\u2081 \u2295 \u03b2\nH : Sum.Lex r\u2081 s a b\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) a) (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) b)\n[PROOFSTEP]\ncases H\n[GOAL]\ncase mpr.inl\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na\u2081\u271d a\u2082\u271d : \u03b1\u2081\nh\u271d : r\u2081 a\u2081\u271d a\u2082\u271d\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a\u2081\u271d))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a\u2082\u271d))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.inr\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nb\u2081\u271d b\u2082\u271d : \u03b2\nh\u271d : s b\u2081\u271d b\u2082\u271d\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr b\u2081\u271d))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr b\u2082\u271d))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.sep\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na\u271d : \u03b1\u2081\nb\u271d : \u03b2\n\u22a2 Sum.Lex r\u2082 s (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inl a\u271d))\n (\u2191(Embedding.sumMap f (Embedding.refl \u03b2)) (Sum.inr b\u271d))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.inl.h\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\na\u2081\u271d a\u2082\u271d : \u03b1\u2081\nh\u271d : r\u2081 a\u2081\u271d a\u2082\u271d\n\u22a2 r\u2082 (\u2191f a\u2081\u271d) (\u2191f a\u2082\u271d)\n[PROOFSTEP]\nrwa [fo]\n[GOAL]\ncase mpr.inr.h\n\u03b1 : Type ?u.140748\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u}\n\u03b1\u2081 : Type u\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\nx\u271d\u00b2 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type u\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\nx\u271d\u00b9 : IsWellOrder \u03b1\u2082 r\u2082\nc : Ordinal.{u}\nf : \u03b1\u2081 \u21aa \u03b1\u2082\nfo : \u2200 {a b : \u03b1\u2081}, r\u2082 (\u2191f a) (\u2191f b) \u2194 r\u2081 a b\nfi :\n \u2200 (a : \u03b1\u2081) (b : \u03b1\u2082),\n r\u2082 b (\u2191{ toEmbedding := f, map_rel_iff' := fo } a) \u2192 \u2203 a', \u2191{ toEmbedding := f, map_rel_iff' := fo } a' = b\n\u03b2 : Type u\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nb\u2081\u271d b\u2082\u271d : \u03b2\nh\u271d : s b\u2081\u271d b\u2082\u271d\n\u22a2 s (\u2191(Embedding.refl \u03b2) b\u2081\u271d) (\u2191(Embedding.refl \u03b2) b\u2082\u271d)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type ?u.163180\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u_3}\n\u22a2 a \u2264 a + b\n[PROOFSTEP]\nsimpa only [add_zero] using add_le_add_left (Ordinal.zero_le b) a\n[GOAL]\n\u03b1 : Type ?u.163578\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u_3}\n\u22a2 a \u2264 b + a\n[PROOFSTEP]\nsimpa only [zero_add] using add_le_add_right (Ordinal.zero_le b) a\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\nh : b = a + b\nx\u271d : a < a + b \u2228 a = a + b\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\nh : b = a + b\nx\u271d : a < a + b \u2228 a = a + b\n\u22a2 a \u2264 a + b \u2228 a + b \u2264 a\n[PROOFSTEP]\nexact Or.inl (le_add_right _ _)\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\nx\u271d : b < a + b \u2228 b = a + b\nh : a = a + b\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\nx\u271d : b < a + b \u2228 b = a + b\nh : a = a + b\n\u22a2 a + b \u2264 b \u2228 b \u2264 a + b\n[PROOFSTEP]\nexact Or.inr (le_add_left _ _)\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\nh\u2081 : b < a + b\nh\u2082 : a < a + b\n\u22a2 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrevert h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u22a2 b < a + b \u2192 a < a + b \u2192 a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrefine inductionOn a ?_\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u22a2 \u2200 (\u03b1 : Type ?u.163980) (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [inst : IsWellOrder \u03b1 r],\n b < type r + b \u2192 type r < type r + b \u2192 type r \u2264 b \u2228 b \u2264 type r\n[PROOFSTEP]\nintro \u03b1\u2081 r\u2081 _\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2081 r\u2081\n\u22a2 b < type r\u2081 + b \u2192 type r\u2081 < type r\u2081 + b \u2192 type r\u2081 \u2264 b \u2228 b \u2264 type r\u2081\n[PROOFSTEP]\nrefine inductionOn b ?_\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2081 r\u2081\n\u22a2 \u2200 (\u03b1 : Type ?u.163980) (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [inst : IsWellOrder \u03b1 r],\n type r < type r\u2081 + type r \u2192 type r\u2081 < type r\u2081 + type r \u2192 type r\u2081 \u2264 type r \u2228 type r \u2264 type r\u2081\n[PROOFSTEP]\nintro \u03b1\u2082 r\u2082 _ \u27e8f\u27e9 \u27e8g\u27e9\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\n\u22a2 type r\u2081 \u2264 type r\u2082 \u2228 type r\u2082 \u2264 type r\u2081\n[PROOFSTEP]\nrw [\u2190 typein_top f, \u2190 typein_top g, le_iff_lt_or_eq, le_iff_lt_or_eq, typein_lt_typein, typein_lt_typein]\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\n\u22a2 (Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top) \u2228\n Sum.Lex r\u2081 r\u2082 f.top g.top \u2228 typein (Sum.Lex r\u2081 r\u2082) f.top = typein (Sum.Lex r\u2081 r\u2082) g.top\n[PROOFSTEP]\nrcases trichotomous_of (Sum.Lex r\u2081 r\u2082) g.top f.top with (h | h | h) <;> [exact Or.inl (Or.inl h); (left; right; rw [h]);\n exact Or.inr (Or.inl h)]\n[GOAL]\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\n\u22a2 (Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top) \u2228\n Sum.Lex r\u2081 r\u2082 f.top g.top \u2228 typein (Sum.Lex r\u2081 r\u2082) f.top = typein (Sum.Lex r\u2081 r\u2082) g.top\n[PROOFSTEP]\nrcases trichotomous_of (Sum.Lex r\u2081 r\u2082) g.top f.top with (h | h | h)\n[GOAL]\ncase inl\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\nh : Sum.Lex r\u2081 r\u2082 g.top f.top\n\u22a2 (Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top) \u2228\n Sum.Lex r\u2081 r\u2082 f.top g.top \u2228 typein (Sum.Lex r\u2081 r\u2082) f.top = typein (Sum.Lex r\u2081 r\u2082) g.top\n[PROOFSTEP]\nexact Or.inl (Or.inl h)\n[GOAL]\ncase inr.inl\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\nh : g.top = f.top\n\u22a2 (Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top) \u2228\n Sum.Lex r\u2081 r\u2082 f.top g.top \u2228 typein (Sum.Lex r\u2081 r\u2082) f.top = typein (Sum.Lex r\u2081 r\u2082) g.top\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr.inl.h\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\nh : g.top = f.top\n\u22a2 Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.inl.h.h\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\nh : g.top = f.top\n\u22a2 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inr.inr\n\u03b1 : Type ?u.163946\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nsrc\u271d : PartialOrder Ordinal.{?u.163980} := inferInstanceAs (PartialOrder Ordinal.{?u.163980})\na b : Ordinal.{?u.163980}\n\u03b1\u2081 : Type ?u.163980\nr\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1\u2081 r\u2081\n\u03b1\u2082 : Type ?u.163980\nr\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1\u2082 r\u2082\nf : r\u2082 \u227ai Sum.Lex r\u2081 r\u2082\ng : r\u2081 \u227ai Sum.Lex r\u2081 r\u2082\nh : Sum.Lex r\u2081 r\u2082 f.top g.top\n\u22a2 (Sum.Lex r\u2081 r\u2082 g.top f.top \u2228 typein (Sum.Lex r\u2081 r\u2082) g.top = typein (Sum.Lex r\u2081 r\u2082) f.top) \u2228\n Sum.Lex r\u2081 r\u2082 f.top g.top \u2228 typein (Sum.Lex r\u2081 r\u2082) f.top = typein (Sum.Lex r\u2081 r\u2082) g.top\n[PROOFSTEP]\nexact Or.inr (Or.inl h)\n[GOAL]\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\n\u22a2 type r + 1 \u2264 type s\n[PROOFSTEP]\nhaveI := hs\n[GOAL]\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\n\u22a2 type r + 1 \u2264 type s\n[PROOFSTEP]\nrefine' \u27e8\u27e8RelEmbedding.ofMonotone (Sum.rec f fun _ => t) (fun a b \u21a6 _), fun a b \u21a6 _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na b : \u03b1 \u2295 PUnit\n\u22a2 Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)\n[PROOFSTEP]\nrcases a with (a | _)\n[GOAL]\ncase refine'_1.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b1 \u2295 PUnit\na : \u03b1\n\u22a2 Sum.Lex r EmptyRelation (Sum.inl a) b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl a)) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase refine'_1.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b1 \u2295 PUnit\nval\u271d : PUnit\n\u22a2 Sum.Lex r EmptyRelation (Sum.inr val\u271d) b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d)) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)\n[PROOFSTEP]\nrcases b with (b | _)\n[GOAL]\ncase refine'_1.inl.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na b : \u03b1\n\u22a2 Sum.Lex r EmptyRelation (Sum.inl a) (Sum.inl b) \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl a)) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl b))\n[PROOFSTEP]\nsimpa only [Sum.lex_inl_inl] using f.map_rel_iff.2\n[GOAL]\ncase refine'_1.inl.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na : \u03b1\nval\u271d : PUnit\n\u22a2 Sum.Lex r EmptyRelation (Sum.inl a) (Sum.inr val\u271d) \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl a))\n ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d))\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_1.inl.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na : \u03b1\nval\u271d : PUnit\na\u271d : Sum.Lex r EmptyRelation (Sum.inl a) (Sum.inr val\u271d)\n\u22a2 s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl a)) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d))\n[PROOFSTEP]\nrw [hf]\n[GOAL]\ncase refine'_1.inl.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d\u00b9 b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na : \u03b1\nval\u271d : PUnit\na\u271d : Sum.Lex r EmptyRelation (Sum.inl a) (Sum.inr val\u271d)\n\u22a2 \u2203 a_1, \u2191f a_1 = (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl a)\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\ncase refine'_1.inr.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nval\u271d : PUnit\nb : \u03b1\n\u22a2 Sum.Lex r EmptyRelation (Sum.inr val\u271d) (Sum.inl b) \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d))\n ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inl b))\n[PROOFSTEP]\nexact False.elim \u2218 Sum.lex_inr_inl\n[GOAL]\ncase refine'_1.inr.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nval\u271d\u00b9 val\u271d : PUnit\n\u22a2 Sum.Lex r EmptyRelation (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d) \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d\u00b9))\n ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) (Sum.inr val\u271d))\n[PROOFSTEP]\nexact False.elim \u2218 Sum.lex_inr_inr.1\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\na : \u03b1 \u2295 PUnit\nb : \u03b2\n\u22a2 s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a) \u2192\n \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nrcases a with (a | _)\n[GOAL]\ncase refine'_2.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\na : \u03b1\n\u22a2 s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inl a)) \u2192\n \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_2.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\na : \u03b1\nh :\n s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inl a))\n\u22a2 \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nhave := @PrincipalSeg.init _ _ _ _ _ \u27e8f, t, hf\u27e9 _ _ h\n[GOAL]\ncase refine'_2.inl\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis\u271d : IsWellOrder \u03b2 s\nb : \u03b2\na : \u03b1\nh :\n s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inl a))\nthis : \u2203 a', \u2191{ toRelEmbedding := f, top := t, down' := hf }.toRelEmbedding a' = b\n\u22a2 \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\ncases' this with w h\n[GOAL]\ncase refine'_2.inl.intro\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na\u271d b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\na : \u03b1\nh\u271d :\n s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inl a))\nw : \u03b1\nh : \u2191{ toRelEmbedding := f, top := t, down' := hf }.toRelEmbedding w = b\n\u22a2 \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nexact \u27e8Sum.inl w, h\u27e9\n[GOAL]\ncase refine'_2.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\nval\u271d : PUnit\n\u22a2 s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inr val\u271d)) \u2192\n \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_2.inr\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\nval\u271d : PUnit\nh :\n s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inr val\u271d))\n\u22a2 \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\ncases' (hf b).1 h with w h\n[GOAL]\ncase refine'_2.inr.intro\n\u03b1\u271d : Type ?u.169275\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt\u271d : \u03b3 \u2192 \u03b3 \u2192 Prop\na b\u271d : Ordinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : IsWellOrder \u03b1 r\n\u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nhs : IsWellOrder \u03b2 s\nx\u271d : type r < type s\nf : r \u21aar s\nt : \u03b2\nhf : \u2200 (b : \u03b2), s b t \u2194 \u2203 a, \u2191f a = b\nthis : IsWellOrder \u03b2 s\nb : \u03b2\nval\u271d : PUnit\nh\u271d :\n s b\n (\u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n (Sum.inr val\u271d))\nw : \u03b1\nh : \u2191f w = b\n\u22a2 \u2203 a',\n \u2191(RelEmbedding.ofMonotone (fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1)\n (_ :\n \u2200 (a b : \u03b1 \u2295 PUnit),\n Sum.Lex r EmptyRelation a b \u2192\n s ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) a) ((fun t_1 => Sum.rec (\u2191f) (fun x => t) t_1) b)))\n a' =\n b\n[PROOFSTEP]\nexact \u27e8Sum.inl w, h\u27e9\n[GOAL]\n\u03b1 : Type ?u.179391\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 succ 1 = 2\n[PROOFSTEP]\nunfold instOfNat OfNat.ofNat\n[GOAL]\n\u03b1 : Type ?u.179391\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 succ One.toOfNat1.1 = { ofNat := \u21912 }.1\n[PROOFSTEP]\nsimpa using by rfl\n[GOAL]\n\u03b1 : Type ?u.179391\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 One.toOfNat1.1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.180733\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 1 \u2264 o \u2194 0 < o\n[PROOFSTEP]\nrw [\u2190 succ_zero, succ_le_iff]\n[GOAL]\n\u03b1 : Type ?u.181133\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 1 \u2264 o \u2194 o \u2260 0\n[PROOFSTEP]\nrw [one_le_iff_pos, Ordinal.pos_iff_ne_zero]\n[GOAL]\n\u03b1 : Type ?u.181722\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal.{u_3}\n\u22a2 a < 1 \u2194 a = 0\n[PROOFSTEP]\nsimpa using @lt_succ_bot_iff _ _ _ a _ _\n[GOAL]\n\u03b1 : Type ?u.182584\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal.{u_3}\n\u22a2 a \u2264 1 \u2194 a = 0 \u2228 a = 1\n[PROOFSTEP]\nsimpa using @le_succ_bot_iff _ _ _ a _\n[GOAL]\n\u03b1 : Type ?u.183756\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 card (succ o) = card o + 1\n[PROOFSTEP]\nsimp only [\u2190 add_one_eq_succ, card_add, card_one]\n[GOAL]\n\u03b1 : Type ?u.184700\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 0 \u2208 Iio 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 0 < type fun x x_1 => x < x_1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : (Quotient.out 1).\u03b1\n\u22a2 a = default\n[PROOFSTEP]\nunfold default\n[GOAL]\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : (Quotient.out 1).\u03b1\n\u22a2 a = { default := enum (fun x x_1 => x < x_1) 0 (_ : 0 < type fun x x_1 => x < x_1) }.1\n[PROOFSTEP]\nrw [\u2190 @enum_typein _ (\u00b7 < \u00b7) (isWellOrder_out_lt _) a]\n[GOAL]\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : (Quotient.out 1).\u03b1\n\u22a2 enum (fun x x_1 => x < x_1) (typein (fun x x_1 => x < x_1) a)\n (_ : typein (fun x x_1 => x < x_1) a < type fun x x_1 => x < x_1) =\n { default := enum (fun x x_1 => x < x_1) 0 (_ : 0 < type fun x x_1 => x < x_1) }.1\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_o\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : (Quotient.out 1).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) a = 0\n[PROOFSTEP]\nrw [\u2190 lt_one_iff_zero]\n[GOAL]\ncase e_o\n\u03b1 : Type ?u.185352\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : (Quotient.out 1).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) a < 1\n[PROOFSTEP]\napply typein_lt_self\n[GOAL]\n\u03b1 : Type ?u.187460\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nx : (Quotient.out 1).\u03b1\n\u22a2 0 < type fun x x_1 => x < x_1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.188563\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nx : (Quotient.out 1).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) x = 0\n[PROOFSTEP]\nrw [one_out_eq x, typein_enum]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nx x' : \u03b1\n\u22a2 typein r x \u2264 typein r x' \u2194 \u00acr x' x\n[PROOFSTEP]\nrw [\u2190 not_lt, typein_lt_typein]\n[GOAL]\n\u03b1 : Type ?u.190090\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nx x' : (Quotient.out o).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) x \u2264 typein (fun x x_1 => x < x_1) x' \u2194 x \u2264 x'\n[PROOFSTEP]\nrw [typein_le_typein]\n[GOAL]\n\u03b1 : Type ?u.190090\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nx x' : (Quotient.out o).\u03b1\n\u22a2 \u00acx' < x \u2194 x \u2264 x'\n[PROOFSTEP]\nexact not_lt\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\no o' : Ordinal.{u_3}\nho : o < type r\nho' : o' < type r\n\u22a2 \u00acr (enum r o' ho') (enum r o ho) \u2194 o \u2264 o'\n[PROOFSTEP]\nrw [\u2190 @not_lt _ _ o' o, enum_lt_enum ho']\n[GOAL]\n\u03b1 : Type ?u.192624\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na o o' : Ordinal.{u_3}\nho : o < type fun x x_1 => x < x_1\nho' : o' < type fun x x_1 => x < x_1\n\u22a2 enum (fun x x_1 => x < x_1) o ho \u2264 enum (fun x x_1 => x < x_1) o' ho' \u2194 o \u2264 o'\n[PROOFSTEP]\nrw [\u2190 @enum_le_enum _ (\u00b7 < \u00b7) (isWellOrder_out_lt _), \u2190 not_lt]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nh0 : 0 < type r\na : \u03b1\n\u22a2 \u00acr a (enum r 0 h0)\n[PROOFSTEP]\nrw [\u2190 enum_typein r a, enum_le_enum r]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nh0 : 0 < type r\na : \u03b1\n\u22a2 0 \u2264 typein r a\n[PROOFSTEP]\napply Ordinal.zero_le\n[GOAL]\n\u03b1 : Type ?u.196508\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.196642}\nh0 : 0 < o\na : (Quotient.out o).\u03b1\n\u22a2 0 < type fun x x_1 => x < x_1\n[PROOFSTEP]\nrwa [type_lt]\n[GOAL]\n\u03b1 : Type ?u.196508\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nh0 : 0 < o\na : (Quotient.out o).\u03b1\n\u22a2 enum (fun x x_1 => x < x_1) 0 (_ : 0 < type fun x x_1 => x < x_1) \u2264 a\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\n\u03b1 : Type ?u.196508\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nh0 : 0 < o\na : (Quotient.out o).\u03b1\n\u22a2 \u00aca < enum (fun x x_1 => x < x_1) 0 (_ : 0 < type fun x x_1 => x < x_1)\n[PROOFSTEP]\napply enum_zero_le\n[GOAL]\n\u03b1 : Type ?u.197714\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.197839}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < type fun x x_1 => x < x_1\n[PROOFSTEP]\nrw [type_lt]\n[GOAL]\n\u03b1 : Type ?u.197714\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.197839}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < succ o\n[PROOFSTEP]\nexact lt_succ o\n[GOAL]\n\u03b1 : Type ?u.197714\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 a \u2264 enum (fun x x_1 => x < x_1) o (_ : o < type fun x x_1 => x < x_1)\n[PROOFSTEP]\nrw [\u2190 @enum_typein _ (\u00b7 < \u00b7) (isWellOrder_out_lt _) a, enum_le_enum', \u2190 lt_succ_iff]\n[GOAL]\n\u03b1 : Type ?u.197714\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) a < succ o\n[PROOFSTEP]\napply typein_lt_self\n[GOAL]\n\u03b1 : Type ?u.201785\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.203623}\nx : \u2191(Iio o)\n\u22a2 \u2191x < type fun x x_1 => x < x_1\n[PROOFSTEP]\nrw [type_lt]\n[GOAL]\n\u03b1 : Type ?u.201785\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.203623}\nx : \u2191(Iio o)\n\u22a2 \u2191x < o\n[PROOFSTEP]\nexact x.2\n[GOAL]\n\u03b1 : Type ?u.201785\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.203623}\n\u22a2 \u2200 {a b : \u2191(Iio o)},\n \u2191{ toFun := fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1),\n invFun := fun x =>\n { val := typein (fun x x_1 => x < x_1) x, property := (_ : typein (fun x x_1 => x < x_1) x < o) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iio o)),\n (fun x =>\n { val := typein (fun x x_1 => x < x_1) x,\n property := (_ : typein (fun x x_1 => x < x_1) x < o) })\n ((fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1)) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (h : (Quotient.out o).\u03b1),\n enum (fun x x_1 => x < x_1) (typein (fun x x_1 => x < x_1) h)\n (_ : typein (fun x x_1 => x < x_1) h < type fun x x_1 => x < x_1) =\n h) }\n a \u2264\n \u2191{ toFun := fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1),\n invFun := fun x =>\n { val := typein (fun x x_1 => x < x_1) x, property := (_ : typein (fun x x_1 => x < x_1) x < o) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iio o)),\n (fun x =>\n { val := typein (fun x x_1 => x < x_1) x,\n property := (_ : typein (fun x x_1 => x < x_1) x < o) })\n ((fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1)) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (h : (Quotient.out o).\u03b1),\n enum (fun x x_1 => x < x_1) (typein (fun x x_1 => x < x_1) h)\n (_ : typein (fun x x_1 => x < x_1) h < type fun x x_1 => x < x_1) =\n h) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nrintro \u27e8a, _\u27e9 \u27e8b, _\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type ?u.201785\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no a : Ordinal.{?u.203623}\nproperty\u271d\u00b9 : a \u2208 Iio o\nb : Ordinal.{?u.203623}\nproperty\u271d : b \u2208 Iio o\n\u22a2 \u2191{ toFun := fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1),\n invFun := fun x =>\n { val := typein (fun x x_1 => x < x_1) x, property := (_ : typein (fun x x_1 => x < x_1) x < o) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iio o)),\n (fun x =>\n { val := typein (fun x x_1 => x < x_1) x,\n property := (_ : typein (fun x x_1 => x < x_1) x < o) })\n ((fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1)) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (h : (Quotient.out o).\u03b1),\n enum (fun x x_1 => x < x_1) (typein (fun x x_1 => x < x_1) h)\n (_ : typein (fun x x_1 => x < x_1) h < type fun x x_1 => x < x_1) =\n h) }\n { val := a, property := property\u271d\u00b9 } \u2264\n \u2191{ toFun := fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1),\n invFun := fun x =>\n { val := typein (fun x x_1 => x < x_1) x, property := (_ : typein (fun x x_1 => x < x_1) x < o) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Iio o)),\n (fun x =>\n { val := typein (fun x x_1 => x < x_1) x,\n property := (_ : typein (fun x x_1 => x < x_1) x < o) })\n ((fun x => enum (fun x x_1 => x < x_1) \u2191x (_ : \u2191x < type fun x x_1 => x < x_1)) x) =\n x),\n right_inv :=\n (_ :\n \u2200 (h : (Quotient.out o).\u03b1),\n enum (fun x x_1 => x < x_1) (typein (fun x x_1 => x < x_1) h)\n (_ : typein (fun x x_1 => x < x_1) h < type fun x x_1 => x < x_1) =\n h) }\n { val := b, property := property\u271d } \u2194\n { val := a, property := property\u271d\u00b9 } \u2264 { val := b, property := property\u271d }\n[PROOFSTEP]\napply enum_le_enum'\n[GOAL]\n\u03b1 : Type ?u.208873\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.208900}\nho : 0 < o\n\u22a2 0 < type fun x x_1 => x < x_1\n[PROOFSTEP]\nrwa [type_lt]\n[GOAL]\n\u03b1 : Type ?u.211339\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 \u2200 (b : Ordinal.{max (u + 1) v}), b < univ \u2194 \u2203 a, \u2191initialSeg.toRelEmbedding a = b\n[PROOFSTEP]\nrefine' fun b => inductionOn b _\n[GOAL]\n\u03b1 : Type ?u.211339\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u22a2 \u2200 (\u03b1 : Type (max (u + 1) v)) (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [inst : IsWellOrder \u03b1 r],\n type r < univ \u2194 \u2203 a, \u2191initialSeg.toRelEmbedding a = type r\n[PROOFSTEP]\nintro \u03b2 s _\n[GOAL]\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\n\u22a2 type s < univ \u2194 \u2203 a, \u2191initialSeg.toRelEmbedding a = type s\n[PROOFSTEP]\nrw [univ, \u2190 lift_umax]\n[GOAL]\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\n\u22a2 type s < lift (type fun x x_1 => x < x_1) \u2194 \u2203 a, \u2191initialSeg.toRelEmbedding a = type s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\n\u22a2 type s < lift (type fun x x_1 => x < x_1) \u2192 \u2203 a, \u2191initialSeg.toRelEmbedding a = type s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\n\u22a2 (\u2203 a, \u2191initialSeg.toRelEmbedding a = type s) \u2192 type s < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : type s < lift (type fun x x_1 => x < x_1)\n\u22a2 \u2203 a, \u2191initialSeg.toRelEmbedding a = type s\n[PROOFSTEP]\nrw [\u2190 lift_id (type s)] at h \u22a2\n[GOAL]\ncase mp\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\n\u22a2 \u2203 a, \u2191initialSeg.toRelEmbedding a = lift (type s)\n[PROOFSTEP]\ncases' lift_type_lt.{_, _, v}.1 h with f\n[GOAL]\ncase mp.intro\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u227ai fun x x_1 => x < x_1\n\u22a2 \u2203 a, \u2191initialSeg.toRelEmbedding a = lift (type s)\n[PROOFSTEP]\ncases' f with f a hf\n[GOAL]\ncase mp.intro.mk\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\nhf : \u2200 (b : Ordinal.{u}), b < a \u2194 \u2203 a, \u2191f a = b\n\u22a2 \u2203 a, \u2191initialSeg.toRelEmbedding a = lift (type s)\n[PROOFSTEP]\nexists a\n[GOAL]\ncase mp.intro.mk\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\nhf : \u2200 (b : Ordinal.{u}), b < a \u2194 \u2203 a, \u2191f a = b\n\u22a2 \u2191initialSeg.toRelEmbedding a = lift (type s)\n[PROOFSTEP]\nrevert hf\n[GOAL]\ncase mp.intro.mk\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u22a2 (\u2200 (b : Ordinal.{u}), b < a \u2194 \u2203 a, \u2191f a = b) \u2192 \u2191initialSeg.toRelEmbedding a = lift (type s)\n[PROOFSTEP]\nrefine inductionOn a ?_\n[GOAL]\ncase mp.intro.mk\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u22a2 \u2200 (\u03b1 : Type u) (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [inst : IsWellOrder \u03b1 r],\n (\u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b) \u2192 \u2191initialSeg.toRelEmbedding (type r) = lift (type s)\n[PROOFSTEP]\nintro \u03b1 r _ hf\n[GOAL]\ncase mp.intro.mk\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\n\u22a2 \u2191initialSeg.toRelEmbedding (type r) = lift (type s)\n[PROOFSTEP]\nrefine' lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2 \u27e8(RelIso.ofSurjective (RelEmbedding.ofMonotone _ _) _).symm\u27e9\n[GOAL]\ncase mp.intro.mk.refine'_1\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\n\u22a2 \u03b2 \u2192 \u03b1\n[PROOFSTEP]\nexact fun b => enum r (f b) ((hf _).2 \u27e8_, rfl\u27e9)\n[GOAL]\ncase mp.intro.mk.refine'_2\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\n\u22a2 \u2200 (a b : \u03b2), s a b \u2192 r (enum r (\u2191f a) (_ : \u2191f a < type r)) (enum r (\u2191f b) (_ : \u2191f b < type r))\n[PROOFSTEP]\nrefine' fun a b h => (typein_lt_typein r).1 _\n[GOAL]\ncase mp.intro.mk.refine'_2\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb\u271d : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh\u271d : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na\u271d : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na b : \u03b2\nh : s a b\n\u22a2 typein r (enum r (\u2191f a) (_ : \u2191f a < type r)) < typein r (enum r (\u2191f b) (_ : \u2191f b < type r))\n[PROOFSTEP]\nrw [typein_enum, typein_enum]\n[GOAL]\ncase mp.intro.mk.refine'_2\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb\u271d : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh\u271d : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na\u271d : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na b : \u03b2\nh : s a b\n\u22a2 \u2191f a < \u2191f b\n[PROOFSTEP]\nexact f.map_rel_iff.2 h\n[GOAL]\ncase mp.intro.mk.refine'_3\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\n\u22a2 Surjective\n \u2191(RelEmbedding.ofMonotone (fun b => enum r (\u2191f b) (_ : \u2191f b < type r))\n (_ : \u2200 (a b : \u03b2), s a b \u2192 r (enum r (\u2191f a) (_ : \u2191f a < type r)) (enum r (\u2191f b) (_ : \u2191f b < type r))))\n[PROOFSTEP]\nintro a'\n[GOAL]\ncase mp.intro.mk.refine'_3\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na' : \u03b1\n\u22a2 \u2203 a,\n \u2191(RelEmbedding.ofMonotone (fun b => enum r (\u2191f b) (_ : \u2191f b < type r))\n (_ : \u2200 (a b : \u03b2), s a b \u2192 r (enum r (\u2191f a) (_ : \u2191f a < type r)) (enum r (\u2191f b) (_ : \u2191f b < type r))))\n a =\n a'\n[PROOFSTEP]\ncases' (hf _).1 (typein_lt_type _ a') with b e\n[GOAL]\ncase mp.intro.mk.refine'_3.intro\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb\u271d : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na' : \u03b1\nb : \u03b2\ne : \u2191f b = typein r a'\n\u22a2 \u2203 a,\n \u2191(RelEmbedding.ofMonotone (fun b => enum r (\u2191f b) (_ : \u2191f b < type r))\n (_ : \u2200 (a b : \u03b2), s a b \u2192 r (enum r (\u2191f a) (_ : \u2191f a < type r)) (enum r (\u2191f b) (_ : \u2191f b < type r))))\n a =\n a'\n[PROOFSTEP]\nexists b\n[GOAL]\ncase mp.intro.mk.refine'_3.intro\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb\u271d : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na' : \u03b1\nb : \u03b2\ne : \u2191f b = typein r a'\n\u22a2 \u2191(RelEmbedding.ofMonotone (fun b => enum r (\u2191f b) (_ : \u2191f b < type r))\n (_ : \u2200 (a b : \u03b2), s a b \u2192 r (enum r (\u2191f a) (_ : \u2191f a < type r)) (enum r (\u2191f b) (_ : \u2191f b < type r))))\n b =\n a'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.intro.mk.refine'_3.intro\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb\u271d : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\nh : lift (type s) < lift (type fun x x_1 => x < x_1)\nf : s \u21aar fun x x_1 => x < x_1\na : Ordinal.{u}\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhf : \u2200 (b : Ordinal.{u}), b < type r \u2194 \u2203 a, \u2191f a = b\na' : \u03b1\nb : \u03b2\ne : \u2191f b = typein r a'\n\u22a2 enum r (\u2191f b) (_ : \u2191f b < type r) = a'\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase mpr\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\nh : \u2203 a, \u2191initialSeg.toRelEmbedding a = type s\n\u22a2 type s < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\ncases' h with a e\n[GOAL]\ncase mpr.intro\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\na : Ordinal.{u}\ne : \u2191initialSeg.toRelEmbedding a = type s\n\u22a2 type s < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\nrw [\u2190 e]\n[GOAL]\ncase mpr.intro\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\na : Ordinal.{u}\ne : \u2191initialSeg.toRelEmbedding a = type s\n\u22a2 \u2191initialSeg.toRelEmbedding a < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\nrefine inductionOn a ?_\n[GOAL]\ncase mpr.intro\n\u03b1 : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsWellOrder \u03b2 s\na : Ordinal.{u}\ne : \u2191initialSeg.toRelEmbedding a = type s\n\u22a2 \u2200 (\u03b1 : Type u) (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [inst : IsWellOrder \u03b1 r],\n \u2191initialSeg.toRelEmbedding (type r) < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\nintro \u03b1 r _\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type ?u.211339\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nb : Ordinal.{max (u + 1) v}\n\u03b2 : Type (max (u + 1) v)\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b2 s\na : Ordinal.{u}\ne : \u2191initialSeg.toRelEmbedding a = type s\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 \u2191initialSeg.toRelEmbedding (type r) < lift (type fun x x_1 => x < x_1)\n[PROOFSTEP]\nexact lift_type_lt.{u, u + 1, max (u + 1) v}.2 \u27e8typein.principalSeg r\u27e9\n[GOAL]\n\u03b1 : Type ?u.221999\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 principalSeg.top = type fun x x_1 => x < x_1\n[PROOFSTEP]\nsimp only [lift.principalSeg_top, univ_id]\n[GOAL]\n\u03b1 : Type ?u.222301\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\n\u22a2 card (type (?m.222405 o)) = card o\n[PROOFSTEP]\nrw [Ordinal.type_lt]\n[GOAL]\n\u03b1 : Type ?u.222528\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u22a2 \u2200 (a b : Type u), Setoid.r a b \u2192 F a = F b\n[PROOFSTEP]\nsuffices : \u2200 {\u03b1 \u03b2}, \u03b1 \u2248 \u03b2 \u2192 F \u03b1 \u2264 F \u03b2\n[GOAL]\n\u03b1 : Type ?u.222528\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\nthis : \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2248 \u03b2 \u2192 F \u03b1 \u2264 F \u03b2\n\u22a2 \u2200 (a b : Type u), Setoid.r a b \u2192 F a = F b\ncase this\n\u03b1 : Type ?u.222528\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2248 \u03b2 \u2192 F \u03b1 \u2264 F \u03b2\n[PROOFSTEP]\nexact fun \u03b1 \u03b2 h => (this h).antisymm (this (Setoid.symm h))\n[GOAL]\ncase this\n\u03b1 : Type ?u.222528\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u22a2 \u2200 {\u03b1 \u03b2 : Type u}, \u03b1 \u2248 \u03b2 \u2192 F \u03b1 \u2264 F \u03b2\n[PROOFSTEP]\nrintro \u03b1 \u03b2 \u27e8f\u27e9\n[GOAL]\ncase this.intro\n\u03b1\u271d : Type ?u.222528\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2243 \u03b2\n\u22a2 F \u03b1 \u2264 F \u03b2\n[PROOFSTEP]\nrefine' le_ciInf_iff'.2 fun i => _\n[GOAL]\ncase this.intro\n\u03b1\u271d : Type ?u.222528\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2243 \u03b2\ni : { r // IsWellOrder \u03b2 r }\n\u22a2 F \u03b1 \u2264 type \u2191i\n[PROOFSTEP]\nhaveI := @RelEmbedding.isWellOrder _ _ (f \u207b\u00b9'o i.1) _ (\u2191(RelIso.preimage f i.1)) i.2\n[GOAL]\ncase this.intro\n\u03b1\u271d : Type ?u.222528\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nF : Type u \u2192 Ordinal.{u} := fun \u03b1 => \u2a05 (r : { r // IsWellOrder \u03b1 r }), type \u2191r\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2243 \u03b2\ni : { r // IsWellOrder \u03b2 r }\nthis : IsWellOrder \u03b1 (\u2191f \u207b\u00b9'o \u2191i)\n\u22a2 F \u03b1 \u2264 type \u2191i\n[PROOFSTEP]\nexact\n (ciInf_le' _ (Subtype.mk (f \u207b\u00b9'o i.val) (@RelEmbedding.isWellOrder _ _ _ _ (\u2191(RelIso.preimage f i.1)) i.2))).trans_eq\n (Quot.sound \u27e8RelIso.preimage f i.1\u27e9)\n[GOAL]\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\n\u22a2 ord #\u03b1 \u2264 type s \u2194 #\u03b1 \u2264 card (type s)\n[PROOFSTEP]\nlet \u27e8r, _, e\u27e9 := ord_eq \u03b1\n[GOAL]\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 ord #\u03b1 \u2264 type s \u2194 #\u03b1 \u2264 card (type s)\n[PROOFSTEP]\nskip\n[GOAL]\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 ord #\u03b1 \u2264 type s \u2194 #\u03b1 \u2264 card (type s)\n[PROOFSTEP]\nsimp only [card_type]\n[GOAL]\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 ord #\u03b1 \u2264 type s \u2194 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 ord #\u03b1 \u2264 type s \u2192 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 #\u03b1 \u2264 #\u03b2 \u2192 ord #\u03b1 \u2264 type s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nh : ord #\u03b1 \u2264 type s\n\u22a2 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\ncase mp\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nh : type r \u2264 type s\n\u22a2 #\u03b1 \u2264 #\u03b2\n[PROOFSTEP]\nexact\n let \u27e8f\u27e9 := h\n \u27e8f.toEmbedding\u27e9\n[GOAL]\ncase mpr\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nh : #\u03b1 \u2264 #\u03b2\n\u22a2 ord #\u03b1 \u2264 type s\n[PROOFSTEP]\ncases' h with f\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nf : \u03b1 \u21aa \u03b2\n\u22a2 ord #\u03b1 \u2264 type s\n[PROOFSTEP]\nhave g := RelEmbedding.preimage f s\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nf : \u03b1 \u21aa \u03b2\ng : \u2191f \u207b\u00b9'o s \u21aar s\n\u22a2 ord #\u03b1 \u2264 type s\n[PROOFSTEP]\nhaveI := RelEmbedding.isWellOrder g\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type ?u.225629\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\no : Ordinal.{u_3}\n\u03b1 \u03b2 : Type u_3\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nx\u271d : IsWellOrder \u03b2 s\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\nf : \u03b1 \u21aa \u03b2\ng : \u2191f \u207b\u00b9'o s \u21aar s\nthis : IsWellOrder \u03b1 (\u2191f \u207b\u00b9'o s)\n\u22a2 ord #\u03b1 \u2264 type s\n[PROOFSTEP]\nexact le_trans (ord_le_type _) g.ordinal_type_le\n[GOAL]\n\u03b1\u271d : Type ?u.226993\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\n\u03b1 : Type u_3\n\u22a2 card (ord (Quotient.mk isEquivalent \u03b1)) = Quotient.mk isEquivalent \u03b1\n[PROOFSTEP]\nlet \u27e8r, _, e\u27e9 := ord_eq \u03b1\n[GOAL]\n\u03b1\u271d : Type ?u.226993\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nw\u271d : IsWellOrder \u03b1 r\ne : ord #\u03b1 = type r\n\u22a2 card (ord (Quotient.mk isEquivalent \u03b1)) = Quotient.mk isEquivalent \u03b1\n[PROOFSTEP]\nsimp only [mk'_def, e, card_type]\n[GOAL]\n\u03b1 : Type ?u.229821\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn : \u2115\n\u22a2 \u2191n \u2264 ord \u2191n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type ?u.229821\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 \u2191Nat.zero \u2264 ord \u2191Nat.zero\n[PROOFSTEP]\napply Ordinal.zero_le\n[GOAL]\ncase succ\n\u03b1 : Type ?u.229821\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn : \u2115\nIH : \u2191n \u2264 ord \u2191n\n\u22a2 \u2191(Nat.succ n) \u2264 ord \u2191(Nat.succ n)\n[PROOFSTEP]\nexact succ_le_of_lt (IH.trans_lt <| ord_lt_ord.2 <| natCast_lt.2 (Nat.lt_succ_self n))\n[GOAL]\n\u03b1 : Type ?u.230856\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 ord 1 = 1\n[PROOFSTEP]\nsimpa using ord_nat 1\n[GOAL]\n\u03b1 : Type ?u.231632\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{v}\n\u22a2 Ordinal.lift (ord c) = ord (lift c)\n[PROOFSTEP]\nrefine' le_antisymm (le_of_forall_lt fun a ha => _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.231632\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{v}\na : Ordinal.{max v u}\nha : a < Ordinal.lift (ord c)\n\u22a2 a < ord (lift c)\n[PROOFSTEP]\nrcases Ordinal.lt_lift_iff.1 ha with \u27e8a, rfl, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type ?u.231632\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{v}\na : Ordinal.{v}\nright\u271d : a < ord c\nha : Ordinal.lift a < Ordinal.lift (ord c)\n\u22a2 Ordinal.lift a < ord (lift c)\n[PROOFSTEP]\nrwa [lt_ord, \u2190 lift_card, lift_lt, \u2190 lt_ord, \u2190 Ordinal.lift_lt]\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.231632\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{v}\n\u22a2 ord (lift c) \u2264 Ordinal.lift (ord c)\n[PROOFSTEP]\nrw [ord_le, \u2190 lift_card, card_ord]\n[GOAL]\n\u03b1 : Type ?u.232210\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\n\u22a2 #(Quotient.out (ord c)).\u03b1 = c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nx : \u03b1\nh : ord #\u03b1 = type r\n\u22a2 card (typein r x) < #\u03b1\n[PROOFSTEP]\nrw [\u2190 lt_ord, h]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nx : \u03b1\nh : ord #\u03b1 = type r\n\u22a2 typein r x < type r\n[PROOFSTEP]\napply typein_lt_type\n[GOAL]\n\u03b1 : Type ?u.232559\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\nx : (Quotient.out (ord c)).\u03b1\n\u22a2 card (typein (fun x x_1 => x < x_1) x) < c\n[PROOFSTEP]\nrw [\u2190 lt_ord]\n[GOAL]\n\u03b1 : Type ?u.232559\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u_3}\nx : (Quotient.out (ord c)).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) x < ord c\n[PROOFSTEP]\napply typein_lt_self\n[GOAL]\n\u03b1 : Type ?u.233731\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 Injective ord\n[PROOFSTEP]\nintro c c' h\n[GOAL]\n\u03b1 : Type ?u.233731\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc c' : Cardinal.{u_3}\nh : ord c = ord c'\n\u22a2 c = c'\n[PROOFSTEP]\nrw [\u2190 card_ord c, \u2190 card_ord c', h]\n[GOAL]\n\u03b1 : Type ?u.236029\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\n\u22a2 lift c < univ\n[PROOFSTEP]\nsimpa only [lift.principalSeg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using\n le_of_lt (lift.principalSeg.{u, u + 1}.lt_top (succ c).ord)\n[GOAL]\n\u03b1 : Type ?u.236679\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\n\u22a2 lift c < univ\n[PROOFSTEP]\nhave := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n[GOAL]\n\u03b1 : Type ?u.236679\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nthis : lift (lift c) < lift univ\n\u22a2 lift c < univ\n[PROOFSTEP]\nrw [lift_lift, lift_univ, univ_umax.{u, v}] at this \n[GOAL]\n\u03b1 : Type ?u.236679\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u}\nthis : lift c < univ\n\u22a2 lift c < univ\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u22a2 ord univ = Ordinal.univ\n[PROOFSTEP]\nrefine' le_antisymm (ord_card_le _) <| le_of_forall_lt fun o h => lt_ord.2 ?_\n[GOAL]\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{max (u + 1) v}\nh : o < Ordinal.univ\n\u22a2 card o < univ\n[PROOFSTEP]\nhave := lift.principalSeg.{u, v}.down.1 (by simpa only [lift.principalSeg_coe] using h)\n[GOAL]\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{max (u + 1) v}\nh : o < Ordinal.univ\n\u22a2 ?m.237120 < lift.principalSeg.top\n[PROOFSTEP]\nsimpa only [lift.principalSeg_coe] using h\n[GOAL]\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{max (u + 1) v}\nh : o < Ordinal.univ\nthis : \u2203 a, \u2191lift.principalSeg.toRelEmbedding a = o\n\u22a2 card o < univ\n[PROOFSTEP]\nrcases this with \u27e8o, h'\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no\u271d : Ordinal.{max (u + 1) v}\nh : o\u271d < Ordinal.univ\no : Ordinal.{u}\nh' : \u2191lift.principalSeg.toRelEmbedding o = o\u271d\n\u22a2 card o\u271d < univ\n[PROOFSTEP]\nrw [\u2190 h', lift.principalSeg_coe, \u2190 lift_card]\n[GOAL]\ncase intro\n\u03b1 : Type ?u.236823\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no\u271d : Ordinal.{max (u + 1) v}\nh : o\u271d < Ordinal.univ\no : Ordinal.{u}\nh' : \u2191lift.principalSeg.toRelEmbedding o = o\u271d\n\u22a2 lift (card o) < univ\n[PROOFSTEP]\napply lift_lt_univ'\n[GOAL]\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nhave := ord_lt_ord.2 h\n[GOAL]\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis : ord c < ord univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nrw [ord_univ] at this \n[GOAL]\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis : ord c < Ordinal.univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\ncases' lift.principalSeg.{u, u + 1}.down.1 (by simpa only [lift.principalSeg_top]) with o e\n[GOAL]\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis : ord c < Ordinal.univ\n\u22a2 ?m.237589 < lift.principalSeg.top\n[PROOFSTEP]\nsimpa only [lift.principalSeg_top]\n[GOAL]\ncase intro\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis : ord c < Ordinal.univ\no : Ordinal.{u}\ne : \u2191lift.principalSeg.toRelEmbedding o = ord c\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nhave := card_ord c\n[GOAL]\ncase intro\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis\u271d : ord c < Ordinal.univ\no : Ordinal.{u}\ne : \u2191lift.principalSeg.toRelEmbedding o = ord c\nthis : card (ord c) = c\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nrw [\u2190 e, lift.principalSeg_coe, \u2190 lift_card] at this \n[GOAL]\ncase intro\n\u03b1 : Type ?u.237323\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{u + 1}\nh : c < univ\nthis\u271d : ord c < Ordinal.univ\no : Ordinal.{u}\ne : \u2191lift.principalSeg.toRelEmbedding o = ord c\nthis : lift (card o) = c\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nexact \u27e8_, this.symm\u27e9\n[GOAL]\n\u03b1 : Type ?u.237859\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{max (u + 1) v}\nh : c < univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nlet \u27e8a, e, h'\u27e9 := lt_lift_iff.1 h\n[GOAL]\n\u03b1 : Type ?u.237859\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{max (u + 1) v}\nh : c < univ\na : Cardinal.{u + 1}\ne : lift a = c\nh' : a < #Ordinal.{u}\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nrw [\u2190 univ_id] at h' \n[GOAL]\n\u03b1 : Type ?u.237859\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{max (u + 1) v}\nh : c < univ\na : Cardinal.{u + 1}\ne : lift a = c\nh' : a < univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nrcases lt_univ.{u}.1 h' with \u27e8c', rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type ?u.237859\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{max (u + 1) v}\nh : c < univ\nc' : Cardinal.{u}\ne : lift (lift c') = c\nh' : lift c' < univ\n\u22a2 \u2203 c', c = lift c'\n[PROOFSTEP]\nexact \u27e8c', by simp only [e.symm, lift_lift]\u27e9\n[GOAL]\n\u03b1 : Type ?u.237859\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nc : Cardinal.{max (u + 1) v}\nh : c < univ\nc' : Cardinal.{u}\ne : lift (lift c') = c\nh' : lift c' < univ\n\u22a2 c = lift c'\n[PROOFSTEP]\nsimp only [e.symm, lift_lift]\n[GOAL]\n\u03b1\u271d : Type ?u.238384\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u22a2 Small.{v, u} \u03b1 \u2194 lift #\u03b1 < univ\n[PROOFSTEP]\nrw [lt_univ']\n[GOAL]\n\u03b1\u271d : Type ?u.238384\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u22a2 Small.{v, u} \u03b1 \u2194 \u2203 c', lift #\u03b1 = lift c'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d : Type ?u.238384\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u22a2 Small.{v, u} \u03b1 \u2192 \u2203 c', lift #\u03b1 = lift c'\n[PROOFSTEP]\nrintro \u27e8\u03b2, e\u27e9\n[GOAL]\ncase mp.mk.intro\n\u03b1\u271d : Type ?u.238384\n\u03b2\u271d : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u03b2 : Type v\ne : Nonempty (\u03b1 \u2243 \u03b2)\n\u22a2 \u2203 c', lift #\u03b1 = lift c'\n[PROOFSTEP]\nexact \u27e8#\u03b2, lift_mk_eq.{u, _, v + 1}.2 e\u27e9\n[GOAL]\ncase mpr\n\u03b1\u271d : Type ?u.238384\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\n\u22a2 (\u2203 c', lift #\u03b1 = lift c') \u2192 Small.{v, u} \u03b1\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1\u271d : Type ?u.238384\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b1 : Type u\nc : Cardinal.{v}\nhc : lift #\u03b1 = lift c\n\u22a2 Small.{v, u} \u03b1\n[PROOFSTEP]\nexact \u27e8\u27e8c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))\u27e9\u27e9\n[GOAL]\n\u03b1 : Type ?u.238720\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nn : \u2115\n\u22a2 \u2191n \u2264 card o \u2194 \u2191n \u2264 o\n[PROOFSTEP]\nrw [\u2190 Cardinal.ord_le, Cardinal.ord_nat]\n[GOAL]\n\u03b1 : Type ?u.239208\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nn : \u2115\n\u22a2 \u2191n < card o \u2194 \u2191n < o\n[PROOFSTEP]\nrw [\u2190 succ_le_iff, \u2190 succ_le_iff, \u2190 nat_succ, nat_le_card]\n[GOAL]\n\u03b1 : Type ?u.239208\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nn : \u2115\n\u22a2 \u2191(Nat.succ n) \u2264 o \u2194 succ \u2191n \u2264 o\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.241773\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nn : \u2115\n\u22a2 card o = \u2191n \u2194 o = \u2191n\n[PROOFSTEP]\nsimp only [le_antisymm_iff, card_le_nat, nat_le_card]\n[GOAL]\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : Fintype \u03b1\n\u22a2 type r = \u2191(Fintype.card \u03b1)\n[PROOFSTEP]\nrw [\u2190 card_eq_nat, card_type, mk_fintype]\n[GOAL]\n\u03b1 : Type ?u.243192\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nn : \u2115\n\u22a2 (type fun x x_1 => x < x_1) = \u2191n\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Ordinal.Basic", "llama_tokens": 78206, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.26774922489378633}} {"text": "[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\n\u22a2 Inhabited (ColimitType F)\n[PROOFSTEP]\ndsimp [ColimitType]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\n\u22a2 Inhabited (Quotient (colimitSetoid F))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\n\u22a2 Monoid (ColimitType F)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f \u226b coconeMorphism F j' = coconeMorphism F j\n[PROOFSTEP]\next\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\nj j' : J\nf : j \u27f6 j'\nx\u271d : \u2191(F.obj j)\n\u22a2 \u2191(F.map f \u226b coconeMorphism F j') x\u271d = \u2191(coconeMorphism F j) x\u271d\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase w.a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\nj j' : J\nf : j \u27f6 j'\nx\u271d : \u2191(F.obj j)\n\u22a2 Setoid.r (Prequotient.of j' (\u2191(F.map f) x\u271d)) (Prequotient.of j x\u271d)\n[PROOFSTEP]\napply Relation.map\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\nj j' : J\nf : j \u27f6 j'\nx : \u2191(F.obj j)\n\u22a2 \u2191(coconeMorphism F j') (\u2191(F.map f) x) = \u2191(coconeMorphism F j) x\n[PROOFSTEP]\nrw [\u2190 cocone_naturality F f]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\nj j' : J\nf : j \u27f6 j'\nx : \u2191(F.obj j)\n\u22a2 \u2191(coconeMorphism F j') (\u2191(F.map f) x) = \u2191(F.map f \u226b coconeMorphism F j') x\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\n\u22a2 ColimitType F \u2192 \u2191s.pt\n[PROOFSTEP]\nfapply Quot.lift\n[GOAL]\ncase f\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\n\u22a2 Prequotient F \u2192 \u2191s.pt\n[PROOFSTEP]\nexact descFunLift F s\n[GOAL]\ncase a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\n\u22a2 \u2200 (a b : Prequotient F), Setoid.r a b \u2192 descFunLift F s a = descFunLift F s b\n[PROOFSTEP]\nintro x y r\n[GOAL]\ncase a\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : Prequotient F\nr : Setoid.r x y\n\u22a2 descFunLift F s x = descFunLift F s y\n[PROOFSTEP]\ninduction' r with _ _ _ _ h _ _ _ _ _ h\u2081 h\u2082 _ _ f x _ _ _ _ _ _ _ _ h _ _ _ _ h\n[GOAL]\ncase a.refl\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s x\u271d = descFunLift F s x\u271d\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.refl\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s x\u271d = descFunLift F s x\u271d\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.symm\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 y\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s y\u271d\n\u22a2 descFunLift F s y\u271d = descFunLift F s x\u271d\u00b9\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.symm\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 y\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s y\u271d\n\u22a2 descFunLift F s y\u271d = descFunLift F s x\u271d\u00b9\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.trans\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b2 y\u271d z\u271d : Prequotient F\nx\u271d\u00b9 : Relation F x\u271d\u00b2 y\u271d\nx\u271d : Relation F y\u271d z\u271d\nh\u2081 : descFunLift F s x\u271d\u00b2 = descFunLift F s y\u271d\nh\u2082 : descFunLift F s y\u271d = descFunLift F s z\u271d\n\u22a2 descFunLift F s x\u271d\u00b2 = descFunLift F s z\u271d\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.trans\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b2 y\u271d z\u271d : Prequotient F\nx\u271d\u00b9 : Relation F x\u271d\u00b2 y\u271d\nx\u271d : Relation F y\u271d z\u271d\nh\u2081 : descFunLift F s x\u271d\u00b2 = descFunLift F s y\u271d\nh\u2082 : descFunLift F s y\u271d = descFunLift F s z\u271d\n\u22a2 descFunLift F s x\u271d\u00b2 = descFunLift F s z\u271d\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.map\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx\u271d y : Prequotient F\nj\u271d j'\u271d : J\nf : j\u271d \u27f6 j'\u271d\nx : \u2191(F.obj j\u271d)\n\u22a2 descFunLift F s (Prequotient.of j'\u271d (\u2191(F.map f) x)) = descFunLift F s (Prequotient.of j\u271d x)\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.map\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx\u271d y : Prequotient F\nj\u271d j'\u271d : J\nf : j\u271d \u27f6 j'\u271d\nx : \u2191(F.obj j\u271d)\n\u22a2 descFunLift F s (Prequotient.of j'\u271d (\u2191(F.map f) x)) = descFunLift F s (Prequotient.of j\u271d x)\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : Prequotient F\nj\u271d : J\nx\u271d y\u271d : \u2191(F.obj j\u271d)\n\u22a2 descFunLift F s (Prequotient.of j\u271d (x\u271d * y\u271d)) = descFunLift F s (mul (Prequotient.of j\u271d x\u271d) (Prequotient.of j\u271d y\u271d))\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : Prequotient F\nj\u271d : J\nx\u271d y\u271d : \u2191(F.obj j\u271d)\n\u22a2 descFunLift F s (Prequotient.of j\u271d (x\u271d * y\u271d)) = descFunLift F s (mul (Prequotient.of j\u271d x\u271d) (Prequotient.of j\u271d y\u271d))\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : Prequotient F\nj\u271d : J\n\u22a2 descFunLift F s (Prequotient.of j\u271d 1) = descFunLift F s one\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : Prequotient F\nj\u271d : J\n\u22a2 descFunLift F s (Prequotient.of j\u271d 1) = descFunLift F s one\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.mul_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 x'\u271d y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 x'\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s x'\u271d\n\u22a2 descFunLift F s (mul x\u271d\u00b9 y\u271d) = descFunLift F s (mul x'\u271d y\u271d)\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.mul_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 x'\u271d y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 x'\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s x'\u271d\n\u22a2 descFunLift F s (mul x\u271d\u00b9 y\u271d) = descFunLift F s (mul x'\u271d y\u271d)\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.mul_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d y'\u271d : Prequotient F\nx\u271d : Relation F y\u271d y'\u271d\nh : descFunLift F s y\u271d = descFunLift F s y'\u271d\n\u22a2 descFunLift F s (mul x\u271d\u00b9 y\u271d) = descFunLift F s (mul x\u271d\u00b9 y'\u271d)\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.mul_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d y'\u271d : Prequotient F\nx\u271d : Relation F y\u271d y'\u271d\nh : descFunLift F s y\u271d = descFunLift F s y'\u271d\n\u22a2 descFunLift F s (mul x\u271d\u00b9 y\u271d) = descFunLift F s (mul x\u271d\u00b9 y'\u271d)\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.mul_assoc\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d y\u271d z\u271d : Prequotient F\n\u22a2 descFunLift F s (mul (mul x\u271d y\u271d) z\u271d) = descFunLift F s (mul x\u271d (mul y\u271d z\u271d))\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.mul_assoc\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d y\u271d z\u271d : Prequotient F\n\u22a2 descFunLift F s (mul (mul x\u271d y\u271d) z\u271d) = descFunLift F s (mul x\u271d (mul y\u271d z\u271d))\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.one_mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s (mul one x\u271d) = descFunLift F s x\u271d\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.one_mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s (mul one x\u271d) = descFunLift F s x\u271d\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.mul_one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s (mul x\u271d one) = descFunLift F s x\u271d\n[PROOFSTEP]\ntry\n simp\n -- symm\n[GOAL]\ncase a.mul_one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d : Prequotient F\n\u22a2 descFunLift F s (mul x\u271d one) = descFunLift F s x\u271d\n[PROOFSTEP]\nsimp\n -- symm\n[GOAL]\ncase a.symm\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 y\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s y\u271d\n\u22a2 descFunLift F s y\u271d = descFunLift F s x\u271d\u00b9\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase a.trans\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b2 y\u271d z\u271d : Prequotient F\nx\u271d\u00b9 : Relation F x\u271d\u00b2 y\u271d\nx\u271d : Relation F y\u271d z\u271d\nh\u2081 : descFunLift F s x\u271d\u00b2 = descFunLift F s y\u271d\nh\u2082 : descFunLift F s y\u271d = descFunLift F s z\u271d\n\u22a2 descFunLift F s x\u271d\u00b2 = descFunLift F s z\u271d\n[PROOFSTEP]\nexact h\u2081.trans h\u2082\n[GOAL]\ncase a.map\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx\u271d y : Prequotient F\nj\u271d j'\u271d : J\nf : j\u271d \u27f6 j'\u271d\nx : \u2191(F.obj j\u271d)\n\u22a2 \u2191(NatTrans.app s.\u03b9 j'\u271d) (\u2191(F.map f) x) = \u2191(NatTrans.app s.\u03b9 j\u271d) x\n[PROOFSTEP]\nexact s.w_apply f x\n[GOAL]\ncase a.mul_1\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 x'\u271d y\u271d : Prequotient F\nx\u271d : Relation F x\u271d\u00b9 x'\u271d\nh : descFunLift F s x\u271d\u00b9 = descFunLift F s x'\u271d\n\u22a2 descFunLift F s x\u271d\u00b9 * descFunLift F s y\u271d = descFunLift F s x'\u271d * descFunLift F s y\u271d\n[PROOFSTEP]\nrw [h]\n -- mul_2\n[GOAL]\ncase a.mul_2\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d\u00b9 y\u271d y'\u271d : Prequotient F\nx\u271d : Relation F y\u271d y'\u271d\nh : descFunLift F s y\u271d = descFunLift F s y'\u271d\n\u22a2 descFunLift F s x\u271d\u00b9 * descFunLift F s y\u271d = descFunLift F s x\u271d\u00b9 * descFunLift F s y'\u271d\n[PROOFSTEP]\nrw [h]\n -- mul_assoc\n[GOAL]\ncase a.mul_assoc\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y x\u271d y\u271d z\u271d : Prequotient F\n\u22a2 descFunLift F s x\u271d * descFunLift F s y\u271d * descFunLift F s z\u271d =\n descFunLift F s x\u271d * (descFunLift F s y\u271d * descFunLift F s z\u271d)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nx y : \u2191(colimit F)\n\u22a2 OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } (x * y) =\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } x *\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } y\n[PROOFSTEP]\ninduction x using Quot.inductionOn\n[GOAL]\ncase h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\ny : \u2191(colimit F)\na\u271d : Prequotient F\n\u22a2 OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } (Quot.mk Setoid.r a\u271d * y) =\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } (Quot.mk Setoid.r a\u271d) *\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } y\n[PROOFSTEP]\ninduction y using Quot.inductionOn\n[GOAL]\ncase h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\na\u271d\u00b9 a\u271d : Prequotient F\n\u22a2 OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) }\n (Quot.mk Setoid.r a\u271d\u00b9 * Quot.mk Setoid.r a\u271d) =\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } (Quot.mk Setoid.r a\u271d\u00b9) *\n OneHom.toFun { toFun := descFun F s, map_one' := (_ : descFun F s 1 = descFun F s 1) } (Quot.mk Setoid.r a\u271d)\n[PROOFSTEP]\ndsimp [descFun]\n[GOAL]\ncase h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\na\u271d\u00b9 a\u271d : Prequotient F\n\u22a2 Quot.lift (descFunLift F s) (_ : \u2200 (x y : Prequotient F), Setoid.r x y \u2192 descFunLift F s x = descFunLift F s y)\n (Quot.mk Setoid.r a\u271d\u00b9 * Quot.mk Setoid.r a\u271d) =\n descFunLift F s a\u271d\u00b9 * descFunLift F s a\u271d\n[PROOFSTEP]\nrw [\u2190 quot_mul]\n[GOAL]\ncase h.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\na\u271d\u00b9 a\u271d : Prequotient F\n\u22a2 Quot.lift (descFunLift F s) (_ : \u2200 (x y : Prequotient F), Setoid.r x y \u2192 descFunLift F s x = descFunLift F s y)\n (Quot.mk Setoid.r (mul a\u271d\u00b9 a\u271d)) =\n descFunLift F s a\u271d\u00b9 * descFunLift F s a\u271d\n[PROOFSTEP]\nsimp only [descFunLift]\n[GOAL]\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s => descMorphism F s) s\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx : \u2191(colimitCocone F).pt\n\u22a2 \u2191m x = \u2191((fun s => descMorphism F s) s) x\n[PROOFSTEP]\ninduction' x using Quot.inductionOn with x\n[GOAL]\ncase w.h\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx : Prequotient F\n\u22a2 \u2191m (Quot.mk Setoid.r x) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x)\n[PROOFSTEP]\ninduction' x with j x x y hx hy\n[GOAL]\ncase w.h.of\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nj : J\nx : \u2191(F.obj j)\n\u22a2 \u2191m (Quot.mk Setoid.r (Prequotient.of j x)) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r (Prequotient.of j x))\n[PROOFSTEP]\nchange _ = s.\u03b9.app j _\n[GOAL]\ncase w.h.of\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nj : J\nx : \u2191(F.obj j)\n\u22a2 \u2191m (Quot.mk Setoid.r (Prequotient.of j x)) = \u2191(NatTrans.app s.\u03b9 j) x\n[PROOFSTEP]\nrw [\u2190 w j]\n[GOAL]\ncase w.h.of\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nj : J\nx : \u2191(F.obj j)\n\u22a2 \u2191m (Quot.mk Setoid.r (Prequotient.of j x)) = \u2191(NatTrans.app (colimitCocone F).\u03b9 j \u226b m) x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase w.h.one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 \u2191m (Quot.mk Setoid.r one) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r one)\n[PROOFSTEP]\nrw [quot_one, map_one]\n[GOAL]\ncase w.h.one\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 1 = \u2191((fun s => descMorphism F s) s) 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase w.h.mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx y : Prequotient F\nhx : \u2191m (Quot.mk Setoid.r x) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x)\nhy : \u2191m (Quot.mk Setoid.r y) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r y)\n\u22a2 \u2191m (Quot.mk Setoid.r (mul x y)) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r (mul x y))\n[PROOFSTEP]\nrw [quot_mul, map_mul, hx, hy]\n[GOAL]\ncase w.h.mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx y : Prequotient F\nhx : \u2191m (Quot.mk Setoid.r x) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x)\nhy : \u2191m (Quot.mk Setoid.r y) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r y)\n\u22a2 \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x) * \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r y) =\n \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x * Quot.mk Setoid.r y)\n[PROOFSTEP]\ndsimp [descMorphism, FunLike.coe, descFun]\n[GOAL]\ncase w.h.mul\nJ : Type v\ninst\u271d : SmallCategory J\nF : J \u2964 MonCat\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx y : Prequotient F\nhx : \u2191m (Quot.mk Setoid.r x) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r x)\nhy : \u2191m (Quot.mk Setoid.r y) = \u2191((fun s => descMorphism F s) s) (Quot.mk Setoid.r y)\n\u22a2 descFunLift F s x * descFunLift F s y =\n Quot.lift (descFunLift F s) (_ : \u2200 (x y : Prequotient F), Setoid.r x y \u2192 descFunLift F s x = descFunLift F s y)\n (Quot.mk Setoid.r x * Quot.mk Setoid.r y)\n[PROOFSTEP]\nsimp only [\u2190 quot_mul, descFunLift]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.MonCat.Colimits", "llama_tokens": 7968, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.26767739484299075}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nS : Sieve X\ng : Y \u27f6 X\n\u22a2 S \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) X \u2192\n Sieve.pullback g S \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) Y\n[PROOFSTEP]\nrintro \u27e8R, hR, RS\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nS : Sieve X\ng : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 Sieve.pullback g S \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) Y\n[PROOFSTEP]\nrefine' \u27e8_, K.pullbacks g _ hR, _\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nS : Sieve X\ng : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 pullbackArrows g R \u2264 (Sieve.pullback g S).arrows\n[PROOFSTEP]\nrw [\u2190 Sieve.sets_iff_generate, Sieve.pullbackArrows_comm]\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nS : Sieve X\ng : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 Sieve.pullback g (Sieve.generate R) \u2264 Sieve.pullback g S\n[PROOFSTEP]\napply Sieve.pullback_monotone\n[GOAL]\ncase intro.intro.a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nS : Sieve X\ng : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 Sieve.generate R \u2264 S\n[PROOFSTEP]\nrwa [Sieve.giGenerate.gc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 \u2200 \u2983X : C\u2984 \u2983S : Sieve X\u2984,\n S \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) X \u2192\n \u2200 (R : Sieve X),\n (\u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n S.arrows f \u2192 Sieve.pullback f R \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) Y) \u2192\n R \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) X\n[PROOFSTEP]\nrintro X S \u27e8R', hR', RS\u27e9 R t\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX : C\nS : Sieve X\nR' : Presieve X\nhR' : R' \u2208 coverings K X\nRS : R' \u2264 S.arrows\nR : Sieve X\nt : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984, S.arrows f \u2192 Sieve.pullback f R \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) Y\n\u22a2 R \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) X\n[PROOFSTEP]\nchoose t\u2081 t\u2082 t\u2083 using t\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX : C\nS : Sieve X\nR' : Presieve X\nhR' : R' \u2208 coverings K X\nRS : R' \u2264 S.arrows\nR : Sieve X\nt\u2081 : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S.arrows f \u2192 Presieve Y\nt\u2082 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2208 coverings K Y\nt\u2083 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2264 (Sieve.pullback f R).arrows\n\u22a2 R \u2208 (fun X S => \u2203 R, R \u2208 coverings K X \u2227 R \u2264 S.arrows) X\n[PROOFSTEP]\nrefine' \u27e8_, K.Transitive _ _ hR' fun _ f hf => t\u2082 (RS _ hf), _\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX : C\nS : Sieve X\nR' : Presieve X\nhR' : R' \u2208 coverings K X\nRS : R' \u2264 S.arrows\nR : Sieve X\nt\u2081 : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S.arrows f \u2192 Presieve Y\nt\u2082 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2208 coverings K Y\nt\u2083 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2264 (Sieve.pullback f R).arrows\n\u22a2 (Presieve.bind R' fun x f hf => t\u2081 (_ : f \u2208 S.arrows)) \u2264 R.arrows\n[PROOFSTEP]\nrintro Y _ \u27e8Z, g, f, hg, hf, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX : C\nS : Sieve X\nR' : Presieve X\nhR' : R' \u2208 coverings K X\nRS : R' \u2264 S.arrows\nR : Sieve X\nt\u2081 : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S.arrows f \u2192 Presieve Y\nt\u2082 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2208 coverings K Y\nt\u2083 : \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984 (a : S.arrows f), t\u2081 a \u2264 (Sieve.pullback f R).arrows\nY Z : C\ng : Y \u27f6 Z\nf : Z \u27f6 X\nhg : R' f\nhf : t\u2081 (_ : f \u2208 S.arrows) g\n\u22a2 g \u226b f \u2208 R.arrows\n[PROOFSTEP]\napply t\u2083 (RS _ hg) _ hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX Y : C\nf : Y \u27f6 X\ni : IsIso f\n\u22a2 Sieve.generate (Presieve.singleton f) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\n\u22a2 pullbackArrows f R \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\n[PROOFSTEP]\nsimp only [Set.mem_def, Sieve.pullbackArrows_comm]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nhR : R \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\n\u22a2 GrothendieckTopology.sieves J Y (Sieve.pullback f (Sieve.generate R))\n[PROOFSTEP]\napply J.pullback_stable f hR\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\n\u22a2 Presieve.bind S Ti \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\n[PROOFSTEP]\napply J.transitive hS\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 X\u2984,\n (Sieve.generate S).arrows f \u2192\n Sieve.pullback f (Sieve.generate (Presieve.bind S Ti)) \u2208 GrothendieckTopology.sieves J Y\n[PROOFSTEP]\nintro Y f\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY : C\nf : Y \u27f6 X\n\u22a2 (Sieve.generate S).arrows f \u2192 Sieve.pullback f (Sieve.generate (Presieve.bind S Ti)) \u2208 GrothendieckTopology.sieves J Y\n[PROOFSTEP]\nrintro \u27e8Z, g, f, hf, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY Z : C\ng : Y \u27f6 Z\nf : Z \u27f6 X\nhf : S f\n\u22a2 Sieve.pullback (g \u226b f) (Sieve.generate (Presieve.bind S Ti)) \u2208 GrothendieckTopology.sieves J Y\n[PROOFSTEP]\nrw [Sieve.pullback_comp]\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY Z : C\ng : Y \u27f6 Z\nf : Z \u27f6 X\nhf : S f\n\u22a2 Sieve.pullback g (Sieve.pullback f (Sieve.generate (Presieve.bind S Ti))) \u2208 GrothendieckTopology.sieves J Y\n[PROOFSTEP]\napply J.pullback_stable g\n[GOAL]\ncase h.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY Z : C\ng : Y \u27f6 Z\nf : Z \u27f6 X\nhf : S f\n\u22a2 Sieve.pullback f (Sieve.generate (Presieve.bind S Ti)) \u2208 GrothendieckTopology.sieves J Z\n[PROOFSTEP]\napply J.superset_covering _ (hTi _ hf)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY Z : C\ng : Y \u27f6 Z\nf : Z \u27f6 X\nhf : S f\n\u22a2 Sieve.generate (Ti f hf) \u2264 Sieve.pullback f (Sieve.generate (Presieve.bind S Ti))\n[PROOFSTEP]\nrintro Y g \u27e8W, h, g, hg, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY\u271d Z : C\ng\u271d : Y\u271d \u27f6 Z\nf : Z \u27f6 X\nhf : S f\nY W : C\nh : Y \u27f6 W\ng : W \u27f6 Z\nhg : Ti f hf g\n\u22a2 (Sieve.pullback f (Sieve.generate (Presieve.bind S Ti))).arrows (h \u226b g)\n[PROOFSTEP]\nexact \u27e8_, h, _, \u27e8_, _, _, hf, hg, rfl\u27e9, by simp\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nJ : GrothendieckTopology C\nX : C\nS : Presieve X\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y\nhS : S \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) X\nhTi : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X R => Sieve.generate R \u2208 GrothendieckTopology.sieves J X) Y\nY\u271d Z : C\ng\u271d : Y\u271d \u27f6 Z\nf : Z \u27f6 X\nhf : S f\nY W : C\nh : Y \u27f6 W\ng : W \u27f6 Z\nhg : Ti f hf g\n\u22a2 h \u226b g \u226b f = (h \u226b g) \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\n\u22a2 toGrothendieck C K \u2264 J \u2194 K \u2264 ofGrothendieck C J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\n\u22a2 toGrothendieck C K \u2264 J \u2192 K \u2264 ofGrothendieck C J\n[PROOFSTEP]\nintro h X R hR\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\nh : toGrothendieck C K \u2264 J\nX : C\nR : Presieve X\nhR : R \u2208 coverings K X\n\u22a2 R \u2208 coverings (ofGrothendieck C J) X\n[PROOFSTEP]\nexact h _ \u27e8_, hR, Sieve.le_generate R\u27e9\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\n\u22a2 K \u2264 ofGrothendieck C J \u2192 toGrothendieck C K \u2264 J\n[PROOFSTEP]\nrintro h X S \u27e8R, hR, RS\u27e9\n[GOAL]\ncase mpr.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\nh : K \u2264 ofGrothendieck C J\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 S \u2208 GrothendieckTopology.sieves J X\n[PROOFSTEP]\napply J.superset_covering _ (h _ hR)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nJ : GrothendieckTopology C\nh : K \u2264 ofGrothendieck C J\nX : C\nS : Sieve X\nR : Presieve X\nhR : R \u2208 coverings K X\nRS : R \u2264 S.arrows\n\u22a2 Sieve.generate R \u2264 S\n[PROOFSTEP]\nrwa [Sieve.giGenerate.gc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nS : Presieve X\n\u22a2 S \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) X \u2192\n pullbackArrows f S \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\n[PROOFSTEP]\nrintro \u27e8Z, g, i, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 pullbackArrows f (Presieve.singleton g) \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\n[PROOFSTEP]\nrefine' \u27e8pullback g f, pullback.snd, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 IsIso pullback.snd\n[PROOFSTEP]\nrefine' \u27e8\u27e8pullback.lift (f \u226b inv g) (\ud835\udfd9 _) (by simp), \u27e8_, by aesop_cat\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 (f \u226b inv g) \u226b g = \ud835\udfd9 Y \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 pullback.lift (f \u226b inv g) (\ud835\udfd9 Y) (_ : (f \u226b inv g) \u226b g = \ud835\udfd9 Y \u226b f) \u226b pullback.snd = \ud835\udfd9 Y\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase intro.intro.intro.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 pullback.snd \u226b pullback.lift (f \u226b inv g) (\ud835\udfd9 Y) (_ : (f \u226b inv g) \u226b g = \ud835\udfd9 Y \u226b f) = \ud835\udfd9 (pullback g f)\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.intro.refine'_1.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 (pullback.snd \u226b pullback.lift (f \u226b inv g) (\ud835\udfd9 Y) (_ : (f \u226b inv g) \u226b g = \ud835\udfd9 Y \u226b f)) \u226b pullback.fst =\n \ud835\udfd9 (pullback g f) \u226b pullback.fst\n[PROOFSTEP]\nrw [assoc, pullback.lift_fst, \u2190 pullback.condition_assoc]\n[GOAL]\ncase intro.intro.intro.refine'_1.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 pullback.fst \u226b g \u226b inv g = \ud835\udfd9 (pullback g f) \u226b pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.refine'_1.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 (pullback.snd \u226b pullback.lift (f \u226b inv g) (\ud835\udfd9 Y) (_ : (f \u226b inv g) \u226b g = \ud835\udfd9 Y \u226b f)) \u226b pullback.snd =\n \ud835\udfd9 (pullback g f) \u226b pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\ni : IsIso g\n\u22a2 pullbackArrows f (Presieve.singleton g) = Presieve.singleton pullback.snd\n[PROOFSTEP]\napply pullback_singleton\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\n\u22a2 \u2200 \u2983X : C\u2984 (S : Presieve X) (Ti : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 S f \u2192 Presieve Y),\n S \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) X \u2192\n (\u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : S f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y) \u2192\n Presieve.bind S Ti \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) X\n[PROOFSTEP]\nrintro X S Ti \u27e8Z, g, i, rfl\u27e9 hS\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\n\u22a2 Presieve.bind (Presieve.singleton g) Ti \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) X\n[PROOFSTEP]\nrcases hS g (singleton_self g) with \u27e8Y, f, i, hTi\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\n\u22a2 Presieve.bind (Presieve.singleton g) Ti \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) X\n[PROOFSTEP]\nrefine' \u27e8_, f \u226b g, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\n\u22a2 IsIso (f \u226b g)\n[PROOFSTEP]\ninfer_instance\n -- Porting note: the next four lines were just \"ext (W k)\"\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\n\u22a2 Presieve.bind (Presieve.singleton g) Ti = Presieve.singleton (f \u226b g)\n[PROOFSTEP]\napply funext\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\n\u22a2 \u2200 (x : C), Presieve.bind (Presieve.singleton g) Ti = Presieve.singleton (f \u226b g)\n[PROOFSTEP]\nrintro W\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW : C\n\u22a2 Presieve.bind (Presieve.singleton g) Ti = Presieve.singleton (f \u226b g)\n[PROOFSTEP]\napply Set.ext\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW : C\n\u22a2 \u2200 (x : W \u27f6 X), x \u2208 Presieve.bind (Presieve.singleton g) Ti \u2194 x \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\nrintro k\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW : C\nk : W \u27f6 X\n\u22a2 k \u2208 Presieve.bind (Presieve.singleton g) Ti \u2194 k \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW : C\nk : W \u27f6 X\n\u22a2 k \u2208 Presieve.bind (Presieve.singleton g) Ti \u2192 k \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\nrintro \u27e8V, h, k, \u27e8_\u27e9, hh, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mp.intro.intro.intro.intro.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW Y : C\nh : W \u27f6 Z\nhh : Ti g (_ : singleton' g g) h\n\u22a2 h \u226b g \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\nrw [hTi] at hh \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mp.intro.intro.intro.intro.mk.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW Y : C\nh : W \u27f6 Z\nhh : Presieve.singleton f h\n\u22a2 h \u226b g \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\ncases hh\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mp.intro.intro.intro.intro.mk.intro.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nY : C\n\u22a2 f \u226b g \u2208 Presieve.singleton (f \u226b g)\n[PROOFSTEP]\napply singleton.mk\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY : C\nf : Y \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nW : C\nk : W \u27f6 X\n\u22a2 k \u2208 Presieve.singleton (f \u226b g) \u2192 k \u2208 Presieve.bind (Presieve.singleton g) Ti\n[PROOFSTEP]\nrintro \u27e8_\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mpr.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nY : C\n\u22a2 f \u226b g \u2208 Presieve.bind (Presieve.singleton g) Ti\n[PROOFSTEP]\nrefine' bind_comp g singleton.mk _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mpr.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nY : C\n\u22a2 Ti g (_ : Presieve.singleton g g) f\n[PROOFSTEP]\nrw [hTi]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2.h.h.mpr.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nX Z : C\ng : Z \u27f6 X\ni\u271d : IsIso g\nTi : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 Presieve.singleton g f \u2192 Presieve Y\nhS : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X) (H : Presieve.singleton g f), Ti f H \u2208 (fun X S => \u2203 Y f x, S = Presieve.singleton f) Y\nY\u271d : C\nf : Y\u271d \u27f6 Z\ni : IsIso f\nhTi : Ti g (_ : Presieve.singleton g g) = Presieve.singleton f\nY : C\n\u22a2 Presieve.singleton f f\n[PROOFSTEP]\napply singleton.mk\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX : C\nR : Presieve X\n\u22a2 R \u2208 coverings \u22a5 X \u2192 R \u2208 coverings K X\n[PROOFSTEP]\nrintro \u27e8Y, f, hf, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasPullbacks C\nK : Pretopology C\nX Y : C\nf : Y \u27f6 X\nhf : IsIso f\n\u22a2 Presieve.singleton f \u2208 coverings K X\n[PROOFSTEP]\nexact K.has_isos f\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Pretopology", "llama_tokens": 11802, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.26740164063736316}} {"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\n\u22a2 \u2200 {s : Set H} {x : H} {u : Set H} {f : H \u2192 H'},\n IsOpen u \u2192 x \u2208 u \u2192 (DifferentiableWithinAtProp I I' f s x \u2194 DifferentiableWithinAtProp I I' f (s \u2229 u) x)\n[PROOFSTEP]\nintro s x u f u_open xu\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\n\u22a2 DifferentiableWithinAtProp I I' f s x \u2194 DifferentiableWithinAtProp I I' f (s \u2229 u) x\n[PROOFSTEP]\nhave : I.symm \u207b\u00b9' (s \u2229 u) \u2229 Set.range I = I.symm \u207b\u00b9' s \u2229 Set.range I \u2229 I.symm \u207b\u00b9' u := by\n simp only [Set.inter_right_comm, Set.preimage_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n[PROOFSTEP]\nsimp only [Set.inter_right_comm, Set.preimage_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 DifferentiableWithinAtProp I I' f s x \u2194 DifferentiableWithinAtProp I I' f (s \u2229 u) x\n[PROOFSTEP]\nrw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x) \u2194\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u) (\u2191I x)\n[PROOFSTEP]\nsymm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u) (\u2191I x) \u2194\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\n[PROOFSTEP]\napply differentiableWithinAt_inter\n[GOAL]\ncase ht\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2208 \ud835\udcdd (\u2191I x)\n[PROOFSTEP]\nhave : u \u2208 \ud835\udcdd (I.symm (I x)) := by\n rw [ModelWithCorners.left_inv]\n exact IsOpen.mem_nhds u_open xu\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 u \u2208 \ud835\udcdd (\u2191(ModelWithCorners.symm I) (\u2191I x))\n[PROOFSTEP]\nrw [ModelWithCorners.left_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\n\u22a2 u \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact IsOpen.mem_nhds u_open xu\n[GOAL]\ncase ht\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H'\nu_open : IsOpen u\nxu : x \u2208 u\nthis\u271d :\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (s \u2229 u) \u2229 range \u2191I =\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u\nthis : u \u2208 \ud835\udcdd (\u2191(ModelWithCorners.symm I) (\u2191I x))\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2208 \ud835\udcdd (\u2191I x)\n[PROOFSTEP]\napply I.continuous_symm.continuousAt this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\n\u22a2 \u2200 {s : Set H} {x : H} {f : H \u2192 H'} {e : LocalHomeomorph H H},\n e \u2208 contDiffGroupoid \u22a4 I \u2192\n x \u2208 e.source \u2192\n DifferentiableWithinAtProp I I' f s x \u2192\n DifferentiableWithinAtProp I I' (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x)\n[PROOFSTEP]\nintro s x f e he hx h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh : DifferentiableWithinAtProp I I' f s x\n\u22a2 DifferentiableWithinAtProp I I' (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x)\n[PROOFSTEP]\nrw [DifferentiableWithinAtProp] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) (\u2191I (\u2191e x))\n[PROOFSTEP]\nhave : I x = (I \u2218 e.symm \u2218 I.symm) (I (e x)) := by simp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\n\u22a2 \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nthis : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) (\u2191I (\u2191e x))\n[PROOFSTEP]\nrw [this] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) (\u2191I (\u2191e x))\n[PROOFSTEP]\nhave : I (e x) \u2208 I.symm \u207b\u00b9' e.target \u2229 Set.range I := by simp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\n\u22a2 \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) (\u2191I (\u2191e x))\n[PROOFSTEP]\nhave := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) (\u2191I (\u2191e x))\n[PROOFSTEP]\nconvert (h.comp' _ (this.differentiableWithinAt le_top)).mono_of_mem _ using 1\n[GOAL]\ncase h.e'_9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\n\u22a2 \u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I) =\n (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) \u2218 \u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_9.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\ny : E\n\u22a2 (\u2191I' \u2218 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I)) y =\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) \u2218 \u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) y\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I \u2229\n \u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) \u2208\n \ud835\udcdd[\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I] \u2191I (\u2191e x)\n[PROOFSTEP]\nrefine'\n mem_nhdsWithin.mpr\n \u27e8I.symm \u207b\u00b9' e.target, e.open_target.preimage I.continuous_symm, by\n simp_rw [Set.mem_preimage, I.left_inv, e.mapsTo hx], _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\n\u22a2 \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target\n[PROOFSTEP]\nsimp_rw [Set.mem_preimage, I.left_inv, e.mapsTo hx]\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne : LocalHomeomorph H H\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhx : x \u2208 e.source\nh :\n DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n ((\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x)))\nthis\u271d\u00b9 : \u2191I x = (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I (\u2191e x))\nthis\u271d : \u2191I (\u2191e x) \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) (\u2191I (\u2191e x))\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2229 range \u2191I) \u2286\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I \u2229\n \u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\n\u22a2 \u2200 {s : Set H} {x : H} {f g : H \u2192 H'},\n (\u2200 (y : H), y \u2208 s \u2192 f y = g y) \u2192\n f x = g x \u2192 DifferentiableWithinAtProp I I' f s x \u2192 DifferentiableWithinAtProp I I' g s x\n[PROOFSTEP]\nintro s x f g h hx hf\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf g : H \u2192 H'\nh : \u2200 (y : H), y \u2208 s \u2192 f y = g y\nhx : f x = g x\nhf : DifferentiableWithinAtProp I I' f s x\n\u22a2 DifferentiableWithinAtProp I I' g s x\n[PROOFSTEP]\napply hf.congr\n[GOAL]\ncase ht\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf g : H \u2192 H'\nh : \u2200 (y : H), y \u2208 s \u2192 f y = g y\nhx : f x = g x\nhf : DifferentiableWithinAtProp I I' f s x\n\u22a2 \u2200 (x : E),\n x \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I \u2192\n (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) x = (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) x\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase ht\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf g : H \u2192 H'\nh : \u2200 (y : H), y \u2208 s \u2192 f y = g y\nhx : f x = g x\nhf : DifferentiableWithinAtProp I I' f s x\ny : E\nhy : y \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I\n\u22a2 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) y = (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) y\n[PROOFSTEP]\nsimp only [mfld_simps] at hy \n[GOAL]\ncase ht\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf g : H \u2192 H'\nh : \u2200 (y : H), y \u2208 s \u2192 f y = g y\nhx : f x = g x\nhf : DifferentiableWithinAtProp I I' f s x\ny : E\nhy : \u2191(ModelWithCorners.symm I) y \u2208 s \u2227 y \u2208 range \u2191I\n\u22a2 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) y = (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) y\n[PROOFSTEP]\nsimp only [h, hy, mfld_simps]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf g : H \u2192 H'\nh : \u2200 (y : H), y \u2208 s \u2192 f y = g y\nhx : f x = g x\nhf : DifferentiableWithinAtProp I I' f s x\n\u22a2 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) = (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x)\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\n\u22a2 \u2200 {s : Set H} {x : H} {f : H \u2192 H'} {e' : LocalHomeomorph H' H'},\n e' \u2208 contDiffGroupoid \u22a4 I' \u2192\n s \u2286 f \u207b\u00b9' e'.source \u2192\n f x \u2208 e'.source \u2192 DifferentiableWithinAtProp I I' f s x \u2192 DifferentiableWithinAtProp I I' (\u2191e' \u2218 f) s x\n[PROOFSTEP]\nintro s x f e' he' hs hx h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAtProp I I' f s x\n\u22a2 DifferentiableWithinAtProp I I' (\u2191e' \u2218 f) s x\n[PROOFSTEP]\nrw [DifferentiableWithinAtProp] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (\u2191e' \u2218 f) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191I x)\n[PROOFSTEP]\nhave A : (I' \u2218 f \u2218 I.symm) (I x) \u2208 I'.symm \u207b\u00b9' e'.source \u2229 Set.range I' := by simp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\n\u22a2 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (\u2191e' \u2218 f) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191I x)\n[PROOFSTEP]\nhave := (mem_groupoid_of_pregroupoid.2 he').1.contDiffWithinAt A\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 (\u2191e' \u2218 f) \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191I x)\n[PROOFSTEP]\nconvert (this.differentiableWithinAt le_top).comp _ h _\n[GOAL]\ncase h.e'_9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\n\u22a2 \u2191I' \u2218 (\u2191e' \u2218 f) \u2218 \u2191(ModelWithCorners.symm I) =\n (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) \u2218 \u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_9.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\ny : E\n\u22a2 (\u2191I' \u2218 (\u2191e' \u2218 f) \u2218 \u2191(ModelWithCorners.symm I)) y =\n ((\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) \u2218 \u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) y\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\n\u22a2 MapsTo (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\ny : E\nhy : y \u2208 \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I\n\u22a2 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) y \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\n[PROOFSTEP]\nsimp only [mfld_simps] at hy \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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'\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhs : s \u2286 f \u207b\u00b9' e'.source\nhx : f x \u2208 e'.source\nh : DifferentiableWithinAt \ud835\udd5c (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I x)\nA : (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x) \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\nthis :\n ContDiffWithinAt \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\n ((\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I x))\ny : E\nhy : \u2191(ModelWithCorners.symm I) y \u2208 s \u2227 y \u2208 range \u2191I\n\u22a2 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) y \u2208 \u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'\n[PROOFSTEP]\nsimpa only [hy, mfld_simps] using hs hy.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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 : M \u2192 M'\ns : Set M\nx : M\n\u22a2 MDifferentiableWithinAt I I' f s x \u2194 LiftPropWithinAt (DifferentiableWithinAtProp I I') f s x\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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 : M \u2192 M'\nx : M\n\u22a2 MDifferentiableAt I I' f x \u2194 LiftPropAt (DifferentiableWithinAtProp I I') f x\n[PROOFSTEP]\napply Iff.and\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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 : M \u2192 M'\nx : M\n\u22a2 ContinuousAt f x \u2194 ContinuousWithinAt f univ x\n[PROOFSTEP]\nrw [continuousWithinAt_univ]\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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 : M \u2192 M'\nx : M\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I' x f) (range \u2191I) (\u2191(extChartAt I x) x) \u2194\n DifferentiableWithinAtProp I I' (\u2191(chartAt H' (f x)) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)))\n (\u2191(LocalHomeomorph.symm (chartAt H x)) \u207b\u00b9' univ) (\u2191(chartAt H x) x)\n[PROOFSTEP]\nsimp [DifferentiableWithinAtProp, Set.univ_inter]\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\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 : M \u2192 M'\nx : M\n\u22a2 DifferentiableWithinAt \ud835\udd5c\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I)) (range \u2191I)\n (\u2191I (\u2191(chartAt H x) x)) \u2194\n DifferentiableWithinAt \ud835\udd5c\n (\u2191I' \u2218 (\u2191(chartAt H' (f x)) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x))) \u2218 \u2191(ModelWithCorners.symm I)) (range \u2191I)\n (\u2191I (\u2191(chartAt H x) x))\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\n\u22a2 UniqueMDiffWithinAt I univ x\n[PROOFSTEP]\nunfold UniqueMDiffWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' univ \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [preimage_univ, univ_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nexact I.unique_diff _ (mem_range_self _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t : Set M\ng : M' \u2192 M''\nu : Set M'\ns : Set M\nx : M\n\u22a2 UniqueMDiffWithinAt I s x \u2194\n UniqueDiffWithinAt \ud835\udd5c (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 (extChartAt I x).target) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\napply uniqueDiffWithinAt_congr\n[GOAL]\ncase st\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t : Set M\ng : M' \u2192 M''\nu : Set M'\ns : Set M\nx : M\n\u22a2 \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x =\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 (extChartAt I x).target] \u2191(extChartAt I x) x\n[PROOFSTEP]\nrw [nhdsWithin_inter, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t\u271d : Set M\ng : M' \u2192 M''\nu : Set M'\ns t : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nht : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\n\u22a2 \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x \u2264\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2229 range \u2191I] \u2191(extChartAt I x) x\n[PROOFSTEP]\nsimpa only [\u2190 map_extChartAt_nhdsWithin] using Filter.map_mono ht\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nU : UniqueMDiffWithinAt I s x\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : HasMFDerivWithinAt I I' f s x f\u2081'\n\u22a2 f' = f\u2081'\n[PROOFSTEP]\nconvert U.eq h.2 h\u2081.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t\u271d : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\ny : M'\ns : Set M\nt : Set M'\nhs : UniqueMDiffWithinAt I s x\nht : UniqueMDiffWithinAt I' t y\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I I') (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrefine (hs.prod ht).mono ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t\u271d : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\ny : M'\ns : Set M\nt : Set M'\nhs : UniqueMDiffWithinAt I s x\nht : UniqueMDiffWithinAt I' t y\n\u22a2 (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) \u00d7\u02e2 (\u2191(LocalEquiv.symm (extChartAt I' y)) \u207b\u00b9' t \u2229 range \u2191I') \u2286\n \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') (x, y))) \u207b\u00b9' s \u00d7\u02e2 t \u2229 range \u2191(ModelWithCorners.prod I I')\n[PROOFSTEP]\nrw [ModelWithCorners.range_prod, \u2190 prod_inter_prod]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t\u271d : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\ny : M'\ns : Set M\nt : Set M'\nhs : UniqueMDiffWithinAt I s x\nht : UniqueMDiffWithinAt I' t y\n\u22a2 (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s) \u00d7\u02e2 (\u2191(LocalEquiv.symm (extChartAt I' y)) \u207b\u00b9' t) \u2229 range \u2191I \u00d7\u02e2 range \u2191I' \u2286\n \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') (x, y))) \u207b\u00b9' s \u00d7\u02e2 t \u2229 range \u2191I \u00d7\u02e2 range \u2191I'\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf\u271d f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f\u271d x\u271d)\ng' : TangentSpace I' (f\u271d x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f\u271d x\u271d))\nf : M \u2192 M'\ns : Set M\nx : M\n\u22a2 MDifferentiableWithinAt I I' f s x \u2194\n ContinuousWithinAt f s x \u2227\n DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I' x f)\n ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrefine' and_congr Iff.rfl (exists_congr fun f' => _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf\u271d f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f\u271d x\u271d)\ng' : TangentSpace I' (f\u271d x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f\u271d x\u271d))\nf : M \u2192 M'\ns : Set M\nx : M\nf' : E \u2192L[\ud835\udd5c] E'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x) \u2194\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf\u271d f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns\u271d t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f\u271d x\u271d)\ng' : TangentSpace I' (f\u271d x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f\u271d x\u271d))\nf : M \u2192 M'\ns : Set M\nx : M\nf' : E \u2192L[\ud835\udd5c] E'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s)\n (\u2191(extChartAt I x) x) \u2194\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [HasFDerivWithinAt, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f s x = 0\n[PROOFSTEP]\nsimp only [mfderivWithin, h, if_neg, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : \u00acMDifferentiableAt I I' f x\n\u22a2 mfderiv I I' f x = 0\n[PROOFSTEP]\nsimp only [mfderiv, h, if_neg, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 HasMFDerivWithinAt I I' f univ x f' \u2194 HasMFDerivAt I I' f x f'\n[PROOFSTEP]\nsimp only [HasMFDerivWithinAt, HasMFDerivAt, continuousWithinAt_univ, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2080 : HasMFDerivAt I I' f x f\u2080'\nh\u2081 : HasMFDerivAt I I' f x f\u2081'\n\u22a2 f\u2080' = f\u2081'\n[PROOFSTEP]\nrw [\u2190 hasMFDerivWithinAt_univ] at h\u2080 h\u2081 \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2080 : HasMFDerivWithinAt I I' f univ x f\u2080'\nh\u2081 : HasMFDerivWithinAt I I' f univ x f\u2081'\n\u22a2 f\u2080' = f\u2081'\n[PROOFSTEP]\nexact (uniqueMDiffWithinAt_univ I).eq h\u2080 h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : t \u2208 \ud835\udcdd[s] x\n\u22a2 HasMFDerivWithinAt I I' f (s \u2229 t) x f' \u2194 HasMFDerivWithinAt I I' f s x f'\n[PROOFSTEP]\nrw [HasMFDerivWithinAt, HasMFDerivWithinAt, extChartAt_preimage_inter_eq, hasFDerivWithinAt_inter',\n continuousWithinAt_inter' h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : t \u2208 \ud835\udcdd[s] x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n[PROOFSTEP]\nexact extChartAt_preimage_mem_nhdsWithin I x h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : t \u2208 \ud835\udcdd x\n\u22a2 HasMFDerivWithinAt I I' f (s \u2229 t) x f' \u2194 HasMFDerivWithinAt I I' f s x f'\n[PROOFSTEP]\nrw [HasMFDerivWithinAt, HasMFDerivWithinAt, extChartAt_preimage_inter_eq, hasFDerivWithinAt_inter,\n continuousWithinAt_inter h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : t \u2208 \ud835\udcdd x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2208 \ud835\udcdd (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nexact extChartAt_preimage_mem_nhds I x h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : HasMFDerivWithinAt I I' f s x f'\nht : HasMFDerivWithinAt I I' f t x f'\n\u22a2 HasMFDerivWithinAt I I' f (s \u222a t) x f'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : HasMFDerivWithinAt I I' f s x f'\nht : HasMFDerivWithinAt I I' f t x f'\n\u22a2 ContinuousWithinAt f (s \u222a t) x\n[PROOFSTEP]\nexact ContinuousWithinAt.union hs.1 ht.1\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : HasMFDerivWithinAt I I' f s x f'\nht : HasMFDerivWithinAt I I' f t x f'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u222a t) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nconvert HasFDerivWithinAt.union hs.2 ht.2 using 1\n[GOAL]\ncase h.e'_11\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : HasMFDerivWithinAt I I' f s x f'\nht : HasMFDerivWithinAt I I' f t x f'\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u222a t) \u2229 range \u2191I =\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I \u222a \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2229 range \u2191I\n[PROOFSTEP]\nsimp only [union_inter_distrib_right, preimage_union]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nhs : s \u2208 \ud835\udcdd x\n\u22a2 HasMFDerivAt I I' f x f'\n[PROOFSTEP]\nrwa [\u2190 univ_inter s, hasMFDerivWithinAt_inter hs, hasMFDerivWithinAt_univ] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 HasMFDerivWithinAt I I' f s x (mfderivWithin I I' f s x)\n[PROOFSTEP]\nrefine' \u27e8h.1, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f) (mfderivWithin I I' f s x)\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfderivWithin, h, if_pos, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 HasFDerivWithinAt\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I))\n (fderivWithin \ud835\udd5c\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I (\u2191(chartAt H x) x)))\n (\u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I) (\u2191I (\u2191(chartAt H x) x))\n[PROOFSTEP]\nexact DifferentiableWithinAt.hasFDerivWithinAt h.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f s x =\n fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfderivWithin, h, if_pos]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\n\u22a2 HasMFDerivAt I I' f x (mfderiv I I' f x)\n[PROOFSTEP]\nrefine' \u27e8h.1, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f) (mfderiv I I' f x) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfderiv, h, if_pos, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\n\u22a2 HasFDerivWithinAt\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I))\n (fderivWithin \ud835\udd5c\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I)) (range \u2191I)\n (\u2191I (\u2191(chartAt H x) x)))\n (range \u2191I) (\u2191I (\u2191(chartAt H x) x))\n[PROOFSTEP]\nexact DifferentiableWithinAt.hasFDerivWithinAt h.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\n\u22a2 mfderiv I I' f x = fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfderiv, h, if_pos]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 _root_.mfderivWithin I I' f s x = f'\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nhxs : UniqueMDiffWithinAt I s x\nx\u271d : TangentSpace I x\n\u22a2 \u2191(_root_.mfderivWithin I I' f s x) x\u271d = \u2191f' x\u271d\n[PROOFSTEP]\nrw [hxs.eq h h.mdifferentiableWithinAt.hasMFDerivWithinAt]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 _root_.mfderivWithin I I' f s x = mfderiv I I' f x\n[PROOFSTEP]\napply HasMFDerivWithinAt.mfderivWithin _ hxs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableAt I I' f x\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 HasMFDerivWithinAt I I' f s x (mfderiv I I' f x)\n[PROOFSTEP]\nexact h.hasMFDerivAt.hasMFDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 MDifferentiableWithinAt I I' f univ x \u2194 MDifferentiableAt I I' f x\n[PROOFSTEP]\nsimp only [MDifferentiableWithinAt, MDifferentiableAt, continuousWithinAt_univ, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nht : t \u2208 \ud835\udcdd x\n\u22a2 MDifferentiableWithinAt I I' f (s \u2229 t) x \u2194 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nrw [MDifferentiableWithinAt, MDifferentiableWithinAt, extChartAt_preimage_inter_eq, differentiableWithinAt_inter,\n continuousWithinAt_inter ht]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nht : t \u2208 \ud835\udcdd x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2208 \ud835\udcdd (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nexact extChartAt_preimage_mem_nhds I x ht\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nht : t \u2208 \ud835\udcdd[s] x\n\u22a2 MDifferentiableWithinAt I I' f (s \u2229 t) x \u2194 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nrw [MDifferentiableWithinAt, MDifferentiableWithinAt, extChartAt_preimage_inter_eq, differentiableWithinAt_inter',\n continuousWithinAt_inter' ht]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nht : t \u2208 \ud835\udcdd[s] x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' t \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n[PROOFSTEP]\nexact extChartAt_preimage_mem_nhdsWithin I x ht\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\nhs : s \u2208 \ud835\udcdd x\n\u22a2 MDifferentiableAt I I' f x\n[PROOFSTEP]\nhave : s = univ \u2229 s := by rw [univ_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\nhs : s \u2208 \ud835\udcdd x\n\u22a2 s = univ \u2229 s\n[PROOFSTEP]\nrw [univ_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : MDifferentiableWithinAt I I' f s x\nhs : s \u2208 \ud835\udcdd x\nthis : s = univ \u2229 s\n\u22a2 MDifferentiableAt I I' f x\n[PROOFSTEP]\nrwa [this, mdifferentiableWithinAt_inter hs, mdifferentiableWithinAt_univ] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 MDifferentiableOn I I' f univ \u2194 MDifferentiable I I' f\n[PROOFSTEP]\nsimp only [MDifferentiableOn, mdifferentiableWithinAt_univ, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 (\u2200 (x : M), MDifferentiableAt I I' f x) \u2194 MDifferentiable I I' f\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : \u2200 (x : M), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 MDifferentiableOn I I' f (s \u2229 u)\n\u22a2 MDifferentiableOn I I' f s\n[PROOFSTEP]\nintro x xs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nh : \u2200 (x : M), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 MDifferentiableOn I I' f (s \u2229 u)\nx : M\nxs : x \u2208 s\n\u22a2 MDifferentiableWithinAt I I' 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\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t\u271d : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nh : \u2200 (x : M), x \u2208 s \u2192 \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 MDifferentiableOn I I' f (s \u2229 u)\nx : M\nxs : x \u2208 s\nt : Set M\nt_open : IsOpen t\nxt : x \u2208 t\nht : MDifferentiableOn I I' f (s \u2229 t)\n\u22a2 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nexact (mdifferentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 (ht x \u27e8xs, xt\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 mfderivWithin I I' f univ = mfderiv I I' f\n[PROOFSTEP]\next x : 1\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\n\u22a2 mfderivWithin I I' f univ x = mfderiv I I' f x\n[PROOFSTEP]\nsimp only [mfderivWithin, mfderiv, mfld_simps]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\n\u22a2 (if MDifferentiableWithinAt I I' f univ x then\n fderivWithin \ud835\udd5c\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I))\n (range \u2191I) (\u2191I (\u2191(chartAt H x) x))\n else 0) =\n if MDifferentiableAt I I' f x then\n fderivWithin \ud835\udd5c\n ((\u2191I' \u2218 \u2191(chartAt H' (f x))) \u2218 f \u2218 \u2191(LocalHomeomorph.symm (chartAt H x)) \u2218 \u2191(ModelWithCorners.symm I))\n (range \u2191I) (\u2191I (\u2191(chartAt H x) x))\n else 0\n[PROOFSTEP]\nrw [mdifferentiableWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nht : t \u2208 \ud835\udcdd x\n\u22a2 mfderivWithin I I' f (s \u2229 t) x = mfderivWithin I I' f s x\n[PROOFSTEP]\nrw [mfderivWithin, mfderivWithin, extChartAt_preimage_inter_eq, mdifferentiableWithinAt_inter ht,\n fderivWithin_inter (extChartAt_preimage_mem_nhds I x ht)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nx' : M\ny : M'\nhx : x' \u2208 (chartAt H x).toLocalEquiv.source\nhy : f x' \u2208 (chartAt H' y).toLocalEquiv.source\n\u22a2 ContinuousWithinAt f univ x' \u2227\n DifferentiableWithinAt \ud835\udd5c (\u2191(extChartAt I' y) \u2218 f \u2218 \u2191(LocalEquiv.symm (extChartAt I x)))\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' univ \u2229 range \u2191I) (\u2191(extChartAt I x) x') \u2194\n ContinuousAt f x' \u2227\n DifferentiableWithinAt \ud835\udd5c (\u2191(extChartAt I' y) \u2218 f \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) (range \u2191I)\n (\u2191(extChartAt I x) x')\n[PROOFSTEP]\nrw [continuousWithinAt_univ, Set.preimage_univ, Set.univ_inter]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\n\u22a2 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nsuffices h : MDifferentiableWithinAt I I' f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\nh : MDifferentiableWithinAt I I' f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nrwa [mdifferentiableWithinAt_inter'] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\nh : MDifferentiableWithinAt I I' f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 f \u207b\u00b9' (extChartAt I' (f x)).source \u2208 \ud835\udcdd[s] x\n[PROOFSTEP]\napply hf.1.preimage_mem_nhdsWithin\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\nh : MDifferentiableWithinAt I I' f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 (extChartAt I' (f x)).source \u2208 \ud835\udcdd (f x)\n[PROOFSTEP]\nexact extChartAt_source_mem_nhds I' (f x)\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\n\u22a2 MDifferentiableWithinAt I I' f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n[PROOFSTEP]\nrw [mdifferentiableWithinAt_iff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\n\u22a2 ContinuousWithinAt f (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x \u2227\n DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I' x f)\n ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source))\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nexact \u27e8hf.1.mono (inter_subset_left _ _), (hf.2.differentiableWithinAt hn).mono (by mfld_set_tac)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nn : \u2115\u221e\nhf : ContMDiffWithinAt I I' n f s x\nhn : 1 \u2264 n\n\u22a2 (extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2286\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm (chartAt H x)) \u207b\u00b9' s) \u2229 range \u2191I\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nst : s \u2286 t\nhs : UniqueMDiffWithinAt I s p.proj\nh : MDifferentiableWithinAt I I' f t p.proj\n\u22a2 tangentMapWithin I I' f s p = tangentMapWithin I I' f t p\n[PROOFSTEP]\nsimp only [tangentMapWithin, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nst : s \u2286 t\nhs : UniqueMDiffWithinAt I s p.proj\nh : MDifferentiableWithinAt I I' f t p.proj\n\u22a2 \u2191(mfderivWithin I I' f s p.proj) p.snd = \u2191(mfderivWithin I I' f t p.proj) p.snd\n[PROOFSTEP]\nrw [mfderivWithin_subset st hs h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 tangentMapWithin I I' f univ = tangentMap I I' f\n[PROOFSTEP]\next p : 1\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\n\u22a2 tangentMapWithin I I' f univ p = tangentMap I I' f p\n[PROOFSTEP]\nsimp only [tangentMapWithin, tangentMap, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\nh : MDifferentiableAt I I' f p.proj\n\u22a2 tangentMapWithin I I' f s p = tangentMap I I' f p\n[PROOFSTEP]\nrw [\u2190 mdifferentiableWithinAt_univ] at h \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\nh : MDifferentiableWithinAt I I' f univ p.proj\n\u22a2 tangentMapWithin I I' f s p = tangentMap I I' f p\n[PROOFSTEP]\nrw [\u2190 tangentMapWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\nh : MDifferentiableWithinAt I I' f univ p.proj\n\u22a2 tangentMapWithin I I' f s p = tangentMapWithin I I' f univ p\n[PROOFSTEP]\nexact tangentMapWithin_subset (subset_univ _) hs h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 HasMFDerivWithinAt I I' f\u2081 s x f'\n[PROOFSTEP]\nrefine' \u27e8ContinuousWithinAt.congr_of_eventuallyEq h.1 h\u2081 hx, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x f\u2081) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\napply HasFDerivWithinAt.congr_of_eventuallyEq h.2\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 writtenInExtChartAt I I' x f\u2081 =\u1da0[\ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x]\n writtenInExtChartAt I I' x f\n[PROOFSTEP]\nhave : (extChartAt I x).symm \u207b\u00b9' {y | f\u2081 y = f y} \u2208 \ud835\udcdd[(extChartAt I x).symm \u207b\u00b9' s \u2229 range I] (extChartAt I x) x :=\n extChartAt_preimage_mem_nhdsWithin I x h\u2081\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' {y | f\u2081 y = f y} \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 writtenInExtChartAt I I' x f\u2081 =\u1da0[\ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x]\n writtenInExtChartAt I I' x f\n[PROOFSTEP]\napply Filter.mem_of_superset this fun y => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' {y | f\u2081 y = f y} \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 \u2200 (y : E),\n y \u2208 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' {y | f\u2081 y = f y} \u2192\n y \u2208 {x_1 | (fun x_2 => writtenInExtChartAt I I' x f\u2081 x_2 = writtenInExtChartAt I I' x f x_2) x_1}\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [hx, mfld_simps]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f s x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 writtenInExtChartAt I I' x f\u2081 (\u2191(extChartAt I x) x) = writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivAt I I' f x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd x] f\n\u22a2 HasMFDerivAt I I' f\u2081 x f'\n[PROOFSTEP]\nrw [\u2190 hasMFDerivWithinAt_univ] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f univ x f'\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd x] f\n\u22a2 HasMFDerivWithinAt I I' f\u2081 univ x f'\n[PROOFSTEP]\napply h.congr_of_eventuallyEq _ (mem_of_mem_nhds h\u2081 : _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : HasMFDerivWithinAt I I' f univ x f'\nh\u2081 : 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_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 MDifferentiableWithinAt I I' f s x \u2194 MDifferentiableWithinAt I I' f\u2081 s x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 MDifferentiableWithinAt I I' f s x \u2192 MDifferentiableWithinAt I I' f\u2081 s x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 MDifferentiableWithinAt I I' f\u2081 s x\n[PROOFSTEP]\napply h.congr_of_eventuallyEq h\u2081 hx\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 MDifferentiableWithinAt I I' f\u2081 s x \u2192 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f\u2081 s x\n\u22a2 MDifferentiableWithinAt I I' f s x\n[PROOFSTEP]\napply h.congr_of_eventuallyEq _ hx.symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f\u2081 s x\n\u22a2 f =\u1da0[\ud835\udcdd[s] x] f\u2081\n[PROOFSTEP]\napply h\u2081.mono\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f\u2081 s x\n\u22a2 \u2200 (x : M), f\u2081 x = f x \u2192 f x = f\u2081 x\n[PROOFSTEP]\nintro y\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f\u2081 s x\ny : M\n\u22a2 f\u2081 y = f y \u2192 f y = f\u2081 y\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x\n[PROOFSTEP]\nby_cases h : MDifferentiableWithinAt I I' f s x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : MDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x\n[PROOFSTEP]\nexact (h.hasMFDerivWithinAt.congr_of_eventuallyEq hL hx).mfderivWithin hs\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 mfderivWithin I I' f\u2081 s x = mfderivWithin I I' f s x\n[PROOFSTEP]\nunfold mfderivWithin\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 (if MDifferentiableWithinAt I I' f\u2081 s x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f\u2081) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n else 0) =\n if MDifferentiableWithinAt I I' f s x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n else 0\n[PROOFSTEP]\nrw [if_neg h, if_neg]\n[GOAL]\ncase neg.hnc\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhs : UniqueMDiffWithinAt I s x\nhL : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nhx : f\u2081 x = f x\nh : \u00acMDifferentiableWithinAt I I' f s x\n\u22a2 \u00acMDifferentiableWithinAt I I' f\u2081 s x\n[PROOFSTEP]\nrwa [\u2190 hL.mdifferentiableWithinAt_iff I I' hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : \u2200 (x : M), x \u2208 s \u2192 f x = f\u2081 x\np : TangentBundle I M\nhp : p.proj \u2208 s\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 tangentMapWithin I I' f s p = tangentMapWithin I I' f\u2081 s p\n[PROOFSTEP]\nrefine TotalSpace.ext _ _ (h p.1 hp) ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : \u2200 (x : M), x \u2208 s \u2192 f x = f\u2081 x\np : TangentBundle I M\nhp : p.proj \u2208 s\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 HEq (tangentMapWithin I I' f s p).snd (tangentMapWithin I I' f\u2081 s p).snd\n[PROOFSTEP]\nsimp only [tangentMapWithin, h p.1 hp, mfderivWithin_congr hs h (h _ hp), HEq.refl]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhL : f\u2081 =\u1da0[\ud835\udcdd x] f\n\u22a2 mfderiv I I' f\u2081 x = mfderiv I I' f x\n[PROOFSTEP]\nhave A : f\u2081 x = f x := (mem_of_mem_nhds hL : _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhL : f\u2081 =\u1da0[\ud835\udcdd x] f\nA : f\u2081 x = f x\n\u22a2 mfderiv I I' f\u2081 x = mfderiv I I' f x\n[PROOFSTEP]\nrw [\u2190 mfderivWithin_univ, \u2190 mfderivWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhL : f\u2081 =\u1da0[\ud835\udcdd x] f\nA : f\u2081 x = f x\n\u22a2 mfderivWithin I I' f\u2081 univ x = mfderivWithin I I' f univ x\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_univ] at hL \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhL : f\u2081 =\u1da0[\ud835\udcdd[univ] x] f\nA : f\u2081 x = f x\n\u22a2 mfderivWithin I I' f\u2081 univ x = mfderivWithin I I' f univ x\n[PROOFSTEP]\nexact hL.mfderivWithin_eq (uniqueMDiffWithinAt_univ I) A\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nx' : M\nh : x = x'\n\u22a2 mfderiv I I' f x = mfderiv I I' f x'\n[PROOFSTEP]\nsubst h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 mfderiv I I' f x = mfderiv I I' f x\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nf' : M \u2192 M'\nh : f = f'\n\u22a2 mfderiv I I' f x = mfderiv I I' f' x\n[PROOFSTEP]\nsubst h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\n\u22a2 mfderiv I I' f x = mfderiv I I' f x\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : ContinuousWithinAt f s x\n\u22a2 {y | writtenInExtChartAt I I'' x (g \u2218 f) y = (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) y} \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n[PROOFSTEP]\napply\n @Filter.mem_of_superset _ _ (f \u2218 (extChartAt I x).symm \u207b\u00b9' (extChartAt I' (f x)).source) _\n (extChartAt_preimage_mem_nhdsWithin I x (h.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _ _)))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nh : ContinuousWithinAt f s x\n\u22a2 f \u2218 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (extChartAt I' (f x)).source \u2286\n {y | writtenInExtChartAt I I'' x (g \u2218 f) y = (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) y}\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\n\u22a2 HasMFDerivWithinAt I I'' (g \u2218 f) s x (ContinuousLinearMap.comp g' f')\n[PROOFSTEP]\nrefine' \u27e8ContinuousWithinAt.comp hg.1 hf.1 hst, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I'' x (g \u2218 f)) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave A :\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f)\n (ContinuousLinearMap.comp g' f' : E \u2192L[\ud835\udd5c] E'') ((extChartAt I x).symm \u207b\u00b9' s \u2229 range I) ((extChartAt I x) x) :=\n by\n have :\n (extChartAt I x).symm \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[(extChartAt I x).symm \u207b\u00b9' s \u2229 range I] (extChartAt I x) x :=\n extChartAt_preimage_mem_nhdsWithin I x (hf.1.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _ _))\n unfold HasMFDerivWithinAt at *\n rw [\u2190 hasFDerivWithinAt_inter' this, \u2190 extChartAt_preimage_inter_eq] at hf \u22a2\n have : writtenInExtChartAt I I' x f ((extChartAt I x) x) = (extChartAt I' (f x)) (f x) := by simp only [mfld_simps]\n rw [\u2190 this] at hg \n apply HasFDerivWithinAt.comp ((extChartAt I x) x) hg.2 hf.2 _\n intro y hy\n simp only [mfld_simps] at hy \n have : f (((chartAt H x).symm : H \u2192 M) (I.symm y)) \u2208 u := hst hy.1.1\n simp only [hy, this, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave :\n (extChartAt I x).symm \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[(extChartAt I x).symm \u207b\u00b9' s \u2229 range I] (extChartAt I x) x :=\n extChartAt_preimage_mem_nhdsWithin I x (hf.1.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _ _))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nunfold HasMFDerivWithinAt at *\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (\u2191(extChartAt I' (f x)) (f x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrw [\u2190 hasFDerivWithinAt_inter' this, \u2190 extChartAt_preimage_inter_eq] at hf \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (\u2191(extChartAt I' (f x)) (f x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave : writtenInExtChartAt I I' x f ((extChartAt I x) x) = (extChartAt I' (f x)) (f x) := by simp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (\u2191(extChartAt I' (f x)) (f x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\n\u22a2 writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (\u2191(extChartAt I' (f x)) (f x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrw [\u2190 this] at hg \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\napply HasFDerivWithinAt.comp ((extChartAt I x) x) hg.2 hf.2 _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n\u22a2 MapsTo (writtenInExtChartAt I I' x f)\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I')\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\ny : E\nhy : y \u2208 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I\n\u22a2 writtenInExtChartAt I I' x f y \u2208 \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I'\n[PROOFSTEP]\nsimp only [mfld_simps] at hy \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\ny : E\nhy :\n (\u2191(LocalHomeomorph.symm (chartAt H x)) (\u2191(ModelWithCorners.symm I) y) \u2208 s \u2227\n f (\u2191(LocalHomeomorph.symm (chartAt H x)) (\u2191(ModelWithCorners.symm I) y)) \u2208\n (chartAt H' (f x)).toLocalEquiv.source) \u2227\n y \u2208 range \u2191I\n\u22a2 writtenInExtChartAt I I' x f y \u2208 \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I'\n[PROOFSTEP]\nhave : f (((chartAt H x).symm : H \u2192 M) (I.symm y)) \u2208 u := hst hy.1.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg :\n ContinuousWithinAt g u (f x) \u2227\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'\n (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I') (writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x))\nhf :\n ContinuousWithinAt f s x \u2227\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nhst : s \u2286 f \u207b\u00b9' u\nthis\u271d\u00b9 :\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' (f x)).source) \u2208\n \ud835\udcdd[\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(extChartAt I x) x\nthis\u271d : writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\ny : E\nhy :\n (\u2191(LocalHomeomorph.symm (chartAt H x)) (\u2191(ModelWithCorners.symm I) y) \u2208 s \u2227\n f (\u2191(LocalHomeomorph.symm (chartAt H x)) (\u2191(ModelWithCorners.symm I) y)) \u2208\n (chartAt H' (f x)).toLocalEquiv.source) \u2227\n y \u2208 range \u2191I\nthis : f (\u2191(LocalHomeomorph.symm (chartAt H x)) (\u2191(ModelWithCorners.symm I) y)) \u2208 u\n\u22a2 writtenInExtChartAt I I' x f y \u2208 \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I'\n[PROOFSTEP]\nsimp only [hy, this, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\nA :\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I'' x (g \u2218 f)) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\napply A.congr_of_eventuallyEq (writtenInExtChartAt_comp hf.1)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g u (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\nhst : s \u2286 f \u207b\u00b9' u\nA :\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (ContinuousLinearMap.comp g' f')\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x)\n\u22a2 writtenInExtChartAt I I'' x (g \u2218 f) (\u2191(extChartAt I x) x) =\n (writtenInExtChartAt I' I'' (f x) g \u2218 writtenInExtChartAt I I' x f) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivAt I' I'' g (f x) g'\nhf : HasMFDerivAt I I' f x f'\n\u22a2 HasMFDerivAt I I'' (g \u2218 f) x (ContinuousLinearMap.comp g' f')\n[PROOFSTEP]\nrw [\u2190 hasMFDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g univ (f x) g'\nhf : HasMFDerivWithinAt I I' f univ x f'\n\u22a2 HasMFDerivWithinAt I I'' (g \u2218 f) univ x (ContinuousLinearMap.comp g' f')\n[PROOFSTEP]\nexact HasMFDerivWithinAt.comp x (hg.mono (subset_univ _)) hf subset_preimage_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivAt I' I'' g (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\n\u22a2 HasMFDerivWithinAt I I'' (g \u2218 f) s x (ContinuousLinearMap.comp g' f')\n[PROOFSTEP]\nrw [\u2190 hasMFDerivWithinAt_univ] at *\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : HasMFDerivWithinAt I' I'' g univ (f x) g'\nhf : HasMFDerivWithinAt I I' f s x f'\n\u22a2 HasMFDerivWithinAt I I'' (g \u2218 f) s x (ContinuousLinearMap.comp g' f')\n[PROOFSTEP]\nexact HasMFDerivWithinAt.comp x (hg.mono (subset_univ _)) hf subset_preimage_univ\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\n\u22a2 MDifferentiableWithinAt I I'' (g \u2218 f) s x\n[PROOFSTEP]\nrcases hf.2 with \u27e8f', hf'\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nf' : E \u2192L[\ud835\udd5c] E'\nhf' :\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\n\u22a2 MDifferentiableWithinAt I I'' (g \u2218 f) s x\n[PROOFSTEP]\nhave F : HasMFDerivWithinAt I I' f s x f' := \u27e8hf.1, hf'\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nf' : E \u2192L[\ud835\udd5c] E'\nhf' :\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nF : HasMFDerivWithinAt I I' f s x f'\n\u22a2 MDifferentiableWithinAt I I'' (g \u2218 f) s x\n[PROOFSTEP]\nrcases hg.2 with \u27e8g', hg'\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng'\u271d : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nf' : E \u2192L[\ud835\udd5c] E'\nhf' :\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nF : HasMFDerivWithinAt I I' f s x f'\ng' : E' \u2192L[\ud835\udd5c] E''\nhg' :\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g' (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I')\n (\u2191(extChartAt I' (f x)) (f x))\n\u22a2 MDifferentiableWithinAt I I'' (g \u2218 f) s x\n[PROOFSTEP]\nhave G : HasMFDerivWithinAt I' I'' g u (f x) g' := \u27e8hg.1, hg'\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf'\u271d f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng'\u271d : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nf' : E \u2192L[\ud835\udd5c] E'\nhf' :\n HasFDerivWithinAt (writtenInExtChartAt I I' x f) f' (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I)\n (\u2191(extChartAt I x) x)\nF : HasMFDerivWithinAt I I' f s x f'\ng' : E' \u2192L[\ud835\udd5c] E''\nhg' :\n HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g' (\u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' u \u2229 range \u2191I')\n (\u2191(extChartAt I' (f x)) (f x))\nG : HasMFDerivWithinAt I' I'' g u (f x) g'\n\u22a2 MDifferentiableWithinAt I I'' (g \u2218 f) s x\n[PROOFSTEP]\nexact (HasMFDerivWithinAt.comp x G F h).mdifferentiableWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 mfderivWithin I I'' (g \u2218 f) s x = ContinuousLinearMap.comp (mfderivWithin I' I'' g u (f x)) (mfderivWithin I I' f s x)\n[PROOFSTEP]\napply HasMFDerivWithinAt.mfderivWithin _ hxs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableWithinAt I' I'' g u (f x)\nhf : MDifferentiableWithinAt I I' f s x\nh : s \u2286 f \u207b\u00b9' u\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 HasMFDerivWithinAt I I'' (g \u2218 f) s x\n (ContinuousLinearMap.comp (mfderivWithin I' I'' g u (f x)) (mfderivWithin I I' f s x))\n[PROOFSTEP]\nexact HasMFDerivWithinAt.comp x hg.hasMFDerivWithinAt hf.hasMFDerivWithinAt h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableAt I' I'' g (f x)\nhf : MDifferentiableAt I I' f x\n\u22a2 mfderiv I I'' (g \u2218 f) x = ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x)\n[PROOFSTEP]\napply HasMFDerivAt.mfderiv\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiableAt I' I'' g (f x)\nhf : MDifferentiableAt I I' f x\n\u22a2 HasMFDerivAt I I'' (g \u2218 f) x (ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x))\n[PROOFSTEP]\nexact HasMFDerivAt.comp x hg.hasMFDerivAt hf.hasMFDerivAt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\ny : M'\nhg : MDifferentiableAt I' I'' g y\nhf : MDifferentiableAt I I' f x\nhy : f x = y\n\u22a2 mfderiv I I'' (g \u2218 f) x = ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x)\n[PROOFSTEP]\nsubst hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx\u271d : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x\u271d \u2192L[\ud835\udd5c] TangentSpace I' (f x\u271d)\ng' : TangentSpace I' (f x\u271d) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x\u271d))\nx : M\nhf : MDifferentiableAt I I' f x\nhg : MDifferentiableAt I' I'' g (f x)\n\u22a2 mfderiv I I'' (g \u2218 f) x = ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x)\n[PROOFSTEP]\nexact mfderiv_comp x hg hf\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 tangentMapWithin I I'' (g \u2218 f) s p = tangentMapWithin I' I'' g u (tangentMapWithin I I' f s p)\n[PROOFSTEP]\nsimp only [tangentMapWithin, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(mfderivWithin I I'' (g \u2218 f) s p.proj) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)\n[PROOFSTEP]\nrw [mfderivWithin_comp p.1 hg hf h hps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(ContinuousLinearMap.comp (mfderivWithin I' I'' g u (f p.proj)) (mfderivWithin I I' f s p.proj)) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableAt I' I'' g (f p.proj)\nhf : MDifferentiableAt I I' f p.proj\n\u22a2 tangentMap I I'' (g \u2218 f) p = tangentMap I' I'' g (tangentMap I I' f p)\n[PROOFSTEP]\nsimp only [tangentMap, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableAt I' I'' g (f p.proj)\nhf : MDifferentiableAt I I' f p.proj\n\u22a2 \u2191(mfderiv I I'' (g \u2218 f) p.proj) p.snd = \u2191(mfderiv I' I'' g (f p.proj)) (\u2191(mfderiv I I' f p.proj) p.snd)\n[PROOFSTEP]\nrw [mfderiv_comp p.1 hg hf]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableAt I' I'' g (f p.proj)\nhf : MDifferentiableAt I I' f p.proj\n\u22a2 \u2191(ContinuousLinearMap.comp (mfderiv I' I'' g (f p.proj)) (mfderiv I I' f p.proj)) p.snd =\n \u2191(mfderiv I' I'' g (f p.proj)) (\u2191(mfderiv I I' f p.proj) p.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiable I' I'' g\nhf : MDifferentiable I I' f\n\u22a2 tangentMap I I'' (g \u2218 f) = tangentMap I' I'' g \u2218 tangentMap I I' f\n[PROOFSTEP]\next p : 1\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\nhg : MDifferentiable I' I'' g\nhf : MDifferentiable I I' f\np : TangentBundle I M\n\u22a2 tangentMap I I'' (g \u2218 f) p = (tangentMap I' I'' g \u2218 tangentMap I I' f) p\n[PROOFSTEP]\nexact tangentMap_comp_at _ (hg _) (hf _)\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 UniqueMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) s x \u2194 UniqueDiffWithinAt \ud835\udd5c s x\n[PROOFSTEP]\nsimp only [UniqueMDiffWithinAt, mfld_simps]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 UniqueMDiffOn \ud835\udcd8(\ud835\udd5c, E) s \u2194 UniqueDiffOn \ud835\udd5c s\n[PROOFSTEP]\nsimp [UniqueMDiffOn, UniqueDiffOn, uniqueMDiffWithinAt_iff_uniqueDiffWithinAt]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nf' : TangentSpace \ud835\udcd8(\ud835\udd5c, E) x \u2192L[\ud835\udd5c] TangentSpace \ud835\udcd8(\ud835\udd5c, E') (f x)\n\u22a2 HasMFDerivWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x f' \u2194 HasFDerivWithinAt f f' s x\n[PROOFSTEP]\nsimpa only [HasMFDerivWithinAt, and_iff_right_iff_imp, mfld_simps] using HasFDerivWithinAt.continuousWithinAt\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nf' : TangentSpace \ud835\udcd8(\ud835\udd5c, E) x \u2192L[\ud835\udd5c] TangentSpace \ud835\udcd8(\ud835\udd5c, E') (f x)\n\u22a2 HasMFDerivAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f x f' \u2194 HasFDerivAt f f' x\n[PROOFSTEP]\nrw [\u2190 hasMFDerivWithinAt_univ, hasMFDerivWithinAt_iff_hasFDerivWithinAt, hasFDerivWithinAt_univ]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 MDifferentiableWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x \u2194 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [MDifferentiableWithinAt, mfld_simps]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 ContinuousWithinAt f s x \u2227 DifferentiableWithinAt \ud835\udd5c f s x \u2194 DifferentiableWithinAt \ud835\udd5c f s x\n[PROOFSTEP]\nexact \u27e8fun H => H.2, fun H => \u27e8H.continuousWithinAt, H\u27e9\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 MDifferentiableAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f x \u2194 DifferentiableAt \ud835\udd5c f x\n[PROOFSTEP]\nsimp only [MDifferentiableAt, differentiableWithinAt_univ, mfld_simps]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 ContinuousAt f x \u2227 DifferentiableAt \ud835\udd5c f x \u2194 DifferentiableAt \ud835\udd5c f x\n[PROOFSTEP]\nexact \u27e8fun H => H.2, fun H => \u27e8H.continuousAt, H\u27e9\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 MDifferentiableOn \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s \u2194 DifferentiableOn \ud835\udd5c f s\n[PROOFSTEP]\nsimp only [MDifferentiableOn, DifferentiableOn, mdifferentiableWithinAt_iff_differentiableWithinAt]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 MDifferentiable \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f \u2194 Differentiable \ud835\udd5c f\n[PROOFSTEP]\nsimp only [MDifferentiable, Differentiable, mdifferentiableAt_iff_differentiableAt]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 mfderivWithin \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nby_cases h : MDifferentiableWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x\n[GOAL]\ncase pos\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nh : MDifferentiableWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x\n\u22a2 mfderivWithin \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [mfderivWithin, h, if_pos, mfld_simps]\n[GOAL]\ncase neg\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nh : \u00acMDifferentiableWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x\n\u22a2 mfderivWithin \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nsimp only [mfderivWithin, h, if_neg, not_false_iff]\n[GOAL]\ncase neg\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nh : \u00acMDifferentiableWithinAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f s x\n\u22a2 0 = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nrw [mdifferentiableWithinAt_iff_differentiableWithinAt] at h \n[GOAL]\ncase neg\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\nh : \u00acDifferentiableWithinAt \ud835\udd5c f s x\n\u22a2 0 = fderivWithin \ud835\udd5c f s x\n[PROOFSTEP]\nexact (fderivWithin_zero_of_not_differentiableWithinAt h).symm\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 mfderiv \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f x = fderiv \ud835\udd5c f x\n[PROOFSTEP]\nrw [\u2190 mfderivWithin_univ, \u2190 fderivWithin_univ]\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\nE' : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : NormedSpace \ud835\udd5c E'\nf : E \u2192 E'\ns : Set E\nx : E\n\u22a2 mfderivWithin \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f univ x = fderivWithin \ud835\udd5c f univ x\n[PROOFSTEP]\nexact mfderivWithin_eq_fderivWithin\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\n\u22a2 HasMFDerivAt I I id x (ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x))\n[PROOFSTEP]\nrefine' \u27e8continuousAt_id, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I x id) (ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) (range \u2191I)\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave : \u2200\u1da0 y in \ud835\udcdd[range I] (extChartAt I x) x, (extChartAt I x \u2218 (extChartAt I x).symm) y = y\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\n\u22a2 \u2200\u1da0 (y : E) in \ud835\udcdd[range \u2191I] \u2191(extChartAt I x) x, (\u2191(extChartAt I x) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) y = y\n[PROOFSTEP]\napply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin I x)\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\n\u22a2 (extChartAt I x).target \u2286 {x_1 | (fun y => (\u2191(extChartAt I x) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) y = y) x_1}\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\nthis : \u2200\u1da0 (y : E) in \ud835\udcdd[range \u2191I] \u2191(extChartAt I x) x, (\u2191(extChartAt I x) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) y = y\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I x id) (ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) (range \u2191I)\n (\u2191(extChartAt I x) x)\n[PROOFSTEP]\napply HasFDerivWithinAt.congr_of_eventuallyEq (hasFDerivWithinAt_id _ _) this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\nthis : \u2200\u1da0 (y : E) in \ud835\udcdd[range \u2191I] \u2191(extChartAt I x) x, (\u2191(extChartAt I x) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) y = y\n\u22a2 (\u2191(extChartAt I x) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) (\u2191(extChartAt I x) x) = id (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 mfderivWithin I I id s x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nrw [MDifferentiable.mfderivWithin (mdifferentiableAt_id I) hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nhxs : UniqueMDiffWithinAt I s x\n\u22a2 mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nexact mfderiv_id I\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\n\u22a2 tangentMap I I id = id\n[PROOFSTEP]\next1 \u27e8x, v\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d x : M\nv : TangentSpace I x\n\u22a2 tangentMap I I id { proj := x, snd := v } = id { proj := x, snd := v }\n[PROOFSTEP]\nsimp [tangentMap]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 tangentMapWithin I I id s p = p\n[PROOFSTEP]\nsimp only [tangentMapWithin, id.def]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 { proj := p.proj, snd := \u2191(mfderivWithin I I id s p.proj) p.snd } = p\n[PROOFSTEP]\nrw [mfderivWithin_id]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 { proj := p.proj, snd := \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I p.proj)) p.snd } = p\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx proj\u271d : M\nsnd\u271d : TangentSpace I proj\u271d\nhs : UniqueMDiffWithinAt I s { proj := proj\u271d, snd := snd\u271d }.proj\n\u22a2 { proj := { proj := proj\u271d, snd := snd\u271d }.proj,\n snd :=\n \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I { proj := proj\u271d, snd := snd\u271d }.proj))\n { proj := proj\u271d, snd := snd\u271d }.snd } =\n { proj := proj\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hxs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle I M\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 UniqueMDiffWithinAt I s p.proj\n[PROOFSTEP]\nexact hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nc\u271d c : M'\nx : M\n\u22a2 HasMFDerivAt I I' (fun x => c) x 0\n[PROOFSTEP]\nrefine' \u27e8continuous_const.continuousAt, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nc\u271d c : M'\nx : M\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I I' x fun x => c) 0 (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [writtenInExtChartAt, (\u00b7 \u2218 \u00b7), hasFDerivWithinAt_const]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 HasMFDerivAt (ModelWithCorners.prod I I') I Prod.fst x\n (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd))\n[PROOFSTEP]\nrefine' \u27e8continuous_fst.continuousAt, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt (ModelWithCorners.prod I I') I x Prod.fst)\n (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)) (range \u2191(ModelWithCorners.prod I I'))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x)\n[PROOFSTEP]\nhave :\n \u2200\u1da0 y in \ud835\udcdd[range (I.prod I')] extChartAt (I.prod I') x x,\n (extChartAt I x.1 \u2218 Prod.fst \u2218 (extChartAt (I.prod I') x).symm) y = y.1 :=\n by\n /- porting note: was\n apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)\n mfld_set_tac\n -/\n filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy\n rw [extChartAt_prod] at hy \n exact (extChartAt I x.1).right_inv hy.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.fst\n[PROOFSTEP]\nfilter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\ny : E \u00d7 E'\nhy : y \u2208 (extChartAt (ModelWithCorners.prod I I') x).target\n\u22a2 (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.fst\n[PROOFSTEP]\nrw [extChartAt_prod] at hy \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\ny : E \u00d7 E'\nhy : y \u2208 (LocalEquiv.prod (extChartAt I x.fst) (extChartAt I' x.snd)).target\n\u22a2 (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.fst\n[PROOFSTEP]\nexact (extChartAt I x.1).right_inv hy.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\nthis :\n \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.fst\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt (ModelWithCorners.prod I I') I x Prod.fst)\n (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)) (range \u2191(ModelWithCorners.prod I I'))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x)\n[PROOFSTEP]\napply HasFDerivWithinAt.congr_of_eventuallyEq hasFDerivWithinAt_fst this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\nthis :\n \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.fst\n\u22a2 (\u2191(extChartAt I x.fst) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x)))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x) =\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x).fst\n[PROOFSTEP]\nexact (extChartAt I x.1).right_inv <| (extChartAt I x.1).map_source (mem_extChartAt_source _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx\u271d : M\ns : Set (M \u00d7 M')\nx : M \u00d7 M'\nhxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x\n\u22a2 mfderivWithin (ModelWithCorners.prod I I') I Prod.fst s x =\n ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)\n[PROOFSTEP]\nrw [MDifferentiable.mfderivWithin (mdifferentiableAt_fst I I') hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx\u271d : M\ns : Set (M \u00d7 M')\nx : M \u00d7 M'\nhxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x\n\u22a2 mfderiv (ModelWithCorners.prod I I') I Prod.fst x =\n ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)\n[PROOFSTEP]\nexact mfderiv_fst I I'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\n\u22a2 tangentMap (ModelWithCorners.prod I I') I Prod.fst p = { proj := p.proj.fst, snd := p.snd.fst }\n[PROOFSTEP]\nsimp [tangentMap]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\n\u22a2 \u2191(ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I p.proj.fst) (TangentSpace I' p.proj.snd)) p.snd = p.snd.fst\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 tangentMapWithin (ModelWithCorners.prod I I') I Prod.fst s p = { proj := p.proj.fst, snd := p.snd.fst }\n[PROOFSTEP]\nsimp only [tangentMapWithin]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 { proj := p.proj.fst, snd := \u2191(mfderivWithin (ModelWithCorners.prod I I') I Prod.fst s p.proj) p.snd } =\n { proj := p.proj.fst, snd := p.snd.fst }\n[PROOFSTEP]\nrw [mfderivWithin_fst]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 { proj := p.proj.fst,\n snd := \u2191(ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I p.proj.fst) (TangentSpace I' p.proj.snd)) p.snd } =\n { proj := p.proj.fst, snd := p.snd.fst }\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\nproj\u271d : M \u00d7 M'\nsnd\u271d : TangentSpace (ModelWithCorners.prod I I') proj\u271d\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s { proj := proj\u271d, snd := snd\u271d }.proj\n\u22a2 { proj := { proj := proj\u271d, snd := snd\u271d }.proj.fst,\n snd :=\n \u2191(ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I { proj := proj\u271d, snd := snd\u271d }.proj.fst)\n (TangentSpace I' { proj := proj\u271d, snd := snd\u271d }.proj.snd))\n { proj := proj\u271d, snd := snd\u271d }.snd } =\n { proj := { proj := proj\u271d, snd := snd\u271d }.proj.fst, snd := { proj := proj\u271d, snd := snd\u271d }.snd.fst }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hxs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n[PROOFSTEP]\nexact hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 HasMFDerivAt (ModelWithCorners.prod I I') I' Prod.snd x\n (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd))\n[PROOFSTEP]\nrefine' \u27e8continuous_snd.continuousAt, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt (ModelWithCorners.prod I I') I' x Prod.snd)\n (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)) (range \u2191(ModelWithCorners.prod I I'))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x)\n[PROOFSTEP]\nhave :\n \u2200\u1da0 y in \ud835\udcdd[range (I.prod I')] extChartAt (I.prod I') x x,\n (extChartAt I' x.2 \u2218 Prod.snd \u2218 (extChartAt (I.prod I') x).symm) y = y.2 :=\n by\n /- porting note: was\n apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin (I.prod I') x)\n mfld_set_tac\n -/\n filter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy\n rw [extChartAt_prod] at hy \n exact (extChartAt I' x.2).right_inv hy.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\n\u22a2 \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.snd\n[PROOFSTEP]\nfilter_upwards [extChartAt_target_mem_nhdsWithin (I.prod I') x] with y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\ny : E \u00d7 E'\nhy : y \u2208 (extChartAt (ModelWithCorners.prod I I') x).target\n\u22a2 (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.snd\n[PROOFSTEP]\nrw [extChartAt_prod] at hy \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\ny : E \u00d7 E'\nhy : y \u2208 (LocalEquiv.prod (extChartAt I x.fst) (extChartAt I' x.snd)).target\n\u22a2 (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.snd\n[PROOFSTEP]\nexact (extChartAt I' x.2).right_inv hy.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\nthis :\n \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.snd\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt (ModelWithCorners.prod I I') I' x Prod.snd)\n (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)) (range \u2191(ModelWithCorners.prod I I'))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x)\n[PROOFSTEP]\napply HasFDerivWithinAt.congr_of_eventuallyEq hasFDerivWithinAt_snd this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nx : M \u00d7 M'\nthis :\n \u2200\u1da0 (y : E \u00d7 E') in \ud835\udcdd[range \u2191(ModelWithCorners.prod I I')] \u2191(extChartAt (ModelWithCorners.prod I I') x) x,\n (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x))) y = y.snd\n\u22a2 (\u2191(extChartAt I' x.snd) \u2218 Prod.snd \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I I') x)))\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x) =\n (\u2191(extChartAt (ModelWithCorners.prod I I') x) x).snd\n[PROOFSTEP]\nexact (extChartAt I' x.2).right_inv <| (extChartAt I' x.2).map_source (mem_extChartAt_source _ _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx\u271d : M\ns : Set (M \u00d7 M')\nx : M \u00d7 M'\nhxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x\n\u22a2 mfderivWithin (ModelWithCorners.prod I I') I' Prod.snd s x =\n ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)\n[PROOFSTEP]\nrw [MDifferentiable.mfderivWithin (mdifferentiableAt_snd I I') hxs]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx\u271d : M\ns : Set (M \u00d7 M')\nx : M \u00d7 M'\nhxs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s x\n\u22a2 mfderiv (ModelWithCorners.prod I I') I' Prod.snd x =\n ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.fst) (TangentSpace I' x.snd)\n[PROOFSTEP]\nexact mfderiv_snd I I'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\n\u22a2 tangentMap (ModelWithCorners.prod I I') I' Prod.snd p = { proj := p.proj.snd, snd := p.snd.snd }\n[PROOFSTEP]\nsimp [tangentMap]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\n\u22a2 \u2191(ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I p.proj.fst) (TangentSpace I' p.proj.snd)) p.snd = p.snd.snd\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 tangentMapWithin (ModelWithCorners.prod I I') I' Prod.snd s p = { proj := p.proj.snd, snd := p.snd.snd }\n[PROOFSTEP]\nsimp only [tangentMapWithin]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 { proj := p.proj.snd, snd := \u2191(mfderivWithin (ModelWithCorners.prod I I') I' Prod.snd s p.proj) p.snd } =\n { proj := p.proj.snd, snd := p.snd.snd }\n[PROOFSTEP]\nrw [mfderivWithin_snd]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 { proj := p.proj.snd,\n snd := \u2191(ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I p.proj.fst) (TangentSpace I' p.proj.snd)) p.snd } =\n { proj := p.proj.snd, snd := p.snd.snd }\n[PROOFSTEP]\nrcases p with \u27e8\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\nproj\u271d : M \u00d7 M'\nsnd\u271d : TangentSpace (ModelWithCorners.prod I I') proj\u271d\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s { proj := proj\u271d, snd := snd\u271d }.proj\n\u22a2 { proj := { proj := proj\u271d, snd := snd\u271d }.proj.snd,\n snd :=\n \u2191(ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I { proj := proj\u271d, snd := snd\u271d }.proj.fst)\n (TangentSpace I' { proj := proj\u271d, snd := snd\u271d }.proj.snd))\n { proj := proj\u271d, snd := snd\u271d }.snd } =\n { proj := { proj := proj\u271d, snd := snd\u271d }.proj.snd, snd := { proj := proj\u271d, snd := snd\u271d }.snd.snd }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hxs\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns\u271d : Set M\nx : M\ns : Set (M \u00d7 M')\np : TangentBundle (ModelWithCorners.prod I I') (M \u00d7 M')\nhs : UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I I') s p.proj\n[PROOFSTEP]\nexact hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nf : M \u2192 M'\ng : M \u2192 M''\nx : M\nhf : MDifferentiableAt I I' f x\nhg : MDifferentiableAt I I'' g x\n\u22a2 mfderiv I (ModelWithCorners.prod I' I'') (fun x => (f x, g x)) x =\n ContinuousLinearMap.prod (mfderiv I I' f x) (mfderiv I I'' g x)\n[PROOFSTEP]\nclassical\nsimp_rw [mfderiv, if_pos (hf.prod_mk hg), if_pos hf, if_pos hg]\nexact hf.2.fderivWithin_prod hg.2 (I.unique_diff _ (mem_range_self _))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nf : M \u2192 M'\ng : M \u2192 M''\nx : M\nhf : MDifferentiableAt I I' f x\nhg : MDifferentiableAt I I'' g x\n\u22a2 mfderiv I (ModelWithCorners.prod I' I'') (fun x => (f x, g x)) x =\n ContinuousLinearMap.prod (mfderiv I I' f x) (mfderiv I I'' g x)\n[PROOFSTEP]\nsimp_rw [mfderiv, if_pos (hf.prod_mk hg), if_pos hf, if_pos hg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\nf : M \u2192 M'\ng : M \u2192 M''\nx : M\nhf : MDifferentiableAt I I' f x\nhg : MDifferentiableAt I I'' g x\n\u22a2 fderivWithin \ud835\udd5c (writtenInExtChartAt I (ModelWithCorners.prod I' I'') x fun x => (f x, g x)) (range \u2191I)\n (\u2191(extChartAt I x) x) =\n ContinuousLinearMap.prod (fderivWithin \ud835\udd5c (writtenInExtChartAt I I' x f) (range \u2191I) (\u2191(extChartAt I x) x))\n (fderivWithin \ud835\udd5c (writtenInExtChartAt I I'' x g) (range \u2191I) (\u2191(extChartAt I x) x))\n[PROOFSTEP]\nexact hf.2.fderivWithin_prod hg.2 (I.unique_diff _ (mem_range_self _))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx x\u2080 : M\ny\u2080 : M'\n\u22a2 mfderiv I (ModelWithCorners.prod I I') (fun x => (x, y\u2080)) x\u2080 =\n ContinuousLinearMap.inl \ud835\udd5c (TangentSpace I x\u2080) (TangentSpace I' y\u2080)\n[PROOFSTEP]\nrefine' ((mdifferentiableAt_id I).mfderiv_prod (mdifferentiableAt_const I I')).trans _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx x\u2080 : M\ny\u2080 : M'\n\u22a2 ContinuousLinearMap.prod (mfderiv I I id x\u2080) (mfderiv I I' (fun x => y\u2080) x\u2080) =\n ContinuousLinearMap.inl \ud835\udd5c (TangentSpace I x\u2080) (TangentSpace I' y\u2080)\n[PROOFSTEP]\nrw [mfderiv_id, mfderiv_const, ContinuousLinearMap.inl]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx x\u2080 : M\ny\u2080 : M'\n\u22a2 mfderiv I' (ModelWithCorners.prod I I') (fun y => (x\u2080, y)) y\u2080 =\n ContinuousLinearMap.inr \ud835\udd5c (TangentSpace I x\u2080) (TangentSpace I' y\u2080)\n[PROOFSTEP]\nrefine' ((mdifferentiableAt_const I' I).mfderiv_prod (mdifferentiableAt_id I')).trans _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx x\u2080 : M\ny\u2080 : M'\n\u22a2 ContinuousLinearMap.prod (mfderiv I' I (fun x => x\u2080) y\u2080) (mfderiv I' I' id y\u2080) =\n ContinuousLinearMap.inr \ud835\udd5c (TangentSpace I x\u2080) (TangentSpace I' y\u2080)\n[PROOFSTEP]\nrw [mfderiv_id, mfderiv_const, ContinuousLinearMap.inr]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f p\n\u22a2 mfderiv (ModelWithCorners.prod I I') I'' f p =\n let_fun this :=\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (z.fst, p.snd)) p +\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (p.fst, z.snd)) p;\n this\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f p\n\u22a2 mfderiv (ModelWithCorners.prod I I') I'' f p =\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (z.fst, p.snd)) p +\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (p.fst, z.snd)) p\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ p] at hf \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f (p.fst, p.snd)\n\u22a2 mfderiv (ModelWithCorners.prod I I') I'' f p =\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (z.fst, p.snd)) p +\n mfderiv (ModelWithCorners.prod I I') I'' (fun z => f (p.fst, z.snd)) p\n[PROOFSTEP]\nerw [mfderiv_comp_of_eq hf ((mdifferentiableAt_fst I I').prod_mk (mdifferentiableAt_const _ _)) rfl,\n mfderiv_comp_of_eq hf ((mdifferentiableAt_const _ _).prod_mk (mdifferentiableAt_snd I I')) rfl, \u2190\n ContinuousLinearMap.comp_add, (mdifferentiableAt_fst I I').mfderiv_prod (mdifferentiableAt_const (I.prod I') I'),\n (mdifferentiableAt_const (I.prod I') I).mfderiv_prod (mdifferentiableAt_snd I I'), mfderiv_fst, mfderiv_snd,\n mfderiv_const, mfderiv_const]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f (p.fst, p.snd)\n\u22a2 mfderiv (ModelWithCorners.prod I I') I'' f p =\n ContinuousLinearMap.comp (mfderiv (ModelWithCorners.prod I I') I'' f (p.fst, p.snd))\n (ContinuousLinearMap.prod (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd)) 0 +\n ContinuousLinearMap.prod 0 (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd)))\n[PROOFSTEP]\nsymm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f (p.fst, p.snd)\n\u22a2 ContinuousLinearMap.comp (mfderiv (ModelWithCorners.prod I I') I'' f (p.fst, p.snd))\n (ContinuousLinearMap.prod (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd)) 0 +\n ContinuousLinearMap.prod 0 (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd))) =\n mfderiv (ModelWithCorners.prod I I') I'' f p\n[PROOFSTEP]\nconvert ContinuousLinearMap.comp_id <| mfderiv (.prod I I') I'' f (p.1, p.2)\n[GOAL]\ncase h.e'_2.h.e'_24\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\nf : M \u00d7 M' \u2192 M''\np : M \u00d7 M'\nhf : MDifferentiableAt (ModelWithCorners.prod I I') I'' f (p.fst, p.snd)\n\u22a2 ContinuousLinearMap.prod (ContinuousLinearMap.fst \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd)) 0 +\n ContinuousLinearMap.prod 0 (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I p.fst) (TangentSpace I' p.snd)) =\n ContinuousLinearMap.id \ud835\udd5c (TangentSpace (ModelWithCorners.prod I I') (p.fst, p.snd))\n[PROOFSTEP]\nexact ContinuousLinearMap.coprod_inl_inr\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nf g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivAt I \ud835\udcd8(\ud835\udd5c, E') (-f) z (-f')\n\u22a2 HasMFDerivAt I \ud835\udcd8(\ud835\udd5c, E') f z f'\n[PROOFSTEP]\nconvert hf.neg\n[GOAL]\ncase h.e'_23\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nf g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivAt I \ud835\udcd8(\ud835\udd5c, E') (-f) z (-f')\n\u22a2 f = - -f\n[PROOFSTEP]\nrw [neg_neg]\n[GOAL]\ncase h.e'_25\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nf g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivAt I \ud835\udcd8(\ud835\udd5c, E') (-f) z (-f')\ne_23\u271d : f = - -f\n\u22a2 f' = - -f'\n[PROOFSTEP]\nrw [neg_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nf g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nhf : MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) z\n\u22a2 MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f z\n[PROOFSTEP]\nconvert hf.neg\n[GOAL]\ncase h.e'_21\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nf g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nhf : MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) z\n\u22a2 f = - -f\n[PROOFSTEP]\nrw [neg_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\n\u22a2 mfderiv I \ud835\udcd8(\ud835\udd5c, E') (-f) x = -mfderiv I \ud835\udcd8(\ud835\udd5c, E') f x\n[PROOFSTEP]\nsimp_rw [mfderiv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\n\u22a2 (if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x (-f)) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0) =\n -if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x f) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0\n[PROOFSTEP]\nby_cases hf : MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\nhf : MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x\n\u22a2 (if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x (-f)) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0) =\n -if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x f) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0\n[PROOFSTEP]\nexact hf.hasMFDerivAt.neg.mfderiv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\nhf : \u00acMDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x\n\u22a2 (if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x (-f)) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0) =\n -if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x f) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0\n[PROOFSTEP]\nrw [if_neg hf]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\nhf : \u00acMDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f x\n\u22a2 (if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x (-f)) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0) =\n -0\n[PROOFSTEP]\nrw [\u2190 mdifferentiableAt_neg] at hf \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d z : M\nf\u271d g : M \u2192 E'\nf' g' : TangentSpace I z \u2192L[\ud835\udd5c] E'\nf : M \u2192 E'\nx : M\nhf : \u00acMDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x\n\u22a2 (if MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (-f) x then\n fderivWithin \ud835\udd5c (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, E') x (-f)) (range \u2191I) (\u2191(extChartAt I x) x)\n else 0) =\n -0\n[PROOFSTEP]\nrw [if_neg hf, neg_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 HasFDerivWithinAt (writtenInExtChartAt I \ud835\udcd8(\ud835\udd5c, F') z (p * q)) (p z \u2022 q' + ContinuousLinearMap.smulRight p' (q z))\n (\u2191(LocalEquiv.symm (extChartAt I z)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I z) z)\n[PROOFSTEP]\nsimpa only [mfld_simps] using hp.2.mul' hq.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') (p * q) s z (p z \u2022 q' + q z \u2022 p')\n[PROOFSTEP]\nconvert hp.mul' hq\n[GOAL]\ncase h.e'_26.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 q z \u2022 p' = ContinuousLinearMap.smulRight p' (q z)\n[PROOFSTEP]\next _\n[GOAL]\ncase h.e'_26.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\nx\u271d : TangentSpace I z\n\u22a2 \u2191(q z \u2022 p') x\u271d = \u2191(ContinuousLinearMap.smulRight p' (q z)) x\u271d\n[PROOFSTEP]\napply mul_comm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\n\u22a2 MDifferentiableAt I I (\u2191e) x\n[PROOFSTEP]\nrefine' \u27e8(e.continuousOn x hx).continuousAt (IsOpen.mem_nhds e.open_source hx), _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave mem : I ((chartAt H x : M \u2192 H) x) \u2208 I.symm \u207b\u00b9' ((chartAt H x).symm \u226b\u2095 e).source \u2229 range I := by\n simp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\n\u22a2 \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave : (chartAt H x).symm.trans e \u2208 contDiffGroupoid \u221e I := HasGroupoid.compatible (chart_mem_atlas H x) h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave A :\n ContDiffOn \ud835\udd5c \u221e (I \u2218 (chartAt H x).symm.trans e \u2218 I.symm) (I.symm \u207b\u00b9' ((chartAt H x).symm.trans e).source \u2229 range I) :=\n this.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave B := A.differentiableOn le_top (I ((chartAt H x : M \u2192 H) x)) mem\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\n (\u2191I (\u2191(chartAt H x) x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfld_simps] at B \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c (\u2191I \u2218 (\u2191e \u2218 \u2191(LocalHomeomorph.symm (chartAt H x))) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (chartAt H x).toLocalEquiv.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm (chartAt H x)) \u207b\u00b9' e.source) \u2229\n range \u2191I)\n (\u2191I (\u2191(chartAt H x) x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrw [inter_comm, differentiableWithinAt_inter] at B \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c (\u2191I \u2218 (\u2191e \u2218 \u2191(LocalHomeomorph.symm (chartAt H x))) \u2218 \u2191(ModelWithCorners.symm I)) (range \u2191I)\n (\u2191I (\u2191(chartAt H x) x))\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191e) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimpa only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : M\nhx : x \u2208 e.source\nmem :\n \u2191I (\u2191(chartAt H x) x) \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I\nthis : LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (LocalHomeomorph.symm (chartAt H x) \u226b\u2095 e).toLocalEquiv.source \u2229 range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c (\u2191I \u2218 (\u2191e \u2218 \u2191(LocalHomeomorph.symm (chartAt H x))) \u2218 \u2191(ModelWithCorners.symm I))\n (range \u2191I \u2229\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' (chartAt H x).toLocalEquiv.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm (chartAt H x)) \u207b\u00b9' e.source)))\n (\u2191I (\u2191(chartAt H x) x))\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' (chartAt H x).toLocalEquiv.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9' (\u2191(LocalHomeomorph.symm (chartAt H x)) \u207b\u00b9' e.source) \u2208\n \ud835\udcdd (\u2191I (\u2191(chartAt H x) x))\n[PROOFSTEP]\napply IsOpen.mem_nhds ((LocalHomeomorph.open_source _).preimage I.continuous_symm) mem.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\n\u22a2 MDifferentiableAt I I (\u2191(LocalHomeomorph.symm e)) x\n[PROOFSTEP]\nrefine' \u27e8(e.continuousOn_symm x hx).continuousAt (IsOpen.mem_nhds e.open_target hx), _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave mem : I x \u2208 I.symm \u207b\u00b9' (e.symm \u226b\u2095 chartAt H (e.symm x)).source \u2229 range I := by simp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\n\u22a2 \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\n[PROOFSTEP]\nsimp only [hx, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave : e.symm.trans (chartAt H (e.symm x)) \u2208 contDiffGroupoid \u221e I := HasGroupoid.compatible h (chart_mem_atlas H _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave A :\n ContDiffOn \ud835\udd5c \u221e (I \u2218 e.symm.trans (chartAt H (e.symm x)) \u2218 I.symm)\n (I.symm \u207b\u00b9' (e.symm.trans (chartAt H (e.symm x))).source \u2229 range I) :=\n this.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nhave B := A.differentiableOn le_top (I x) mem\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\n (\u2191I x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimp only [mfld_simps] at B \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c\n (\u2191I \u2218 (\u2191(chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' (chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source) \u2229\n range \u2191I)\n (\u2191I x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nrw [inter_comm, differentiableWithinAt_inter] at B \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c\n (\u2191I \u2218 (\u2191(chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (range \u2191I) (\u2191I x)\n\u22a2 DifferentiableWithinAt \ud835\udd5c (writtenInExtChartAt I I x \u2191(LocalHomeomorph.symm e)) (range \u2191I) (\u2191(extChartAt I x) x)\n[PROOFSTEP]\nsimpa only [mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx\u271d : M\ne : LocalHomeomorph M H\nh : e \u2208 atlas H M\nx : H\nhx : x \u2208 e.target\nmem :\n \u2191I x \u2208\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I\nthis : LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x) \u2208 contDiffGroupoid \u22a4 I\nA :\n ContDiffOn \ud835\udd5c \u22a4\n (\u2191I \u2218 \u2191(LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(ModelWithCorners.symm I))\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (LocalHomeomorph.symm e \u226b\u2095 chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source \u2229\n range \u2191I)\nB :\n DifferentiableWithinAt \ud835\udd5c\n (\u2191I \u2218 (\u2191(chartAt H (\u2191(LocalHomeomorph.symm e) x)) \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(ModelWithCorners.symm I))\n (range \u2191I \u2229\n (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' (chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source)))\n (\u2191I x)\n\u22a2 \u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229\n \u2191(ModelWithCorners.symm I) \u207b\u00b9'\n (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' (chartAt H (\u2191(LocalHomeomorph.symm e) x)).toLocalEquiv.source) \u2208\n \ud835\udcdd (\u2191I x)\n[PROOFSTEP]\napply IsOpen.mem_nhds ((LocalHomeomorph.open_source _).preimage I.continuous_symm) mem.1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np q : TangentBundle I M\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.source\n\u22a2 tangentMap I I (\u2191(chartAt H p.proj)) q = \u2191(TotalSpace.toProd H E).symm (\u2191(chartAt (ModelProd H E) p) q)\n[PROOFSTEP]\ndsimp [tangentMap]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np q : TangentBundle I M\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.source\n\u22a2 { proj := \u2191(chartAt H p.proj) q.proj, snd := \u2191(mfderiv I I (\u2191(chartAt H p.proj)) q.proj) q.snd } =\n \u2191(TotalSpace.toProd H E).symm (\u2191(chartAt (ModelProd H E) p) q)\n[PROOFSTEP]\nrw [MDifferentiableAt.mfderiv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np q : TangentBundle I M\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.source\n\u22a2 { proj := \u2191(chartAt H p.proj) q.proj,\n snd :=\n \u2191(fderivWithin \ud835\udd5c (writtenInExtChartAt I I q.proj \u2191(chartAt H p.proj)) (range \u2191I)\n (\u2191(extChartAt I q.proj) q.proj))\n q.snd } =\n \u2191(TotalSpace.toProd H E).symm (\u2191(chartAt (ModelProd H E) p) q)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np q : TangentBundle I M\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.source\n\u22a2 MDifferentiableAt I I (\u2191(chartAt H p.proj)) q.proj\n[PROOFSTEP]\nexact mdifferentiableAt_atlas _ (chart_mem_atlas _ _) h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np : TangentBundle I M\nq : TangentBundle I H\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.target\n\u22a2 tangentMap I I (\u2191(LocalHomeomorph.symm (chartAt H p.proj))) q =\n \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p)) (\u2191(TotalSpace.toProd H E) q)\n[PROOFSTEP]\ndsimp only [tangentMap]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np : TangentBundle I M\nq : TangentBundle I H\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.target\n\u22a2 { proj := \u2191(LocalHomeomorph.symm (chartAt H p.proj)) q.proj,\n snd := \u2191(mfderiv I I (\u2191(LocalHomeomorph.symm (chartAt H p.proj))) q.proj) q.snd } =\n \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p)) (\u2191(TotalSpace.toProd H E) q)\n[PROOFSTEP]\nrw [MDifferentiableAt.mfderiv (mdifferentiableAt_atlas_symm _ (chart_mem_atlas _ _) h)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np : TangentBundle I M\nq : TangentBundle I H\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.target\n\u22a2 { proj := \u2191(LocalHomeomorph.symm (chartAt H p.proj)) q.proj,\n snd :=\n \u2191(fderivWithin \ud835\udd5c (writtenInExtChartAt I I q.proj \u2191(LocalHomeomorph.symm (chartAt H p.proj))) (range \u2191I)\n (\u2191(extChartAt I q.proj) q.proj))\n q.snd } =\n \u2191(LocalHomeomorph.symm (chartAt (ModelProd H E) p)) (\u2191(TotalSpace.toProd H E) q)\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_coe, TangentBundle.chartAt, h, tangentBundleCore, mfld_simps, (\u00b7 \u2218 \u00b7)]\n -- `simp` fails to apply `LocalEquiv.prod_symm` with `ModelProd`\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np : TangentBundle I M\nq : TangentBundle I H\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.target\n\u22a2 { proj := \u2191(LocalHomeomorph.symm (chartAt H p.proj)) q.proj,\n snd :=\n \u2191(fderivWithin \ud835\udd5c\n (fun x =>\n \u2191I\n (\u2191(chartAt H (\u2191(LocalHomeomorph.symm (chartAt H p.proj)) q.proj))\n (\u2191(LocalHomeomorph.symm (chartAt H p.proj)) (\u2191(ModelWithCorners.symm I) x))))\n (range \u2191I) (\u2191I q.proj))\n q.snd } =\n \u2191(LocalHomeomorph.symm\n (FiberBundleCore.localTriv\n (VectorBundleCore.toFiberBundleCore\n { baseSet := fun i => (\u2191i).source, isOpen_baseSet := (_ : \u2200 (i : \u2191(atlas H M)), IsOpen (\u2191i).source),\n indexAt := achart H, mem_baseSet_at := (_ : \u2200 (x : M), x \u2208 (chartAt H x).toLocalEquiv.source),\n coordChange := fun i j x =>\n fderivWithin \ud835\udd5c (fun x => \u2191I (\u2191\u2191j (\u2191(LocalHomeomorph.symm \u2191i) (\u2191(ModelWithCorners.symm I) x))))\n (range \u2191I) (\u2191I (\u2191\u2191i x)),\n coordChange_self :=\n (_ :\n \u2200 (i : \u2191(atlas H M)) (x : M),\n x \u2208 (fun i => (\u2191i).source) i \u2192\n \u2200 (v : E),\n \u2191((fun i j x =>\n fderivWithin \ud835\udd5c\n (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218\n \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 continuousOn_coordChange :=\n (_ :\n \u2200 (i j : \u2191(atlas H M)),\n ContinuousOn\n (fderivWithin \ud835\udd5c\n (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) \u2218\n fun x => \u2191(LocalHomeomorph.extend (\u2191i) I) x)\n ((fun i => (\u2191i).source) i \u2229 (fun i => (\u2191i).source) j)),\n coordChange_comp :=\n (_ :\n \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\n (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218\n \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\n \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\n (\u2191(LocalHomeomorph.extend (\u2191j) I) \u2218\n \u2191(LocalEquiv.symm (LocalHomeomorph.extend (\u2191i) I)))\n (range \u2191I) (\u2191(LocalHomeomorph.extend (\u2191i) I) x))\n i k x)\n v) })\n (achart H p.proj)).toLocalHomeomorph)\n (\u2191(LocalHomeomorph.symm (LocalHomeomorph.prod (chartAt H p.proj) (LocalHomeomorph.refl E))) (q.proj, q.snd))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_snd.e_a.e_x.e_a\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\ninst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2079 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2078 : TopologicalSpace M'\ninst\u271d\u2077 : ChartedSpace H' M'\ninst\u271d\u2076 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b3 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2 : TopologicalSpace M''\ninst\u271d\u00b9 : ChartedSpace H'' M''\ninst\u271d : SmoothManifoldWithCorners I'' M''\ns : Set M\nx : M\ne : LocalHomeomorph M H\np : TangentBundle I M\nq : TangentBundle I H\nh : q.proj \u2208 (chartAt H p.proj).toLocalEquiv.target\n\u22a2 q.proj =\n \u2191\u2191(achart H p.proj)\n (\u2191(LocalHomeomorph.symm (LocalHomeomorph.prod (chartAt H p.proj) (LocalHomeomorph.refl E))) (q.proj, q.snd)).fst\n[PROOFSTEP]\nexact ((chartAt H (TotalSpace.proj p)).right_inv h).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\n\u22a2 ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x) =\n ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nhave : mfderiv I I (e.symm \u2218 e) x = (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) :=\n mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\n\u22a2 ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x) =\n ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\n\u22a2 mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nhave : mfderiv I I (_root_.id : M \u2192 M) x = ContinuousLinearMap.id _ _ := mfderiv_id I\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis\u271d :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\nthis : mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n\u22a2 mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis\u271d :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\nthis : mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n\u22a2 mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x = mfderiv I I id x\n[PROOFSTEP]\napply Filter.EventuallyEq.mfderiv_eq\n[GOAL]\ncase hL\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis\u271d :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\nthis : mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\n\u22a2 \u2191(LocalHomeomorph.symm e) \u2218 \u2191e =\u1da0[\ud835\udcdd x] id\n[PROOFSTEP]\nhave : e.source \u2208 \ud835\udcdd x := IsOpen.mem_nhds e.open_source hx\n[GOAL]\ncase hL\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis\u271d\u00b9 :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\nthis\u271d : mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\nthis : e.source \u2208 \ud835\udcdd x\n\u22a2 \u2191(LocalHomeomorph.symm e) \u2218 \u2191e =\u1da0[\ud835\udcdd x] id\n[PROOFSTEP]\nexact Filter.mem_of_superset this (by mfld_set_tac)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nthis\u271d\u00b9 :\n mfderiv I I (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x =\n ContinuousLinearMap.comp (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (mfderiv I I' (\u2191e) x)\nthis\u271d : mfderiv I I id x = ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)\nthis : e.source \u2208 \ud835\udcdd x\n\u22a2 e.source \u2286 {x | (fun x => (\u2191(LocalHomeomorph.symm e) \u2218 \u2191e) x = id x) x}\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I x\n\u22a2 \u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (AddHom.toFun src\u271d.toAddHom y) = y\n[PROOFSTEP]\nhave : (ContinuousLinearMap.id _ _ : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I x) y = y := rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I x\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) y = y\n\u22a2 \u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) (AddHom.toFun src\u271d.toAddHom y) = y\n[PROOFSTEP]\nconv_rhs => rw [\u2190 this, \u2190 he.symm_comp_deriv hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I x\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.symm_comp_deriv hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I x\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.symm_comp_deriv hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I x\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I x)) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.symm_comp_deriv hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) y) = y\n[PROOFSTEP]\nhave : (ContinuousLinearMap.id \ud835\udd5c _ : TangentSpace I' (e x) \u2192L[\ud835\udd5c] TangentSpace I' (e x)) y = y := rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) y) = y\n[PROOFSTEP]\nconv_rhs => rw [\u2190 this, \u2190 he.comp_symm_deriv (e.map_source hx)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.comp_symm_deriv (e.map_source hx)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.comp_symm_deriv (e.map_source hx)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n| y\n[PROOFSTEP]\nrw [\u2190 this, \u2190 he.comp_symm_deriv (e.map_source hx)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) y) =\n \u2191(ContinuousLinearMap.comp (mfderiv I I' (\u2191e) (\u2191(LocalHomeomorph.symm e) (\u2191e x)))\n (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)))\n y\n[PROOFSTEP]\nrw [e.left_inv hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nx : M\nhx : x \u2208 e.source\nsrc\u271d : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (\u2191e x) := mfderiv I I' (\u2191e) x\ny : TangentSpace I' (\u2191e x)\nthis : \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' (\u2191e x))) y = y\n\u22a2 AddHom.toFun src\u271d.toAddHom (\u2191(mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x)) y) =\n \u2191(ContinuousLinearMap.comp (mfderiv I I' (\u2191e) x) (mfderiv I' I (\u2191(LocalHomeomorph.symm e)) (\u2191e x))) y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\n\u22a2 MDifferentiable I I'' (e \u226b\u2095 e')\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\n\u22a2 MDifferentiableOn I I'' (\u2191(e \u226b\u2095 e')) (e \u226b\u2095 e').toLocalEquiv.source\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\nx : M\nhx : x \u2208 (e \u226b\u2095 e').toLocalEquiv.source\n\u22a2 MDifferentiableWithinAt I I'' (\u2191(e \u226b\u2095 e')) (e \u226b\u2095 e').toLocalEquiv.source x\n[PROOFSTEP]\nsimp only [mfld_simps] at hx \n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\nx : M\nhx : x \u2208 e.source \u2227 \u2191e x \u2208 e'.source\n\u22a2 MDifferentiableWithinAt I I'' (\u2191(e \u226b\u2095 e')) (e \u226b\u2095 e').toLocalEquiv.source x\n[PROOFSTEP]\nexact ((he'.mdifferentiableAt hx.2).comp _ (he.mdifferentiableAt hx.1)).mdifferentiableWithinAt\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\n\u22a2 MDifferentiableOn I'' I (\u2191(LocalHomeomorph.symm (e \u226b\u2095 e'))) (e \u226b\u2095 e').toLocalEquiv.target\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\nx : M''\nhx : x \u2208 (e \u226b\u2095 e').toLocalEquiv.target\n\u22a2 MDifferentiableWithinAt I'' I (\u2191(LocalHomeomorph.symm (e \u226b\u2095 e'))) (e \u226b\u2095 e').toLocalEquiv.target x\n[PROOFSTEP]\nsimp only [mfld_simps] at hx \n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2075 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2074 : TopologicalSpace M\ninst\u271d\u00b9\u00b3 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u2070 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2079 : TopologicalSpace M'\ninst\u271d\u2078 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ne : LocalHomeomorph M M'\nhe : MDifferentiable I I' e\ne' : LocalHomeomorph M' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I' M'\ninst\u271d : SmoothManifoldWithCorners I'' M''\nhe' : MDifferentiable I' I'' e'\nx : M''\nhx : x \u2208 e'.target \u2227 \u2191(LocalHomeomorph.symm e') x \u2208 e.target\n\u22a2 MDifferentiableWithinAt I'' I (\u2191(LocalHomeomorph.symm (e \u226b\u2095 e'))) (e \u226b\u2095 e').toLocalEquiv.target x\n[PROOFSTEP]\nexact ((he.symm.mdifferentiableAt hx.2).comp _ (he'.symm.mdifferentiableAt hx.1)).mdifferentiableWithinAt\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\n\u22a2 UniqueMDiffWithinAt I' (f '' s) (f x)\n[PROOFSTEP]\nhave := hs.inter' <| hf.1 (extChartAt_source_mem_nhds I' (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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 UniqueMDiffWithinAt I' (f '' s) (f x)\n[PROOFSTEP]\nrefine (((hf.2.mono ?sub1).uniqueDiffWithinAt this hd).mono ?sub2).congr_pt ?pt\n[GOAL]\ncase sub1\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I \u2286\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I\ncase sub2\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f ''\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) \u2286\n \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\ncase pt\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n[PROOFSTEP]\ncase pt => simp only [mfld_simps]\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n[PROOFSTEP]\ncase pt => simp only [mfld_simps]\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f (\u2191(extChartAt I x) x) = \u2191(extChartAt I' (f x)) (f x)\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\ncase sub1\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I \u2286\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I\ncase sub2\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f ''\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) \u2286\n \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\n[PROOFSTEP]\ncase sub1 => mfld_set_tac\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I \u2286\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I\n[PROOFSTEP]\ncase sub1 => mfld_set_tac\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I \u2286\n \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I\n[PROOFSTEP]\nmfld_set_tac\n[GOAL]\ncase sub2\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f ''\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) \u2286\n \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\n[PROOFSTEP]\ncase sub2 =>\n rintro _ \u27e8y, \u27e8\u27e8hys, hfy\u27e9, -\u27e9, rfl\u27e9\n exact \u27e8\u27e8_, hys, ((extChartAt I' (f x)).left_inv hfy).symm\u27e9, mem_range_self _\u27e9\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f ''\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) \u2286\n \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\n[PROOFSTEP]\ncase sub2 =>\n rintro _ \u27e8y, \u27e8\u27e8hys, hfy\u27e9, -\u27e9, rfl\u27e9\n exact \u27e8\u27e8_, hys, ((extChartAt I' (f x)).left_inv hfy).symm\u27e9, mem_range_self _\u27e9\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\n\u22a2 writtenInExtChartAt I I' x f ''\n (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) \u2229 range \u2191I) \u2286\n \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\n[PROOFSTEP]\nrintro _ \u27e8y, \u27e8\u27e8hys, hfy\u27e9, -\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\nf : M \u2192 M'\nf' : E \u2192L[\ud835\udd5c] E'\nhf : HasMFDerivWithinAt I I' f s x f'\nhd : DenseRange \u2191f'\nthis : UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' (f x)).source) x\ny : E\nhys : \u2191(LocalEquiv.symm (extChartAt I x)) y \u2208 s\nhfy : \u2191(LocalEquiv.symm (extChartAt I x)) y \u2208 f \u207b\u00b9' (extChartAt I' (f x)).source\n\u22a2 writtenInExtChartAt I I' x f y \u2208 \u2191(LocalEquiv.symm (extChartAt I' (f x))) \u207b\u00b9' (f '' s) \u2229 range \u2191I'\n[PROOFSTEP]\nexact \u27e8\u27e8_, hys, ((extChartAt I' (f x)).left_inv hfy).symm\u27e9, mem_range_self _\u27e9\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\ne : LocalHomeomorph M M'\nhe : LocalHomeomorph.MDifferentiable I I' e\nhx : x \u2208 e.source\n\u22a2 UniqueMDiffWithinAt I' (e.target \u2229 \u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x)\n[PROOFSTEP]\nrw [\u2190 e.image_source_inter_eq', inter_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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nx : M\nhs : UniqueMDiffWithinAt I s x\ne : LocalHomeomorph M M'\nhe : LocalHomeomorph.MDifferentiable I I' e\nhx : x \u2208 e.source\n\u22a2 UniqueMDiffWithinAt I' (\u2191e '' (s \u2229 e.source)) (\u2191e x)\n[PROOFSTEP]\nexact\n (hs.inter (e.open_source.mem_nhds hx)).image_denseRange (he.mdifferentiableAt hx).hasMFDerivAt.hasMFDerivWithinAt\n (he.mfderiv_surjective hx).denseRange\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\n\u22a2 UniqueDiffOn \ud835\udd5c ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s)\n[PROOFSTEP]\napply UniqueMDiffOn.uniqueDiffOn\n[GOAL]\ncase a\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\n\u22a2 UniqueMDiffOn \ud835\udcd8(\ud835\udd5c, E) ((extChartAt I x).target \u2229 \u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 LocalEquiv.image_source_inter_eq', inter_comm, extChartAt_source]\n[GOAL]\ncase a\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\n\u22a2 UniqueMDiffOn \ud835\udcd8(\ud835\udd5c, E) (\u2191(extChartAt I x) '' (s \u2229 (chartAt H x).toLocalEquiv.source))\n[PROOFSTEP]\nexact\n (hs.inter (chartAt H x).open_source).image_denseRange' (fun y hy \u21a6 hasMFDerivWithinAt_extChartAt I hy.2) fun y hy \u21a6\n ((mdifferentiable_chart _ _).mfderiv_surjective hy.2).denseRange\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\ny : M'\nf : M \u2192 M'\nhf : ContinuousOn f s\n\u22a2 UniqueMDiffOn I (s \u2229 f \u207b\u00b9' (extChartAt I' y).source)\n[PROOFSTEP]\nintro z hz\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\ny : M'\nf : M \u2192 M'\nhf : ContinuousOn f s\nz : M\nhz : z \u2208 s \u2229 f \u207b\u00b9' (extChartAt I' y).source\n\u22a2 UniqueMDiffWithinAt I (s \u2229 f \u207b\u00b9' (extChartAt I' y).source) z\n[PROOFSTEP]\napply (hs z hz.1).inter'\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\ny : M'\nf : M \u2192 M'\nhf : ContinuousOn f s\nz : M\nhz : z \u2208 s \u2229 f \u207b\u00b9' (extChartAt I' y).source\n\u22a2 f \u207b\u00b9' (extChartAt I' y).source \u2208 \ud835\udcdd[s] z\n[PROOFSTEP]\napply (hf z hz.1).preimage_mem_nhdsWithin\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\nH : Type u_3\ninst\u271d\u2079 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\ninst\u271d\u2077 : ChartedSpace H M\ninst\u271d\u2076 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b3 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : SmoothManifoldWithCorners I' M'\ns : Set M\nhs : UniqueMDiffOn I s\nx : M\ny : M'\nf : M \u2192 M'\nhf : ContinuousOn f s\nz : M\nhz : z \u2208 s \u2229 f \u207b\u00b9' (extChartAt I' y).source\n\u22a2 (extChartAt I' y).source \u2208 \ud835\udcdd (f z)\n[PROOFSTEP]\nexact (isOpen_extChartAt_source I' y).mem_nhds hz.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p\n[PROOFSTEP]\nset e := trivializationAt F Z p.proj\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p\n[PROOFSTEP]\nhave hp : p \u2208 e.source := FiberBundle.mem_trivializationAt_proj_source\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p\n[PROOFSTEP]\nhave : UniqueMDiffWithinAt (I.prod \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (e p)\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n[PROOFSTEP]\nrw [\u2190 Prod.mk.eta (p := e p), FiberBundle.trivializationAt_proj_fst]\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (p.proj, (\u2191e p).snd)\n[PROOFSTEP]\nexact hs.prod (uniqueMDiffWithinAt_univ _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s) p\n[PROOFSTEP]\nrw [\u2190 e.left_inv hp]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (TotalSpace.proj \u207b\u00b9' s)\n (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (\u2191e.toLocalHomeomorph p))\n[PROOFSTEP]\nrefine (this.preimage_localHomeomorph e.mdifferentiable.symm (e.map_source hp)).mono ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\n\u22a2 (LocalHomeomorph.symm e.toLocalHomeomorph).toLocalEquiv.target \u2229\n \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e.toLocalHomeomorph)) \u207b\u00b9' s \u00d7\u02e2 univ \u2286\n TotalSpace.proj \u207b\u00b9' s\n[PROOFSTEP]\nrintro y \u27e8hy, hys, -\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2078 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2077 : TopologicalSpace M\ninst\u271d\u00b9\u2076 : ChartedSpace H M\ninst\u271d\u00b9\u2075 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\ninst\u271d\u00b9\u2070 : ChartedSpace H' M'\ninst\u271d\u2079 : SmoothManifoldWithCorners I' M'\ns : Set M\nF : Type u_8\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nZ : M \u2192 Type u_9\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F Z)\ninst\u271d\u2075 : (b : M) \u2192 TopologicalSpace (Z b)\ninst\u271d\u2074 : (b : M) \u2192 AddCommMonoid (Z b)\ninst\u271d\u00b3 : (b : M) \u2192 Module \ud835\udd5c (Z b)\ninst\u271d\u00b2 : FiberBundle F Z\ninst\u271d\u00b9 : VectorBundle \ud835\udd5c F Z\ninst\u271d : SmoothVectorBundle F Z I\np : TotalSpace F Z\nhs : UniqueMDiffWithinAt I s p.proj\ne : Trivialization F TotalSpace.proj := trivializationAt F Z p.proj\nhp : p \u2208 e.source\nthis : UniqueMDiffWithinAt (ModelWithCorners.prod I \ud835\udcd8(\ud835\udd5c, F)) (s \u00d7\u02e2 univ) (\u2191e p)\ny : TotalSpace F Z\nhy : y \u2208 (LocalHomeomorph.symm e.toLocalHomeomorph).toLocalEquiv.target\nhys : (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e.toLocalHomeomorph)) y).fst \u2208 s\n\u22a2 y \u2208 TotalSpace.proj \u207b\u00b9' s\n[PROOFSTEP]\nrwa [LocalHomeomorph.symm_symm, e.coe_coe, e.coe_fst hy] at hys \n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.MFDeriv", "llama_tokens": 208905, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.41489884579676883, "lm_q1q2_score": 0.2672882474464243}} {"text": "[GOAL]\n\u22a2 |exp 1 - 2244083 / 825552| \u2264 1 / 10 ^ 10\n[PROOFSTEP]\napply exp_approx_start\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 0 1 (2244083 / 825552)| \u2264 |1| ^ 0 / \u2191(Nat.factorial 0) * (1 / 10 ^ 10)\n[PROOFSTEP]\niterate 13 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 0 1 (2244083 / 825552)| \u2264 |1| ^ 0 / \u2191(Nat.factorial 0) * (1 / 10 ^ 10)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 0 + 1 = ?m.664\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 1 = ?m.664\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21911 = ?m.675\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 1 1 ((2244083 / 825552 - 1) * 1)| \u2264 |1| ^ 1 / \u2191(Nat.factorial 1) * (1 / 10 ^ 10 * 1)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 1 + 1 = ?m.1007\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 2 = ?m.1007\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21912 = ?m.1009\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 2 1 (((2244083 / 825552 - 1) * 1 - 1) * 2)| \u2264 |1| ^ 2 / \u2191(Nat.factorial 2) * (1 / 10 ^ 10 * 1 * 2)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 2 + 1 = ?m.1077\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 3 = ?m.1077\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21913 = ?m.1079\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 3 1 ((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3)| \u2264\n |1| ^ 3 / \u2191(Nat.factorial 3) * (1 / 10 ^ 10 * 1 * 2 * 3)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 3 + 1 = ?m.1147\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 4 = ?m.1147\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21914 = ?m.1149\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 4 1 (((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4)| \u2264\n |1| ^ 4 / \u2191(Nat.factorial 4) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 4 + 1 = ?m.1217\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 5 = ?m.1217\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21915 = ?m.1219\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 5 1 ((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5)| \u2264\n |1| ^ 5 / \u2191(Nat.factorial 5) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 5 + 1 = ?m.1287\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 6 = ?m.1287\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21916 = ?m.1289\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 6 1 (((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6)| \u2264\n |1| ^ 6 / \u2191(Nat.factorial 6) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 6 + 1 = ?m.1357\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 7 = ?m.1357\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21917 = ?m.1359\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 7 1 ((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7)| \u2264\n |1| ^ 7 / \u2191(Nat.factorial 7) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 7 + 1 = ?m.1427\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 8 = ?m.1427\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21918 = ?m.1429\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 8 1\n (((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8)| \u2264\n |1| ^ 8 / \u2191(Nat.factorial 8) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 8 + 1 = ?m.1497\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 9 = ?m.1497\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21919 = ?m.1499\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 9 1\n ((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) *\n 9)| \u2264\n |1| ^ 9 / \u2191(Nat.factorial 9) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 9 + 1 = ?m.1567\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 10 = ?m.1567\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219110 = ?m.1569\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 10 1\n (((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) * 9 -\n 1) *\n 10)| \u2264\n |1| ^ 10 / \u2191(Nat.factorial 10) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 10 + 1 = ?m.1637\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 11 = ?m.1637\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219111 = ?m.1639\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 11 1\n ((((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11)| \u2264\n |1| ^ 11 / \u2191(Nat.factorial 11) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 11 + 1 = ?m.1707\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 12 = ?m.1707\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219112 = ?m.1709\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 12 1\n (((((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12)| \u2264\n |1| ^ 12 / \u2191(Nat.factorial 12) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 12 + 1 = ?m.1777\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 13 = ?m.1777\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219113 = ?m.1779\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 13 1\n ((((((((((((((2244083 / 825552 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) * 8 - 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13)| \u2264\n |1| ^ 13 / \u2191(Nat.factorial 13) * (1 / 10 ^ 10 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 13 1 (5 / 7)| \u2264 |1| ^ 13 / \u2191(Nat.factorial 13) * (243243 / 390625)\n[PROOFSTEP]\nrefine' exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) _\n[GOAL]\n\u22a2 13 + 1 = ?m.2354\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 14 = ?m.2354\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219114 = ?m.2356\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u22a2 |1| \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u22a2 |1 - 5 / 7| \u2264 243243 / 390625 - |1| / 14 * ((14 + 1) / 14)\n[PROOFSTEP]\nrw [_root_.abs_one, abs_of_pos]\n[GOAL]\ncase h\n\u22a2 1 - 5 / 7 \u2264 243243 / 390625 - 1 / 14 * ((14 + 1) / 14)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u22a2 0 < 1 - 5 / 7\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 |exp 1 - 363916618873 / 133877442384| \u2264 1 / 10 ^ 20\n[PROOFSTEP]\napply exp_approx_start\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 0 1 (363916618873 / 133877442384)| \u2264 |1| ^ 0 / \u2191(Nat.factorial 0) * (1 / 10 ^ 20)\n[PROOFSTEP]\niterate 21 refine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 0 1 (363916618873 / 133877442384)| \u2264 |1| ^ 0 / \u2191(Nat.factorial 0) * (1 / 10 ^ 20)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 0 + 1 = ?m.3624\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 1 = ?m.3624\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21911 = ?m.3635\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 1 1 ((363916618873 / 133877442384 - 1) * 1)| \u2264 |1| ^ 1 / \u2191(Nat.factorial 1) * (1 / 10 ^ 20 * 1)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 1 + 1 = ?m.3967\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 2 = ?m.3967\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21912 = ?m.3969\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 2 1 (((363916618873 / 133877442384 - 1) * 1 - 1) * 2)| \u2264\n |1| ^ 2 / \u2191(Nat.factorial 2) * (1 / 10 ^ 20 * 1 * 2)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 2 + 1 = ?m.4037\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 3 = ?m.4037\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21913 = ?m.4039\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 3 1 ((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3)| \u2264\n |1| ^ 3 / \u2191(Nat.factorial 3) * (1 / 10 ^ 20 * 1 * 2 * 3)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 3 + 1 = ?m.4107\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 4 = ?m.4107\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21914 = ?m.4109\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 4 1 (((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4)| \u2264\n |1| ^ 4 / \u2191(Nat.factorial 4) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 4 + 1 = ?m.4177\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 5 = ?m.4177\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21915 = ?m.4179\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 5 1 ((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5)| \u2264\n |1| ^ 5 / \u2191(Nat.factorial 5) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 5 + 1 = ?m.4247\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 6 = ?m.4247\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21916 = ?m.4249\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 6 1 (((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6)| \u2264\n |1| ^ 6 / \u2191(Nat.factorial 6) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 6 + 1 = ?m.4317\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 7 = ?m.4317\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21917 = ?m.4319\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 7 1\n ((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7)| \u2264\n |1| ^ 7 / \u2191(Nat.factorial 7) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 7 + 1 = ?m.4387\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 8 = ?m.4387\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21918 = ?m.4389\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 8 1\n (((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *\n 8)| \u2264\n |1| ^ 8 / \u2191(Nat.factorial 8) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 8 + 1 = ?m.4457\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 9 = ?m.4457\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u21919 = ?m.4459\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 9 1\n ((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *\n 8 -\n 1) *\n 9)| \u2264\n |1| ^ 9 / \u2191(Nat.factorial 9) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 9 + 1 = ?m.4527\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 10 = ?m.4527\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219110 = ?m.4529\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 10 1\n (((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10)| \u2264\n |1| ^ 10 / \u2191(Nat.factorial 10) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 10 + 1 = ?m.4597\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 11 = ?m.4597\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219111 = ?m.4599\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 11 1\n ((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11)| \u2264\n |1| ^ 11 / \u2191(Nat.factorial 11) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 11 + 1 = ?m.4667\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 12 = ?m.4667\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219112 = ?m.4669\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 12 1\n (((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 - 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12)| \u2264\n |1| ^ 12 / \u2191(Nat.factorial 12) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 12 + 1 = ?m.4737\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 13 = ?m.4737\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219113 = ?m.4739\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 13 1\n ((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13)| \u2264\n |1| ^ 13 / \u2191(Nat.factorial 13) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 13 + 1 = ?m.4807\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 14 = ?m.4807\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219114 = ?m.4809\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 14 1\n (((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14)| \u2264\n |1| ^ 14 / \u2191(Nat.factorial 14) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 14 + 1 = ?m.4877\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 15 = ?m.4877\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219115 = ?m.4879\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 15 1\n ((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15)| \u2264\n |1| ^ 15 / \u2191(Nat.factorial 15) * (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 15 + 1 = ?m.4947\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 16 = ?m.4947\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219116 = ?m.4949\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 16 1\n (((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16)| \u2264\n |1| ^ 16 / \u2191(Nat.factorial 16) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 16 + 1 = ?m.5017\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 17 = ?m.5017\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219117 = ?m.5019\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 17 1\n ((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) * 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16 -\n 1) *\n 17)| \u2264\n |1| ^ 17 / \u2191(Nat.factorial 17) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 17 + 1 = ?m.5087\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 18 = ?m.5087\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219118 = ?m.5089\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 18 1\n (((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *\n 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16 -\n 1) *\n 17 -\n 1) *\n 18)| \u2264\n |1| ^ 18 / \u2191(Nat.factorial 18) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 18 + 1 = ?m.5157\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 19 = ?m.5157\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219119 = ?m.5159\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 19 1\n ((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *\n 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16 -\n 1) *\n 17 -\n 1) *\n 18 -\n 1) *\n 19)| \u2264\n |1| ^ 19 / \u2191(Nat.factorial 19) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 19 + 1 = ?m.5227\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 20 = ?m.5227\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219120 = ?m.5229\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 20 1\n (((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *\n 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16 -\n 1) *\n 17 -\n 1) *\n 18 -\n 1) *\n 19 -\n 1) *\n 20)| \u2264\n |1| ^ 20 / \u2191(Nat.factorial 20) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20)\n[PROOFSTEP]\nrefine' exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) _\n[GOAL]\n\u22a2 20 + 1 = ?m.5297\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 21 = ?m.5297\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219121 = ?m.5299\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u22a2 |exp 1 -\n expNear 21 1\n ((((((((((((((((((((((363916618873 / 133877442384 - 1) * 1 - 1) * 2 - 1) * 3 - 1) * 4 - 1) * 5 - 1) * 6 - 1) *\n 7 -\n 1) *\n 8 -\n 1) *\n 9 -\n 1) *\n 10 -\n 1) *\n 11 -\n 1) *\n 12 -\n 1) *\n 13 -\n 1) *\n 14 -\n 1) *\n 15 -\n 1) *\n 16 -\n 1) *\n 17 -\n 1) *\n 18 -\n 1) *\n 19 -\n 1) *\n 20 -\n 1) *\n 21)| \u2264\n |1| ^ 21 / \u2191(Nat.factorial 21) *\n (1 / 10 ^ 20 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 * 21)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u22a2 |exp 1 - expNear 21 1 (36295539 / 44271641)| \u2264 |1| ^ 21 / \u2191(Nat.factorial 21) * (311834363841 / 610351562500)\n[PROOFSTEP]\nrefine' exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) _\n[GOAL]\n\u22a2 21 + 1 = ?m.5994\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 22 = ?m.5994\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u219122 = ?m.5996\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u22a2 |1| \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u22a2 |1 - 36295539 / 44271641| \u2264 311834363841 / 610351562500 - |1| / 22 * ((22 + 1) / 22)\n[PROOFSTEP]\nrw [_root_.abs_one, abs_of_pos]\n[GOAL]\ncase h\n\u22a2 1 - 36295539 / 44271641 \u2264 311834363841 / 610351562500 - 1 / 22 * ((22 + 1) / 22)\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase h\n\u22a2 0 < 1 - 36295539 / 44271641\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 2.7182818283 < 2244083 / 825552 - 1 / 10 ^ 10\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 1 / 10 ^ 10 + 2244083 / 825552 < 2.7182818286\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0.36787944116 < exp (-1)\n[PROOFSTEP]\nrw [exp_neg, lt_inv _ (exp_pos _)]\n[GOAL]\n\u22a2 exp 1 < 0.36787944116\u207b\u00b9\n\u22a2 0 < 0.36787944116\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) _\n[GOAL]\n\u22a2 1 / 10 ^ 10 + 2244083 / 825552 < 0.36787944116\u207b\u00b9\n\u22a2 0 < 0.36787944116\n[PROOFSTEP]\nall_goals norm_num\n[GOAL]\n\u22a2 1 / 10 ^ 10 + 2244083 / 825552 < 0.36787944116\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 0.36787944116\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 exp (-1) < 0.3678794412\n[PROOFSTEP]\nrw [exp_neg, inv_lt (exp_pos _)]\n[GOAL]\n\u22a2 0.3678794412\u207b\u00b9 < exp 1\n\u22a2 0 < 0.3678794412\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)\n[GOAL]\n\u22a2 0.3678794412\u207b\u00b9 < 2244083 / 825552 - 1 / 10 ^ 10\n\u22a2 0 < 0.3678794412\n[PROOFSTEP]\nall_goals norm_num\n[GOAL]\n\u22a2 0.3678794412\u207b\u00b9 < 2244083 / 825552 - 1 / 10 ^ 10\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0 < 0.3678794412\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 10 ^ 10\n[PROOFSTEP]\nsuffices |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n by\n norm_num1 at *\n assumption\n[GOAL]\nthis : |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 10 ^ 10\n[PROOFSTEP]\nnorm_num1 at *\n[GOAL]\nthis : |log 2 - 287209 / 414355| \u2264 1 / 10000000000\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 10000000000\n[PROOFSTEP]\nassumption\n[GOAL]\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\nhave t : |(2\u207b\u00b9 : \u211d)| = 2\u207b\u00b9 := by rw [abs_of_pos]; norm_num\n[GOAL]\n\u22a2 |2\u207b\u00b9| = 2\u207b\u00b9\n[PROOFSTEP]\nrw [abs_of_pos]\n[GOAL]\n\u22a2 0 < 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\nhave z := Real.abs_log_sub_add_sum_range_le (show |(2\u207b\u00b9 : \u211d)| < 1 by rw [t]; norm_num) 34\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\n\u22a2 |2\u207b\u00b9| < 1\n[PROOFSTEP]\nrw [t]\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\n\u22a2 2\u207b\u00b9 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |(Finset.sum (range 34) fun i => 2\u207b\u00b9 ^ (i + 1) / (\u2191i + 1)) + log (1 - 2\u207b\u00b9)| \u2264 |2\u207b\u00b9| ^ (34 + 1) / (1 - |2\u207b\u00b9|)\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\nrw [t] at z \n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |(Finset.sum (range 34) fun i => 2\u207b\u00b9 ^ (i + 1) / (\u2191i + 1)) + log (1 - 2\u207b\u00b9)| \u2264 2\u207b\u00b9 ^ (34 + 1) / (1 - 2\u207b\u00b9)\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\nnorm_num1 at z \n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |(Finset.sum (range 34) fun x => (1 / 2) ^ (x + 1) / (\u2191x + 1)) + log (1 / 2)| \u2264 1 / 17179869184\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\nrw [one_div (2 : \u211d), log_inv, \u2190 sub_eq_add_neg, _root_.abs_sub_comm] at z \n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 |log 2 - 287209 / 414355| \u2264 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)\n[PROOFSTEP]\napply le_trans (_root_.abs_sub_le _ _ _) (add_le_add z _)\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 |(Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)) - 287209 / 414355| \u2264 1 / 10 ^ 10 - 1 / 2 ^ 34\n[PROOFSTEP]\nsimp_rw [sum_range_succ]\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 |(Finset.sum (range 0) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)) + 2\u207b\u00b9 ^ (0 + 1) / (\u21910 + 1) + 2\u207b\u00b9 ^ (1 + 1) / (\u21911 + 1) +\n 2\u207b\u00b9 ^ (2 + 1) / (\u21912 + 1) +\n 2\u207b\u00b9 ^ (3 + 1) / (\u21913 + 1) +\n 2\u207b\u00b9 ^ (4 + 1) / (\u21914 + 1) +\n 2\u207b\u00b9 ^ (5 + 1) / (\u21915 + 1) +\n 2\u207b\u00b9 ^ (6 + 1) / (\u21916 + 1) +\n 2\u207b\u00b9 ^ (7 + 1) / (\u21917 + 1) +\n 2\u207b\u00b9 ^ (8 + 1) / (\u21918 + 1) +\n 2\u207b\u00b9 ^ (9 + 1) / (\u21919 + 1) +\n 2\u207b\u00b9 ^ (10 + 1) / (\u219110 + 1) +\n 2\u207b\u00b9 ^ (11 + 1) / (\u219111 + 1) +\n 2\u207b\u00b9 ^ (12 + 1) / (\u219112 + 1) +\n 2\u207b\u00b9 ^ (13 + 1) / (\u219113 + 1) +\n 2\u207b\u00b9 ^ (14 + 1) / (\u219114 + 1) +\n 2\u207b\u00b9 ^ (15 + 1) / (\u219115 + 1) +\n 2\u207b\u00b9 ^ (16 + 1) / (\u219116 + 1) +\n 2\u207b\u00b9 ^ (17 + 1) / (\u219117 + 1) +\n 2\u207b\u00b9 ^ (18 + 1) / (\u219118 + 1) +\n 2\u207b\u00b9 ^ (19 + 1) / (\u219119 + 1) +\n 2\u207b\u00b9 ^ (20 + 1) / (\u219120 + 1) +\n 2\u207b\u00b9 ^ (21 + 1) / (\u219121 + 1) +\n 2\u207b\u00b9 ^ (22 + 1) / (\u219122 + 1) +\n 2\u207b\u00b9 ^ (23 + 1) / (\u219123 + 1) +\n 2\u207b\u00b9 ^ (24 + 1) / (\u219124 + 1) +\n 2\u207b\u00b9 ^ (25 + 1) / (\u219125 + 1) +\n 2\u207b\u00b9 ^ (26 + 1) / (\u219126 + 1) +\n 2\u207b\u00b9 ^ (27 + 1) / (\u219127 + 1) +\n 2\u207b\u00b9 ^ (28 + 1) / (\u219128 + 1) +\n 2\u207b\u00b9 ^ (29 + 1) / (\u219129 + 1) +\n 2\u207b\u00b9 ^ (30 + 1) / (\u219130 + 1) +\n 2\u207b\u00b9 ^ (31 + 1) / (\u219131 + 1) +\n 2\u207b\u00b9 ^ (32 + 1) / (\u219132 + 1) +\n 2\u207b\u00b9 ^ (33 + 1) / (\u219133 + 1) -\n 287209 / 414355| \u2264\n 1 / 10 ^ 10 - 1 / 2 ^ 34\n[PROOFSTEP]\nnorm_num\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 |30417026706710207 / 51397301678363663775930777600| \u2264 7011591 / 167772160000000000\n[PROOFSTEP]\nrw [abs_of_pos]\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 30417026706710207 / 51397301678363663775930777600 \u2264 7011591 / 167772160000000000\n[PROOFSTEP]\nnorm_num\n[GOAL]\nt : |2\u207b\u00b9| = 2\u207b\u00b9\nz : |log 2 - Finset.sum (range 34) fun x => 2\u207b\u00b9 ^ (x + 1) / (\u2191x + 1)| \u2264 1 / 17179869184\n\u22a2 0 < 30417026706710207 / 51397301678363663775930777600\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 0.6931471803 < 287209 / 414355 - 1 / 10 ^ 10\n[PROOFSTEP]\nnorm_num1\n[GOAL]\n\u22a2 1 / 10 ^ 10 + 287209 / 414355 < 0.6931471808\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.Data.Complex.ExponentialBounds", "llama_tokens": 16630, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.2669367346441609}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nW : (Cover J X)\u1d52\u1d56\n\u22a2 (diagram J P X \u22d9 F).obj W \u2245 (diagram J (P \u22d9 F) X).obj W\n[PROOFSTEP]\nrefine' _ \u226a\u226b HasLimit.isoOfNatIso (W.unop.multicospanComp _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nW : (Cover J X)\u1d52\u1d56\n\u22a2 (diagram J P X \u22d9 F).obj W \u2245 limit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)\n[PROOFSTEP]\nrefine' (isLimitOfPreserves F (limit.isLimit _)).conePointUniqueUpToIso (limit.isLimit _)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\n\u22a2 \u2200 {X_1 Y : (Cover J X)\u1d52\u1d56} (f : X_1 \u27f6 Y),\n (diagram J P X \u22d9 F).map f \u226b\n ((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n Y).hom =\n ((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n X_1).hom \u226b\n (diagram J (P \u22d9 F) X).map f\n[PROOFSTEP]\nintro A B f\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nA B : (Cover J X)\u1d52\u1d56\nf : A \u27f6 B\n\u22a2 (diagram J P X \u22d9 F).map f \u226b\n ((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n B).hom =\n ((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n A).hom \u226b\n (diagram J (P \u22d9 F) X).map f\n[PROOFSTEP]\napply Multiequalizer.hom_ext\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nA B : (Cover J X)\u1d52\u1d56\nf : A \u27f6 B\n\u22a2 \u2200 (a : (Cover.index B.unop (P \u22d9 F)).L),\n ((diagram J P X \u22d9 F).map f \u226b\n ((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n B).hom) \u226b\n Multiequalizer.\u03b9 (Cover.index B.unop (P \u22d9 F)) a =\n (((fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)\n A).hom \u226b\n (diagram J (P \u22d9 F) X).map f) \u226b\n Multiequalizer.\u03b9 (Cover.index B.unop (P \u22d9 F)) a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nA B : (Cover J X)\u1d52\u1d56\nf : A \u27f6 B\n\u22a2 \u2200 (a : (Cover.index B.unop (P \u22d9 F)).L),\n (F.map\n (Multiequalizer.lift (Cover.index B.unop P) (multiequalizer (Cover.index A.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index A.unop P) (Cover.Arrow.map I f.unop))\n (_ :\n \u2200 (I : (Cover.index B.unop P).R),\n Multiequalizer.\u03b9 (Cover.index A.unop P)\n (MulticospanIndex.fstTo (Cover.index A.unop P) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.fst (Cover.index A.unop P) (Cover.Relation.map I f.unop) =\n Multiequalizer.\u03b9 (Cover.index A.unop P)\n (MulticospanIndex.sndTo (Cover.index A.unop P) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.snd (Cover.index A.unop P) (Cover.Relation.map I f.unop))) \u226b\n (IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index B.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index B.unop P) \u22d9 F))).hom \u226b\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P B.unop).symm).hom) \u226b\n Multiequalizer.\u03b9 (Cover.index B.unop (P \u22d9 F)) a =\n (((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index A.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index A.unop P) \u22d9 F))).hom \u226b\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P A.unop).symm).hom) \u226b\n Multiequalizer.lift (Cover.index B.unop (P \u22d9 F)) (multiequalizer (Cover.index A.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index A.unop (P \u22d9 F)) (Cover.Arrow.map I f.unop))\n (_ :\n \u2200 (I : (Cover.index B.unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index A.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index A.unop (P \u22d9 F)) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.fst (Cover.index A.unop (P \u22d9 F)) (Cover.Relation.map I f.unop) =\n Multiequalizer.\u03b9 (Cover.index A.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index A.unop (P \u22d9 F)) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.snd (Cover.index A.unop (P \u22d9 F)) (Cover.Relation.map I f.unop))) \u226b\n Multiequalizer.\u03b9 (Cover.index B.unop (P \u22d9 F)) a\n[PROOFSTEP]\nsimp only [Functor.mapCone_\u03c0_app, Multiequalizer.multifork_\u03c0_app_left, Iso.symm_hom, Multiequalizer.lift_\u03b9,\n eqToHom_refl, Category.comp_id, limit.conePointUniqueUpToIso_hom_comp,\n GrothendieckTopology.Cover.multicospanComp_hom_inv_left, HasLimit.isoOfNatIso_hom_\u03c0, Category.assoc]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nA B : (Cover J X)\u1d52\u1d56\nf : A \u27f6 B\n\u22a2 \u2200 (a : (Cover.index B.unop (P \u22d9 F)).L),\n F.map\n (Multiequalizer.lift (Cover.index B.unop P) (multiequalizer (Cover.index A.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index A.unop P) (Cover.Arrow.map I f.unop))\n (_ :\n \u2200 (I : (Cover.index B.unop P).R),\n Multiequalizer.\u03b9 (Cover.index A.unop P)\n (MulticospanIndex.fstTo (Cover.index A.unop P) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.fst (Cover.index A.unop P) (Cover.Relation.map I f.unop) =\n Multiequalizer.\u03b9 (Cover.index A.unop P)\n (MulticospanIndex.sndTo (Cover.index A.unop P) (Cover.Relation.map I f.unop)) \u226b\n MulticospanIndex.snd (Cover.index A.unop P) (Cover.Relation.map I f.unop))) \u226b\n F.map (Multiequalizer.\u03b9 (Cover.index B.unop P) a) =\n F.map (Multiequalizer.\u03b9 (Cover.index A.unop P) (Cover.Arrow.map a f.unop))\n[PROOFSTEP]\nsimp only [\u2190 F.map_comp, limit.lift_\u03c0, Multifork.of\u03b9_\u03c0_app, implies_true]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nW : (Cover J X)\u1d52\u1d56\ni : Cover.Arrow W.unop\n\u22a2 NatTrans.app (diagramCompIso J F P X).hom W \u226b Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i =\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop P) i)\n[PROOFSTEP]\ndelta diagramCompIso\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nW : (Cover J X)\u1d52\u1d56\ni : Cover.Arrow W.unop\n\u22a2 NatTrans.app\n (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom\n W \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i =\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop P) i)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2074 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b3 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u00b2 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u00b9 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\nX : C\nW : (Cover J X)\u1d52\u1d56\ni : Cover.Arrow W.unop\n\u22a2 ((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F))).hom \u226b\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i =\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop P) i)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\n\u22a2 (plusObj J P \u22d9 F).obj X \u2245 (plusObj J (P \u22d9 F)).obj X\n[PROOFSTEP]\nrefine' _ \u226a\u226b HasColimit.isoOfNatIso (J.diagramCompIso F P X.unop)\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\n\u22a2 (plusObj J P \u22d9 F).obj X \u2245 colimit (diagram J P X.unop \u22d9 F)\n[PROOFSTEP]\nrefine'\n (isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).coconePointUniqueUpToIso (colimit.isColimit _)\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\n\u22a2 \u2200 {X Y : C\u1d52\u1d56} (f : X \u27f6 Y),\n (plusObj J P \u22d9 F).map f \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n Y).hom =\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n X).hom \u226b\n (plusObj J (P \u22d9 F)).map f\n[PROOFSTEP]\nintro X Y f\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (plusObj J P \u22d9 F).map f \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n Y).hom =\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n X).hom \u226b\n (plusObj J (P \u22d9 F)).map f\n[PROOFSTEP]\napply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 \u2200 (j : (Cover J X.unop)\u1d52\u1d56),\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n (plusObj J P \u22d9 F).map f \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n Y).hom =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n X).hom \u226b\n (plusObj J (P \u22d9 F)).map f\n[PROOFSTEP]\nintro W\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n (plusObj J P \u22d9 F).map f \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n Y).hom =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n ((fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso (diagramCompIso J F P X.unop))\n X).hom \u226b\n (plusObj J (P \u22d9 F)).map f\n[PROOFSTEP]\ndsimp [plusObj, plusMap]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop) \u226b colimit.pre (diagram J P Y.unop) (pullback J f.unop).op) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n ((IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom) \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop) \u226b colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nsimp only [Functor.map_comp, Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop)) \u226b\n F.map (colimit.pre (diagram J P Y.unop) (pullback J f.unop).op) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop) \u226b colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nslice_rhs 1 2 => erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]\n[GOAL]\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimMap (diagramPullback J (P \u22d9 F) f.unop)\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]\n[GOAL]\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimMap (diagramPullback J (P \u22d9 F) f.unop)\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]\n[GOAL]\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom\ncase a.a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimMap (diagramPullback J (P \u22d9 F) f.unop)\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).fac]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop)) \u226b\n F.map (colimit.pre (diagram J P Y.unop) (pullback J f.unop).op) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n ((NatTrans.app (colimit.cocone (diagram J P X.unop \u22d9 F)).\u03b9 W \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom) \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop)) \u226b\n colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nslice_lhs 1 3 =>\n simp only [\u2190 F.map_comp]\n dsimp [colimMap, IsColimit.map, colimit.pre]\n simp only [colimit.\u03b9_desc_assoc, colimit.\u03b9_desc]\n dsimp [Cocones.precompose]\n simp only [Category.assoc, colimit.\u03b9_desc]\n dsimp [Cocone.whisker]\n rw [F.map_comp]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop)) \u226b F.map (colimit.pre (diagram J P Y.unop) (pullback J f.unop).op)\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\n simp only [\u2190 F.map_comp]\n dsimp [colimMap, IsColimit.map, colimit.pre]\n simp only [colimit.\u03b9_desc_assoc, colimit.\u03b9_desc]\n dsimp [Cocones.precompose]\n simp only [Category.assoc, colimit.\u03b9_desc]\n dsimp [Cocone.whisker]\n rw [F.map_comp]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop)) \u226b F.map (colimit.pre (diagram J P Y.unop) (pullback J f.unop).op)\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\n simp only [\u2190 F.map_comp]\n dsimp [colimMap, IsColimit.map, colimit.pre]\n simp only [colimit.\u03b9_desc_assoc, colimit.\u03b9_desc]\n dsimp [Cocones.precompose]\n simp only [Category.assoc, colimit.\u03b9_desc]\n dsimp [Cocone.whisker]\n rw [F.map_comp]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramPullback J P f.unop)) \u226b F.map (colimit.pre (diagram J P Y.unop) (pullback J f.unop).op)\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\nsimp only [\u2190 F.map_comp]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (colimit.\u03b9 (diagram J P X.unop) W \u226b\n colimMap (diagramPullback J P f.unop) \u226b colimit.pre (diagram J P Y.unop) (pullback J f.unop).op)\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\ndsimp [colimMap, IsColimit.map, colimit.pre]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (colimit.\u03b9 (diagram J P X.unop) W \u226b\n colimit.desc (diagram J P X.unop)\n ((Cocones.precompose (diagramPullback J P f.unop)).obj\n (colimit.cocone ((pullback J f.unop).op \u22d9 diagram J P Y.unop))) \u226b\n colimit.desc ((pullback J f.unop).op \u22d9 diagram J P Y.unop)\n (Cocone.whisker (pullback J f.unop).op (colimit.cocone (diagram J P Y.unop))))\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\nsimp only [colimit.\u03b9_desc_assoc, colimit.\u03b9_desc]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (NatTrans.app\n ((Cocones.precompose (diagramPullback J P f.unop)).obj\n (colimit.cocone ((pullback J f.unop).op \u22d9 diagram J P Y.unop))).\u03b9\n W \u226b\n colimit.desc ((pullback J f.unop).op \u22d9 diagram J P Y.unop)\n (Cocone.whisker (pullback J f.unop).op (colimit.cocone (diagram J P Y.unop))))\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\ndsimp [Cocones.precompose]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n ((Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n colimit.\u03b9 ((pullback J f.unop).op \u22d9 diagram J P Y.unop) W) \u226b\n colimit.desc ((pullback J f.unop).op \u22d9 diagram J P Y.unop)\n (Cocone.whisker (pullback J f.unop).op (colimit.cocone (diagram J P Y.unop))))\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\nsimp only [Category.assoc, colimit.\u03b9_desc]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n NatTrans.app (Cocone.whisker (pullback J f.unop).op (colimit.cocone (diagram J P Y.unop))).\u03b9 W)\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\ndsimp [Cocone.whisker]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop)))\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 ((F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n F.map (colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop)))) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom) \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n ((NatTrans.app (colimit.cocone (diagram J P X.unop \u22d9 F)).\u03b9 W \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom) \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop)) \u226b\n colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n F.map (colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop))) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n NatTrans.app (colimit.cocone (diagram J P X.unop \u22d9 F)).\u03b9 W \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop) \u226b colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nslice_lhs 2 3 => erw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P Y.unop))).fac]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop))) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)))\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P Y.unop))).fac]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop))) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)))\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P Y.unop))).fac]\n[GOAL]\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map (colimit.\u03b9 (diagram J P Y.unop) (op (Cover.pullback W.unop f.unop))) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P Y.unop)))\n (colimit.isColimit (diagram J P Y.unop \u22d9 F))).hom\ncase a.a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom\ncase a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n| F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I)))\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P Y.unop))).fac]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (colimit.cocone (diagram J P Y.unop \u22d9 F)).\u03b9 (op (Cover.pullback W.unop f.unop)) \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n NatTrans.app (colimit.cocone (diagram J P X.unop \u22d9 F)).\u03b9 W \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop) \u226b colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n colimit.\u03b9 (diagram J P Y.unop \u22d9 F) (op (Cover.pullback W.unop f.unop)) \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P Y.unop)).hom =\n colimit.\u03b9 (diagram J P X.unop \u22d9 F) W \u226b\n (HasColimit.isoOfNatIso (diagramCompIso J F P X.unop)).hom \u226b\n colimMap (diagramPullback J (P \u22d9 F) f.unop) \u226b colimit.pre (diagram J (P \u22d9 F) Y.unop) (pullback J f.unop).op\n[PROOFSTEP]\nsimp only [HasColimit.isoOfNatIso_\u03b9_hom_assoc, GrothendieckTopology.diagramPullback_app, colimit.\u03b9_pre,\n HasColimit.isoOfNatIso_\u03b9_hom, \u03b9_colimMap_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop)) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) (op (Cover.pullback W.unop f.unop)) =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) ((pullback J f.unop).op.obj W)\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop))) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) (op (Cover.pullback W.unop f.unop)) =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I))) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) ((pullback J f.unop).op.obj W)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop))) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) (op (Cover.pullback W.unop f.unop)) =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F))\n (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I))) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) Y.unop) (op (Cover.pullback W.unop f.unop))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop)) =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F))\n (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I))\n[PROOFSTEP]\next\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\na\u271d : (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)).L\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop))) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F))\n (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I))) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\na\u271d : (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)).L\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop))) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F))\n (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I))) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nW : (Cover J X.unop)\u1d52\u1d56\na\u271d : (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)).L\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) P) (multiequalizer (Cover.index W.unop P))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop P) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop P).R),\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.fstTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop P) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop P)\n (MulticospanIndex.sndTo (Cover.index W.unop P) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop P) (Cover.Relation.base I))) \u226b\n NatTrans.app (diagramCompIso J F P Y.unop).hom (op (Cover.pullback W.unop f.unop)) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F))\n (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun I => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) (Cover.Arrow.base I))\n (_ :\n \u2200 (I : (Cover.index ((pullback J f.unop).op.obj W).unop (P \u22d9 F)).R),\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I) =\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 F)) (Cover.Relation.base I)) \u226b\n Multiequalizer.\u03b9 (Cover.index (Cover.pullback W.unop f.unop) (P \u22d9 F)) a\u271d\n[PROOFSTEP]\nerw [Multiequalizer.lift_\u03b9, diagramCompIso_hom_\u03b9, diagramCompIso_hom_\u03b9, \u2190 F.map_comp, Multiequalizer.lift_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b NatTrans.app (plusCompIso J F P).hom X =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b colimit.\u03b9 (diagram J (P \u22d9 F) X.unop) W\n[PROOFSTEP]\ndelta diagramCompIso plusCompIso\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F)) \u226a\u226b\n HasColimit.isoOfNatIso\n (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)).hom\n X =\n NatTrans.app\n (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom\n W \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [IsColimit.descCoconeMorphism_Hom, IsColimit.uniqueUpToIso_hom, Cocones.forget_map, Iso.trans_hom,\n NatIso.ofComponents_hom_app, Functor.mapIso_hom, \u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n (IsColimit.coconePointUniqueUpToIso (isColimitOfPreserves F (colimit.isColimit (diagram J P X.unop)))\n (colimit.isColimit (diagram J P X.unop \u22d9 F))).hom) \u226b\n (HasColimit.isoOfNatIso\n (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)).hom =\n ((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F))).hom \u226b\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) X.unop) W\n[PROOFSTEP]\nerw [(isColimitOfPreserves F (colimit.isColimit (J.diagram P (unop X)))).fac]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (colimit.cocone (diagram J P X.unop \u22d9 F)).\u03b9 W \u226b\n (HasColimit.isoOfNatIso\n (NatIso.ofComponents fun W =>\n IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F)) \u226a\u226b\n HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm)).hom =\n ((IsLimit.conePointUniqueUpToIso\n (isLimitOfPreserves F (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P))))\n (limit.isLimit (MulticospanIndex.multicospan (Cover.index W.unop P) \u22d9 F))).hom \u226b\n (HasLimit.isoOfNatIso (Cover.multicospanComp F P W.unop).symm).hom) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [Category.assoc, HasLimit.isoOfNatIso_hom_\u03c0, Iso.symm_hom, Cover.multicospanComp_hom_inv_left, eqToHom_refl,\n Category.comp_id, limit.conePointUniqueUpToIso_hom_comp, Functor.mapCone_\u03c0_app, Multiequalizer.multifork_\u03c0_app_left,\n Multiequalizer.lift_\u03b9, Functor.map_comp, eq_self_iff_true, Category.assoc, Iso.trans_hom, Iso.cancel_iso_hom_left,\n NatIso.ofComponents_hom_app, colimit.cocone_\u03b9, Category.assoc, HasColimit.isoOfNatIso_\u03b9_hom]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\n\u22a2 whiskerLeft (plusObj J P) \u03b7 \u226b (plusCompIso J G P).hom = (plusCompIso J F P).hom \u226b plusMap J (whiskerLeft P \u03b7)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\n\u22a2 NatTrans.app (whiskerLeft (plusObj J P) \u03b7 \u226b (plusCompIso J G P).hom) X =\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerLeft P \u03b7)) X\n[PROOFSTEP]\napply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\n\u22a2 \u2200 (j : (Cover J X.unop)\u1d52\u1d56),\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n NatTrans.app (whiskerLeft (plusObj J P) \u03b7 \u226b (plusCompIso J G P).hom) X =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerLeft P \u03b7)) X\n[PROOFSTEP]\nintro W\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n NatTrans.app (whiskerLeft (plusObj J P) \u03b7 \u226b (plusCompIso J G P).hom) X =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerLeft P \u03b7)) X\n[PROOFSTEP]\ndsimp [plusObj, plusMap]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n NatTrans.app \u03b7 (colimit (diagram J P X.unop)) \u226b NatTrans.app (plusCompIso J G P).hom X =\n F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n NatTrans.app (plusCompIso J F P).hom X \u226b colimMap (diagramNatTrans J (whiskerLeft P \u03b7) X.unop)\n[PROOFSTEP]\nsimp only [\u03b9_plusCompIso_hom, \u03b9_colimMap, whiskerLeft_app, \u03b9_plusCompIso_hom_assoc, NatTrans.naturality_assoc,\n GrothendieckTopology.diagramNatTrans_app]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b\n NatTrans.app (diagramCompIso J G P X.unop).hom W \u226b colimit.\u03b9 (diagram J (P \u22d9 G) X.unop) W =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) b) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 G) X.unop) W\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b NatTrans.app (diagramCompIso J G P X.unop).hom W) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 G) X.unop) W =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) b)) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 G) X.unop) W\n[PROOFSTEP]\ncongr 1\n -- porting note: this used to work with `ext`\n -- See https://github.com/leanprover-community/mathlib4/issues/5229\n[GOAL]\ncase w.h.e_a\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b NatTrans.app (diagramCompIso J G P X.unop).hom W =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) b)\n[PROOFSTEP]\napply Multiequalizer.hom_ext\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 \u2200 (a : (Cover.index W.unop (P \u22d9 G)).L),\n (NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b NatTrans.app (diagramCompIso J G P X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) b)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (P \u22d9 G)).L\n\u22a2 (NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b NatTrans.app (diagramCompIso J G P X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) b)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (P \u22d9 G)).L\n\u22a2 (NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b NatTrans.app (diagramCompIso J G P X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (P \u22d9 G)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app \u03b7 (P.obj (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (P \u22d9 G)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerLeft P \u03b7) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (P \u22d9 G)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (P \u22d9 G)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerLeft P \u03b7) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (P \u22d9 G)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (P \u22d9 G)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 G)) a\n[PROOFSTEP]\nsimp\n -- Porting note: in mathlib3 `simp` managed to apply this.\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u00b9\u00b9 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u00b9\u2070 : Category.{max v u, w\u2082} E\nF\u271d : D \u2964 E\ninst\u271d\u2079 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2078 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u2077 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\u271d\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u2076 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u2075 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d\u2074 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\u271d\nF G : D \u2964 E\n\u03b7 : F \u27f6 G\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b3 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\ninst\u271d\u00b2 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 G\ninst\u271d : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) G\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (P \u22d9 G)).L\n\u22a2 NatTrans.app \u03b7 (multiequalizer (Cover.index W.unop P)) \u226b G.map (Multiequalizer.\u03b9 (Cover.index W.unop P) a) =\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop P) a) \u226b NatTrans.app \u03b7 (P.obj (op a.Y))\n[PROOFSTEP]\nerw [\u03b7.naturality]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u22a2 whiskerRight (plusMap J \u03b7) F \u226b (plusCompIso J F Q).hom = (plusCompIso J F P).hom \u226b plusMap J (whiskerRight \u03b7 F)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\n\u22a2 NatTrans.app (whiskerRight (plusMap J \u03b7) F \u226b (plusCompIso J F Q).hom) X =\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerRight \u03b7 F)) X\n[PROOFSTEP]\napply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\n\u22a2 \u2200 (j : (Cover J X.unop)\u1d52\u1d56),\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n NatTrans.app (whiskerRight (plusMap J \u03b7) F \u226b (plusCompIso J F Q).hom) X =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 j \u226b\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerRight \u03b7 F)) X\n[PROOFSTEP]\nintro W\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n NatTrans.app (whiskerRight (plusMap J \u03b7) F \u226b (plusCompIso J F Q).hom) X =\n NatTrans.app (F.mapCocone (colimit.cocone (diagram J P X.unop))).\u03b9 W \u226b\n NatTrans.app ((plusCompIso J F P).hom \u226b plusMap J (whiskerRight \u03b7 F)) X\n[PROOFSTEP]\ndsimp [plusObj, plusMap]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramNatTrans J \u03b7 X.unop)) \u226b NatTrans.app (plusCompIso J F Q).hom X =\n F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n NatTrans.app (plusCompIso J F P).hom X \u226b colimMap (diagramNatTrans J (whiskerRight \u03b7 F) X.unop)\n[PROOFSTEP]\nsimp only [\u03b9_colimMap, whiskerRight_app, \u03b9_plusCompIso_hom_assoc, GrothendieckTopology.diagramNatTrans_app]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W) \u226b\n F.map (colimMap (diagramNatTrans J \u03b7 X.unop)) \u226b NatTrans.app (plusCompIso J F Q).hom X =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) b) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, \u2190 F.map_comp]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map (colimit.\u03b9 (diagram J P X.unop) W \u226b colimMap (diagramNatTrans J \u03b7 X.unop)) \u226b\n NatTrans.app (plusCompIso J F Q).hom X =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) ((diagram J (P \u22d9 F) X.unop).obj W)\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (b : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) b) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) b =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) b) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) b)) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\ndsimp [colimMap, IsColimit.map]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (colimit.\u03b9 (diagram J P X.unop) W \u226b\n colimit.desc (diagram J P X.unop)\n ((Cocones.precompose (diagramNatTrans J \u03b7 X.unop)).obj (colimit.cocone (diagram J Q X.unop)))) \u226b\n NatTrans.app (plusCompIso J F Q).hom X =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [colimit.\u03b9_desc]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (NatTrans.app ((Cocones.precompose (diagramNatTrans J \u03b7 X.unop)).obj (colimit.cocone (diagram J Q X.unop))).\u03b9\n W) \u226b\n NatTrans.app (plusCompIso J F Q).hom X =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\ndsimp [Cocones.precompose]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i) \u226b\n colimit.\u03b9 (diagram J Q X.unop) W) \u226b\n NatTrans.app (plusCompIso J F Q).hom X =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [Functor.map_comp, Category.assoc, \u03b9_plusCompIso_hom]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W \u226b colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n colimit.\u03b9 (diagram J (Q \u22d9 F) X.unop) W\n[PROOFSTEP]\ncongr 1\n -- porting note: this used to work with `ext`\n -- See https://github.com/leanprover-community/mathlib4/issues/5229\n[GOAL]\ncase w.h.e_a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W =\n NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)\n[PROOFSTEP]\napply Multiequalizer.hom_ext\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\n\u22a2 \u2200 (a : (Cover.index W.unop (Q \u22d9 F)).L),\n (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (Q \u22d9 F)).L\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (Q \u22d9 F)).L\n\u22a2 (F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n NatTrans.app (diagramCompIso J F Q X.unop).hom W) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a =\n (NatTrans.app (diagramCompIso J F P X.unop).hom W \u226b\n Multiequalizer.lift (Cover.index W.unop (Q \u22d9 F)) (multiequalizer (Cover.index W.unop (P \u22d9 F)))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b F.map (NatTrans.app \u03b7 (op i.Y)))\n (_ :\n \u2200 (i : (Cover.index W.unop (Q \u22d9 F)).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop (Q \u22d9 F)) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop (P \u22d9 F)) i \u226b NatTrans.app (whiskerRight \u03b7 F) (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop (Q \u22d9 F)) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop (Q \u22d9 F)) i)) \u226b\n Multiequalizer.\u03b9 (Cover.index W.unop (Q \u22d9 F)) a\n[PROOFSTEP]\nsimp only [diagramCompIso_hom_\u03b9_assoc, Multiequalizer.lift_\u03b9, diagramCompIso_hom_\u03b9, Category.assoc]\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nX : C\u1d52\u1d56\nW : (Cover J X.unop)\u1d52\u1d56\na : (Cover.index W.unop (Q \u22d9 F)).L\n\u22a2 F.map\n (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (_ :\n \u2200 (i : (Cover.index W.unop Q).R),\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.fstTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.fst (Cover.index W.unop Q) i =\n (fun i => Multiequalizer.\u03b9 (Cover.index W.unop P) i \u226b NatTrans.app \u03b7 (op i.Y))\n (MulticospanIndex.sndTo (Cover.index W.unop Q) i) \u226b\n MulticospanIndex.snd (Cover.index W.unop Q) i)) \u226b\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop Q) a) =\n F.map (Multiequalizer.\u03b9 (Cover.index W.unop P) a) \u226b F.map (NatTrans.app \u03b7 (op a.Y))\n[PROOFSTEP]\nsimp only [\u2190 F.map_comp, Multiequalizer.lift_\u03b9]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\n\u22a2 whiskerRight (toPlus J P) F \u226b (plusCompIso J F P).hom = toPlus J (P \u22d9 F)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app (whiskerRight (toPlus J P) F \u226b (plusCompIso J F P).hom) x\u271d = NatTrans.app (toPlus J (P \u22d9 F)) x\u271d\n[PROOFSTEP]\ndsimp [toPlus]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 F.map (Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P x\u271d.unop) (op \u22a4)) \u226b\n NatTrans.app (plusCompIso J F P).hom x\u271d =\n Cover.toMultiequalizer \u22a4 (P \u22d9 F) \u226b colimit.\u03b9 (diagram J (P \u22d9 F) x\u271d.unop) (op \u22a4)\n[PROOFSTEP]\nsimp only [\u03b9_plusCompIso_hom, Functor.map_comp, Category.assoc]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 F.map (Cover.toMultiequalizer \u22a4 P) \u226b\n NatTrans.app (diagramCompIso J F P x\u271d.unop).hom (op \u22a4) \u226b colimit.\u03b9 (diagram J (P \u22d9 F) x\u271d.unop) (op \u22a4) =\n Cover.toMultiequalizer \u22a4 (P \u22d9 F) \u226b colimit.\u03b9 (diagram J (P \u22d9 F) x\u271d.unop) (op \u22a4)\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 (F.map (Cover.toMultiequalizer \u22a4 P) \u226b NatTrans.app (diagramCompIso J F P x\u271d.unop).hom (op \u22a4)) \u226b\n colimit.\u03b9 (diagram J (P \u22d9 F) x\u271d.unop) (op \u22a4) =\n Cover.toMultiequalizer \u22a4 (P \u22d9 F) \u226b colimit.\u03b9 (diagram J (P \u22d9 F) x\u271d.unop) (op \u22a4)\n[PROOFSTEP]\ncongr 1\n -- porting note: this used to work with `ext`\n -- See https://github.com/leanprover-community/mathlib4/issues/5229\n[GOAL]\ncase w.h.e_a\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 F.map (Cover.toMultiequalizer \u22a4 P) \u226b NatTrans.app (diagramCompIso J F P x\u271d.unop).hom (op \u22a4) =\n Cover.toMultiequalizer \u22a4 (P \u22d9 F)\n[PROOFSTEP]\napply Multiequalizer.hom_ext\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 \u2200 (a : (Cover.index (op \u22a4).unop (P \u22d9 F)).L),\n (F.map (Cover.toMultiequalizer \u22a4 P) \u226b NatTrans.app (diagramCompIso J F P x\u271d.unop).hom (op \u22a4)) \u226b\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop (P \u22d9 F)) a =\n Cover.toMultiequalizer \u22a4 (P \u22d9 F) \u226b Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop (P \u22d9 F)) a\n[PROOFSTEP]\ndelta Cover.toMultiequalizer\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 \u2200 (a : (Cover.index (op \u22a4).unop (P \u22d9 F)).L),\n (F.map\n (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op x\u271d.unop)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I)) \u226b\n NatTrans.app (diagramCompIso J F P x\u271d.unop).hom (op \u22a4)) \u226b\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop (P \u22d9 F)) a =\n Multiequalizer.lift (Cover.index \u22a4 (P \u22d9 F)) ((P \u22d9 F).obj (op x\u271d.unop)) (fun I => (P \u22d9 F).map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 (P \u22d9 F)).R),\n (fun I => (P \u22d9 F).map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 (P \u22d9 F)) I =\n (fun I => (P \u22d9 F).map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop (P \u22d9 F)) a\n[PROOFSTEP]\nsimp only [diagramCompIso_hom_\u03b9, Category.assoc, \u2190 F.map_comp]\n[GOAL]\ncase w.h.e_a.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nx\u271d : C\u1d52\u1d56\n\u22a2 \u2200 (a : (Cover.index (op \u22a4).unop (P \u22d9 F)).L),\n F.map\n (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op x\u271d.unop)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I) \u226b\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P) a) =\n Multiequalizer.lift (Cover.index \u22a4 (P \u22d9 F)) ((P \u22d9 F).obj (op x\u271d.unop)) (fun I => (P \u22d9 F).map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 (P \u22d9 F)).R),\n (fun I => (P \u22d9 F).map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 (P \u22d9 F)) I =\n (fun I => (P \u22d9 F).map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 (P \u22d9 F)) I) \u226b\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop (P \u22d9 F)) a\n[PROOFSTEP]\nsimp only [unop_op, limit.lift_\u03c0, Multifork.of\u03b9_\u03c0_app, Functor.comp_obj, Functor.comp_map, implies_true]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\n\u22a2 toPlus J (P \u22d9 F) \u226b (plusCompIso J F P).inv = whiskerRight (toPlus J P) F\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nhP : Presheaf.IsSheaf J (plusObj J P \u22d9 F)\n\u22a2 (plusCompIso J F P).inv = plusLift J (whiskerRight (toPlus J P) F) hP\n[PROOFSTEP]\napply J.plusLift_unique\n[GOAL]\ncase h\u03b3\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2077 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2076 : Category.{max v u, w\u2082} E\nF : D \u2964 E\ninst\u271d\u2075 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) D\ninst\u271d\u2074 : \u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1), HasLimitsOfShape (WalkingMulticospan fst snd) E\ninst\u271d\u00b3 : (X : C) \u2192 (W : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F\nP : C\u1d52\u1d56 \u2964 D\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 E\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 F\nhP : Presheaf.IsSheaf J (plusObj J P \u22d9 F)\n\u22a2 toPlus J (P \u22d9 F) \u226b (plusCompIso J F P).inv = whiskerRight (toPlus J P) F\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.CompatiblePlus", "llama_tokens": 85421, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.399811640739795, "lm_q1q2_score": 0.26603888195251135}} {"text": "[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\n\u22a2 \u00acIsUnit p\n[PROOFSTEP]\nsimpa only [Associates.isUnit_iff_eq_one] using hp.1\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = a * b\nha : a = p\n\u22a2 IsUnit b\n[PROOFSTEP]\nrw [ha] at h \n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n\u22a2 IsUnit b\n[PROOFSTEP]\napply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => rw [h]) h\u2081\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n\u22a2 Associated (p * b) p\n[PROOFSTEP]\nconv_rhs => rw [h]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n| p\n[PROOFSTEP]\nrw [h]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n| p\n[PROOFSTEP]\nrw [h]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : IsAtom p\na b : Associates M\nh : p = p * b\nha : a = p\n| p\n[PROOFSTEP]\nrw [h]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\n\u22a2 p \u2260 \u22a5\n[PROOFSTEP]\nsimpa only [Associates.isUnit_iff_eq_one, Associates.bot_eq_one] using hp.1\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\nb : Associates M\nx\u271d : b < p\na : Associates M\nhab : p = b * a\nhb\u271d : \u00acp \u2223 b\nhb : IsUnit b\n\u22a2 b = \u22a5\n[PROOFSTEP]\nrwa [Associates.isUnit_iff_eq_one, \u2190 Associates.bot_eq_one] at hb \n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\nb : Associates M\nx\u271d : b < p\na : Associates M\nhab : p = b * a\nhb : \u00acp \u2223 b\nha : IsUnit a\n\u22a2 b = p * \u2191(IsUnit.unit ha)\u207b\u00b9\n[PROOFSTEP]\nsimp [hab]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\nb : Associates M\nx\u271d : b < p\na : Associates M\nhab : p = b * a\nhb : \u00acp \u2223 b\nha : IsUnit a\n\u22a2 b = b * a * \u2191(IsUnit.unit ha)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\nb : Associates M\nx\u271d : b < p\na : Associates M\nhab : p = b * a\nhb : \u00acp \u2223 b\nha : IsUnit a\n\u22a2 b = b * (a * \u2191(IsUnit.unit ha)\u207b\u00b9)\n[PROOFSTEP]\nrw [IsUnit.mul_val_inv ha]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nh\u2081 : p \u2260 0\nhp : Irreducible p\nb : Associates M\nx\u271d : b < p\na : Associates M\nhab : p = b * a\nhb : \u00acp \u2223 b\nha : IsUnit a\n\u22a2 b = b * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\n\u22a2 \u2203 c, c 1 = p \u2227 StrictMono c \u2227 \u2200 {r : Associates M}, r \u2264 p ^ n \u2194 \u2203 i, r = c i\n[PROOFSTEP]\nrefine' \u27e8fun i => p ^ (i : \u2115), _, fun n m h => _, @fun y => \u27e8fun h => _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\n\u22a2 (fun i => p ^ \u2191i) 1 = p\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\n\u22a2 p ^ \u21911 = p\n[PROOFSTEP]\nrw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\n\u22a2 1 < n + 1\n[PROOFSTEP]\nexact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)\n[GOAL]\ncase refine'_2\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn\u271d : \u2115\nhn : n\u271d \u2260 0\nhp : Prime p\nn m : Fin (n\u271d + 1)\nh : n < m\n\u22a2 (fun i => p ^ \u2191i) n < (fun i => p ^ \u2191i) m\n[PROOFSTEP]\nexact\n Associates.dvdNotUnit_iff_lt.mp\n \u27e8pow_ne_zero n hp.ne_zero, p ^ (m - n : \u2115),\n not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),\n (pow_mul_pow_sub p h.le).symm\u27e9\n[GOAL]\ncase refine'_3\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\ny : Associates M\nh : y \u2264 p ^ n\n\u22a2 \u2203 i, y = (fun i => p ^ \u2191i) i\n[PROOFSTEP]\nobtain \u27e8i, i_le, hi\u27e9 := (dvd_prime_pow hp n).1 h\n[GOAL]\ncase refine'_3.intro.intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\ny : Associates M\nh : y \u2264 p ^ n\ni : \u2115\ni_le : i \u2264 n\nhi : Associated y (p ^ i)\n\u22a2 \u2203 i, y = (fun i => p ^ \u2191i) i\n[PROOFSTEP]\nrw [associated_iff_eq] at hi \n[GOAL]\ncase refine'_3.intro.intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\ny : Associates M\nh : y \u2264 p ^ n\ni : \u2115\ni_le : i \u2264 n\nhi : y = p ^ i\n\u22a2 \u2203 i, y = (fun i => p ^ \u2191i) i\n[PROOFSTEP]\nexact \u27e8\u27e8i, Nat.lt_succ_of_le i_le\u27e9, hi\u27e9\n[GOAL]\ncase refine'_4\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\ny : Associates M\n\u22a2 (\u2203 i, y = (fun i => p ^ \u2191i) i) \u2192 y \u2264 p ^ n\n[PROOFSTEP]\nrintro \u27e8i, rfl\u27e9\n[GOAL]\ncase refine'_4.intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np : Associates M\nn : \u2115\nhn : n \u2260 0\nhp : Prime p\ni : Fin (n + 1)\n\u22a2 (fun i => p ^ \u2191i) i \u2264 p ^ n\n[PROOFSTEP]\nexact \u27e8p ^ (n - i : \u2115), (pow_mul_pow_sub p (Nat.succ_le_succ_iff.mp i.2)).symm\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 IsUnit (c 0)\n[PROOFSTEP]\nobtain \u27e8i, hr\u27e9 := h\u2082.mp Associates.one_le\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (n + 1)\nhr : 1 = c i\n\u22a2 IsUnit (c 0)\n[PROOFSTEP]\nrw [Associates.isUnit_iff_eq_one, \u2190 Associates.le_one_iff, hr]\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (n + 1)\nhr : 1 = c i\n\u22a2 c 0 \u2264 c i\n[PROOFSTEP]\nexact h\u2081.monotone (Fin.zero_le i)\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\n\u22a2 Irreducible (c 1)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nhn : Nat.zero \u2260 0\nc : Fin (Nat.zero + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 Irreducible (c 1)\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 Irreducible (c 1)\n[PROOFSTEP]\nrefine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h\u2082.2 \u27e81, rfl\u27e9))).mp \u27e8_, fun b hb => _\u27e9\n[GOAL]\ncase succ.refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 c 1 \u2260 \u22a5\n[PROOFSTEP]\nexact ne_bot_of_gt (h\u2081 (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))\n[GOAL]\ncase succ.refine'_2\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nb : Associates M\nhb : b < c 1\n\u22a2 b = \u22a5\n[PROOFSTEP]\nobtain \u27e8\u27e8i, hi\u27e9, rfl\u27e9 := h\u2082.1 (hb.le.trans (h\u2082.2 \u27e81, rfl\u27e9))\n[GOAL]\ncase succ.refine'_2.intro.mk\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : \u2115\nhi : i < Nat.succ n + 1\nhb : c { val := i, isLt := hi } < c 1\n\u22a2 c { val := i, isLt := hi } = \u22a5\n[PROOFSTEP]\ncases i\n[GOAL]\ncase succ.refine'_2.intro.mk.zero\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : Nat.zero < Nat.succ n + 1\nhb : c { val := Nat.zero, isLt := hi } < c 1\n\u22a2 c { val := Nat.zero, isLt := hi } = \u22a5\n[PROOFSTEP]\nexact (Associates.isUnit_iff_eq_one _).mp (first_of_chain_isUnit h\u2081 @h\u2082)\n[GOAL]\ncase succ.refine'_2.intro.mk.succ\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nhq : q \u2260 0\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nn\u271d : \u2115\nhi : Nat.succ n\u271d < Nat.succ n + 1\nhb : c { val := Nat.succ n\u271d, isLt := hi } < c 1\n\u22a2 c { val := Nat.succ n\u271d, isLt := hi } = \u22a5\n[PROOFSTEP]\nsimpa [Fin.lt_iff_val_lt_val] using h\u2081.lt_iff_lt.mp hb\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np q r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhp : Prime p\nhr : r \u2223 q\nhp' : p \u2223 r\n\u22a2 p = c 1\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np q r : Associates M\nhp : Prime p\nhr : r \u2223 q\nhp' : p \u2223 r\nhn : Nat.zero \u2260 0\nc : Fin (Nat.zero + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 p = c 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\np q r : Associates M\nhp : Prime p\nhr : r \u2223 q\nhp' : p \u2223 r\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\n\u22a2 p = c 1\n[PROOFSTEP]\nobtain \u27e8i, rfl\u27e9 := h\u2082.1 (dvd_trans hp' hr)\n[GOAL]\ncase succ.intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (Nat.succ n + 1)\nhp : Prime (c i)\nhp' : c i \u2223 r\n\u22a2 c i = c 1\n[PROOFSTEP]\nrefine' congr_arg c (eq_of_ge_of_not_gt _ fun hi => _)\n[GOAL]\ncase succ.intro.refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (Nat.succ n + 1)\nhp : Prime (c i)\nhp' : c i \u2223 r\n\u22a2 1 \u2264 i\n[PROOFSTEP]\nrw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, \u2190 Fin.val_zero' (n.succ + 1), \u2190 Fin.lt_iff_val_lt_val,\n Fin.pos_iff_ne_zero]\n[GOAL]\ncase succ.intro.refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (Nat.succ n + 1)\nhp : Prime (c i)\nhp' : c i \u2223 r\n\u22a2 i \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase succ.intro.refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhp : Prime (c 0)\nhp' : c 0 \u2223 r\n\u22a2 False\n[PROOFSTEP]\nexact hp.not_unit (first_of_chain_isUnit h\u2081 @h\u2082)\n[GOAL]\ncase succ.intro.refine'_2\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\ni : Fin (Nat.succ n + 1)\nhp : Prime (c i)\nhp' : c i \u2223 r\nhi : 1 < i\n\u22a2 False\n[PROOFSTEP]\nobtain rfl | \u27e8j, rfl\u27e9 := i.eq_zero_or_eq_succ\n[GOAL]\ncase succ.intro.refine'_2.inl\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhp : Prime (c 0)\nhp' : c 0 \u2223 r\nhi : 1 < 0\n\u22a2 False\n[PROOFSTEP]\ncases hi\n[GOAL]\ncase succ.intro.refine'_2.inr.intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nj : Fin (n + 1)\nhp : Prime (c (Fin.succ j))\nhp' : c (Fin.succ j) \u2223 r\nhi : 1 < Fin.succ j\n\u22a2 False\n[PROOFSTEP]\nrefine'\n not_irreducible_of_not_unit_dvdNotUnit\n (DvdNotUnit.not_unit (Associates.dvdNotUnit_iff_lt.2 (h\u2081 (show (0 : Fin (n + 2)) < j from _)))) _ hp.irreducible\n[GOAL]\ncase succ.intro.refine'_2.inr.intro.refine'_1\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nj : Fin (n + 1)\nhp : Prime (c (Fin.succ j))\nhp' : c (Fin.succ j) \u2223 r\nhi : 1 < Fin.succ j\n\u22a2 0 < \u2191\u2191j\n[PROOFSTEP]\nsimpa [Fin.succ_lt_succ_iff, Fin.lt_iff_val_lt_val] using hi\n[GOAL]\ncase succ.intro.refine'_2.inr.intro.refine'_2\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nj : Fin (n + 1)\nhp : Prime (c (Fin.succ j))\nhp' : c (Fin.succ j) \u2223 r\nhi : 1 < Fin.succ j\n\u22a2 DvdNotUnit (c \u2191\u2191j) (c (Fin.succ j))\n[PROOFSTEP]\nrefine' Associates.dvdNotUnit_iff_lt.2 (h\u2081 _)\n[GOAL]\ncase succ.intro.refine'_2.inr.intro.refine'_2\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq r : Associates M\nhr : r \u2223 q\nn : \u2115\nhn : Nat.succ n \u2260 0\nc : Fin (Nat.succ n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nj : Fin (n + 1)\nhp : Prime (c (Fin.succ j))\nhp' : c (Fin.succ j) \u2223 r\nhi : 1 < Fin.succ j\n\u22a2 \u2191\u2191j < Fin.succ j\n[PROOFSTEP]\nsimpa only [Fin.coe_eq_castSucc] using Fin.lt_succ\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\n\u22a2 Finset.card m \u2264 n + 1\n[PROOFSTEP]\nclassical\nhave mem_image : \u2200 r : Associates M, r \u2264 q \u2192 r \u2208 Finset.univ.image c :=\n by\n intro r hr\n obtain \u27e8i, hi\u27e9 := h\u2082.1 hr\n exact Finset.mem_image.2 \u27e8i, Finset.mem_univ _, hi.symm\u27e9\nrw [\u2190 Finset.card_fin (n + 1)]\nexact (Finset.card_le_of_subset fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\n\u22a2 Finset.card m \u2264 n + 1\n[PROOFSTEP]\nhave mem_image : \u2200 r : Associates M, r \u2264 q \u2192 r \u2208 Finset.univ.image c :=\n by\n intro r hr\n obtain \u27e8i, hi\u27e9 := h\u2082.1 hr\n exact Finset.mem_image.2 \u27e8i, Finset.mem_univ _, hi.symm\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\n\u22a2 \u2200 (r : Associates M), r \u2264 q \u2192 r \u2208 Finset.image c Finset.univ\n[PROOFSTEP]\nintro r hr\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\nr : Associates M\nhr : r \u2264 q\n\u22a2 r \u2208 Finset.image c Finset.univ\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := h\u2082.1 hr\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\nr : Associates M\nhr : r \u2264 q\ni : Fin (n + 1)\nhi : r = c i\n\u22a2 r \u2208 Finset.image c Finset.univ\n[PROOFSTEP]\nexact Finset.mem_image.2 \u27e8i, Finset.mem_univ _, hi.symm\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\nmem_image : \u2200 (r : Associates M), r \u2264 q \u2192 r \u2208 Finset.image c Finset.univ\n\u22a2 Finset.card m \u2264 n + 1\n[PROOFSTEP]\nrw [\u2190 Finset.card_fin (n + 1)]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoidWithZero M\nq : Associates M\nn : \u2115\nc : Fin (n + 1) \u2192 Associates M\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nm : Finset (Associates M)\nhm : \u2200 (r : Associates M), r \u2208 m \u2192 r \u2264 q\nmem_image : \u2200 (r : Associates M), r \u2264 q \u2192 r \u2208 Finset.image c Finset.univ\n\u22a2 Finset.card m \u2264 Finset.card Finset.univ\n[PROOFSTEP]\nexact (Finset.card_le_of_subset fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\n\u22a2 \u2203 i, r = c 1 ^ \u2191i\n[PROOFSTEP]\nclassical\nlet i := Multiset.card (normalizedFactors r)\nhave hi : normalizedFactors r = Multiset.replicate i (c 1) :=\n by\n apply Multiset.eq_replicate_of_mem\n intro b hb\n refine'\n eq_second_of_chain_of_prime_dvd hn h\u2081 (@fun r' => h\u2082) (prime_of_normalized_factor b hb) hr\n (dvd_of_mem_normalizedFactors hb)\nhave H : r = c 1 ^ i :=\n by\n have := UniqueFactorizationMonoid.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)\n rw [associated_iff_eq, hi, Multiset.prod_replicate] at this \n rw [this]\nrefine' \u27e8\u27e8i, _\u27e9, H\u27e9\nhave : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115)).card = i + 1 :=\n by\n conv_rhs => rw [\u2190 Finset.card_fin (i + 1)]\n cases n\n \u00b7 contradiction\n rw [Finset.card_image_iff]\n refine' Set.injOn_of_injective (fun m m' h => Fin.ext _) _\n refine' pow_injective_of_not_unit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h\u2081) _ h\n exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h\u2081 (@h\u2082) hq)\nsuffices H' : \u2200 r \u2208 Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115), r \u2264 q\n\u00b7 simp only [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, \u2190 this]\n apply card_subset_divisors_le_length_of_chain (@h\u2082) H'\nsimp only [Finset.mem_image]\nrintro r \u27e8a, _, rfl\u27e9\nrefine' dvd_trans _ hr\nuse c 1 ^ (i - (a : \u2115))\nrw [pow_mul_pow_sub (c 1)]\n\u00b7 exact H\n\u00b7 exact Nat.succ_le_succ_iff.mp a.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\n\u22a2 \u2203 i, r = c 1 ^ \u2191i\n[PROOFSTEP]\nlet i := Multiset.card (normalizedFactors r)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\n\u22a2 \u2203 i, r = c 1 ^ \u2191i\n[PROOFSTEP]\nhave hi : normalizedFactors r = Multiset.replicate i (c 1) :=\n by\n apply Multiset.eq_replicate_of_mem\n intro b hb\n refine'\n eq_second_of_chain_of_prime_dvd hn h\u2081 (@fun r' => h\u2082) (prime_of_normalized_factor b hb) hr\n (dvd_of_mem_normalizedFactors hb)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\n\u22a2 normalizedFactors r = Multiset.replicate i (c 1)\n[PROOFSTEP]\napply Multiset.eq_replicate_of_mem\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\n\u22a2 \u2200 (b : Associates M), b \u2208 normalizedFactors r \u2192 b = c 1\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase a\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nb : Associates M\nhb : b \u2208 normalizedFactors r\n\u22a2 b = c 1\n[PROOFSTEP]\nrefine'\n eq_second_of_chain_of_prime_dvd hn h\u2081 (@fun r' => h\u2082) (prime_of_normalized_factor b hb) hr\n (dvd_of_mem_normalizedFactors hb)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\n\u22a2 \u2203 i, r = c 1 ^ \u2191i\n[PROOFSTEP]\nhave H : r = c 1 ^ i :=\n by\n have := UniqueFactorizationMonoid.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)\n rw [associated_iff_eq, hi, Multiset.prod_replicate] at this \n rw [this]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\n\u22a2 r = c 1 ^ i\n[PROOFSTEP]\nhave := UniqueFactorizationMonoid.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nthis : Associated (Multiset.prod (normalizedFactors r)) r\n\u22a2 r = c 1 ^ i\n[PROOFSTEP]\nrw [associated_iff_eq, hi, Multiset.prod_replicate] at this \n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nthis : c 1 ^ i = r\n\u22a2 r = c 1 ^ i\n[PROOFSTEP]\nrw [this]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 \u2203 i, r = c 1 ^ \u2191i\n[PROOFSTEP]\nrefine' \u27e8\u27e8i, _\u27e9, H\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 i < n + 1\n[PROOFSTEP]\nhave : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115)).card = i + 1 :=\n by\n conv_rhs => rw [\u2190 Finset.card_fin (i + 1)]\n cases n\n \u00b7 contradiction\n rw [Finset.card_image_iff]\n refine' Set.injOn_of_injective (fun m m' h => Fin.ext _) _\n refine' pow_injective_of_not_unit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h\u2081) _ h\n exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h\u2081 (@h\u2082) hq)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Finset.card_fin (i + 1)]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n| i + 1\n[PROOFSTEP]\nrw [\u2190 Finset.card_fin (i + 1)]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n| i + 1\n[PROOFSTEP]\nrw [\u2190 Finset.card_fin (i + 1)]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n| i + 1\n[PROOFSTEP]\nrw [\u2190 Finset.card_fin (i + 1)]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = Finset.card Finset.univ\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhn : Nat.zero \u2260 0\nc : Fin (Nat.zero + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = Finset.card Finset.univ\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2260 0\nc : Fin (Nat.succ n\u271d + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = Finset.card Finset.univ\n[PROOFSTEP]\nrw [Finset.card_image_iff]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2260 0\nc : Fin (Nat.succ n\u271d + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\n\u22a2 Set.InjOn (fun m => c 1 ^ \u2191m) \u2191Finset.univ\n[PROOFSTEP]\nrefine' Set.injOn_of_injective (fun m m' h => Fin.ext _) _\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2260 0\nc : Fin (Nat.succ n\u271d + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nm m' : Fin (i + 1)\nh : c 1 ^ \u2191m = c 1 ^ \u2191m'\n\u22a2 \u2191m = \u2191m'\n[PROOFSTEP]\nrefine' pow_injective_of_not_unit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h\u2081) _ h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2260 0\nc : Fin (Nat.succ n\u271d + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nm m' : Fin (i + 1)\nh : c 1 ^ \u2191m = c 1 ^ \u2191m'\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2260 0\nc : Fin (Nat.succ n\u271d + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nm m' : Fin (i + 1)\nh : c 1 ^ \u2191m = c 1 ^ \u2191m'\n\u22a2 c 1 \u2260 0\n[PROOFSTEP]\nexact Irreducible.ne_zero (second_of_chain_is_irreducible hn h\u2081 (@h\u2082) hq)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\n\u22a2 i < n + 1\n[PROOFSTEP]\nsuffices H' : \u2200 r \u2208 Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115), r \u2264 q\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\nH' : \u2200 (r_1 : Associates M), r_1 \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ \u2192 r_1 \u2264 q\n\u22a2 i < n + 1\n[PROOFSTEP]\nsimp only [\u2190 Nat.succ_le_iff, Nat.succ_eq_add_one, \u2190 this]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\nH' : \u2200 (r_1 : Associates M), r_1 \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ \u2192 r_1 \u2264 q\n\u22a2 Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) \u2264 n + 1\n[PROOFSTEP]\napply card_subset_divisors_le_length_of_chain (@h\u2082) H'\n[GOAL]\ncase H'\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\n\u22a2 \u2200 (r_1 : Associates M), r_1 \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ \u2192 r_1 \u2264 q\n[PROOFSTEP]\nsimp only [Finset.mem_image]\n[GOAL]\ncase H'\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\n\u22a2 \u2200 (r_1 : Associates M), (\u2203 a, a \u2208 Finset.univ \u2227 c 1 ^ \u2191a = r_1) \u2192 r_1 \u2264 q\n[PROOFSTEP]\nrintro r \u27e8a, _, rfl\u27e9\n[GOAL]\ncase H'.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\na : Fin (\u2191Multiset.card (normalizedFactors r) + 1)\nleft\u271d : a \u2208 Finset.univ\n\u22a2 c 1 ^ \u2191a \u2264 q\n[PROOFSTEP]\nrefine' dvd_trans _ hr\n[GOAL]\ncase H'.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\na : Fin (\u2191Multiset.card (normalizedFactors r) + 1)\nleft\u271d : a \u2208 Finset.univ\n\u22a2 c 1 ^ \u2191a \u2223 r\n[PROOFSTEP]\nuse c 1 ^ (i - (a : \u2115))\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\na : Fin (\u2191Multiset.card (normalizedFactors r) + 1)\nleft\u271d : a \u2208 Finset.univ\n\u22a2 r = c 1 ^ \u2191a * c 1 ^ (i - \u2191a)\n[PROOFSTEP]\nrw [pow_mul_pow_sub (c 1)]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\na : Fin (\u2191Multiset.card (normalizedFactors r) + 1)\nleft\u271d : a \u2208 Finset.univ\n\u22a2 r = c 1 ^ i\n[PROOFSTEP]\nexact H\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq r : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhr : r \u2223 q\nhq : q \u2260 0\ni : (fun x => \u2115) (normalizedFactors r) := \u2191Multiset.card (normalizedFactors r)\nhi : normalizedFactors r = Multiset.replicate i (c 1)\nH : r = c 1 ^ i\nthis : Finset.card (Finset.image (fun m => c 1 ^ \u2191m) Finset.univ) = i + 1\na : Fin (\u2191Multiset.card (normalizedFactors r) + 1)\nleft\u271d : a \u2208 Finset.univ\n\u22a2 \u2191a \u2264 i\n[PROOFSTEP]\nexact Nat.succ_le_succ_iff.mp a.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\n\u22a2 q = c 1 ^ n\n[PROOFSTEP]\nclassical\nobtain \u27e8i, hi'\u27e9 := element_of_chain_eq_pow_second_of_chain hn h\u2081 (@fun r => h\u2082) (dvd_refl q) hq\nconvert hi'\nrefine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)\ncalc\n n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm\n _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr (h\u2081.injective.injOn _)).symm\n _ \u2264 (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115)).card := (Finset.card_le_of_subset ?_)\n _ \u2264 (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le\n _ = i + 1 := Finset.card_fin _\nintro r hr\nobtain \u27e8j, -, rfl\u27e9 := Finset.mem_image.1 hr\nhave := h\u2082.2 \u27e8j, rfl\u27e9\nrw [hi'] at this \nhave h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this\nrcases h with \u27e8u, hu, hu'\u27e9\nrefine' Finset.mem_image.mpr \u27e8u, Finset.mem_univ _, _\u27e9\n\u00b7 rw [associated_iff_eq] at hu' \n rw [Fin.val_cast_of_lt (Nat.lt_succ_of_le hu), hu']\n\u00b7 rw [\u2190 irreducible_iff_prime]\n exact second_of_chain_is_irreducible hn h\u2081 (@h\u2082) hq\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\n\u22a2 q = c 1 ^ n\n[PROOFSTEP]\nobtain \u27e8i, hi'\u27e9 := element_of_chain_eq_pow_second_of_chain hn h\u2081 (@fun r => h\u2082) (dvd_refl q) hq\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\n\u22a2 q = c 1 ^ n\n[PROOFSTEP]\nconvert hi'\n[GOAL]\ncase h.e'_3.h.e'_6\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\n\u22a2 n = \u2191i\n[PROOFSTEP]\nrefine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)\n[GOAL]\ncase h.e'_3.h.e'_6\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\n\u22a2 Nat.succ n \u2264 Nat.succ \u2191i\n[PROOFSTEP]\ncalc\n n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm\n _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr (h\u2081.injective.injOn _)).symm\n _ \u2264 (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : \u2115)).card := (Finset.card_le_of_subset ?_)\n _ \u2264 (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le\n _ = i + 1 := Finset.card_fin _\n[GOAL]\ncase h.e'_3.h.e'_6\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\n\u22a2 Finset.image c Finset.univ \u2286 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase h.e'_3.h.e'_6\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nr : Associates M\nhr : r \u2208 Finset.image c Finset.univ\n\u22a2 r \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\n[PROOFSTEP]\nobtain \u27e8j, -, rfl\u27e9 := Finset.mem_image.1 hr\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\n\u22a2 c j \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\n[PROOFSTEP]\nhave := h\u2082.2 \u27e8j, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 q\n\u22a2 c j \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\n[PROOFSTEP]\nrw [hi'] at this \n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\n\u22a2 c j \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\n[PROOFSTEP]\nhave h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_2\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\nh : \u2203 i_1, i_1 \u2264 \u2191i \u2227 Associated (c j) (c 1 ^ i_1)\n\u22a2 c j \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\n\u22a2 Prime (c 1)\n[PROOFSTEP]\nrcases h with \u27e8u, hu, hu'\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\nu : \u2115\nhu : u \u2264 \u2191i\nhu' : Associated (c j) (c 1 ^ u)\n\u22a2 c j \u2208 Finset.image (fun m => c 1 ^ \u2191m) Finset.univ\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\n\u22a2 Prime (c 1)\n[PROOFSTEP]\nrefine' Finset.mem_image.mpr \u27e8u, Finset.mem_univ _, _\u27e9\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\nu : \u2115\nhu : u \u2264 \u2191i\nhu' : Associated (c j) (c 1 ^ u)\n\u22a2 c 1 ^ \u2191\u2191u = c j\n[PROOFSTEP]\nrw [associated_iff_eq] at hu' \n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\nu : \u2115\nhu : u \u2264 \u2191i\nhu' : c j = c 1 ^ u\n\u22a2 c 1 ^ \u2191\u2191u = c j\n[PROOFSTEP]\nrw [Fin.val_cast_of_lt (Nat.lt_succ_of_le hu), hu']\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\n\u22a2 Prime (c 1)\n[PROOFSTEP]\nrw [\u2190 irreducible_iff_prime]\n[GOAL]\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\ninst\u271d : UniqueFactorizationMonoid M\nq : Associates M\nn : \u2115\nhn : n \u2260 0\nc : Fin (n + 1) \u2192 Associates M\nh\u2081 : StrictMono c\nh\u2082 : \u2200 {r : Associates M}, r \u2264 q \u2194 \u2203 i, r = c i\nhq : q \u2260 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ \u2191i\nj : Fin (n + 1)\nhr : c j \u2208 Finset.image c Finset.univ\nthis : c j \u2264 c 1 ^ \u2191i\n\u22a2 Irreducible (c 1)\n[PROOFSTEP]\nexact second_of_chain_is_irreducible hn h\u2081 (@h\u2082) hq\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l \u2264 m } \u2243o { l // l \u2264 n }\n\u22a2 \u2191(\u2191d { val := 1, property := (_ : 1 \u2223 m) }) = 1\n[PROOFSTEP]\nletI : OrderBot { l : Associates M // l \u2264 m } := Subtype.orderBot bot_le\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l \u2264 m } \u2243o { l // l \u2264 n }\nthis : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\n\u22a2 \u2191(\u2191d { val := 1, property := (_ : 1 \u2223 m) }) = 1\n[PROOFSTEP]\nletI : OrderBot { l : Associates N // l \u2264 n } := Subtype.orderBot bot_le\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l \u2264 m } \u2243o { l // l \u2264 n }\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\n\u22a2 \u2191(\u2191d { val := 1, property := (_ : 1 \u2223 m) }) = 1\n[PROOFSTEP]\nsimp only [\u2190 Associates.bot_eq_one, Subtype.mk_bot, bot_le, Subtype.coe_eq_bot_iff]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l \u2264 m } \u2243o { l // l \u2264 n }\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\n\u22a2 \u2191d \u22a5 = \u22a5\n[PROOFSTEP]\nletI : BotHomClass ({ l // l \u2264 m } \u2243o { l // l \u2264 n }) _ _ := OrderIsoClass.toBotHomClass\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm : Associates M\nn : Associates N\nd : { l // l \u2264 m } \u2243o { l // l \u2264 n }\nthis\u271d\u00b9 : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis\u271d : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\nthis : BotHomClass ({ l // l \u2264 m } \u2243o { l // l \u2264 n }) { l // l \u2264 m } { l // l \u2264 n } := OrderIsoClass.toBotHomClass\n\u22a2 \u2191d \u22a5 = \u22a5\n[PROOFSTEP]\nexact map_bot d\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n\u22a2 u = 1\n[PROOFSTEP]\nrw [show u = (d.symm \u27e8d \u27e8u, hu'\u27e9, (d \u27e8u, hu'\u27e9).prop\u27e9) by\n simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk]]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n\u22a2 u =\n \u2191(\u2191(OrderIso.symm d)\n { val := \u2191(\u2191d { val := u, property := hu' }),\n property := (_ : \u2191(\u2191d { val := u, property := hu' }) \u2208 Set.Iic n) })\n[PROOFSTEP]\nsimp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subtype.coe_mk]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n\u22a2 \u2191(\u2191(OrderIso.symm d)\n { val := \u2191(\u2191d { val := u, property := hu' }),\n property := (_ : \u2191(\u2191d { val := u, property := hu' }) \u2208 Set.Iic n) }) =\n 1\n[PROOFSTEP]\nconv_rhs => rw [\u2190 factor_orderIso_map_one_eq_bot d.symm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d.symm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d.symm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d.symm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : \u2191(\u2191d { val := u, property := hu' }) = 1\n\u22a2 \u2191(\u2191(OrderIso.symm d)\n { val := \u2191(\u2191d { val := u, property := hu' }),\n property := (_ : \u2191(\u2191d { val := u, property := hu' }) \u2208 Set.Iic n) }) =\n \u2191(\u2191(OrderIso.symm d) { val := 1, property := (_ : 1 \u2223 n) })\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : u = 1\n\u22a2 \u2191(\u2191d { val := u, property := hu' }) = 1\n[PROOFSTEP]\nsimp_rw [hu]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : u = 1\n\u22a2 \u2191(\u2191d { val := 1, property := (_ : (fun x => x \u2208 Set.Iic m) 1) }) = 1\n[PROOFSTEP]\nconv_rhs => rw [\u2190 factor_orderIso_map_one_eq_bot d]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : u = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : u = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d : CancelCommMonoidWithZero N\nm u : Associates M\nn : Associates N\nhu' : u \u2264 m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhu : u = 1\n| 1\n[PROOFSTEP]\nrw [\u2190 factor_orderIso_map_one_eq_bot d]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s \u2264 n\n[PROOFSTEP]\nby_cases hs : s = 0\n[GOAL]\ncase pos\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\nhs : s = 0\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s \u2264 n\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\nhs : \u00acs = 0\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s \u2264 n\n[PROOFSTEP]\nsuffices (d \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9 : Associates N) ^ s = (d \u27e8p ^ s, hs'\u27e9)\n by\n rw [this]\n apply Subtype.prop (d \u27e8p ^ s, hs'\u27e9)\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\nhs : \u00acs = 0\nthis : \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s = \u2191(\u2191d { val := p ^ s, property := hs' })\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s \u2264 n\n[PROOFSTEP]\nrw [this]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\nhs : \u00acs = 0\nthis : \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s = \u2191(\u2191d { val := p ^ s, property := hs' })\n\u22a2 \u2191(\u2191d { val := p ^ s, property := hs' }) \u2264 n\n[PROOFSTEP]\napply Subtype.prop (d \u27e8p ^ s, hs'\u27e9)\n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs' : p ^ s \u2264 m\nhs : \u00acs = 0\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ s = \u2191(\u2191d { val := p ^ s, property := hs' })\n[PROOFSTEP]\nobtain \u27e8c\u2081, rfl, hc\u2081', hc\u2081''\u27e9 := exists_chain_of_prime_pow hs (prime_of_normalized_factor p hp)\n[GOAL]\ncase neg.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\n\u22a2 \u2191(\u2191d { val := c\u2081 1, property := (_ : c\u2081 1 \u2223 m) }) ^ s = \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nlet c\u2082 : Fin (s + 1) \u2192 Associates N := fun t => d \u27e8c\u2081 t, le_trans (hc\u2081''.2 \u27e8t, by simp\u27e9) hs'\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nt : Fin (s + 1)\n\u22a2 c\u2081 t = c\u2081 t\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\n\u22a2 \u2191(\u2191d { val := c\u2081 1, property := (_ : c\u2081 1 \u2223 m) }) ^ s = \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nhave c\u2082_def : \u2200 t, c\u2082 t = d \u27e8c\u2081 t, _\u27e9 := fun t => rfl\n[GOAL]\ncase neg.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\n\u22a2 \u2191(\u2191d { val := c\u2081 1, property := (_ : c\u2081 1 \u2223 m) }) ^ s = \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nrw [\u2190 c\u2082_def]\n[GOAL]\ncase neg.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\n\u22a2 c\u2082 1 ^ s = \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nrefine' (eq_pow_second_of_chain_of_has_chain hs (fun t u h => _) (@fun r => \u27e8@fun hr => _, _\u27e9) _).symm\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nt u : Fin (s + 1)\nh : t < u\n\u22a2 c\u2082 t < c\u2082 u\n[PROOFSTEP]\nrw [c\u2082_def, c\u2082_def, Subtype.coe_lt_coe, d.lt_iff_lt, Subtype.mk_lt_mk, hc\u2081'.lt_iff_lt]\n[GOAL]\ncase neg.intro.intro.intro.refine'_1\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nt u : Fin (s + 1)\nh : t < u\n\u22a2 t < u\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n\u22a2 \u2203 i, r = c\u2082 i\n[PROOFSTEP]\nhave : r \u2264 n := hr.trans (d \u27e8c\u2081 1 ^ s, _\u27e9).2\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n\u22a2 \u2203 i, r = c\u2082 i\n[PROOFSTEP]\nsuffices d.symm \u27e8r, this\u27e9 \u2264 \u27e8c\u2081 1 ^ s, hs'\u27e9 by\n obtain \u27e8i, hi\u27e9 := hc\u2081''.1 this\n use i\n simp only [c\u2082_def, \u2190 hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis\u271d : r \u2264 n\nthis : \u2191(OrderIso.symm d) { val := r, property := this\u271d } \u2264 { val := c\u2081 1 ^ s, property := hs' }\n\u22a2 \u2203 i, r = c\u2082 i\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := hc\u2081''.1 this\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis\u271d : r \u2264 n\nthis : \u2191(OrderIso.symm d) { val := r, property := this\u271d } \u2264 { val := c\u2081 1 ^ s, property := hs' }\ni : Fin (s + 1)\nhi : \u2191(\u2191(OrderIso.symm d) { val := r, property := this\u271d }) = c\u2081 i\n\u22a2 \u2203 i, r = c\u2082 i\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis\u271d : r \u2264 n\nthis : \u2191(OrderIso.symm d) { val := r, property := this\u271d } \u2264 { val := c\u2081 1 ^ s, property := hs' }\ni : Fin (s + 1)\nhi : \u2191(\u2191(OrderIso.symm d) { val := r, property := this\u271d }) = c\u2081 i\n\u22a2 r = c\u2082 i\n[PROOFSTEP]\nsimp only [c\u2082_def, \u2190 hi, d.apply_symm_apply, Subtype.coe_eta, Subtype.coe_mk]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n\u22a2 \u2191(OrderIso.symm d) { val := r, property := this } \u2264 { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nconv_rhs => rw [\u2190 d.symm_apply_apply \u27e8c\u2081 1 ^ s, hs'\u27e9]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n| { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nrw [\u2190 d.symm_apply_apply \u27e8c\u2081 1 ^ s, hs'\u27e9]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n| { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nrw [\u2190 d.symm_apply_apply \u27e8c\u2081 1 ^ s, hs'\u27e9]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n| { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nrw [\u2190 d.symm_apply_apply \u27e8c\u2081 1 ^ s, hs'\u27e9]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n\u22a2 \u2191(OrderIso.symm d) { val := r, property := this } \u2264 \u2191(OrderIso.symm d) (\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nrw [d.symm.le_iff_le]\n[GOAL]\ncase neg.intro.intro.intro.refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\nhr : r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\nthis : r \u2264 n\n\u22a2 { val := r, property := this } \u2264 \u2191d { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nsimpa only [\u2190 Subtype.coe_le_coe, Subtype.coe_mk] using hr\n[GOAL]\ncase neg.intro.intro.intro.refine'_3\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\n\u22a2 (\u2203 i, r = c\u2082 i) \u2192 r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nrintro \u27e8i, hr\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_3.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\ni : Fin (s + 1)\nhr : r = c\u2082 i\n\u22a2 r \u2264 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' })\n[PROOFSTEP]\nrw [hr, c\u2082_def, Subtype.coe_le_coe, d.le_iff_le]\n[GOAL]\ncase neg.intro.intro.intro.refine'_3.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nr : Associates N\ni : Fin (s + 1)\nhr : r = c\u2082 i\n\u22a2 { val := c\u2081 i, property := (_ : c\u2081 i \u2264 m) } \u2264 { val := c\u2081 1 ^ s, property := hs' }\n[PROOFSTEP]\nsimpa [Subtype.mk_le_mk] using hc\u2081''.2 \u27e8i, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.intro.refine'_4\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\ns : \u2115\nhs : \u00acs = 0\nc\u2081 : Fin (s + 1) \u2192 Associates M\nhp : c\u2081 1 \u2208 normalizedFactors m\nhs' : c\u2081 1 ^ s \u2264 m\nhc\u2081' : StrictMono c\u2081\nhc\u2081'' : \u2200 {r : Associates M}, r \u2264 c\u2081 1 ^ s \u2194 \u2203 i, r = c\u2081 i\nc\u2082 : Fin (s + 1) \u2192 Associates N := fun t => \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\nc\u2082_def : \u2200 (t : Fin (s + 1)), c\u2082 t = \u2191(\u2191d { val := c\u2081 t, property := (_ : c\u2081 t \u2264 m) })\n\u22a2 \u2191(\u2191d { val := c\u2081 1 ^ s, property := hs' }) \u2260 0\n[PROOFSTEP]\nexact ne_zero_of_dvd_ne_zero hn (Subtype.prop (d \u27e8c\u2081 1 ^ s, _\u27e9))\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 Prime \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\n[PROOFSTEP]\nrw [\u2190 irreducible_iff_prime]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 Irreducible \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\n[PROOFSTEP]\nrefine' (Associates.isAtom_iff <| ne_zero_of_dvd_ne_zero hn (d \u27e8p, _\u27e9).prop).mp \u27e8_, fun b hb => _\u27e9\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2260 \u22a5\n[PROOFSTEP]\nrw [Ne.def, \u2190 Associates.isUnit_iff_eq_bot, Associates.isUnit_iff_eq_one, coe_factor_orderIso_map_eq_one_iff _ d]\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 \u00acp = 1\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm : Associates M\nn : Associates N\nhn : n \u2260 0\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhp : 1 \u2208 normalizedFactors m\n\u22a2 False\n[PROOFSTEP]\nexact (prime_of_normalized_factor 1 hp).not_unit isUnit_one\n[GOAL]\ncase refine'_2\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\n\u22a2 b = \u22a5\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := d.surjective \u27e8b, le_trans (le_of_lt hb) (d \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9).prop\u27e9\n[GOAL]\ncase refine'_2.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\n\u22a2 b = \u22a5\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_mk b _, \u2190 hx] at hb \n[GOAL]\ncase refine'_2.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\n\u22a2 b = \u22a5\n[PROOFSTEP]\nletI : OrderBot { l : Associates M // l \u2264 m } := Subtype.orderBot bot_le\n[GOAL]\ncase refine'_2.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\nthis : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\n\u22a2 b = \u22a5\n[PROOFSTEP]\nletI : OrderBot { l : Associates N // l \u2264 n } := Subtype.orderBot bot_le\n[GOAL]\ncase refine'_2.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\n\u22a2 b = \u22a5\n[PROOFSTEP]\nsuffices x = \u22a5 by\n rw [this, OrderIso.map_bot d] at hx \n refine' (Subtype.mk_eq_bot_iff _ _).mp hx.symm\n simp\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\nthis\u271d\u00b9 : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis\u271d : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\nthis : x = \u22a5\n\u22a2 b = \u22a5\n[PROOFSTEP]\nrw [this, OrderIso.map_bot d] at hx \n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u22a5 = { val := b, property := (_ : b \u2264 n) }\nthis\u271d\u00b9 : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis\u271d : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\nthis : x = \u22a5\n\u22a2 b = \u22a5\n[PROOFSTEP]\nrefine' (Subtype.mk_eq_bot_iff _ _).mp hx.symm\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u22a5 = { val := b, property := (_ : b \u2264 n) }\nthis\u271d\u00b9 : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis\u271d : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\nthis : x = \u22a5\n\u22a2 \u22a5 \u2208 Set.Iic n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.intro\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nx : \u2191(Set.Iic m)\nhb : \u2191(\u2191d x) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d x = { val := b, property := (_ : b \u2264 n) }\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\n\u22a2 x = \u22a5\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := x\n[GOAL]\ncase refine'_2.intro.mk\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\na : Associates M\nha : a \u2208 Set.Iic m\nhb : \u2191(\u2191d { val := a, property := ha }) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d { val := a, property := ha } = { val := b, property := (_ : b \u2264 n) }\n\u22a2 { val := a, property := ha } = \u22a5\n[PROOFSTEP]\nrw [Subtype.mk_eq_bot_iff]\n[GOAL]\ncase refine'_2.intro.mk\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\na : Associates M\nha : a \u2208 Set.Iic m\nhb : \u2191(\u2191d { val := a, property := ha }) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d { val := a, property := ha } = { val := b, property := (_ : b \u2264 n) }\n\u22a2 a = \u22a5\n[PROOFSTEP]\nexact\n ((Associates.isAtom_iff <| Prime.ne_zero <| prime_of_normalized_factor p hp).mpr <|\n irreducible_of_normalized_factor p hp).right\n a (Subtype.mk_lt_mk.mp <| d.lt_iff_lt.mp hb)\n[GOAL]\ncase refine'_2.intro.mk.hbot\nM : Type u_1\ninst\u271d\u2074 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero N\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nb : Associates N\nhb\u271d : b < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nthis\u271d : OrderBot { l // l \u2264 m } := Subtype.orderBot (_ : \u22a5 \u2264 m)\nthis : OrderBot { l // l \u2264 n } := Subtype.orderBot (_ : \u22a5 \u2264 n)\na : Associates M\nha : a \u2208 Set.Iic m\nhb : \u2191(\u2191d { val := a, property := ha }) < \u2191(\u2191d { val := p, property := (_ : p \u2223 m) })\nhx : \u2191d { val := a, property := ha } = { val := b, property := (_ : b \u2264 n) }\n\u22a2 \u22a5 \u2208 Set.Iic m\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u2075 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero N\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : UniqueFactorizationMonoid M\ninst\u271d\u00b9 : DecidableEq (Associates M)\ninst\u271d : DecidableEq (Associates N)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nobtain \u27e8q, hq, hq'\u27e9 :=\n exists_mem_normalizedFactors_of_dvd hn (map_prime_of_factor_orderIso hn hp d).irreducible\n (d \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9).prop\n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u2075 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero N\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : UniqueFactorizationMonoid M\ninst\u271d\u00b9 : DecidableEq (Associates M)\ninst\u271d : DecidableEq (Associates N)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nq : Associates N\nhq : q \u2208 normalizedFactors n\nhq' : Associated (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) q\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nrw [associated_iff_eq] at hq' \n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u2075 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero N\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : UniqueFactorizationMonoid M\ninst\u271d\u00b9 : DecidableEq (Associates M)\ninst\u271d : DecidableEq (Associates N)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nq : Associates N\nhq : q \u2208 normalizedFactors n\nhq' : \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) = q\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nrwa [hq']\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 multiplicity p m \u2264 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhn : n = 0\n\u22a2 multiplicity p m \u2264 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhn : \u00acn = 0\n\u22a2 multiplicity p m \u2264 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nby_cases hm : m = 0\n[GOAL]\ncase pos\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 multiplicity p m \u2264 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nsimp [hm] at hp \n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 multiplicity p m \u2264 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nrw [\u2190 PartENat.natCast_get (finite_iff_dom.1 <| finite_prime_left (prime_of_normalized_factor p hp) hm), \u2190\n pow_dvd_iff_le_multiplicity]\n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) ^ Part.get (multiplicity p m) (_ : (multiplicity p m).Dom) \u2223 n\n[PROOFSTEP]\nexact pow_image_of_prime_by_factor_orderIso_dvd hn hp d (pow_multiplicity_dvd _)\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 multiplicity p m = multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nrefine' le_antisymm (multiplicity_prime_le_multiplicity_image_by_factor_orderIso hp d) _\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264 multiplicity p m\n[PROOFSTEP]\nsuffices\n multiplicity (\u2191(d \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9)) n \u2264\n multiplicity (\u2191(d.symm (d \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9))) m\n by\n rw [d.symm_apply_apply \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9, Subtype.coe_mk] at this \n exact this\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nthis :\n multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264\n multiplicity (\u2191(\u2191(OrderIso.symm d) (\u2191d { val := p, property := (_ : p \u2223 m) }))) m\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264 multiplicity p m\n[PROOFSTEP]\nrw [d.symm_apply_apply \u27e8p, dvd_of_mem_normalizedFactors hp\u27e9, Subtype.coe_mk] at this \n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nthis :\n multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264 multiplicity (\u2191{ val := p, property := (_ : p \u2223 m) }) m\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264 multiplicity p m\n[PROOFSTEP]\nexact this\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264\n multiplicity (\u2191(\u2191(OrderIso.symm d) (\u2191d { val := p, property := (_ : p \u2223 m) }))) m\n[PROOFSTEP]\nletI := Classical.decEq (Associates N)\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero N\ninst\u271d\u2074 : UniqueFactorizationMonoid N\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableEq (Associates M)\nm p : Associates M\nn : Associates N\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : \u2191(Set.Iic m) \u2243o \u2191(Set.Iic n)\nthis : DecidableEq (Associates N) := Classical.decEq (Associates N)\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n \u2264\n multiplicity (\u2191(\u2191(OrderIso.symm d) (\u2191d { val := p, property := (_ : p \u2223 m) }))) m\n[PROOFSTEP]\nsimpa only [Subtype.coe_eta] using\n multiplicity_prime_le_multiplicity_image_by_factor_orderIso\n (mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors hn hp d) d.symm\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nl : \u2191(Set.Iic (Associates.mk m))\n\u22a2 \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := l\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx : Associates M\nhx : x \u2208 Set.Iic (Associates.mk m)\n\u22a2 \u2191associatesEquivOfUniqueUnits \u2191{ val := x, property := hx } \u2223 m\n[PROOFSTEP]\nrw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx : Associates M\nhx : x \u2208 Set.Iic (Associates.mk m)\n\u22a2 x \u2264 Associates.mk m\n[PROOFSTEP]\nexact hx\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nl : \u2191(Set.Iic (Associates.mk n))\n\u22a2 \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := l\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx : Associates N\nhx : x \u2208 Set.Iic (Associates.mk n)\n\u22a2 \u2191associatesEquivOfUniqueUnits \u2191{ val := x, property := hx } \u2223 n\n[PROOFSTEP]\nrw [Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_dvd_iff]\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx : Associates N\nhx : x \u2208 Set.Iic (Associates.mk n)\n\u22a2 x \u2264 Associates.mk n\n[PROOFSTEP]\nexact hx\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx\u271d : \u2191(Set.Iic (Associates.mk m))\nl : Associates M\nhl : l \u2208 Set.Iic (Associates.mk m)\n\u22a2 (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l, property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n { val := l, property := hl }) =\n { val := l, property := hl }\n[PROOFSTEP]\nsimp only [Subtype.coe_eta, Equiv.symm_apply_apply, Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, mk_out, out_mk,\n normalize_eq]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nx\u271d : \u2191(Set.Iic (Associates.mk n))\nl : Associates N\nhl : l \u2208 Set.Iic (Associates.mk n)\n\u22a2 (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d { val := \u2191associatesEquivOfUniqueUnits \u2191l, property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n { val := l, property := hl }) =\n { val := l, property := hl }\n[PROOFSTEP]\nsimp only [Subtype.coe_eta, Equiv.apply_symm_apply, Subtype.coe_mk, associatesEquivOfUniqueUnits_apply, out_mk,\n normalize_eq, mk_out]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2200 {a b : \u2191(Set.Iic (Associates.mk m))},\n \u2191{\n toFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) },\n invFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk m))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk n))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n x) =\n x) }\n a \u2264\n \u2191{\n toFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) },\n invFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk m))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk n))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n x) =\n x) }\n b \u2194\n a \u2264 b\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase mk.mk\nM : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero N\ninst\u271d\u00b9 : Unique M\u02e3\ninst\u271d : Unique N\u02e3\nm : M\nn : N\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\na : Associates M\nha : a \u2208 Set.Iic (Associates.mk m)\nb : Associates M\nhb : b \u2208 Set.Iic (Associates.mk m)\n\u22a2 \u2191{\n toFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) },\n invFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk m))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk n))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n x) =\n x) }\n { val := a, property := ha } \u2264\n \u2191{\n toFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) },\n invFun := fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) },\n left_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk m))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : \u2191(Set.Iic (Associates.mk n))),\n (fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 m) }) \u2264\n Associates.mk n) })\n ((fun l =>\n {\n val :=\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }),\n property :=\n (_ :\n Associates.mk\n \u2191(\u2191d.symm\n { val := \u2191associatesEquivOfUniqueUnits \u2191l,\n property := (_ : \u2191associatesEquivOfUniqueUnits \u2191l \u2223 n) }) \u2264\n Associates.mk m) })\n x) =\n x) }\n { val := b, property := hb } \u2194\n { val := a, property := ha } \u2264 { val := b, property := hb }\n[PROOFSTEP]\nsimp only [Equiv.coe_fn_mk, Subtype.mk_le_mk, Associates.mk_le_mk_iff_dvd_iff, hd, Subtype.coe_mk,\n associatesEquivOfUniqueUnits_apply, out_dvd_iff, mk_out]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nsuffices\n Prime\n (d \u27e8associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by simp [dvd_of_mem_normalizedFactors hp]\u27e9 :\n N)\n by\n simp only [associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq, associatesEquivOfUniqueUnits_symm_apply] at this \n obtain \u27e8q, hq, hq'\u27e9 :=\n exists_mem_normalizedFactors_of_dvd hn this.irreducible\n (d \u27e8p, by apply dvd_of_mem_normalizedFactors; convert hp\u27e9).prop\n rwa [associated_iff_eq.mp hq']\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p) \u2223 m\n[PROOFSTEP]\nsimp [dvd_of_mem_normalizedFactors hp]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n Prime\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) })\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nsimp only [associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq, associatesEquivOfUniqueUnits_symm_apply] at this \n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis : Prime \u2191(\u2191d { val := p, property := (_ : (fun l => l \u2223 m) p) })\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nobtain \u27e8q, hq, hq'\u27e9 :=\n exists_mem_normalizedFactors_of_dvd hn this.irreducible\n (d \u27e8p, by apply dvd_of_mem_normalizedFactors; convert hp\u27e9).prop\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis : Prime \u2191(\u2191d { val := p, property := (_ : (fun l => l \u2223 m) p) })\n\u22a2 p \u2223 m\n[PROOFSTEP]\napply dvd_of_mem_normalizedFactors\n[GOAL]\ncase H\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis : Prime \u2191(\u2191d { val := p, property := (_ : (fun l => l \u2223 m) p) })\n\u22a2 p \u2208 normalizedFactors m\n[PROOFSTEP]\nconvert hp\n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis : Prime \u2191(\u2191d { val := p, property := (_ : (fun l => l \u2223 m) p) })\nq : N\nhq : q \u2208 normalizedFactors n\nhq' : Associated (\u2191(\u2191d { val := p, property := (_ : (fun l => l \u2223 m) p) })) q\n\u22a2 \u2191(\u2191d { val := p, property := (_ : p \u2223 m) }) \u2208 normalizedFactors n\n[PROOFSTEP]\nrwa [associated_iff_eq.mp hq']\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 Prime\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) })\n[PROOFSTEP]\nhave :\n Associates.mk\n (d\n \u27e8associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by\n simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq,\n associatesEquivOfUniqueUnits_symm_apply]\u27e9 :\n N) =\n \u2191(mkFactorOrderIsoOfFactorDvdEquiv hd\n \u27e8associatesEquivOfUniqueUnits.symm p,\n by\n simp only [associatesEquivOfUniqueUnits_symm_apply]\n exact mk_dvd_mk.mpr (dvd_of_mem_normalizedFactors hp)\u27e9) :=\n by rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p) \u2223 m\n[PROOFSTEP]\nsimp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq,\n associatesEquivOfUniqueUnits_symm_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)\n[PROOFSTEP]\nsimp only [associatesEquivOfUniqueUnits_symm_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 Associates.mk p \u2208 Set.Iic (Associates.mk m)\n[PROOFSTEP]\nexact mk_dvd_mk.mpr (dvd_of_mem_normalizedFactors hp)\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n[PROOFSTEP]\nrw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n\u22a2 Prime\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) })\n[PROOFSTEP]\nrw [\u2190 Associates.prime_mk, this]\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n\u22a2 Prime\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n[PROOFSTEP]\nletI := Classical.decEq (Associates M)\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\n\u22a2 Prime\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n[PROOFSTEP]\nrefine' map_prime_of_factor_orderIso (mk_ne_zero.mpr hn) _ _\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\n\u22a2 \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 normalizedFactors (Associates.mk m)\n[PROOFSTEP]\nobtain \u27e8q, hq, hq'\u27e9 :=\n exists_mem_normalizedFactors_of_dvd (mk_ne_zero.mpr hm)\n ((prime_mk p).mpr (prime_of_normalized_factor p (by convert hp))).irreducible\n (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))\n[GOAL]\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\n\u22a2 p \u2208 normalizedFactors ?m.1306674\n[PROOFSTEP]\nconvert hp\n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2076 : CancelCommMonoidWithZero N\ninst\u271d\u2075 : Unique M\u02e3\ninst\u271d\u2074 : Unique N\u02e3\ninst\u271d\u00b3 : UniqueFactorizationMonoid M\ninst\u271d\u00b2 : UniqueFactorizationMonoid N\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : DecidableEq N\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\nq : Associates M\nhq : q \u2208 normalizedFactors (Associates.mk m)\nhq' : Associated (Associates.mk p) q\n\u22a2 \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 normalizedFactors (Associates.mk m)\n[PROOFSTEP]\nsimpa only [associated_iff_eq.mp hq', associatesEquivOfUniqueUnits_symm_apply] using hq\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n = multiplicity p m\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 multiplicity p m = multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nsuffices\n multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (Associates.mk\n \u2191(d\n \u27e8associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by\n simp [dvd_of_mem_normalizedFactors hp]\u27e9))\n (Associates.mk n)\n by\n simpa only [multiplicity_mk_eq_multiplicity, associatesEquivOfUniqueUnits_symm_apply,\n associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq] using this\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p) \u2223 m\n[PROOFSTEP]\nsimp [dvd_of_mem_normalizedFactors hp]\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) }))\n (Associates.mk n)\n\u22a2 multiplicity p m = multiplicity (\u2191(\u2191d { val := p, property := (_ : p \u2223 m) })) n\n[PROOFSTEP]\nsimpa only [multiplicity_mk_eq_multiplicity, associatesEquivOfUniqueUnits_symm_apply,\n associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq] using this\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) }))\n (Associates.mk n)\n[PROOFSTEP]\nhave :\n Associates.mk\n (d\n \u27e8associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by\n simp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_symm_apply,\n associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq]\u27e9 :\n N) =\n \u2191(mkFactorOrderIsoOfFactorDvdEquiv hd\n \u27e8associatesEquivOfUniqueUnits.symm p,\n by\n rw [associatesEquivOfUniqueUnits_symm_apply]\n exact mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp)\u27e9) :=\n by rw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p) \u2223 m\n[PROOFSTEP]\nsimp only [dvd_of_mem_normalizedFactors hp, associatesEquivOfUniqueUnits_symm_apply, associatesEquivOfUniqueUnits_apply,\n out_mk, normalize_eq]\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)\n[PROOFSTEP]\nrw [associatesEquivOfUniqueUnits_symm_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 Associates.mk p \u2208 Set.Iic (Associates.mk m)\n[PROOFSTEP]\nexact mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp)\n[GOAL]\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\n\u22a2 Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n[PROOFSTEP]\nrw [mkFactorOrderIsoOfFactorDvdEquiv_apply_coe]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n\u22a2 multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : p * 1 \u2223 m) }))\n (Associates.mk n)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\n\u22a2 multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (\u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) }))\n (Associates.mk n)\n[PROOFSTEP]\nletI := Classical.decEq (Associates M)\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\n\u22a2 multiplicity (Associates.mk p) (Associates.mk m) =\n multiplicity\n (\u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) }))\n (Associates.mk n)\n[PROOFSTEP]\nrefine'\n multiplicity_prime_eq_multiplicity_image_by_factor_orderIso (mk_ne_zero.mpr hn) _\n (mkFactorOrderIsoOfFactorDvdEquiv hd)\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\n\u22a2 Associates.mk p \u2208 normalizedFactors (Associates.mk m)\n[PROOFSTEP]\nobtain \u27e8q, hq, hq'\u27e9 :=\n exists_mem_normalizedFactors_of_dvd (mk_ne_zero.mpr hm)\n ((prime_mk p).mpr (prime_of_normalized_factor p hp)).irreducible (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))\n[GOAL]\ncase h.intro.intro\nM : Type u_1\ninst\u271d\u2078 : CancelCommMonoidWithZero M\nN : Type u_2\ninst\u271d\u2077 : CancelCommMonoidWithZero N\ninst\u271d\u2076 : Unique M\u02e3\ninst\u271d\u2075 : Unique N\u02e3\ninst\u271d\u2074 : UniqueFactorizationMonoid M\ninst\u271d\u00b3 : UniqueFactorizationMonoid N\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nm p : M\nn : N\nhm : m \u2260 0\nhn : n \u2260 0\nhp : p \u2208 normalizedFactors m\nd : { l // l \u2223 m } \u2243 { l // l \u2223 n }\nhd : \u2200 (l l' : { l // l \u2223 m }), \u2191(\u2191d l) \u2223 \u2191(\u2191d l') \u2194 \u2191l \u2223 \u2191l'\nthis\u271d :\n Associates.mk\n \u2191(\u2191d\n { val := \u2191associatesEquivOfUniqueUnits (\u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p),\n property := (_ : \u2191normalize p \u2223 m) }) =\n \u2191(\u2191(mkFactorOrderIsoOfFactorDvdEquiv hd)\n { val := \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p,\n property := (_ : \u2191(MulEquiv.symm associatesEquivOfUniqueUnits) p \u2208 Set.Iic (Associates.mk m)) })\nthis : DecidableEq (Associates M) := Classical.decEq (Associates M)\nq : Associates M\nhq : q \u2208 normalizedFactors (Associates.mk m)\nhq' : Associated (Associates.mk p) q\n\u22a2 Associates.mk p \u2208 normalizedFactors (Associates.mk m)\n[PROOFSTEP]\nrwa [associated_iff_eq.mp hq']\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.ChainOfDivisors", "llama_tokens": 67327, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.2658476869407446}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nx\u271d : OpensLeCover U\n\u22a2 {\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n (\ud835\udfd9 x\u271d) =\n \ud835\udfd9\n ({\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.obj\n x\u271d)\n[PROOFSTEP]\nrefine Over.OverMorphism.ext ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nx\u271d : OpensLeCover U\n\u22a2 ({\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n (\ud835\udfd9 x\u271d)).left =\n (\ud835\udfd9\n ({\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264\n (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.obj\n x\u271d)).left\n[PROOFSTEP]\nsimp only [Functor.id_obj, Sieve.generate_apply, Functor.const_obj_obj, Over.homMk_left, eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : OpensLeCover U\nf : x\u271d\u00b2 \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 {\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n (f \u226b g) =\n {\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n f \u226b\n {\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n g\n[PROOFSTEP]\nrefine Over.OverMorphism.ext ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : OpensLeCover U\nf : x\u271d\u00b2 \u27f6 x\u271d\u00b9\ng : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 ({\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n (f \u226b g)).left =\n ({\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264\n (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n f \u226b\n {\n obj := fun V =>\n {\n obj :=\n { left := V.obj, right := { as := PUnit.unit },\n hom := homOfLE (_ : (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264 (Functor.fromPUnit Y).obj { as := PUnit.unit }) },\n property :=\n (_ :\n \u2203 Y_1 h g,\n presieveOfCoveringAux U Y g \u2227\n h \u226b g =\n { left := V.obj, right := { as := PUnit.unit },\n hom :=\n homOfLE\n (_ :\n (\ud835\udfed (Opens \u2191X)).obj V.obj \u2264\n (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },\n map := fun {x x_1} g => Over.homMk g }.map\n g).left\n[PROOFSTEP]\nsimp only [Functor.id_obj, Sieve.generate_apply, Functor.const_obj_obj, Over.homMk_left, eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 \u2200 (X_1 : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom),\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X_1 =\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj X_1\n[PROOFSTEP]\nrintro \u27e8\u27e8_, _\u27e9, _\u27e9\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nleft\u271d : Opens \u2191X\nright\u271d : Discrete PUnit\nhom\u271d : (\ud835\udfed (Opens \u2191X)).obj left\u271d \u27f6 (Functor.fromPUnit Y).obj right\u271d\nproperty\u271d : (Sieve.generate (presieveOfCoveringAux U Y)).arrows { left := left\u271d, right := right\u271d, hom := hom\u271d }.hom\n\u22a2 (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj\n { obj := { left := left\u271d, right := right\u271d, hom := hom\u271d }, property := property\u271d } =\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj\n { obj := { left := left\u271d, right := right\u271d, hom := hom\u271d }, property := property\u271d }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nleft\u271d : Opens \u2191X\nright\u271d : Discrete PUnit\nhom\u271d : (\ud835\udfed (Opens \u2191X)).obj left\u271d \u27f6 (Functor.fromPUnit Y).obj right\u271d\nproperty\u271d : (Sieve.generate (presieveOfCoveringAux U Y)).arrows { left := left\u271d, right := right\u271d, hom := hom\u271d }.hom\n\u22a2 { obj := { left := left\u271d, right := right\u271d, hom := hom\u271d }, property := property\u271d } =\n (generateEquivalenceOpensLe_inverse' U ?m.44608).obj\n ((generateEquivalenceOpensLe_functor' U ?m.44561).obj\n { obj := { left := left\u271d, right := right\u271d, hom := hom\u271d }, property := property\u271d })\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 \u2200 (X_1 Y_1 : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) (f : X_1 \u27f6 Y_1),\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f =\n eqToHom\n (_ :\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X_1 =\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj\n X_1) \u226b\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).map f \u226b\n eqToHom\n (_ :\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj\n Y_1 =\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y_1)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nX\u271d Y\u271d : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f\u271d =\n eqToHom\n (_ :\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X\u271d =\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj X\u271d) \u226b\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).map f\u271d \u226b\n eqToHom\n (_ :\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj Y\u271d =\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y\u271d)\n[PROOFSTEP]\nrefine Over.OverMorphism.ext ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nX\u271d Y\u271d : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f\u271d).left =\n (eqToHom\n (_ :\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X\u271d =\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj\n X\u271d) \u226b\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).map f\u271d \u226b\n eqToHom\n (_ :\n (generateEquivalenceOpensLe_functor' U ?m.44561 \u22d9 generateEquivalenceOpensLe_inverse' U ?m.44608).obj Y\u271d =\n (\ud835\udfed (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y\u271d)).left\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 \u2200 (X_1 : OpensLeCover U),\n (generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) \u22d9\n generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj\n X_1 =\n (\ud835\udfed (OpensLeCover U)).obj X_1\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nX\u271d : OpensLeCover U\n\u22a2 (generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) \u22d9 generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj\n X\u271d =\n (\ud835\udfed (OpensLeCover U)).obj X\u271d\n[PROOFSTEP]\nrefine FullSubcategory.ext _ _ ?_\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nX\u271d : OpensLeCover U\n\u22a2 ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) \u22d9 generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj\n X\u271d).obj =\n ((\ud835\udfed (OpensLeCover U)).obj X\u271d).obj\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 \u2200 (X_1 Y_1 : OpensLeCover U) (f : X_1 \u27f6 Y_1),\n HEq\n ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) \u22d9\n generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).map\n f)\n ((\ud835\udfed (OpensLeCover U)).map f)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nX\u271d Y\u271d : OpensLeCover U\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 HEq\n ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) \u22d9\n generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).map\n f\u271d)\n ((\ud835\udfed (OpensLeCover U)).map f\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt)) \u226b\n NatTrans.app (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).\u03c0 j =\n NatTrans.app\n (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor\n (F.mapCone (Cocone.op (opensLeCoverCocone U)))).\u03c0\n j\n[PROOFSTEP]\nerw [\u2190 F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt) \u226b\n NatTrans.app (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)).\u03c0 j) =\n NatTrans.app\n (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor\n (F.mapCone (Cocone.op (opensLeCoverCocone U)))).\u03c0\n j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map (eqToHom (_ : op (opensLeCoverCocone U).pt = op Y) \u226b j.unop.obj.hom.op) =\n F.map\n (NatTrans.app (opensLeCoverCocone U).\u03b9 ((generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj j.unop)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt)) \u226b\n NatTrans.app\n (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor\n (F.mapCone (Cocone.op (opensLeCoverCocone U)))).\u03c0\n j =\n NatTrans.app (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).\u03c0 j\n[PROOFSTEP]\nerw [\u2190 F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt) \u226b\n NatTrans.app (Cocone.op (opensLeCoverCocone U)).\u03c0\n ((Equivalence.op (generateEquivalenceOpensLe U hY)).functor.obj j)) =\n NatTrans.app (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).\u03c0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nj : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)\u1d52\u1d56\n\u22a2 F.map\n (eqToHom (_ : op Y = op (opensLeCoverCocone U).pt) \u226b\n (NatTrans.app (opensLeCoverCocone U).\u03b9\n ((generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj j.unop)).op) =\n F.map j.unop.obj.hom.op\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) \u226b\n ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt))) =\n \ud835\udfd9\n (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor\n (F.mapCone (Cocone.op (opensLeCoverCocone U))))\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) \u226b\n ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt)))).Hom =\n (\ud835\udfd9\n (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor\n (F.mapCone (Cocone.op (opensLeCoverCocone U))))).Hom\n[PROOFSTEP]\nsimp [eqToHom_map]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt))) \u226b\n ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) =\n \ud835\udfd9 (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)))\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =\n op (opensLeCoverCocone U).pt))) \u226b\n ConeMorphism.mk\n (F.map\n (eqToHom\n (_ :\n op (opensLeCoverCocone U).pt =\n op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt)))).Hom =\n (\ud835\udfd9 (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)))).Hom\n[PROOFSTEP]\nsimp [eqToHom_map]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nR : Presieve Y\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y\n\u22a2 IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y R)))) \u2243\n IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate R).arrows)))\n[PROOFSTEP]\nconvert\n isLimitOpensLeEquivGenerate\u2081 F (coveringOfPresieve Y R)\n (coveringOfPresieve.iSup_eq_of_mem_grothendieck Y R hR).symm using\n 1\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\nR : Presieve Y\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y\n\u22a2 IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate R).arrows))) =\n IsLimit\n (F.mapCone\n (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux (coveringOfPresieve Y R) Y)).arrows)))\n[PROOFSTEP]\nrw [covering_presieve_eq_self R]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 IsSheaf F \u2194 IsSheafOpensLeCover F\n[PROOFSTEP]\nrefine' (Presheaf.isSheaf_iff_isLimit _ _).trans _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) \u2194\n IsSheafOpensLeCover F\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) \u2192\n IsSheafOpensLeCover 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\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n \u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone U))))\n[PROOFSTEP]\nrw [(isLimitOpensLeEquivGenerate\u2081 F U rfl).nonempty_congr]\n[GOAL]\ncase mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n \u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty\n (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows))))\n[PROOFSTEP]\napply h\n[GOAL]\ncase mp.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n \u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Sieve.generate (presieveOfCoveringAux U (iSup U)) \u2208\n GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) (iSup U)\n[PROOFSTEP]\napply presieveOfCovering.mem_grothendieckTopology\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 IsSheafOpensLeCover F \u2192\n \u2200 \u2983X_1 : Opens \u2191X\u2984 (S : Sieve X_1),\n S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) X_1 \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))\n[PROOFSTEP]\nintro h Y S\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : IsSheafOpensLeCover F\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 S \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))\n[PROOFSTEP]\nrw [\u2190 Sieve.generate_sieve S]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : IsSheafOpensLeCover F\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 Sieve.generate S.arrows \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y \u2192\n Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows))))\n[PROOFSTEP]\nintro hS\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : IsSheafOpensLeCover F\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y\n\u22a2 Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate S.arrows).arrows))))\n[PROOFSTEP]\nrw [\u2190 (isLimitOpensLeEquivGenerate\u2082 F S.1 hS).nonempty_congr]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : IsSheafOpensLeCover F\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) Y\n\u22a2 Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y S.arrows)))))\n[PROOFSTEP]\napply h\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover", "llama_tokens": 11635, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.40356685373537454, "lm_q1q2_score": 0.26571547914598764}} {"text": "[GOAL]\n\u03b2 : Type v\nf : \u03b2 \u2192 Type v\nP : Type v\ns : (b : \u03b2) \u2192 P \u27f6 f b\nb : \u03b2\nx : P\n\u22a2 Pi.\u03c0 f b (Pi.lift s x) = s b x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b2 : Type v\nf g : \u03b2 \u2192 Type v\n\u03b1 : (j : \u03b2) \u2192 f j \u27f6 g j\nb : \u03b2\nx : \u220f fun b => f b\n\u22a2 Pi.\u03c0 g b (Pi.map \u03b1 x) = \u03b1 b (Pi.\u03c0 f b x)\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u271d : Cone (Functor.empty (Type u))\n\u22a2 \u2200 (j : Discrete PEmpty),\n (fun x x => PUnit.unit) x\u271d \u226b\n NatTrans.app { pt := PUnit, \u03c0 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PUnit)).hom }.\u03c0\n j =\n NatTrans.app x\u271d.\u03c0 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\nx\u271d\u00b2 : Cone (Functor.empty (Type u))\nx\u271d\u00b9 : x\u271d\u00b2.pt \u27f6 { pt := PUnit, \u03c0 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PUnit)).hom }.pt\nx\u271d :\n \u2200 (j : Discrete PEmpty),\n x\u271d\u00b9 \u226b\n NatTrans.app { pt := PUnit, \u03c0 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PUnit)).hom }.\u03c0\n j =\n NatTrans.app x\u271d\u00b2.\u03c0 j\n\u22a2 x\u271d\u00b9 = (fun x x => PUnit.unit) x\u271d\u00b2\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\nx\u271d\u00b3 : Cone (Functor.empty (Type u))\nx\u271d\u00b2 : x\u271d\u00b3.pt \u27f6 { pt := PUnit, \u03c0 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PUnit)).hom }.pt\nx\u271d\u00b9 :\n \u2200 (j : Discrete PEmpty),\n x\u271d\u00b2 \u226b\n NatTrans.app { pt := PUnit, \u03c0 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PUnit)).hom }.\u03c0\n j =\n NatTrans.app x\u271d\u00b3.\u03c0 j\nx\u271d : x\u271d\u00b3.pt\n\u22a2 x\u271d\u00b2 x\u271d = (fun x x => PUnit.unit) x\u271d\u00b3 x\u271d\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\nX : Type u\n\u22a2 IsTerminal X \u2243 (X \u2245 PUnit)\n[PROOFSTEP]\ncalc\n IsTerminal X \u2243 Unique X := isTerminalEquivUnique _\n _ \u2243 (X \u2243 PUnit.{u + 1}) := (uniqueEquivEquivUnique _ _)\n _ \u2243 (X \u2245 PUnit) := equivEquivIso\n[GOAL]\nx\u271d : Cocone (Functor.empty (Type u))\n\u22a2 { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.pt \u27f6 x\u271d.pt\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\nx\u271d : Cocone (Functor.empty (Type u))\n\u22a2 \u2200 (j : Discrete PEmpty),\n NatTrans.app { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.\u03b9\n j \u226b\n (fun x a => PEmpty.casesOn (fun x_1 => x.pt) a) x\u271d =\n NatTrans.app x\u271d.\u03b9 j\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9\u27e9\n[GOAL]\nx\u271d\u00b2 : Cocone (Functor.empty (Type u))\nx\u271d\u00b9 : { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.pt \u27f6 x\u271d\u00b2.pt\nx\u271d :\n \u2200 (j : Discrete PEmpty),\n NatTrans.app { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.\u03b9\n j \u226b\n x\u271d\u00b9 =\n NatTrans.app x\u271d\u00b2.\u03b9 j\n\u22a2 x\u271d\u00b9 = (fun x a => PEmpty.casesOn (fun x_1 => x.pt) a) x\u271d\u00b2\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nx\u271d\u00b2 : Cocone (Functor.empty (Type u))\nx\u271d\u00b9 : { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.pt \u27f6 x\u271d\u00b2.pt\nx\u271d :\n \u2200 (j : Discrete PEmpty),\n NatTrans.app { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.\u03b9\n j \u226b\n x\u271d\u00b9 =\n NatTrans.app x\u271d\u00b2.\u03b9 j\nx : { pt := PEmpty, \u03b9 := (Functor.uniqueFromEmpty ((Functor.const (Discrete PEmpty)).obj PEmpty)).inv }.pt\n\u22a2 x\u271d\u00b9 x = (fun x a => PEmpty.casesOn (fun x_1 => x.pt) a) x\u271d\u00b2 x\n[PROOFSTEP]\ncases x\n[GOAL]\n\u22a2 binaryProductFunctor \u2245 prod.functor\n[PROOFSTEP]\nrefine' NatIso.ofComponents (fun X => _) (fun _ => _)\n[GOAL]\ncase refine'_1\nX : Type u\n\u22a2 binaryProductFunctor.obj X \u2245 prod.functor.obj X\n[PROOFSTEP]\nrefine' NatIso.ofComponents (fun Y => _) (fun _ => _)\n[GOAL]\ncase refine'_1.refine'_1\nX Y : Type u\n\u22a2 (binaryProductFunctor.obj X).obj Y \u2245 (prod.functor.obj X).obj Y\n[PROOFSTEP]\nexact ((limit.isLimit _).conePointUniqueUpToIso (binaryProductLimit X Y)).symm\n[GOAL]\ncase refine'_1.refine'_2\nX X\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (binaryProductFunctor.obj X).map x\u271d \u226b\n ((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y\u271d).hom =\n ((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) X\u271d).hom \u226b\n (prod.functor.obj X).map x\u271d\n[PROOFSTEP]\napply Limits.prod.hom_ext\n[GOAL]\ncase refine'_1.refine'_2.h\u2081\nX X\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((binaryProductFunctor.obj X).map x\u271d \u226b\n ((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y\u271d).hom) \u226b\n prod.fst =\n (((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) X\u271d).hom \u226b\n (prod.functor.obj X).map x\u271d) \u226b\n prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.refine'_2.h\u2082\nX X\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((binaryProductFunctor.obj X).map x\u271d \u226b\n ((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) Y\u271d).hom) \u226b\n prod.snd =\n (((fun Y => (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm) X\u271d).hom \u226b\n (prod.functor.obj X).map x\u271d) \u226b\n prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.refine'_2.h\u2081\nX X\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (binaryProductFunctor.obj X).map x\u271d \u226b _root_.Prod.fst = _root_.Prod.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.refine'_2.h\u2082\nX X\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (binaryProductFunctor.obj X).map x\u271d \u226b _root_.Prod.snd = _root_.Prod.snd \u226b x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nX\u271d Y\u271d : Type u\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 binaryProductFunctor.map x\u271d \u226b\n ((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n Y\u271d).hom =\n ((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n X\u271d).hom \u226b\n prod.functor.map x\u271d\n[PROOFSTEP]\next : 2\n[GOAL]\ncase refine'_2.w.h\nX\u271d Y\u271d : Type u\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Type u\n\u22a2 NatTrans.app\n (binaryProductFunctor.map x\u271d\u00b9 \u226b\n ((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n Y\u271d).hom)\n x\u271d =\n NatTrans.app\n (((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n X\u271d).hom \u226b\n prod.functor.map x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\napply Limits.prod.hom_ext\n[GOAL]\ncase refine'_2.w.h.h\u2081\nX\u271d Y\u271d : Type u\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Type u\n\u22a2 NatTrans.app\n (binaryProductFunctor.map x\u271d\u00b9 \u226b\n ((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n Y\u271d).hom)\n x\u271d \u226b\n prod.fst =\n NatTrans.app\n (((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n X\u271d).hom \u226b\n prod.functor.map x\u271d\u00b9)\n x\u271d \u226b\n prod.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.w.h.h\u2082\nX\u271d Y\u271d : Type u\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Type u\n\u22a2 NatTrans.app\n (binaryProductFunctor.map x\u271d\u00b9 \u226b\n ((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n Y\u271d).hom)\n x\u271d \u226b\n prod.snd =\n NatTrans.app\n (((fun X =>\n NatIso.ofComponents fun Y =>\n (IsLimit.conePointUniqueUpToIso (limit.isLimit (pair X Y)) (binaryProductLimit X Y)).symm)\n X\u271d).hom \u226b\n prod.functor.map x\u271d\u00b9)\n x\u271d \u226b\n prod.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.w.h.h\u2081\nX\u271d Y\u271d : Type u\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Type u\n\u22a2 NatTrans.app (binaryProductFunctor.map x\u271d\u00b9) x\u271d \u226b _root_.Prod.fst = _root_.Prod.fst \u226b x\u271d\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.w.h.h\u2082\nX\u271d Y\u271d : Type u\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Type u\n\u22a2 NatTrans.app (binaryProductFunctor.map x\u271d\u00b9) x\u271d \u226b _root_.Prod.snd = _root_.Prod.snd\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : Type u\nc : BinaryCofan X Y\n\u22a2 Nonempty (IsColimit c) \u2194\n Injective (BinaryCofan.inl c) \u2227\n Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n[PROOFSTEP]\nclassical\nconstructor\n\u00b7 rintro \u27e8h\u27e9\n rw [\u2190 show _ = c.inl from h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.left\u27e9, \u2190\n show _ = c.inr from h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.right\u27e9]\n dsimp [binaryCoproductCocone]\n refine'\n \u27e8(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp Sum.inl_injective,\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp Sum.inr_injective, _\u27e9\n erw [Set.range_comp, \u2190 eq_compl_iff_isCompl, Set.range_comp _ Sum.inr, \u2190\n Set.image_compl_eq (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.bijective]\n simp\n\u00b7 rintro \u27e8h\u2081, h\u2082, h\u2083\u27e9\n have : \u2200 x, x \u2208 Set.range c.inl \u2228 x \u2208 Set.range c.inr :=\n by\n rw [eq_compl_iff_isCompl.mpr h\u2083.symm]\n exact fun _ => or_not\n refine' \u27e8BinaryCofan.IsColimit.mk _ _ _ _ _\u27e9\n \u00b7 intro T f g x\n exact\n if h : x \u2208 Set.range c.inl then f ((Equiv.ofInjective _ h\u2081).symm \u27e8x, h\u27e9)\n else g ((Equiv.ofInjective _ h\u2082).symm \u27e8x, (this x).resolve_left h\u27e9)\n \u00b7 intro T f g\n funext x\n dsimp\n simp [h\u2081.eq_iff]\n \u00b7 intro T f g\n funext x\n dsimp\n simp only [Set.mem_range, Equiv.ofInjective_symm_apply, dite_eq_right_iff, forall_exists_index]\n intro y e\n have : c.inr x \u2208 Set.range c.inl \u2293 Set.range c.inr := \u27e8\u27e8_, e\u27e9, \u27e8_, rfl\u27e9\u27e9\n rw [disjoint_iff.mp h\u2083.1] at this \n exact this.elim\n \u00b7 rintro T _ _ m rfl rfl\n funext x\n dsimp\n split_ifs <;> exact congr_arg _ (Equiv.apply_ofInjective_symm _ \u27e8_, _\u27e9).symm\n[GOAL]\nX Y : Type u\nc : BinaryCofan X Y\n\u22a2 Nonempty (IsColimit c) \u2194\n Injective (BinaryCofan.inl c) \u2227\n Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX Y : Type u\nc : BinaryCofan X Y\n\u22a2 Nonempty (IsColimit c) \u2192\n Injective (BinaryCofan.inl c) \u2227\n Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n\u22a2 Injective (BinaryCofan.inl c) \u2227\n Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n[PROOFSTEP]\nrw [\u2190 show _ = c.inl from h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.left\u27e9, \u2190\n show _ = c.inr from h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.right\u27e9]\n[GOAL]\ncase mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n\u22a2 Injective\n (NatTrans.app (binaryCoproductCocone X Y).\u03b9 { as := left } \u226b\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227\n Injective\n (NatTrans.app (binaryCoproductCocone X Y).\u03b9 { as := right } \u226b\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227\n IsCompl\n (Set.range\n (NatTrans.app (binaryCoproductCocone X Y).\u03b9 { as := left } \u226b\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range\n (NatTrans.app (binaryCoproductCocone X Y).\u03b9 { as := right } \u226b\n (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n[PROOFSTEP]\ndsimp [binaryCoproductCocone]\n[GOAL]\ncase mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n\u22a2 Injective (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227\n Injective (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227\n IsCompl (Set.range (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n[PROOFSTEP]\nrefine'\n \u27e8(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp Sum.inl_injective,\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp Sum.inr_injective, _\u27e9\n[GOAL]\ncase mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n\u22a2 IsCompl (Set.range (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n (Set.range (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv))\n[PROOFSTEP]\nerw [Set.range_comp, \u2190 eq_compl_iff_isCompl, Set.range_comp _ Sum.inr, \u2190\n Set.image_compl_eq (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.bijective]\n[GOAL]\ncase mp.intro\nX Y : Type u\nc : BinaryCofan X Y\nh : IsColimit c\n\u22a2 (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv '' Set.range Sum.inl =\n \u2191(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).symm.toEquiv '' (Set.range Sum.inr)\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nX Y : Type u\nc : BinaryCofan X Y\n\u22a2 Injective (BinaryCofan.inl c) \u2227\n Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) \u2192\n Nonempty (IsColimit c)\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082, h\u2083\u27e9\n[GOAL]\ncase mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n\u22a2 Nonempty (IsColimit c)\n[PROOFSTEP]\nhave : \u2200 x, x \u2208 Set.range c.inl \u2228 x \u2208 Set.range c.inr :=\n by\n rw [eq_compl_iff_isCompl.mpr h\u2083.symm]\n exact fun _ => or_not\n[GOAL]\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n\u22a2 \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n[PROOFSTEP]\nrw [eq_compl_iff_isCompl.mpr h\u2083.symm]\n[GOAL]\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\n\u22a2 \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 (Set.range (BinaryCofan.inl c))\u1d9c\n[PROOFSTEP]\nexact fun _ => or_not\n[GOAL]\ncase mpr.intro.intro\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n\u22a2 Nonempty (IsColimit c)\n[PROOFSTEP]\nrefine' \u27e8BinaryCofan.IsColimit.mk _ _ _ _ _\u27e9\n[GOAL]\ncase mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n\u22a2 {T : Type u} \u2192 (X \u27f6 T) \u2192 (Y \u27f6 T) \u2192 (c.pt \u27f6 T)\n[PROOFSTEP]\nintro T f g x\n[GOAL]\ncase mpr.intro.intro.refine'_1\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : c.pt\n\u22a2 T\n[PROOFSTEP]\nexact\n if h : x \u2208 Set.range c.inl then f ((Equiv.ofInjective _ h\u2081).symm \u27e8x, h\u27e9)\n else g ((Equiv.ofInjective _ h\u2082).symm \u27e8x, (this x).resolve_left h\u27e9)\n[GOAL]\ncase mpr.intro.intro.refine'_2\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n\u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T),\n (BinaryCofan.inl c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) =\n f\n[PROOFSTEP]\nintro T f g\n[GOAL]\ncase mpr.intro.intro.refine'_2\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\n\u22a2 (BinaryCofan.inl c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) =\n f\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mpr.intro.intro.refine'_2.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := left }\n\u22a2 (BinaryCofan.inl c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }))\n x =\n f x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mpr.intro.intro.refine'_2.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := left }\n\u22a2 (if h : BinaryCofan.inl c x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inl c x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := BinaryCofan.inl c x, property := (_ : BinaryCofan.inl c x \u2208 Set.range (BinaryCofan.inr c)) })) =\n f x\n[PROOFSTEP]\nsimp [h\u2081.eq_iff]\n[GOAL]\ncase mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n\u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T),\n (BinaryCofan.inr c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) =\n g\n[PROOFSTEP]\nintro T f g\n[GOAL]\ncase mpr.intro.intro.refine'_3\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\n\u22a2 (BinaryCofan.inr c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) =\n g\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\n\u22a2 (BinaryCofan.inr c \u226b fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }))\n x =\n g x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\n\u22a2 (if h : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inr c)) })) =\n g x\n[PROOFSTEP]\nsimp only [Set.mem_range, Equiv.ofInjective_symm_apply, dite_eq_right_iff, forall_exists_index]\n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\n\u22a2 \u2200 (x_1 : X) (h : BinaryCofan.inl c x_1 = BinaryCofan.inr c x),\n f\n (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) =\n g x\n[PROOFSTEP]\nintro y e\n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\n\u22a2 f\n (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) =\n g x\n[PROOFSTEP]\nhave : c.inr x \u2208 Set.range c.inl \u2293 Set.range c.inr := \u27e8\u27e8_, e\u27e9, \u27e8_, rfl\u27e9\u27e9\n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis\u271d :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c) \u2293 Set.range (BinaryCofan.inr c)\n\u22a2 f\n (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) =\n g x\n[PROOFSTEP]\nrw [disjoint_iff.mp h\u2083.1] at this \n[GOAL]\ncase mpr.intro.intro.refine'_3.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis\u271d :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nf : X \u27f6 T\ng : Y \u27f6 T\nx : (pair X Y).obj { as := right }\ny : X\ne : BinaryCofan.inl c y = BinaryCofan.inr c x\nthis : BinaryCofan.inr c x \u2208 \u22a5\n\u22a2 f\n (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm\n { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) =\n g x\n[PROOFSTEP]\nexact this.elim\n[GOAL]\ncase mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\n\u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T) (m : c.pt \u27f6 T),\n BinaryCofan.inl c \u226b m = f \u2192\n BinaryCofan.inr c \u226b m = g \u2192\n m = fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n g\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })\n[PROOFSTEP]\nrintro T _ _ m rfl rfl\n[GOAL]\ncase mpr.intro.intro.refine'_4\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt \u27f6 T\n\u22a2 m = fun x =>\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n (BinaryCofan.inr c \u226b m)\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt \u27f6 T\nx : c.pt\n\u22a2 m x =\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n (BinaryCofan.inl c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })\n else\n (BinaryCofan.inr c \u226b m)\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mpr.intro.intro.refine'_4.h\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt \u27f6 T\nx : c.pt\n\u22a2 m x =\n if h : x \u2208 Set.range (BinaryCofan.inl c) then\n m (BinaryCofan.inl c (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }))\n else\n m\n (BinaryCofan.inr c\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt \u27f6 T\nx : c.pt\nh\u271d : x \u2208 Set.range (BinaryCofan.inl c)\n\u22a2 m x = m (BinaryCofan.inl c (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h\u271d }))\n[PROOFSTEP]\nexact congr_arg _ (Equiv.apply_ofInjective_symm _ \u27e8_, _\u27e9).symm\n[GOAL]\ncase neg\nX Y : Type u\nc : BinaryCofan X Y\nh\u2081 : Injective (BinaryCofan.inl c)\nh\u2082 : Injective (BinaryCofan.inr c)\nh\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c))\nthis :\n \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }),\n x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c)\nT : Type u\nm : c.pt \u27f6 T\nx : c.pt\nh\u271d : \u00acx \u2208 Set.range (BinaryCofan.inl c)\n\u22a2 m x =\n m\n (BinaryCofan.inr c\n (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm\n { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }))\n[PROOFSTEP]\nexact congr_arg _ (Equiv.apply_ofInjective_symm _ \u27e8_, _\u27e9).symm\n[GOAL]\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsColimit (BinaryCofan.mk f Subtype.val)\n[PROOFSTEP]\napply Nonempty.some\n[GOAL]\ncase h\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 Nonempty (IsColimit (BinaryCofan.mk f Subtype.val))\n[PROOFSTEP]\nrw [binaryCofan_isColimit_iff]\n[GOAL]\ncase h\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 Injective (BinaryCofan.inl (BinaryCofan.mk f Subtype.val)) \u2227\n Injective (BinaryCofan.inr (BinaryCofan.mk f Subtype.val)) \u2227\n IsCompl (Set.range (BinaryCofan.inl (BinaryCofan.mk f Subtype.val)))\n (Set.range (BinaryCofan.inr (BinaryCofan.mk f Subtype.val)))\n[PROOFSTEP]\nrefine' \u27e8(mono_iff_injective f).mp inferInstance, Subtype.val_injective, _\u27e9\n[GOAL]\ncase h\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsCompl (Set.range (BinaryCofan.inl (BinaryCofan.mk f Subtype.val)))\n (Set.range (BinaryCofan.inr (BinaryCofan.mk f Subtype.val)))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 IsCompl (Set.range (BinaryCofan.inr (BinaryCofan.mk f Subtype.val)))\n (Set.range (BinaryCofan.inl (BinaryCofan.mk f Subtype.val)))\n[PROOFSTEP]\nrw [\u2190 eq_compl_iff_isCompl]\n[GOAL]\ncase h\nX Y : Type u\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 Set.range (BinaryCofan.inr (BinaryCofan.mk f Subtype.val)) =\n (Set.range (BinaryCofan.inl (BinaryCofan.mk f Subtype.val)))\u1d9c\n[PROOFSTEP]\nexact Subtype.range_val\n[GOAL]\nJ : Type v\nF : J \u2192 Type u\ninst\u271d : UnivLE.{v, u}\ns : Cone (Discrete.functor F)\nm :\n s.pt \u27f6\n { pt := Shrink ((j : J) \u2192 F j),\n \u03c0 :=\n Discrete.natTrans fun x f =>\n match x, f with\n | { as := j }, f => \u2191(equivShrink ((j : J) \u2192 F j)).symm f j }.pt\nw :\n \u2200 (j : Discrete J),\n m \u226b\n NatTrans.app\n { pt := Shrink ((j : J) \u2192 F j),\n \u03c0 :=\n Discrete.natTrans fun x f =>\n match x, f with\n | { as := j }, f => \u2191(equivShrink ((j : J) \u2192 F j)).symm f j }.\u03c0\n j =\n NatTrans.app s.\u03c0 j\nx : s.pt\nj : J\n\u22a2 \u2191(equivShrink ((j : J) \u2192 F j)).symm (m x) j =\n \u2191(equivShrink ((j : J) \u2192 F j)).symm\n ((fun s x =>\n \u2191(equivShrink ((j : J) \u2192 (Discrete.functor F).obj { as := j })) fun j => NatTrans.app s.\u03c0 { as := j } x)\n s x)\n j\n[PROOFSTEP]\nsimpa using (congr_fun (w \u27e8j\u27e9) x : _)\n[GOAL]\nJ : Type u\nF : J \u2192 Type u\ns : Cocone (Discrete.functor F)\nm :\n { pt := (j : J) \u00d7 F j,\n \u03b9 :=\n Discrete.natTrans fun x x_1 =>\n match x, x_1 with\n | { as := j }, x => { fst := j, snd := x } }.pt \u27f6\n s.pt\nw :\n \u2200 (j : Discrete J),\n NatTrans.app\n { pt := (j : J) \u00d7 F j,\n \u03b9 :=\n Discrete.natTrans fun x x_1 =>\n match x, x_1 with\n | { as := j }, x => { fst := j, snd := x } }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s x => NatTrans.app s.\u03b9 { as := x.fst } x.snd) s\n[PROOFSTEP]\nfunext \u27e8j, x\u27e9\n[GOAL]\ncase h\nJ : Type u\nF : J \u2192 Type u\ns : Cocone (Discrete.functor F)\nm :\n { pt := (j : J) \u00d7 F j,\n \u03b9 :=\n Discrete.natTrans fun x x_1 =>\n match x, x_1 with\n | { as := j }, x => { fst := j, snd := x } }.pt \u27f6\n s.pt\nw :\n \u2200 (j : Discrete J),\n NatTrans.app\n { pt := (j : J) \u00d7 F j,\n \u03b9 :=\n Discrete.natTrans fun x x_1 =>\n match x, x_1 with\n | { as := j }, x => { fst := j, snd := x } }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nj : J\nx : F j\n\u22a2 m { fst := j, snd := x } = (fun s x => NatTrans.app s.\u03b9 { as := x.fst } x.snd) s { fst := j, snd := x }\n[PROOFSTEP]\nexact congr_fun (w \u27e8j\u27e9) x\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\n\u22a2 { l //\n l \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s \u2227\n \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (Fork.of\u03b9 f w).pt).obj WalkingParallelPair.zero},\n m \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s \u2192 m = l }\n[PROOFSTEP]\nrefine' \u27e8fun i => _, _, _\u27e9\n[GOAL]\ncase refine'_1\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\ni : ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero\n\u22a2 ((Functor.const WalkingParallelPair).obj (Fork.of\u03b9 f w).pt).obj WalkingParallelPair.zero\n[PROOFSTEP]\napply Classical.choose (t (s.\u03b9 i) _)\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\ni : ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero\n\u22a2 g (Fork.\u03b9 s i) = h (Fork.\u03b9 s i)\n[PROOFSTEP]\napply congr_fun s.condition i\n[GOAL]\ncase refine'_2\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\n\u22a2 (fun i => Classical.choose (_ : \u2203! x, f x = Fork.\u03b9 s i)) \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase refine'_2.h\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\ni : ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero\n\u22a2 ((fun i => Classical.choose (_ : \u2203! x, f x = Fork.\u03b9 s i)) \u226b Fork.\u03b9 (Fork.of\u03b9 f w)) i = Fork.\u03b9 s i\n[PROOFSTEP]\nexact (Classical.choose_spec (t (s.\u03b9 i) (congr_fun s.condition i))).1\n[GOAL]\ncase refine'_3\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\n\u22a2 \u2200\n {m :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (Fork.of\u03b9 f w).pt).obj WalkingParallelPair.zero},\n m \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s \u2192 m = fun i => Classical.choose (_ : \u2203! x, f x = Fork.\u03b9 s i)\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase refine'_3\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (Fork.of\u03b9 f w).pt).obj WalkingParallelPair.zero\nhm : m \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s\n\u22a2 m = fun i => Classical.choose (_ : \u2203! x, f x = Fork.\u03b9 s i)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase refine'_3.h\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : \u2200 (y : Y), g y = h y \u2192 \u2203! x, f x = y\ns : Fork g h\nm :\n ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero \u27f6\n ((Functor.const WalkingParallelPair).obj (Fork.of\u03b9 f w).pt).obj WalkingParallelPair.zero\nhm : m \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = Fork.\u03b9 s\ni : ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero\n\u22a2 m i = Classical.choose (_ : \u2203! x, f x = Fork.\u03b9 s i)\n[PROOFSTEP]\nexact (Classical.choose_spec (t (s.\u03b9 i) (congr_fun s.condition i))).2 _ (congr_fun hm i)\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\n\u22a2 \u2203! x, f x = y\n[PROOFSTEP]\nlet y' : PUnit \u27f6 Y := fun _ => y\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\n\u22a2 \u2203! x, f x = y\n[PROOFSTEP]\nhave hy' : y' \u226b g = y' \u226b h := funext fun _ => hy\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\n\u22a2 \u2203! x, f x = y\n[PROOFSTEP]\nrefine' \u27e8(Fork.IsLimit.lift' t _ hy').1 \u27e8\u27e9, congr_fun (Fork.IsLimit.lift' t y' _).2 \u27e8\u27e9, _\u27e9\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\n\u22a2 \u2200 (y_1 : X), (fun x => f x = y) y_1 \u2192 y_1 = \u2191(Fork.IsLimit.lift' t y' hy') PUnit.unit\n[PROOFSTEP]\nintro x' hx'\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\n\u22a2 x' = \u2191(Fork.IsLimit.lift' t y' hy') PUnit.unit\n[PROOFSTEP]\nsuffices : (fun _ : PUnit => x') = (Fork.IsLimit.lift' t y' hy').1\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\nthis : (fun x => x') = \u2191(Fork.IsLimit.lift' t y' hy')\n\u22a2 x' = \u2191(Fork.IsLimit.lift' t y' hy') PUnit.unit\ncase this\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\n\u22a2 (fun x => x') = \u2191(Fork.IsLimit.lift' t y' hy')\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase this\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\n\u22a2 (fun x => x') = \u2191(Fork.IsLimit.lift' t y' hy')\n[PROOFSTEP]\napply Fork.IsLimit.hom_ext t\n[GOAL]\ncase this\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\n\u22a2 (fun x => x') \u226b Fork.\u03b9 (Fork.of\u03b9 f w) = \u2191(Fork.IsLimit.lift' t y' hy') \u226b Fork.\u03b9 (Fork.of\u03b9 f w)\n[PROOFSTEP]\nfunext \u27e8\u27e9\n[GOAL]\ncase this.h\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\nt : IsLimit (Fork.of\u03b9 f w)\ny : Y\nhy : g y = h y\ny' : PUnit \u27f6 Y := fun x => y\nhy' : y' \u226b g = y' \u226b h\nx' : X\nhx' : f x' = y\n\u22a2 ((fun x => x') \u226b Fork.\u03b9 (Fork.of\u03b9 f w)) PUnit.unit =\n (\u2191(Fork.IsLimit.lift' t y' hy') \u226b Fork.\u03b9 (Fork.of\u03b9 f w)) PUnit.unit\n[PROOFSTEP]\napply hx'.trans (congr_fun (Fork.IsLimit.lift' t _ hy').2 \u27e8\u27e9).symm\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\ns : Fork g h\ni : ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.zero\n\u22a2 g (Fork.\u03b9 s i) = h (Fork.\u03b9 s i)\n[PROOFSTEP]\napply congr_fun s.condition i\n[GOAL]\nX Y Z : Type u\nf : X \u27f6 Y\ng h : Y \u27f6 Z\nw : f \u226b g = f \u226b h\n\u22a2 (equalizerIso g h).hom \u226b Subtype.val = equalizer.\u03b9 g h\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\ns : Cofork f g\na b : Y\nh : CoequalizerRel f g a b\n\u22a2 Cofork.\u03c0 s a = Cofork.\u03c0 s b\n[PROOFSTEP]\ncases h\n[GOAL]\ncase Rel\nX Y Z : Type u\nf g : X \u27f6 Y\ns : Cofork f g\nx\u271d : X\n\u22a2 Cofork.\u03c0 s (f x\u271d) = Cofork.\u03c0 s (g x\u271d)\n[PROOFSTEP]\napply congr_fun s.condition\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\n\u22a2 \u03c0 \u207b\u00b9' (\u03c0 '' U) = U\n[PROOFSTEP]\nhave lem : \u2200 x y, CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U) :=\n by\n rintro _ _ \u27e8x\u27e9\n change x \u2208 f \u207b\u00b9' U \u2194 x \u2208 g \u207b\u00b9' U\n rw [H]\n -- porting note: tidy was able to fill the structure automatically\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\n\u22a2 \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\n[PROOFSTEP]\nrintro _ _ \u27e8x\u27e9\n[GOAL]\ncase Rel\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nx : X\n\u22a2 f x \u2208 U \u2194 g x \u2208 U\n[PROOFSTEP]\nchange x \u2208 f \u207b\u00b9' U \u2194 x \u2208 g \u207b\u00b9' U\n[GOAL]\ncase Rel\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nx : X\n\u22a2 x \u2208 f \u207b\u00b9' U \u2194 x \u2208 g \u207b\u00b9' U\n[PROOFSTEP]\nrw [H]\n -- porting note: tidy was able to fill the structure automatically\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\n\u22a2 \u03c0 \u207b\u00b9' (\u03c0 '' U) = U\n[PROOFSTEP]\nhave eqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U :=\n { refl := by tauto\n symm := by tauto\n trans := by tauto }\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\n\u22a2 \u2200 (x : Y), x \u2208 U \u2194 x \u2208 U\n[PROOFSTEP]\ntauto\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\n\u22a2 \u2200 {x y : Y}, (x \u2208 U \u2194 y \u2208 U) \u2192 (y \u2208 U \u2194 x \u2208 U)\n[PROOFSTEP]\ntauto\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\n\u22a2 \u2200 {x y z : Y}, (x \u2208 U \u2194 y \u2208 U) \u2192 (y \u2208 U \u2194 z \u2208 U) \u2192 (x \u2208 U \u2194 z \u2208 U)\n[PROOFSTEP]\ntauto\n[GOAL]\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\n\u22a2 \u03c0 \u207b\u00b9' (\u03c0 '' U) = U\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d : Y\n\u22a2 x\u271d \u2208 \u03c0 \u207b\u00b9' (\u03c0 '' U) \u2194 x\u271d \u2208 U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d : Y\n\u22a2 x\u271d \u2208 \u03c0 \u207b\u00b9' (\u03c0 '' U) \u2192 x\u271d \u2208 U\n[PROOFSTEP]\nrw [\u2190 show _ = \u03c0 from h.comp_coconePointUniqueUpToIso_inv (coequalizerColimit f g).2 WalkingParallelPair.one]\n[GOAL]\ncase h.mp\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d : Y\n\u22a2 x\u271d \u2208\n (NatTrans.app (coequalizerColimit f g).cocone.\u03b9 WalkingParallelPair.one \u226b\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) \u207b\u00b9'\n ((NatTrans.app (coequalizerColimit f g).cocone.\u03b9 WalkingParallelPair.one \u226b\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) ''\n U) \u2192\n x\u271d \u2208 U\n[PROOFSTEP]\nrintro \u27e8y, hy, e'\u27e9\n[GOAL]\ncase h.mp.intro.intro\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d y : Y\nhy : y \u2208 U\ne' :\n (NatTrans.app (coequalizerColimit f g).cocone.\u03b9 WalkingParallelPair.one \u226b\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n y =\n (NatTrans.app (coequalizerColimit f g).cocone.\u03b9 WalkingParallelPair.one \u226b\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n x\u271d\n\u22a2 x\u271d \u2208 U\n[PROOFSTEP]\ndsimp at e' \n[GOAL]\ncase h.mp.intro.intro\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d y : Y\nhy : y \u2208 U\ne' :\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.\u03c0 (coequalizerColimit f g).cocone y) =\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.\u03c0 (coequalizerColimit f g).cocone x\u271d)\n\u22a2 x\u271d \u2208 U\n[PROOFSTEP]\nreplace e' :=\n (mono_iff_injective (h.coconePointUniqueUpToIso (coequalizerColimit f g).isColimit).inv).mp inferInstance e'\n[GOAL]\ncase h.mp.intro.intro\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d y : Y\nhy : y \u2208 U\ne' : Cofork.\u03c0 (coequalizerColimit f g).cocone y = Cofork.\u03c0 (coequalizerColimit f g).cocone x\u271d\n\u22a2 x\u271d \u2208 U\n[PROOFSTEP]\nexact (eqv.eqvGen_iff.mp (EqvGen.mono lem (Quot.exact _ e'))).mp hy\n[GOAL]\ncase h.mpr\nX Y Z : Type u\nf g : X \u27f6 Y\n\u03c0 : Y \u27f6 Z\ne : f \u226b \u03c0 = g \u226b \u03c0\nh : IsColimit (Cofork.of\u03c0 \u03c0 e)\nU : Set Y\nH : f \u207b\u00b9' U = g \u207b\u00b9' U\nlem : \u2200 (x y : Y), CoequalizerRel f g x y \u2192 (x \u2208 U \u2194 y \u2208 U)\neqv : _root_.Equivalence fun x y => x \u2208 U \u2194 y \u2208 U\nx\u271d : Y\n\u22a2 x\u271d \u2208 U \u2192 x\u271d \u2208 \u03c0 \u207b\u00b9' (\u03c0 '' U)\n[PROOFSTEP]\nexact fun hx => \u27e8_, hx, rfl\u27e9\n[GOAL]\nW X Y Z : Type u\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (s : PullbackCone f g),\n (fun s x =>\n { val := (PullbackCone.fst s x, PullbackCone.snd s x),\n property := (_ : (PullbackCone.fst s \u226b f) x = (PullbackCone.snd s \u226b g) x) })\n s \u226b\n PullbackCone.fst (pullbackCone f g) =\n PullbackCone.fst s\n[PROOFSTEP]\naesop\n[GOAL]\nW X Y Z : Type u\nf\u271d : X \u27f6 Z\ng\u271d : Y \u27f6 Z\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 \u2200 (s : PullbackCone f g),\n (fun s x =>\n { val := (PullbackCone.fst s x, PullbackCone.snd s x),\n property := (_ : (PullbackCone.fst s \u226b f) x = (PullbackCone.snd s \u226b g) x) })\n s \u226b\n PullbackCone.snd (pullbackCone f g) =\n PullbackCone.snd s\n[PROOFSTEP]\naesop\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Types", "llama_tokens": 22021, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.2652040369885606}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\nj : Discrete \u03b1\n\u22a2 j \u2208 {j}\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\n\u22a2 m =\n (fun s =>\n colimit.desc (liftToFinset F) { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x })\n s\n[PROOFSTEP]\napply colimit.hom_ext\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\n\u22a2 \u2200 (j : Finset (Discrete \u03b1)),\n colimit.\u03b9 (liftToFinset F) j \u226b m =\n colimit.\u03b9 (liftToFinset F) j \u226b\n (fun s =>\n colimit.desc (liftToFinset F)\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x })\n s\n[PROOFSTEP]\nrintro t\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\n\u22a2 colimit.\u03b9 (liftToFinset F) t \u226b m =\n colimit.\u03b9 (liftToFinset F) t \u226b\n (fun s =>\n colimit.desc (liftToFinset F)\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x })\n s\n[PROOFSTEP]\ndsimp [liftToFinset]\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\n\u22a2 colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n m =\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n colimit.desc\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x }\n[PROOFSTEP]\napply colimit.hom_ext\n[GOAL]\ncase w.w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\n\u22a2 \u2200 (j : Discrete { x // x \u2208 t }),\n colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) j \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n m =\n colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) j \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n colimit.desc\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x }\n[PROOFSTEP]\nrintro \u27e8\u27e8j, hj\u27e9\u27e9\n[GOAL]\ncase w.w.mk.mk\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\nj : Discrete \u03b1\nhj : j \u2208 t\n\u22a2 colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) { as := { val := j, property := hj } } \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n m =\n colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) { as := { val := j, property := hj } } \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n colimit.desc\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x }\n[PROOFSTEP]\nconvert h j using 1\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\nj : Discrete \u03b1\nhj : j \u2208 t\ne_1\u271d : ((Discrete.functor fun x => F.obj \u2191x).obj { as := { val := j, property := hj } } \u27f6 s.pt) = (F.obj j \u27f6 s.pt)\n\u22a2 colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) { as := { val := j, property := hj } } \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n m =\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m\n[PROOFSTEP]\nsimp [\u2190 colimit.w (liftToFinset F) \u27e8\u27e8Finset.singleton_subset_iff.2 hj\u27e9\u27e9]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\nj : Discrete \u03b1\nhj : j \u2208 t\ne_1\u271d : ((Discrete.functor fun x => F.obj \u2191x).obj { as := { val := j, property := hj } } \u27f6 s.pt) = (F.obj j \u27f6 s.pt)\n\u22a2 colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) { as := { val := j, property := hj } } \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n m =\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : \u2191{ val := j, property := (_ : j \u2208 {j}) } \u2208 t) } \u226b\n colimit.\u03b9 (liftToFinset F) t \u226b m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1 : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\nF : Discrete \u03b1 \u2964 C\ns : Cocone F\nm :\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b colimit.\u03b9 (liftToFinset F) {j} }.pt \u27f6\n s.pt\nh :\n \u2200 (j : Discrete \u03b1),\n NatTrans.app\n { pt := colimit (liftToFinset F),\n \u03b9 :=\n Discrete.natTrans fun j =>\n Sigma.\u03b9 (fun x => F.obj \u2191x) { val := j, property := (_ : j \u2208 {j}) } \u226b\n colimit.\u03b9 (liftToFinset F) {j} }.\u03b9\n j \u226b\n m =\n NatTrans.app s.\u03b9 j\nt : Finset (Discrete \u03b1)\nj : Discrete \u03b1\nhj : j \u2208 t\ne_1\u271d : ((Discrete.functor fun x => F.obj \u2191x).obj { as := { val := j, property := hj } } \u27f6 s.pt) = (F.obj j \u27f6 s.pt)\n\u22a2 colimit.\u03b9 (Discrete.functor fun x => F.obj \u2191x) { as := { val := j, property := hj } } \u226b\n colimit.\u03b9\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n t \u226b\n colimit.desc\n (Functor.mk\n { obj := fun s => \u2210 fun x => F.obj \u2191x,\n map := fun {x Y} h =>\n Sigma.desc fun y => Sigma.\u03b9 (fun x => F.obj \u2191x) { val := \u2191y, property := (_ : \u2191y \u2208 Y) } })\n { pt := s.pt, \u03b9 := NatTrans.mk fun t => Sigma.desc fun x => NatTrans.app s.\u03b9 \u2191x } =\n NatTrans.app s.\u03b9 j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1\u271d : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\n\u03b1 : Type w\n\u22a2 HasColimitsOfShape (Discrete \u03b1) C\n[PROOFSTEP]\nclassical exact \u27e8fun F => HasColimit.mk (liftToFinsetColimitCocone F)\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\n\u03b1\u271d : Type w\ninst\u271d\u00b9 : HasFiniteCoproducts C\ninst\u271d : HasFilteredColimitsOfSize.{w, w, v, u} C\n\u03b1 : Type w\n\u22a2 HasColimitsOfShape (Discrete \u03b1) C\n[PROOFSTEP]\nexact \u27e8fun F => HasColimit.mk (liftToFinsetColimitCocone F)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Constructions.Filtered", "llama_tokens": 5663, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.2649253930937334}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 Finset.card (increment hP G \u03b5).parts = stepBound (Finset.card P.parts)\n[PROOFSTEP]\nhave hP\u03b1' : stepBound P.parts.card \u2264 card \u03b1 := (mul_le_mul_left' (pow_le_pow_of_le_left' (by norm_num) _) _).trans hP\u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 4 \u2264 16\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\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\n\u22a2 Finset.card (increment hP G \u03b5).parts = stepBound (Finset.card P.parts)\n[PROOFSTEP]\nhave hPpos : 0 < stepBound P.parts.card := stepBound_pos (nonempty_of_not_uniform hPG).card_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card (increment hP G \u03b5).parts = stepBound (Finset.card P.parts)\n[PROOFSTEP]\nrw [increment, card_bind]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2211 A in attach P.parts, Finset.card (chunk hP G \u03b5 (_ : \u2191A \u2208 P.parts)).parts = stepBound (Finset.card P.parts)\n[PROOFSTEP]\nsimp_rw [chunk, apply_dite Finpartition.parts, apply_dite card, sum_dite]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2211 x in\n attach\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)),\n 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 \u2191\u2191x)).parts +\n \u2211 x in\n attach\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)),\n 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 \u2191\u2191x)).parts =\n stepBound (Finset.card P.parts)\n[PROOFSTEP]\nrw [sum_const_nat, sum_const_nat, card_attach, card_attach]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) *\n ?m.158358 +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) *\n ?m.158415 =\n stepBound (Finset.card P.parts)\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158415\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158358\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nrotate_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158415\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158358\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) *\n ?m.158358 +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) *\n ?m.158415 =\n stepBound (Finset.card P.parts)\n[PROOFSTEP]\nany_goals exact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158415\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2115\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 \u2200\n (x :\n { x //\n x \u2208\n filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts) }),\n x \u2208\n attach\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) \u2192\n 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 \u2191\u2191x)).parts =\n ?m.158358\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) *\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 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) *\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 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) =\n stepBound (Finset.card P.parts)\n[PROOFSTEP]\nexact fun x hx => card_parts_equitabilise _ _ (Nat.div_pos hP\u03b1' hPpos).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) *\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 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts)) +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) *\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 / Finset.card P.parts -\n Fintype.card \u03b1 / stepBound (Finset.card P.parts) * 4 ^ Finset.card P.parts +\n 1)) =\n stepBound (Finset.card P.parts)\n[PROOFSTEP]\nrw [Nat.sub_add_cancel a_add_one_le_four_pow_parts_card,\n Nat.sub_add_cancel ((Nat.le_succ _).trans a_add_one_le_four_pow_parts_card), \u2190 add_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 (Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts))) *\n 4 ^ Finset.card P.parts =\n stepBound (Finset.card P.parts)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nhP\u03b1' : stepBound (Finset.card P.parts) \u2264 Fintype.card \u03b1\nhPpos : 0 < stepBound (Finset.card P.parts)\n\u22a2 Finset.card\n (filter\n (fun x =>\n Finset.card \u2191x =\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 (attach P.parts)) +\n Finset.card\n (filter\n (fun x =>\n \u00acFinset.card \u2191x =\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 (attach P.parts)) =\n Finset.card P.parts\n[PROOFSTEP]\nrw [filter_card_add_filter_neg_card_eq_card, card_attach]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG\u271d : SimpleGraph \u03b1\n\u03b5\u271d : \u211d\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 IsEquipartition (increment hP G \u03b5)\n[PROOFSTEP]\nsimp_rw [IsEquipartition, Set.equitableOn_iff_exists_eq_eq_add_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG\u271d : SimpleGraph \u03b1\n\u03b5\u271d : \u211d\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 \u2203 b, \u2200 (a : Finset \u03b1), a \u2208 \u2191(increment hP G \u03b5).parts \u2192 Finset.card a = b \u2228 Finset.card a = b + 1\n[PROOFSTEP]\nrefine' \u27e8m, fun A hA => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG\u271d : SimpleGraph \u03b1\n\u03b5\u271d : \u211d\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nA : Finset \u03b1\nhA : A \u2208 \u2191(increment hP G \u03b5).parts\n\u22a2 Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nrw [mem_coe, increment, mem_bind] at hA \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG\u271d : SimpleGraph \u03b1\n\u03b5\u271d : \u211d\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nA : Finset \u03b1\nhA : \u2203 A_1 hA, A \u2208 (chunk hP G \u03b5 hA).parts\n\u22a2 Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nobtain \u27e8U, hU, hA\u27e9 := hA\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG\u271d : SimpleGraph \u03b1\n\u03b5\u271d : \u211d\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nA U : Finset \u03b1\nhU : U \u2208 P.parts\nhA : A \u2208 (chunk hP G \u03b5 hU).parts\n\u22a2 Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) \u2228\n Finset.card A = Fintype.card \u03b1 / stepBound (Finset.card P.parts) + 1\n[PROOFSTEP]\nexact card_eq_of_mem_parts_chunk hA\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 (Finset.biUnion (attach (offDiag P.parts)) fun UV =>\n (chunk hP G \u03b5 (_ : (\u2191UV).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191UV).snd \u2208 P.parts)).parts) \u2286\n offDiag (increment hP G \u03b5).parts\n[PROOFSTEP]\nrintro \u27e8Ui, Vj\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nUi Vj : Finset \u03b1\n\u22a2 ((Ui, Vj) \u2208\n Finset.biUnion (attach (offDiag P.parts)) fun UV =>\n (chunk hP G \u03b5 (_ : (\u2191UV).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191UV).snd \u2208 P.parts)).parts) \u2192\n (Ui, Vj) \u2208 offDiag (increment hP G \u03b5).parts\n[PROOFSTEP]\nsimp only [increment, mem_offDiag, bind_parts, mem_biUnion, Prod.exists, exists_and_left, exists_prop, mem_product,\n mem_attach, true_and_iff, Subtype.exists, and_imp, mem_offDiag, forall_exists_index, bex_imp, Ne.def]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nUi Vj : Finset \u03b1\n\u22a2 \u2200 (x x_1 : Finset \u03b1) (x_2 : x \u2208 P.parts \u2227 x_1 \u2208 P.parts \u2227 \u00acx = x_1),\n Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (x, x_1), property := (_ : (x, x_1) \u2208 offDiag P.parts) }).fst \u2208 P.parts)).parts \u2192\n Vj \u2208\n (chunk hP G \u03b5\n (_ : (\u2191{ val := (x, x_1), property := (_ : (x, x_1) \u2208 offDiag P.parts) }).snd \u2208 P.parts)).parts \u2192\n (\u2203 a h, Ui \u2208 (chunk hP G \u03b5 (_ : \u2191{ val := a, property := (_ : a \u2208 P.parts) } \u2208 P.parts)).parts) \u2227\n (\u2203 a h, Vj \u2208 (chunk hP G \u03b5 (_ : \u2191{ val := a, property := (_ : a \u2208 P.parts) } \u2208 P.parts)).parts) \u2227 \u00acUi = Vj\n[PROOFSTEP]\nrefine' fun U V hUV hUi hVj => \u27e8\u27e8_, hUV.1, hUi\u27e9, \u27e8_, hUV.2.1, hVj\u27e9, _\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nUi Vj U V : Finset \u03b1\nhUV : U \u2208 P.parts \u2227 V \u2208 P.parts \u2227 \u00acU = V\nhUi : Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).fst \u2208 P.parts)).parts\nhVj : Vj \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).snd \u2208 P.parts)).parts\n\u22a2 \u00acUi = Vj\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nUi U V : Finset \u03b1\nhUV : U \u2208 P.parts \u2227 V \u2208 P.parts \u2227 \u00acU = V\nhUi : Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).fst \u2208 P.parts)).parts\nhVj : Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).snd \u2208 P.parts)).parts\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := nonempty_of_mem_parts _ hUi\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\nUi U V : Finset \u03b1\nhUV : U \u2208 P.parts \u2227 V \u2208 P.parts \u2227 \u00acU = V\nhUi : Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).fst \u2208 P.parts)).parts\nhVj : Ui \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (U, V), property := (_ : (U, V) \u2208 offDiag P.parts) }).snd \u2208 P.parts)).parts\ni : \u03b1\nhi : i \u2208 Ui\n\u22a2 False\n[PROOFSTEP]\nexact hUV.2.2 (P.disjoint.elim_finset hUV.1 hUV.2.1 i (Finpartition.le _ hUi hi) <| Finpartition.le _ hVj hi)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 \u2211 x in attach (offDiag P.parts),\n SzemerediRegularity.pairContrib G \u03b5 hP x / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u2264\n energy (increment hP G \u03b5) G\n[PROOFSTEP]\nsimp_rw [pairContrib, \u2190 sum_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 (\u2211 x in attach (offDiag P.parts),\n \u2211 i in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n edgeDensity G i.fst i.snd ^ 2) /\n \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u2264\n energy (increment hP G \u03b5) G\n[PROOFSTEP]\nrefine' div_le_div_of_le_of_nonneg (\u03b1 := \u211a) _ (sq_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 \u2211 x in attach (offDiag P.parts),\n \u2211 i in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n edgeDensity G i.fst i.snd ^ 2 \u2264\n \u2211 uv in offDiag (increment hP G \u03b5).parts, edgeDensity G uv.fst uv.snd ^ 2\n[PROOFSTEP]\nrw [\u2190 sum_biUnion]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 \u2211 x in\n Finset.biUnion (attach (offDiag P.parts)) fun x =>\n (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n edgeDensity G x.fst x.snd ^ 2 \u2264\n \u2211 uv in offDiag (increment hP G \u03b5).parts, edgeDensity G uv.fst uv.snd ^ 2\n[PROOFSTEP]\nexact sum_le_sum_of_subset_of_nonneg distinct_pairs_increment fun i _ _ => sq_nonneg _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 Set.PairwiseDisjoint \u2191(attach (offDiag P.parts)) fun x =>\n (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts\n[PROOFSTEP]\nsimp only [Set.PairwiseDisjoint, Function.onFun, disjoint_left, inf_eq_inter, mem_inter, mem_product]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\n\u22a2 Set.Pairwise \u2191(attach (offDiag P.parts)) fun x y =>\n \u2200 \u2983a : Finset \u03b1 \u00d7 Finset \u03b1\u2984,\n a.fst \u2208 (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u2227 a.snd \u2208 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts \u2192\n \u00ac(a.fst \u2208 (chunk hP G \u03b5 (_ : (\u2191y).fst \u2208 P.parts)).parts \u2227 a.snd \u2208 (chunk hP G \u03b5 (_ : (\u2191y).snd \u2208 P.parts)).parts)\n[PROOFSTEP]\nrintro \u27e8\u27e8s\u2081, s\u2082\u27e9, hs\u27e9 _ \u27e8\u27e8t\u2081, t\u2082\u27e9, ht\u27e9 _ hst \u27e8u, v\u27e9 huv\u2081 huv\u2082\n[GOAL]\ncase mk.mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ns\u2081 s\u2082 : Finset \u03b1\nhs : (s\u2081, s\u2082) \u2208 offDiag P.parts\na\u271d\u00b9 : { val := (s\u2081, s\u2082), property := hs } \u2208 \u2191(attach (offDiag P.parts))\nt\u2081 t\u2082 : Finset \u03b1\nht : (t\u2081, t\u2082) \u2208 offDiag P.parts\na\u271d : { val := (t\u2081, t\u2082), property := ht } \u2208 \u2191(attach (offDiag P.parts))\nhst : { val := (s\u2081, s\u2082), property := hs } \u2260 { val := (t\u2081, t\u2082), property := ht }\nu v : Finset \u03b1\nhuv\u2081 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs }).snd \u2208 P.parts)).parts\nhuv\u2082 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht }).snd \u2208 P.parts)).parts\n\u22a2 False\n[PROOFSTEP]\nrw [mem_offDiag] at hs ht \n[GOAL]\ncase mk.mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ns\u2081 s\u2082 : Finset \u03b1\nhs\u271d : (s\u2081, s\u2082) \u2208 offDiag P.parts\nhs : (s\u2081, s\u2082).fst \u2208 P.parts \u2227 (s\u2081, s\u2082).snd \u2208 P.parts \u2227 (s\u2081, s\u2082).fst \u2260 (s\u2081, s\u2082).snd\na\u271d\u00b9 : { val := (s\u2081, s\u2082), property := hs\u271d } \u2208 \u2191(attach (offDiag P.parts))\nt\u2081 t\u2082 : Finset \u03b1\nht\u271d : (t\u2081, t\u2082) \u2208 offDiag P.parts\nht : (t\u2081, t\u2082).fst \u2208 P.parts \u2227 (t\u2081, t\u2082).snd \u2208 P.parts \u2227 (t\u2081, t\u2082).fst \u2260 (t\u2081, t\u2082).snd\na\u271d : { val := (t\u2081, t\u2082), property := ht\u271d } \u2208 \u2191(attach (offDiag P.parts))\nhst : { val := (s\u2081, s\u2082), property := hs\u271d } \u2260 { val := (t\u2081, t\u2082), property := ht\u271d }\nu v : Finset \u03b1\nhuv\u2081 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).snd \u2208 P.parts)).parts\nhuv\u2082 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).snd \u2208 P.parts)).parts\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := Finpartition.nonempty_of_mem_parts _ huv\u2081.1\n[GOAL]\ncase mk.mk.mk.mk.mk.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ns\u2081 s\u2082 : Finset \u03b1\nhs\u271d : (s\u2081, s\u2082) \u2208 offDiag P.parts\nhs : (s\u2081, s\u2082).fst \u2208 P.parts \u2227 (s\u2081, s\u2082).snd \u2208 P.parts \u2227 (s\u2081, s\u2082).fst \u2260 (s\u2081, s\u2082).snd\na\u271d\u00b9 : { val := (s\u2081, s\u2082), property := hs\u271d } \u2208 \u2191(attach (offDiag P.parts))\nt\u2081 t\u2082 : Finset \u03b1\nht\u271d : (t\u2081, t\u2082) \u2208 offDiag P.parts\nht : (t\u2081, t\u2082).fst \u2208 P.parts \u2227 (t\u2081, t\u2082).snd \u2208 P.parts \u2227 (t\u2081, t\u2082).fst \u2260 (t\u2081, t\u2082).snd\na\u271d : { val := (t\u2081, t\u2082), property := ht\u271d } \u2208 \u2191(attach (offDiag P.parts))\nhst : { val := (s\u2081, s\u2082), property := hs\u271d } \u2260 { val := (t\u2081, t\u2082), property := ht\u271d }\nu v : Finset \u03b1\nhuv\u2081 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).snd \u2208 P.parts)).parts\nhuv\u2082 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).snd \u2208 P.parts)).parts\na : \u03b1\nha : a \u2208 (u, v).fst\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := Finpartition.nonempty_of_mem_parts _ huv\u2081.2\n[GOAL]\ncase mk.mk.mk.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ns\u2081 s\u2082 : Finset \u03b1\nhs\u271d : (s\u2081, s\u2082) \u2208 offDiag P.parts\nhs : (s\u2081, s\u2082).fst \u2208 P.parts \u2227 (s\u2081, s\u2082).snd \u2208 P.parts \u2227 (s\u2081, s\u2082).fst \u2260 (s\u2081, s\u2082).snd\na\u271d\u00b9 : { val := (s\u2081, s\u2082), property := hs\u271d } \u2208 \u2191(attach (offDiag P.parts))\nt\u2081 t\u2082 : Finset \u03b1\nht\u271d : (t\u2081, t\u2082) \u2208 offDiag P.parts\nht : (t\u2081, t\u2082).fst \u2208 P.parts \u2227 (t\u2081, t\u2082).snd \u2208 P.parts \u2227 (t\u2081, t\u2082).fst \u2260 (t\u2081, t\u2082).snd\na\u271d : { val := (t\u2081, t\u2082), property := ht\u271d } \u2208 \u2191(attach (offDiag P.parts))\nhst : { val := (s\u2081, s\u2082), property := hs\u271d } \u2260 { val := (t\u2081, t\u2082), property := ht\u271d }\nu v : Finset \u03b1\nhuv\u2081 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (s\u2081, s\u2082), property := hs\u271d }).snd \u2208 P.parts)).parts\nhuv\u2082 :\n (u, v).fst \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).fst \u2208 P.parts)).parts \u2227\n (u, v).snd \u2208 (chunk hP G \u03b5 (_ : (\u2191{ val := (t\u2081, t\u2082), property := ht\u271d }).snd \u2208 P.parts)).parts\na : \u03b1\nha : a \u2208 (u, v).fst\nb : \u03b1\nhb : b \u2208 (u, v).snd\n\u22a2 False\n[PROOFSTEP]\nexact\n hst\n (Subtype.ext_val <|\n Prod.ext (P.disjoint.elim_finset hs.1 ht.1 a (Finpartition.le _ huv\u2081.1 ha) <| Finpartition.le _ huv\u2082.1 ha) <|\n P.disjoint.elim_finset hs.2.1 ht.2.1 b (Finpartition.le _ huv\u2081.2 hb) <| Finpartition.le _ huv\u2082.2 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\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) \u2264\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / 16 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [pairContrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) \u2264\n \u2191(\u2211 i in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n edgeDensity G i.fst i.snd ^ 2) /\n 16 ^ Finset.card P.parts\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\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) \u2264\n (\u2211 x in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n \u2191(edgeDensity G x.fst x.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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 : SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd\n\u22a2 \u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 + 0 \u2264\n (\u2211 x in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n \u2191(edgeDensity G x.fst x.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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 : SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd\n\u22a2 \u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 \u2264\n (\u2211 x in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n \u2191(edgeDensity G x.fst x.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nexact edgeDensity_chunk_uniform hP\u03b1 hP\u03b5 _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nx : { i // i \u2208 offDiag P.parts }\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u03b1 : 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 : \u00acSimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd\n\u22a2 \u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 + \u03b5 ^ 4 / 3 \u2264\n (\u2211 x in (chunk hP G \u03b5 (_ : (\u2191x).fst \u2208 P.parts)).parts \u00d7\u02e2 (chunk hP G \u03b5 (_ : (\u2191x).snd \u2208 P.parts)).parts,\n \u2191(edgeDensity G x.fst x.snd) ^ 2) /\n 16 ^ Finset.card P.parts\n[PROOFSTEP]\nexact edgeDensity_chunk_not_uniform hP\u03b1 hP\u03b5 h\u03b5\u2081 (mem_offDiag.1 x.2).2.2 h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4)) /\n \u2191(Finset.card P.parts) ^ 2 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n[PROOFSTEP]\nconv_rhs =>\n rw [\u2190 sum_div, card_increment hP\u03b1 hPG, stepBound, \u2190 Nat.cast_pow, mul_pow, pow_right_comm, Nat.cast_mul, mul_comm, \u2190\n div_div, show 4 ^ 2 = 16 by norm_num, sum_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n| \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n[PROOFSTEP]\nrw [\u2190 sum_div, card_increment hP\u03b1 hPG, stepBound, \u2190 Nat.cast_pow, mul_pow, pow_right_comm, Nat.cast_mul, mul_comm, \u2190\n div_div, show 4 ^ 2 = 16 by norm_num, sum_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n| \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n[PROOFSTEP]\nrw [\u2190 sum_div, card_increment hP\u03b1 hPG, stepBound, \u2190 Nat.cast_pow, mul_pow, pow_right_comm, Nat.cast_mul, mul_comm, \u2190\n div_div, show 4 ^ 2 = 16 by norm_num, sum_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n| \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n[PROOFSTEP]\nrw [\u2190 sum_div, card_increment hP\u03b1 hPG, stepBound, \u2190 Nat.cast_pow, mul_pow, pow_right_comm, Nat.cast_mul, mul_comm, \u2190\n div_div, show 4 ^ 2 = 16 by norm_num, sum_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 4 ^ 2 = 16\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4)) /\n \u2191(Finset.card P.parts) ^ 2 \u2264\n (\u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(16 ^ Finset.card P.parts)) /\n \u2191(Finset.card P.parts ^ 2)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_pow, Nat.cast_pow 16]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts ^ 2) * (\u03b5 ^ 5 / 4)) /\n \u2191(Finset.card P.parts ^ 2) \u2264\n (\u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u219116 ^ Finset.card P.parts) /\n \u2191(Finset.card P.parts ^ 2)\n[PROOFSTEP]\nrefine' div_le_div_of_le_of_nonneg _ (Nat.cast_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts ^ 2) * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u219116 ^ Finset.card P.parts\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / 16 ^ Finset.card P.parts\n[PROOFSTEP]\ntrans\n \u2211 x in P.parts.offDiag.attach,\n ((G.edgeDensity x.1.1 x.1.2 : \u211d) ^ 2 - \u03b5 ^ 5 / \u219125 + if G.IsUniform \u03b5 x.1.1 x.1.2 then (0 : \u211d) else \u03b5 ^ 4 / 3 : \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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3)\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) \u2264\n \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / 16 ^ Finset.card P.parts\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) \u2264\n \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / 16 ^ Finset.card P.parts\n[PROOFSTEP]\nexact sum_le_sum fun i _ => pairContrib_lower_bound i h\u03b5\u2081 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3)\n[PROOFSTEP]\nhave :\n \u2211 x in P.parts.offDiag.attach,\n ((G.edgeDensity x.1.1 x.1.2 : \u211d) ^ 2 - \u03b5 ^ 5 / \u219125 + if G.IsUniform \u03b5 x.1.1 x.1.2 then (0 : \u211d) else \u03b5 ^ 4 / 3 :\n \u211d) =\n \u2211 x in P.parts.offDiag,\n ((G.edgeDensity x.1 x.2 : \u211d) ^ 2 - \u03b5 ^ 5 / \u219125 + if G.IsUniform \u03b5 x.1 x.2 then (0 : \u211d) else \u03b5 ^ 4 / 3) :=\n by convert sum_attach (\u03b2 := \u211d); 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\n\u22a2 \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n[PROOFSTEP]\nconvert sum_attach (\u03b2 := \u211d)\n[GOAL]\ncase h.e'_2.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nx\u271d : { x // x \u2208 offDiag P.parts }\na\u271d : x\u271d \u2208 attach (offDiag P.parts)\n\u22a2 (\u2191(edgeDensity G (\u2191x\u271d).fst (\u2191x\u271d).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x\u271d).fst (\u2191x\u271d).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2191(edgeDensity G (\u2191x\u271d).fst (\u2191x\u271d).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x\u271d).fst (\u2191x\u271d).snd then 0 else \u03b5 ^ 4 / 3\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3)\n[PROOFSTEP]\nrw [this, sum_add_distrib, sum_sub_distrib, sum_const, nsmul_eq_mul, sum_ite, sum_const_zero, zero_add, sum_const,\n nsmul_eq_mul, \u2190 Finpartition.nonUniforms]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card (nonUniforms P G \u03b5)) * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\nrw [Finpartition.IsUniform, not_le] at hPG \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card (nonUniforms P G \u03b5)) * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\nrefine' le_trans _ (add_le_add_left (mul_le_mul_of_nonneg_right hPG.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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 0 \u2264 \u03b5 ^ 4 / 3\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\nconv_rhs =>\n enter [1, 2]\n rw [offDiag_card]\n conv => enter [1, 1, 2]; rw [\u2190 mul_one P.parts.card]\n rw [\u2190 Nat.mul_sub_left_distrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\n enter [1, 2]\n rw [offDiag_card]\n conv => enter [1, 1, 2]; rw [\u2190 mul_one P.parts.card]\n rw [\u2190 Nat.mul_sub_left_distrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\n enter [1, 2]\n rw [offDiag_card]\n conv => enter [1, 1, 2]; rw [\u2190 mul_one P.parts.card]\n rw [\u2190 Nat.mul_sub_left_distrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 - \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\nenter [1, 2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2191(Finset.card (offDiag P.parts)) * (\u03b5 ^ 5 / 25)\n[PROOFSTEP]\nrw [offDiag_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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2191(Finset.card P.parts * Finset.card P.parts - Finset.card P.parts) * (\u03b5 ^ 5 / 25)\n[PROOFSTEP]\nconv => enter [1, 1, 2]; rw [\u2190 mul_one 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2191(Finset.card P.parts * Finset.card P.parts - Finset.card P.parts) * (\u03b5 ^ 5 / 25)\n[PROOFSTEP]\nenter [1, 1, 2]; rw [\u2190 mul_one 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2191(Finset.card P.parts * Finset.card P.parts - Finset.card P.parts) * (\u03b5 ^ 5 / 25)\n[PROOFSTEP]\nenter [1, 1, 2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| Finset.card P.parts\n[PROOFSTEP]\nrw [\u2190 mul_one 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n| \u2191(Finset.card P.parts * Finset.card P.parts - Finset.card P.parts * 1) * (\u03b5 ^ 5 / 25)\n[PROOFSTEP]\nrw [\u2190 Nat.mul_sub_left_distrib]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) \u2264\n \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 -\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * (\u03b5 ^ 5 / 25) +\n \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 * (\u03b5 ^ 4 / 3)\n[PROOFSTEP]\nsimp_rw [mul_assoc, sub_add_eq_add_sub, add_sub_assoc, \u2190 mul_sub_left_distrib, mul_div_assoc' \u03b5, \u2190 pow_succ,\n show 4 + 1 = 5 by rfl, div_eq_mul_one_div (\u03b5 ^ 5), \u2190 mul_sub_left_distrib, mul_left_comm _ (\u03b5 ^ 5), sq, Nat.cast_mul,\n mul_assoc, \u2190 mul_assoc (\u03b5 ^ 5)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 4 + 1 = 5\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) * \u2191(edgeDensity G x.fst x.snd) +\n \u03b5 ^ 5 * \u2191(Finset.card P.parts) * (\u2191(Finset.card P.parts) * (1 / 4)) \u2264\n \u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) * \u2191(edgeDensity G x.fst x.snd) +\n \u03b5 ^ 5 * \u2191(Finset.card P.parts) * (\u2191(Finset.card P.parts - 1) * (1 / 3 - 1 / 25))\n[PROOFSTEP]\nrefine' add_le_add_left (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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 0 \u2264 \u03b5 ^ 5 * \u2191(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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 \u2191(Finset.card P.parts) * (1 / 4) \u2264 \u2191(Finset.card P.parts - 1) * (1 / 3 - 1 / 25)\n[PROOFSTEP]\nrw [Nat.cast_sub (P.parts_nonempty <| univ_nonempty.ne_empty).card_pos, mul_sub_right_distrib, Nat.cast_one, one_mul,\n le_sub_comm, \u2190 mul_sub_left_distrib, \u2190 div_le_iff (show (0 : \u211d) < 1 / 3 - 1 / 25 - 1 / 4 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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 0 < 1 / 3 - 1 / 25 - 1 / 4\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 (1 / 3 - 1 / 25) / (1 / 3 - 1 / 25 - 1 / 4) \u2264 \u2191(Finset.card P.parts)\n[PROOFSTEP]\nexact le_trans (show _ \u2264 (7 : \u211d) by norm_num) (by exact_mod_cast hP\u2087)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 (1 / 3 - 1 / 25) / (1 / 3 - 1 / 25 - 1 / 4) \u2264 7\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\ninst\u271d : Nonempty \u03b1\nh\u03b5\u2081 : \u03b5 \u2264 1\nhP\u2087 : 7 \u2264 Finset.card P.parts\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhP\u03b5 : 100 \u2264 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhPG : \u2191(Finset.card P.parts * (Finset.card P.parts - 1)) * \u03b5 < \u2191(Finset.card (nonUniforms P G \u03b5))\nthis :\n \u2211 x in attach (offDiag P.parts),\n (\u2191(edgeDensity G (\u2191x).fst (\u2191x).snd) ^ 2 - \u03b5 ^ 5 / 25 +\n if SimpleGraph.IsUniform G \u03b5 (\u2191x).fst (\u2191x).snd then 0 else \u03b5 ^ 4 / 3) =\n \u2211 x in offDiag P.parts,\n (\u2191(edgeDensity G x.fst x.snd) ^ 2 - \u03b5 ^ 5 / 25 + if SimpleGraph.IsUniform G \u03b5 x.fst x.snd then 0 else \u03b5 ^ 4 / 3)\n\u22a2 7 \u2264 \u2191(Finset.card P.parts)\n[PROOFSTEP]\nexact_mod_cast hP\u2087\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 \u2191(energy P G) + \u03b5 ^ 5 / 4 \u2264 \u2191(energy (increment hP G \u03b5) G)\n[PROOFSTEP]\nrw [coe_energy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.fst uv.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264\n \u2191(energy (increment hP G \u03b5) G)\n[PROOFSTEP]\nhave h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) h\u03b5\u2081 hP\u2087 hP\u03b1 h\u03b5.le hPG\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nh :\n (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4)) /\n \u2191(Finset.card P.parts) ^ 2 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n\u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.fst uv.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264\n \u2191(energy (increment hP G \u03b5) G)\n[PROOFSTEP]\nrw [add_div, mul_div_cancel_left] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nh :\n (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n\u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.fst uv.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264\n \u2191(energy (increment hP G \u03b5) G)\ncase ha\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nh :\n (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 +\n \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) / \u2191(Finset.card P.parts) ^ 2 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n\u22a2 \u2191(Finset.card P.parts) ^ 2 \u2260 0\n[PROOFSTEP]\nexact h.trans (by exact_mod_cast offDiag_pairs_le_increment_energy)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nh :\n (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n\u22a2 \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u2264\n \u2191(energy (increment hP G \u03b5) G)\n[PROOFSTEP]\nexact_mod_cast offDiag_pairs_le_increment_energy\n[GOAL]\ncase ha\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\nP : Finpartition univ\nhP\u271d : IsEquipartition P\nG : SimpleGraph \u03b1\n\u03b5 : \u211d\ninst\u271d : Nonempty \u03b1\nhP : IsEquipartition P\nhP\u2087 : 7 \u2264 Finset.card P.parts\nh\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5\nhP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1\nhPG : \u00acFinpartition.IsUniform P G \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nh :\n (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.fst x.snd) ^ 2) / \u2191(Finset.card P.parts) ^ 2 +\n \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) / \u2191(Finset.card P.parts) ^ 2 \u2264\n \u2211 x in attach (offDiag P.parts),\n \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2\n\u22a2 \u2191(Finset.card P.parts) ^ 2 \u2260 0\n[PROOFSTEP]\npositivity\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Regularity.Increment", "llama_tokens": 37618, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.411110869232168, "lm_q1q2_score": 0.2648479491251195}} {"text": "[GOAL]\nx x\u271d : PEmpty\n\u22a2 PEmpty\n[PROOFSTEP]\ncases x\n[GOAL]\nx y z : PEmpty\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\ncases x\n", "meta": {"mathlib_filename": "Mathlib.Algebra.PEmptyInstances", "llama_tokens": 65, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.26482907213575}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\ns t : Set \u03b1\na : \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 Prod.swap x \u2208 mulAntidiagonal s t a \u2194 x \u2208 mulAntidiagonal t s a\n[PROOFSTEP]\nsimp [mul_comm, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\ns t : Set \u03b1\na : \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x.snd \u2208 s \u2227 x.fst \u2208 t \u2227 x.snd * x.fst = a \u2194 x \u2208 mulAntidiagonal t s a\n[PROOFSTEP]\nsimp [mul_comm, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(mulAntidiagonal s t a)\nh : (\u2191x).fst = (\u2191y).fst\n\u22a2 a = (\u2191y).fst * (\u2191x).snd\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(mulAntidiagonal s t a)\nh : (\u2191x).fst = (\u2191y).fst\n\u22a2 a = (\u2191x).fst * (\u2191x).snd\n[PROOFSTEP]\nexact x.2.2.2.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(mulAntidiagonal s t a)\nh : (\u2191x).snd = (\u2191y).snd\n\u22a2 a = (\u2191x).fst * (\u2191y).snd\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(mulAntidiagonal s t a)\nh : (\u2191x).snd = (\u2191y).snd\n\u22a2 a = (\u2191x).fst * (\u2191x).snd\n[PROOFSTEP]\nexact x.2.2.2.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\n\u22a2 Set.Finite (mulAntidiagonal s t a)\n[PROOFSTEP]\nrefine' not_infinite.1 fun h => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\n\u22a2 False\n[PROOFSTEP]\nhave h1 : (mulAntidiagonal s t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o (\u00b7 \u2264 \u00b7)) := fun f hf =>\n hs (Prod.fst \u2218 f) fun n => (mem_mulAntidiagonal.1 (hf n)).1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\n\u22a2 False\n[PROOFSTEP]\nhave h2 : (mulAntidiagonal s t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o (\u00b7 \u2264 \u00b7)) := fun f hf =>\n ht (Prod.snd \u2218 f) fun n => (mem_mulAntidiagonal.1 (hf n)).2.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8m, n, mn, h2'\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 False\n[PROOFSTEP]\nrefine' mn.ne (g.injective <| (h.natEmbedding _).injective _)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(mulAntidiagonal s t a\u271d)\nhs : IsPwo s\nht : IsPwo t\na : \u03b1\nh : Set.Infinite (mulAntidiagonal s t a)\nh1 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : PartiallyWellOrderedOn (mulAntidiagonal s t a) (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n \u2200 (m n : \u2115),\n m \u2264 n \u2192\n (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\nm n : \u2115\nmn : m < n\nh2' :\n (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m))\n \u2191(\u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n))\n\u22a2 \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g m) = \u2191(Infinite.natEmbedding (mulAntidiagonal s t a) h) (\u2191g n)\n[PROOFSTEP]\nexact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2'\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.MulAntidiagonal", "llama_tokens": 2683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5506073802837478, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.26455510620215633}} {"text": "[GOAL]\n\u03b1 : Type u\nl : Thunk (List \u03b1)\nt : List \u03b1\n\u22a2 (fun xs => Thunk.get l ++ xs) t = (fun xs => Thunk.get l ++ xs) [] ++ t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nl : List \u03b1\n\u22a2 toList (ofList l) = l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u22a2 toList (ofList []) = []\ncase cons \u03b1 : Type u head\u271d : \u03b1 tail\u271d : List \u03b1 \u22a2 toList (ofList (head\u271d :: tail\u271d)) = head\u271d :: tail\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 toList (ofList (head\u271d :: tail\u271d)) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp only [DList.toList, DList.ofList, List.cons_append, List.append_nil]\n[GOAL]\n\u03b1 : Type u\nl : DList \u03b1\n\u22a2 ofList (toList l) = l\n[PROOFSTEP]\ncases' l with app inv\n[GOAL]\ncase mk\n\u03b1 : Type u\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 ofList (toList { apply := app, invariant := inv }) = { apply := app, invariant := inv }\n[PROOFSTEP]\nsimp only [ofList, toList, mk.injEq]\n[GOAL]\ncase mk\n\u03b1 : Type u\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 (fun x => app [] ++ x) = app\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mk.h\n\u03b1 : Type u\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\nx : List \u03b1\n\u22a2 app [] ++ x = app x\n[PROOFSTEP]\nrw [(inv x)]\n[GOAL]\n\u03b1 : Type u\n\u22a2 toList empty = []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nx : \u03b1\n\u22a2 toList (singleton x) = [x]\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nl\u2081 l\u2082 : DList \u03b1\n\u22a2 toList (append l\u2081 l\u2082) = toList l\u2081 ++ toList l\u2082\n[PROOFSTEP]\ncases' l\u2081 with _ l\u2081_invariant\n[GOAL]\ncase mk\n\u03b1 : Type u\nl\u2082 : DList \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d, invariant := l\u2081_invariant } l\u2082) =\n toList { apply := apply\u271d, invariant := l\u2081_invariant } ++ toList l\u2082\n[PROOFSTEP]\ncases' l\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (append { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } { apply := apply\u271d, invariant := invariant\u271d }) =\n toList { apply := apply\u271d\u00b9, invariant := l\u2081_invariant } ++ toList { apply := apply\u271d, invariant := invariant\u271d }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\napply\u271d\u00b9 : List \u03b1 \u2192 List \u03b1\nl\u2081_invariant : \u2200 (l : List \u03b1), apply\u271d\u00b9 l = apply\u271d\u00b9 [] ++ l\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d\u00b9 (apply\u271d []) = apply\u271d\u00b9 [] ++ apply\u271d []\n[PROOFSTEP]\nrw [l\u2081_invariant]\n[GOAL]\n\u03b1 : Type u\nx : \u03b1\nl : DList \u03b1\n\u22a2 toList (cons x l) = x :: toList l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\ninvariant\u271d : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (cons x { apply := apply\u271d, invariant := invariant\u271d }) =\n x :: toList { apply := apply\u271d, invariant := invariant\u271d }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nx : \u03b1\nl : DList \u03b1\n\u22a2 toList (push l x) = toList l ++ [x]\n[PROOFSTEP]\ncases' l with _ l_invariant\n[GOAL]\ncase mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 toList (push { apply := apply\u271d, invariant := l_invariant } x) =\n toList { apply := apply\u271d, invariant := l_invariant } ++ [x]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk\n\u03b1 : Type u\nx : \u03b1\napply\u271d : List \u03b1 \u2192 List \u03b1\nl_invariant : \u2200 (l : List \u03b1), apply\u271d l = apply\u271d [] ++ l\n\u22a2 apply\u271d [x] = apply\u271d [] ++ [x]\n[PROOFSTEP]\nrw [l_invariant]\n", "meta": {"mathlib_filename": "Mathlib.Data.DList.Defs", "llama_tokens": 1506, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.264555099115813}} {"text": "[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\n\u22a2 bind\u2081 X = AlgHom.id R (MvPolynomial \u03c3 R)\n[PROOFSTEP]\next1 i\n[GOAL]\ncase hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\ni : \u03c3\n\u22a2 \u2191(bind\u2081 X) (X i) = \u2191(AlgHom.id R (MvPolynomial \u03c3 R)) (X i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\nx : R\n\u22a2 \u2191(bind\u2081 f) (\u2191C x) = \u2191C x\n[PROOFSTEP]\nsimp [bind\u2081, algebraMap_eq]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 bind\u2082 C = RingHom.id (MvPolynomial \u03c3 R)\n[PROOFSTEP]\next : 2\n[GOAL]\ncase hC.a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (bind\u2082 C) C) x\u271d = \u2191(RingHom.comp (RingHom.id (MvPolynomial \u03c3 R)) C) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hX.a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ni\u271d : \u03c3\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(bind\u2082 C) (X i\u271d)) = coeff m\u271d (\u2191(RingHom.id (MvPolynomial \u03c3 R)) (X i\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191join\u2082 (\u2191(map f) \u03c6) = \u2191(bind\u2082 f) \u03c6\n[PROOFSTEP]\nsimp only [join\u2082, bind\u2082, eval\u2082Hom_map_hom, RingHom.id_comp]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\np : MvPolynomial \u03c3 R\n\u22a2 \u2191(aeval id) (\u2191(rename f) p) = \u2191(aeval f) p\n[PROOFSTEP]\nrw [aeval_rename, Function.comp.left_id]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 MvPolynomial \u03c5 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(bind\u2081 g) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(bind\u2081 g) (f i)) \u03c6\n[PROOFSTEP]\nsimp [bind\u2081, \u2190 comp_aeval]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 MvPolynomial \u03c5 R\n\u22a2 AlgHom.comp (bind\u2081 g) (bind\u2081 f) = bind\u2081 fun i => \u2191(bind\u2081 g) (f i)\n[PROOFSTEP]\next1\n[GOAL]\ncase hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 MvPolynomial \u03c5 R\ni\u271d : \u03c3\n\u22a2 \u2191(AlgHom.comp (bind\u2081 g) (bind\u2081 f)) (X i\u271d) = \u2191(bind\u2081 fun i => \u2191(bind\u2081 g) (f i)) (X i\u271d)\n[PROOFSTEP]\napply bind\u2081_bind\u2081\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* MvPolynomial \u03c3 T\n\u22a2 RingHom.comp (bind\u2082 g) (bind\u2082 f) = bind\u2082 (RingHom.comp (bind\u2082 g) f)\n[PROOFSTEP]\next : 2\n[GOAL]\ncase hC.a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* MvPolynomial \u03c3 T\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp (bind\u2082 g) (bind\u2082 f)) C) x\u271d = \u2191(RingHom.comp (bind\u2082 (RingHom.comp (bind\u2082 g) f)) C) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hX.a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* MvPolynomial \u03c3 T\ni\u271d : \u03c3\nm\u271d : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff m\u271d (\u2191(RingHom.comp (bind\u2082 g) (bind\u2082 f)) (X i\u271d)) = coeff m\u271d (\u2191(bind\u2082 (RingHom.comp (bind\u2082 g) f)) (X i\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 \u03c5\n\u22a2 AlgHom.comp (rename g) (bind\u2081 f) = bind\u2081 fun i => \u2191(rename g) (f i)\n[PROOFSTEP]\next1 i\n[GOAL]\ncase hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\ng : \u03c4 \u2192 \u03c5\ni : \u03c3\n\u22a2 \u2191(AlgHom.comp (rename g) (bind\u2081 f)) (X i) = \u2191(bind\u2081 fun i => \u2191(rename g) (f i)) (X i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(map g) (\u2191(bind\u2082 f) \u03c6) = \u2191(bind\u2082 (RingHom.comp (map g) f)) \u03c6\n[PROOFSTEP]\nsimp only [bind\u2082, eval\u2082_comp_right, coe_eval\u2082Hom, eval\u2082_map]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 eval\u2082 (RingHom.comp (map g) f) (\u2191(map g) \u2218 X) \u03c6 = eval\u2082 (RingHom.comp (map g) f) X \u03c6\n[PROOFSTEP]\ncongr 1 with : 1\n[GOAL]\ncase e_g.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\ng : S \u2192+* T\n\u03c6 : MvPolynomial \u03c3 R\nx\u271d : \u03c3\n\u22a2 (\u2191(map g) \u2218 X) x\u271d = X x\u271d\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_X]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c4 \u2192 MvPolynomial \u03c5 R\ng : \u03c3 \u2192 \u03c4\n\u22a2 AlgHom.comp (bind\u2081 f) (rename g) = bind\u2081 (f \u2218 g)\n[PROOFSTEP]\next1 i\n[GOAL]\ncase hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c5 : Type u_6\nf : \u03c4 \u2192 MvPolynomial \u03c5 R\ng : \u03c3 \u2192 \u03c4\ni : \u03c3\n\u22a2 \u2191(AlgHom.comp (bind\u2081 f) (rename g)) (X i) = \u2191(bind\u2081 (f \u2218 g)) (X i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : S \u2192+* MvPolynomial \u03c3 T\ng : R \u2192+* S\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(bind\u2082 f) (\u2191(map g) \u03c6) = \u2191(bind\u2082 (RingHom.comp f g)) \u03c6\n[PROOFSTEP]\nsimp [bind\u2082]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\n\u22a2 RingHom.comp (map f) C = RingHom.comp C f\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (map f) C) x\u271d = \u2191(RingHom.comp C f) x\u271d\n[PROOFSTEP]\napply map_C\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : MvPolynomial \u03c4 R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191f (\u2191(bind\u2081 g) \u03c6) = \u2191(eval\u2082Hom (RingHom.comp f C) fun i => \u2191f (g i)) \u03c6\n[PROOFSTEP]\nrw [bind\u2081, map_aeval, algebraMap_eq]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(map f) (\u2191(bind\u2081 g) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6)\n[PROOFSTEP]\nrw [hom_bind\u2081, map_comp_C, \u2190 eval\u2082Hom_map_hom]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(eval\u2082Hom C fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(map f) (g i)) (\u2191(map f) \u03c6)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 S\n\u22a2 RingHom.comp (eval\u2082Hom f g) C = f\n[PROOFSTEP]\next1 r\n[GOAL]\ncase a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c3 \u2192 S\nr : R\n\u22a2 \u2191(RingHom.comp (eval\u2082Hom f g) C) r = \u2191f r\n[PROOFSTEP]\nexact eval\u2082_C f g r\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* S\ng : \u03c4 \u2192 S\nh : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(eval\u2082Hom f g) (\u2191(bind\u2081 h) \u03c6) = \u2191(eval\u2082Hom f fun i => \u2191(eval\u2082Hom f g) (h i)) \u03c6\n[PROOFSTEP]\nrw [hom_bind\u2081, eval\u2082Hom_comp_C]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : Algebra R S\nf : \u03c4 \u2192 S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 AlgHom.comp (aeval f) (bind\u2081 g) = aeval fun i => \u2191(aeval f) (g i)\n[PROOFSTEP]\next1\n[GOAL]\ncase hf\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : Algebra R S\nf : \u03c4 \u2192 S\ng : \u03c3 \u2192 MvPolynomial \u03c4 R\ni\u271d : \u03c3\n\u22a2 \u2191(AlgHom.comp (aeval f) (bind\u2081 g)) (X i\u271d) = \u2191(aeval fun i => \u2191(aeval f) (g i)) (X i\u271d)\n[PROOFSTEP]\napply aeval_bind\u2081\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : S \u2192+* T\ng : \u03c3 \u2192 T\nh : R \u2192+* MvPolynomial \u03c3 S\n\u22a2 RingHom.comp (eval\u2082Hom f g) (bind\u2082 h) = eval\u2082Hom (RingHom.comp (eval\u2082Hom f g) h) g\n[PROOFSTEP]\next : 2\n[GOAL]\ncase hC.a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : S \u2192+* T\ng : \u03c3 \u2192 T\nh : R \u2192+* MvPolynomial \u03c3 S\nx\u271d : R\n\u22a2 \u2191(RingHom.comp (RingHom.comp (eval\u2082Hom f g) (bind\u2082 h)) C) x\u271d =\n \u2191(RingHom.comp (eval\u2082Hom (RingHom.comp (eval\u2082Hom f g) h) g) C) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hX\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : S \u2192+* T\ng : \u03c3 \u2192 T\nh : R \u2192+* MvPolynomial \u03c3 S\ni\u271d : \u03c3\n\u22a2 \u2191(RingHom.comp (eval\u2082Hom f g) (bind\u2082 h)) (X i\u271d) = \u2191(eval\u2082Hom (RingHom.comp (eval\u2082Hom f g) h) g) (X i\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2191(bind\u2081 f) (\u2191(monomial d) r) = \u2191C r * \u220f i in d.support, f i ^ \u2191d i\n[PROOFSTEP]\nsimp only [monomial_eq, AlgHom.map_mul, bind\u2081_C_right, Finsupp.prod, AlgHom.map_prod, AlgHom.map_pow, bind\u2081_X_right]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) r) = \u2191f r * \u2191(monomial d) 1\n[PROOFSTEP]\nsimp only [monomial_eq, RingHom.map_mul, bind\u2082_C_right, Finsupp.prod, map_prod, map_pow, bind\u2082_X_right, C_1, one_mul]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\nf : R \u2192+* MvPolynomial \u03c3 S\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2191(bind\u2082 f) (\u2191(monomial d) 1) = \u2191(monomial d) 1\n[PROOFSTEP]\nrw [bind\u2082_monomial, f.map_one, one_mul]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(bind\u2081 f) \u03c6) \u2286 Finset.biUnion (vars \u03c6) fun i => vars (f i)\n[PROOFSTEP]\ncalc\n (bind\u2081 f \u03c6).vars = (\u03c6.support.sum fun x : \u03c3 \u2192\u2080 \u2115 => (bind\u2081 f) (monomial x (coeff x \u03c6))).vars := by\n rw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum]\n _ \u2264 \u03c6.support.biUnion fun i : \u03c3 \u2192\u2080 \u2115 => ((bind\u2081 f) (monomial i (coeff i \u03c6))).vars := (vars_sum_subset _ _)\n _ = \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => vars (C (coeff d \u03c6) * \u220f i in d.support, f i ^ d i) := by\n simp only [bind\u2081_monomial]\n _ \u2264 \u03c6.support.biUnion fun d : \u03c3 \u2192\u2080 \u2115 => d.support.biUnion fun i => vars (f i) :=\n ?_\n -- proof below\n _ \u2264 \u03c6.vars.biUnion fun i : \u03c3 => vars (f i) :=\n ?_\n -- proof below\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 vars (\u2191(bind\u2081 f) \u03c6) = vars (\u2211 x in support \u03c6, \u2191(bind\u2081 f) (\u2191(monomial x) (coeff x \u03c6)))\n[PROOFSTEP]\nrw [\u2190 AlgHom.map_sum, \u2190 \u03c6.as_sum]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun i => vars (\u2191(bind\u2081 f) (\u2191(monomial i) (coeff i \u03c6)))) =\n Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)\n[PROOFSTEP]\nsimp only [bind\u2081_monomial]\n[GOAL]\ncase calc_1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i)) \u2264\n Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)\n[PROOFSTEP]\napply Finset.biUnion_mono\n[GOAL]\ncase calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2200 (a : \u03c3 \u2192\u2080 \u2115),\n a \u2208 support \u03c6 \u2192 vars (\u2191C (coeff a \u03c6) * \u220f i in a.support, f i ^ \u2191a i) \u2286 Finset.biUnion a.support fun i => vars (f i)\n[PROOFSTEP]\nintro d _hd\n[GOAL]\ncase calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 vars (\u2191C (coeff d \u03c6) * \u220f i in d.support, f i ^ \u2191d i) \u2286 Finset.biUnion d.support fun i => vars (f i)\n[PROOFSTEP]\ncalc\n vars (C (coeff d \u03c6) * \u220f i : \u03c3 in d.support, f i ^ d i) \u2264\n (C (coeff d \u03c6)).vars \u222a (\u220f i : \u03c3 in d.support, f i ^ d i).vars :=\n vars_mul _ _\n _ \u2264 (\u220f i : \u03c3 in d.support, f i ^ d i).vars := by\n simp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl]\n _ \u2264 d.support.biUnion fun i : \u03c3 => vars (f i ^ d i) := (vars_prod _)\n _ \u2264 d.support.biUnion fun i : \u03c3 => (f i).vars := ?_\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 vars (\u2191C (coeff d \u03c6)) \u222a vars (\u220f i in d.support, f i ^ \u2191d i) \u2264 vars (\u220f i in d.support, f i ^ \u2191d i)\n[PROOFSTEP]\nsimp only [Finset.empty_union, vars_C, Finset.le_iff_subset, Finset.Subset.refl]\n[GOAL]\ncase calc_1.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 (Finset.biUnion d.support fun i => vars (f i ^ \u2191d i)) \u2264 Finset.biUnion d.support fun i => vars (f i)\n[PROOFSTEP]\napply Finset.biUnion_mono\n[GOAL]\ncase calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\n\u22a2 \u2200 (a : \u03c3), a \u2208 d.support \u2192 vars (f a ^ \u2191d a) \u2286 vars (f a)\n[PROOFSTEP]\nintro i _hi\n[GOAL]\ncase calc_1.h.h\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n_hd : d \u2208 support \u03c6\ni : \u03c3\n_hi : i \u2208 d.support\n\u22a2 vars (f i ^ \u2191d i) \u2286 vars (f i)\n[PROOFSTEP]\napply vars_pow\n[GOAL]\ncase calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 (Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2264\n Finset.biUnion (vars \u03c6) fun i => vars (f i)\n[PROOFSTEP]\nintro j\n[GOAL]\ncase calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (j \u2208 Finset.biUnion (support \u03c6) fun d => Finset.biUnion d.support fun i => vars (f i)) \u2192\n j \u2208 Finset.biUnion (vars \u03c6) fun i => vars (f i)\n[PROOFSTEP]\nsimp_rw [Finset.mem_biUnion]\n[GOAL]\ncase calc_2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\n\u22a2 (\u2203 a, a \u2208 support \u03c6 \u2227 \u2203 a_1, a_1 \u2208 a.support \u2227 j \u2208 vars (f a_1)) \u2192 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)\n[PROOFSTEP]\nrintro \u27e8d, hd, \u27e8i, hi, hj\u27e9\u27e9\n[GOAL]\ncase calc_2.intro.intro.intro.intro\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : CommSemiring T\nf\u271d : \u03c3 \u2192 MvPolynomial \u03c4 R\ninst\u271d : DecidableEq \u03c4\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support \u03c6\ni : \u03c3\nhi : i \u2208 d.support\nhj : j \u2208 vars (f i)\n\u22a2 \u2203 a, a \u2208 vars \u03c6 \u2227 j \u2208 vars (f a)\n[PROOFSTEP]\nexact \u27e8i, (mem_vars _).mpr \u27e8d, hd, hi\u27e9, hj\u27e9\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nh : j \u2208 vars (\u2191(bind\u2081 f) \u03c6)\n\u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i)\n[PROOFSTEP]\nclassical simpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nh : j \u2208 vars (\u2191(bind\u2081 f) \u03c6)\n\u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i)\n[PROOFSTEP]\nsimpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1605153}, Functor.mapConst = Functor.map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1605153\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nrfl\n -- porting note: I guess `map_const` no longer has a default implementation?\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 : Type ?u.1605153} (x : MvPolynomial \u03b1 R), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d : Type ?u.1605153\nx\u271d : MvPolynomial \u03b1\u271d R\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 <$> \u00b7)]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.1605153} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : MvPolynomial \u03b1 R), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.1605153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : MvPolynomial \u03b1\u271d R\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 <$> \u00b7)]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (x : MvPolynomial \u03b1 R) (y : MvPolynomial \u03b2 R),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\nx\u271d : MvPolynomial \u03b1\u271d R\ny\u271d : MvPolynomial \u03b2\u271d R\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [SeqLeft.seqLeft, Seq.seq, (\u00b7 <$> \u00b7), bind\u2081_rename]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\nx\u271d : MvPolynomial \u03b1\u271d R\ny\u271d : MvPolynomial \u03b2\u271d R\n\u22a2 \u2191(bind\u2081 fun a => \u2191(bind\u2081 fun x => X a) y\u271d) x\u271d = \u2191(bind\u2081 ((fun y => \u2191(rename y) y\u271d) \u2218 Function.const \u03b2\u271d)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (x : MvPolynomial \u03b1 R) (y : MvPolynomial \u03b2 R),\n (SeqRight.seqRight x fun x => y) = Seq.seq (Function.const \u03b1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\nx\u271d : MvPolynomial \u03b1\u271d R\ny\u271d : MvPolynomial \u03b2\u271d R\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [SeqRight.seqRight, Seq.seq, (\u00b7 <$> \u00b7), bind\u2081_rename]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\nx\u271d : MvPolynomial \u03b1\u271d R\ny\u271d : MvPolynomial \u03b2\u271d R\n\u22a2 \u2191(bind\u2081 fun x => y\u271d) x\u271d = \u2191(bind\u2081 (Function.const \u03b1\u271d y\u271d)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (g : \u03b1 \u2192 \u03b2) (x : MvPolynomial \u03b1 R), (Seq.seq (pure g) fun x_1 => x) = g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : MvPolynomial \u03b1\u271d R\n\u22a2 (Seq.seq (pure g\u271d) fun x => x\u271d) = g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [(\u00b7 <$> \u00b7), pure, Seq.seq]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (f : \u03b1 \u2192 \u03b2) (x : MvPolynomial \u03b1 R),\n (do\n let a \u2190 x\n pure (f a)) =\n f <$> x\n[PROOFSTEP]\naesop\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (f : MvPolynomial (\u03b1 \u2192 \u03b2) R) (x : MvPolynomial \u03b1 R),\n (do\n let x_1 \u2190 f\n x_1 <$> x) =\n Seq.seq f fun x_1 => x\n[PROOFSTEP]\naesop\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.1607152} (x : \u03b1) (f : \u03b1 \u2192 MvPolynomial \u03b2 R), pure x >>= f = f x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d : Type ?u.1607152\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 MvPolynomial \u03b2\u271d R\n\u22a2 pure x\u271d >>= f\u271d = f\u271d x\u271d\n[PROOFSTEP]\nsimp [pure, bind]\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.1607152} (x : MvPolynomial \u03b1 R) (f : \u03b1 \u2192 MvPolynomial \u03b2 R) (g : \u03b2 \u2192 MvPolynomial \u03b3 R),\n x >>= f >>= g = x >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.1607152\nx\u271d : MvPolynomial \u03b1\u271d R\nf\u271d : \u03b1\u271d \u2192 MvPolynomial \u03b2\u271d R\ng\u271d : \u03b2\u271d \u2192 MvPolynomial \u03b3\u271d R\n\u22a2 x\u271d >>= f\u271d >>= g\u271d = x\u271d >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\nsimp [bind, \u2190 bind\u2081_comp_bind\u2081]\n", "meta": {"mathlib_filename": "Mathlib.Data.MvPolynomial.Monad", "llama_tokens": 13801, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.26424934071595463}} {"text": "[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\nf g : \u03b1 \u21aa \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\n{cases f; cases g; congr\n}\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\nf g : \u03b1 \u21aa \u03b2\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\ng : \u03b1 \u21aa \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninj'\u271d : Injective toFun\u271d\nh : { toFun := toFun\u271d, inj' := inj'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, inj' := inj'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Sort u\n\u03b2 : Sort v\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\ninj'\u271d\u00b9 : Injective toFun\u271d\u00b9\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninj'\u271d : Injective toFun\u271d\nh : { toFun := toFun\u271d\u00b9, inj' := inj'\u271d\u00b9 }.toFun = { toFun := toFun\u271d, inj' := inj'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, inj' := inj'\u271d\u00b9 } = { toFun := toFun\u271d, inj' := inj'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\n\u22a2 Embedding.trans (Equiv.toEmbedding e) (Equiv.toEmbedding e.symm) = Embedding.refl \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(Embedding.trans (Equiv.toEmbedding e) (Equiv.toEmbedding e.symm)) x\u271d = \u2191(Embedding.refl \u03b1) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\n\u22a2 Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e) = Embedding.refl \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx\u271d : \u03b2\n\u22a2 \u2191(Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e)) x\u271d = \u2191(Embedding.refl \u03b2) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\n\u22a2 Injective fun a' => if a' = a then b else if \u2191f a' = b then \u2191f a else \u2191f a'\n[PROOFSTEP]\nintro x y\n (h : ite _ _ _ = ite _ _ _)\n -- TODO: once we have `cc` we can avoid all the manual cases below by doing\n -- split_ifs at h <;> (try subst b) <;> (try simp only [f.injective.eq_iff] at *) <;> cc\n[GOAL]\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh : (if x = a then b else if \u2191f x = b then \u2191f a else \u2191f x) = if y = a then b else if \u2191f y = b then \u2191f a else \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsplit_ifs at h with h\u2081 h\u2082 _ _ h\u2085 h\u2086\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : y = a\nh : b = b\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : y = a\nh : b = b\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\n_ : \u2191f y = b\nh : b = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\n_ : \u2191f y = b\nh : b = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\n_ : \u00ac\u2191f y = b\nh : b = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\n_ : \u00ac\u2191f y = b\nh : b = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : y = a\nh : \u2191f a = b\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : y = a\nh : \u2191f a = b\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : \u00acy = a\nh\u2086 : \u2191f y = b\nh : \u2191f a = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : \u00acy = a\nh\u2086 : \u2191f y = b\nh : \u2191f a = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : \u00acy = a\nh\u2086 : \u00ac\u2191f y = b\nh : \u2191f a = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u2191f x = b\nh\u2085 : \u00acy = a\nh\u2086 : \u00ac\u2191f y = b\nh : \u2191f a = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d : y = a\nh : \u2191f x = b\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d : y = a\nh : \u2191f x = b\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u2191f y = b\nh : \u2191f x = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u2191f y = b\nh : \u2191f x = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u00ac\u2191f y = b\nh : \u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry subst b\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u00ac\u2191f y = b\nh : \u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : y = a\nh : b = b\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : y = a\nh : b = b\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\nh : \u2191f y = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\nh : \u2191f y = \u2191f a\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\n_ : \u00ac\u2191f y = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\n_ : \u00ac\u2191f y = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : y = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : \u2191f a = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : y = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : \u2191f a = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\nh : \u2191f a = \u2191f a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh\u2086 : \u2191f y = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\nh : \u2191f a = \u2191f a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh\u2086 : \u2191f y = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\nh : \u2191f a = \u2191f y\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh\u2086 : \u00ac\u2191f y = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\nh : \u2191f a = \u2191f y\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh\u2086 : \u00ac\u2191f y = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : y = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\n_ : \u00ac\u2191f x = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : y = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\n_ : \u00ac\u2191f x = \u2191f x\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : \u00acy = a\nh : \u2191f x = \u2191f a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\n_ : \u00ac\u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : \u00acy = a\nh : \u2191f x = \u2191f a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\n_ : \u00ac\u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u00ac\u2191f y = b\nh : \u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\ntry simp only [f.injective.eq_iff] at *\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u00ac\u2191f y = b\nh : \u2191f x = \u2191f y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [f.injective.eq_iff] at *\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : y = a\nh : True\n\u22a2 x = y\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : x = a\nh\u2082 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\nh : y = a\n\u22a2 x = y\n[PROOFSTEP]\nrw [h\u2081, h]\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : y = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : a = x\n\u22a2 x = y\n[PROOFSTEP]\nrw [h\u2085, \u2190 h]\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : True\nh\u2086 : y = x\n\u22a2 x = y\n[PROOFSTEP]\nexact h\u2086.symm\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : a = y\nh\u2086 : \u00acy = x\n\u22a2 x = y\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u2085 : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f x)\nh : a = y\nh\u2086 : \u00acy = x\n\u22a2 False\n[PROOFSTEP]\nexact h\u2085 h.symm\n[GOAL]\ncase pos\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\nh : x = a\n_ : \u00acx = y\n\u22a2 x = y\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\nx y : \u03b1\nh\u2081 : \u00acx = a\nh\u271d : \u00acy = a\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = \u2191f y)\nh : x = a\n_ : \u00acx = y\n\u22a2 False\n[PROOFSTEP]\nexact h\u2081 h\n[GOAL]\ncase neg\n\u03b1 : Sort ?u.14414\n\u03b2 : Sort ?u.14415\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\nx y : \u03b1\nh\u2081 : \u00acx = a\n_ : \u00ac\u2191f x = b\nh\u271d\u00b9 : \u00acy = a\nh\u271d : \u00ac\u2191f y = b\nh : x = y\n\u22a2 x = y\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : (a' : \u03b1) \u2192 Decidable (a' = a)\ninst\u271d : (a' : \u03b1) \u2192 Decidable (\u2191f a' = b)\n\u22a2 \u2191(setValue f a b) a = b\n[PROOFSTEP]\nsimp [setValue]\n[GOAL]\n\u03b2 : Sort u_1\nb : \u03b2\n\u22a2 Injective fun x => b\n[PROOFSTEP]\nrintro \u27e8\u27e9 \u27e8\u27e9 _\n[GOAL]\ncase unit.unit\n\u03b2 : Sort u_1\nb : \u03b2\na\u271d : (fun x => b) PUnit.unit = (fun x => b) PUnit.unit\n\u22a2 PUnit.unit = PUnit.unit\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ninst\u271d : Inhabited \u03b3\ne : \u03b1 \u21aa \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b3\nh : (fun f => extend (\u2191e) f default) f\u2081 = (fun f => extend (\u2191e) f default) f\u2082\nx : \u03b1\n\u22a2 f\u2081 x = f\u2082 x\n[PROOFSTEP]\nsimpa only [e.injective.extend_apply] using congr_fun h (e x)\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b4 : Sort u_4\nh : \u03b1 \u2243 \u03b2\nh' : \u03b3 \u2243 \u03b4\nx : \u03b1 \u21aa \u03b3\n\u22a2 (fun f => Embedding.congr h.symm h'.symm f) ((fun f => Embedding.congr h h' f) x) = x\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b4 : Sort u_4\nh : \u03b1 \u2243 \u03b2\nh' : \u03b3 \u2243 \u03b4\nx : \u03b1 \u21aa \u03b3\nx\u271d : \u03b1\n\u22a2 \u2191((fun f => Embedding.congr h.symm h'.symm f) ((fun f => Embedding.congr h h' f) x)) x\u271d = \u2191x x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b4 : Sort u_4\nh : \u03b1 \u2243 \u03b2\nh' : \u03b3 \u2243 \u03b4\nx : \u03b2 \u21aa \u03b4\n\u22a2 (fun f => Embedding.congr h h' f) ((fun f => Embedding.congr h.symm h'.symm f) x) = x\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b4 : Sort u_4\nh : \u03b1 \u2243 \u03b2\nh' : \u03b3 \u2243 \u03b4\nx : \u03b2 \u21aa \u03b4\nx\u271d : \u03b2\n\u22a2 \u2191((fun f => Embedding.congr h h' f) ((fun f => Embedding.congr h.symm h'.symm f) x)) x\u271d = \u2191x x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b3\u2081 : Sort u_3\n\u03b1\u2082 : Sort u_4\n\u03b2\u2082 : Sort u_5\n\u03b3\u2082 : Sort u_6\nea : \u03b1\u2081 \u2243 \u03b1\u2082\neb : \u03b2\u2081 \u2243 \u03b2\u2082\nec : \u03b3\u2081 \u2243 \u03b3\u2082\nf : \u03b1\u2081 \u21aa \u03b2\u2081\ng : \u03b2\u2081 \u21aa \u03b3\u2081\n\u22a2 \u2191(embeddingCongr ea ec) (Embedding.trans f g) =\n Embedding.trans (\u2191(embeddingCongr ea eb) f) (\u2191(embeddingCongr eb ec) g)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b3\u2081 : Sort u_3\n\u03b1\u2082 : Sort u_4\n\u03b2\u2082 : Sort u_5\n\u03b3\u2082 : Sort u_6\nea : \u03b1\u2081 \u2243 \u03b1\u2082\neb : \u03b2\u2081 \u2243 \u03b2\u2082\nec : \u03b3\u2081 \u2243 \u03b3\u2082\nf : \u03b1\u2081 \u21aa \u03b2\u2081\ng : \u03b2\u2081 \u21aa \u03b3\u2081\nx\u271d : \u03b1\u2082\n\u22a2 \u2191(\u2191(embeddingCongr ea ec) (Embedding.trans f g)) x\u271d =\n \u2191(Embedding.trans (\u2191(embeddingCongr ea eb) f) (\u2191(embeddingCongr eb ec) g)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 Function.Injective fun x =>\n if h : p \u2191x then Sum.inl { val := \u2191x, property := h } else Sum.inr { val := \u2191x, property := (_ : q \u2191x) }\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\n\u22a2 (fun x => if h : p \u2191x then Sum.inl { val := \u2191x, property := h } else Sum.inr { val := \u2191x, property := (_ : q \u2191x) })\n x =\n (fun x =>\n if h : p \u2191x then Sum.inl { val := \u2191x, property := h } else Sum.inr { val := \u2191x, property := (_ : q \u2191x) })\n y \u2192\n x = y\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\n\u22a2 ((if h : p \u2191x then Sum.inl { val := \u2191x, property := h } else Sum.inr { val := \u2191x, property := (_ : q \u2191x) }) =\n if h : p \u2191y then Sum.inl { val := \u2191y, property := h } else Sum.inr { val := \u2191y, property := (_ : q \u2191y) }) \u2192\n x = y\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\nh\u271d\u00b9 : p \u2191x\nh\u271d : p \u2191y\n\u22a2 Sum.inl { val := \u2191x, property := h\u271d\u00b9 } = Sum.inl { val := \u2191y, property := h\u271d } \u2192 x = y\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\nh\u271d\u00b9 : p \u2191x\nh\u271d : \u00acp \u2191y\n\u22a2 False \u2192 x = y\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\nh\u271d\u00b9 : \u00acp \u2191x\nh\u271d : p \u2191y\n\u22a2 False \u2192 x = y\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nx y : { x // p x \u2228 q x }\nh\u271d\u00b9 : \u00acp \u2191x\nh\u271d : \u00acp \u2191y\n\u22a2 Sum.inr { val := \u2191x, property := (_ : q \u2191x) } = Sum.inr { val := \u2191y, property := (_ : q \u2191y) } \u2192 x = y\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), p x \u2192 q x\nx y : { x // p x }\n\u22a2 (fun x => { val := \u2191x, property := (_ : q \u2191x) }) x = (fun x => { val := \u2191x, property := (_ : q \u2191x) }) y \u2192 x = y\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n", "meta": {"mathlib_filename": "Mathlib.Logic.Embedding.Basic", "llama_tokens": 10570, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804337438502, "lm_q2_score": 0.4263215925474903, "lm_q1q2_score": 0.26379945995090504}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : K\n\u22a2 x\u271d \u2208 {x | \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) x \u2264 1} \u2194\n x\u271d \u2208\n \u2191(\u2a05 (v : HeightOneSpectrum R) (_ : \u00acv \u2208 S),\n (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)\n[PROOFSTEP]\nsimp [SetLike.mem_coe, Subring.mem_iInf]\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 \u2191(Subalgebra.toSubring (integer S K)) =\n \u2191(\u2a05 (v : HeightOneSpectrum R) (_ : \u00acv \u2208 S), (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : K\n\u22a2 x\u271d \u2208 \u2191(Subalgebra.toSubring (integer S K)) \u2194\n x\u271d \u2208\n \u2191(\u2a05 (v : HeightOneSpectrum R) (_ : \u00acv \u2208 S),\n (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : K\u02e3\n\u22a2 x\u271d \u2208 {x | \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191x = 1} \u2194\n x\u271d \u2208\n \u2191(\u2a05 (v : HeightOneSpectrum R) (_ : \u00acv \u2208 S),\n ValuationSubring.unitGroup (Valuation.valuationSubring (HeightOneSpectrum.valuation v)))\n[PROOFSTEP]\nsimp only [mem_setOf, SetLike.mem_coe, Subgroup.mem_iInf, Valuation.mem_unitGroup_iff]\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx : { x // x \u2208 integer S K }\u02e3\nv : HeightOneSpectrum R\nhv : \u00acv \u2208 S\n\u22a2 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) * \u2191(HeightOneSpectrum.valuation v) \u2191x.inv = 1\n[PROOFSTEP]\nrw [Units.val_mk0, \u2190 map_mul, Subtype.mk_eq_mk.mp x.val_inv, v.valuation.map_one]\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : { x // x \u2208 unit S K }\n\u22a2 (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : { x // x \u2208 unit S K }\n\u22a2 \u2191\u2191((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x\u271d)) =\n \u2191\u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : { x // x \u2208 integer S K }\u02e3\n\u22a2 (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d : { x // x \u2208 integer S K }\u02e3\n\u22a2 \u2191\u2191((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x\u271d)) =\n \u2191\u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d\u00b9 x\u271d : { x // x \u2208 unit S K }\n\u22a2 Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n (x\u271d\u00b9 * x\u271d) =\n Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n x\u271d\u00b9 *\n Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nS : Set (HeightOneSpectrum R)\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nx\u271d\u00b9 x\u271d : { x // x \u2208 unit S K }\n\u22a2 \u2191\u2191(Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property := (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192\n \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n (x\u271d\u00b9 * x\u271d)) =\n \u2191\u2191(Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192\n \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192\n \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n x\u271d\u00b9 *\n Equiv.toFun\n {\n toFun := fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ : \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) },\n invFun := fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) },\n left_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 unit S K }),\n (fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192\n \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n ((fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n x) =\n x),\n right_inv :=\n (_ :\n \u2200 (x : { x // x \u2208 integer S K }\u02e3),\n (fun x =>\n {\n val :=\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) },\n inv :=\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R), \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) },\n val_inv :=\n (_ :\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } *\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } =\n 1),\n inv_val :=\n (_ :\n { val := \u2191(\u2191x)\u207b\u00b9,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x\u207b\u00b9 \u2264 1) } *\n { val := \u2191\u2191x,\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192 \u2191(HeightOneSpectrum.valuation v) \u2191\u2191x \u2264 1) } =\n 1) })\n ((fun x =>\n { val := Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False),\n property :=\n (_ :\n \u2200 (v : HeightOneSpectrum R),\n \u00acv \u2208 S \u2192\n \u2191(HeightOneSpectrum.valuation v) \u2191(Units.mk0 \u2191\u2191x (_ : \u2191\u2191x = 0 \u2192 False)) = 1) })\n x) =\n x) }\n x\u271d)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.DedekindDomain.SInteger", "llama_tokens": 15265, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.26348251312249693}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\n\u22a2 Intersecting {a} \u2194 a \u2260 \u22a5\n[PROOFSTEP]\nsimp [Intersecting]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\nhs : Intersecting s\nha : a \u2260 \u22a5\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 \u00acDisjoint a b\n\u22a2 Intersecting (insert a s)\n[PROOFSTEP]\nrintro b (rfl | hb) c (rfl | hc)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\nb c\u271d : \u03b1\nhs : Intersecting s\nc : \u03b1\nha : c \u2260 \u22a5\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 \u00acDisjoint c b\n\u22a2 \u00acDisjoint c c\n[PROOFSTEP]\nrwa [disjoint_self]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\nb\u271d c\u271d : \u03b1\nhs : Intersecting s\nb : \u03b1\nha : b \u2260 \u22a5\nh : \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2192 \u00acDisjoint b b_1\nc : \u03b1\nhc : c \u2208 s\n\u22a2 \u00acDisjoint b c\n[PROOFSTEP]\nexact h _ hc\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\nb\u271d c\u271d : \u03b1\nhs : Intersecting s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nha : c \u2260 \u22a5\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 \u00acDisjoint c b\n\u22a2 \u00acDisjoint b c\n[PROOFSTEP]\nexact fun H => h _ hb H.symm\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b\u271d c\u271d : \u03b1\nhs : Intersecting s\nha : a \u2260 \u22a5\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 \u00acDisjoint a b\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\n\u22a2 \u00acDisjoint b c\n[PROOFSTEP]\nexact hs hb hc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\n\u22a2 Intersecting s \u2194 (Set.Pairwise s fun a b => \u00acDisjoint a b) \u2227 s \u2260 {\u22a5}\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8fun a ha b hb _ => h ha hb, _\u27e9, fun h a ha b hb hab => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\nh : Intersecting s\n\u22a2 s \u2260 {\u22a5}\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\nt : Set \u03b1\na b c : \u03b1\nh : Intersecting {\u22a5}\n\u22a2 False\n[PROOFSTEP]\nexact intersecting_singleton.1 h rfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nh : (Set.Pairwise s fun a b => \u00acDisjoint a b) \u2227 s \u2260 {\u22a5}\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nhab : Disjoint a b\n\u22a2 False\n[PROOFSTEP]\nhave := h.1.eq ha hb (Classical.not_not.2 hab)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nh : (Set.Pairwise s fun a b => \u00acDisjoint a b) \u2227 s \u2260 {\u22a5}\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nhab : Disjoint a b\nthis : a = b\n\u22a2 False\n[PROOFSTEP]\nrw [this, disjoint_self] at hab \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nh : (Set.Pairwise s fun a b => \u00acDisjoint a b) \u2227 s \u2260 {\u22a5}\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nhab : b = \u22a5\nthis : a = b\n\u22a2 False\n[PROOFSTEP]\nrw [hab] at hb \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nh : (Set.Pairwise s fun a b => \u00acDisjoint a b) \u2227 s \u2260 {\u22a5}\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : \u22a5 \u2208 s\nhab : b = \u22a5\nthis : a = b\n\u22a2 False\n[PROOFSTEP]\nexact h.2 (eq_singleton_iff_unique_mem.2 \u27e8hb, fun c hc => not_ne_iff.1 fun H => h.1 hb hc H.symm disjoint_bot_left\u27e9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d t : Set \u03b1\na b c : \u03b1\ninst\u271d : Subsingleton \u03b1\ns : Set \u03b1\n\u22a2 Intersecting s \u2194 s = \u2205\n[PROOFSTEP]\nrefine'\n subsingleton_of_subsingleton.intersecting.trans\n \u27e8not_imp_comm.2 fun h => subsingleton_of_subsingleton.eq_singleton_of_mem _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d t : Set \u03b1\na b c : \u03b1\ninst\u271d : Subsingleton \u03b1\ns : Set \u03b1\nh : \u00acs = \u2205\n\u22a2 \u22a5 \u2208 s\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := nonempty_iff_ne_empty.2 h\n[GOAL]\ncase refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d t : Set \u03b1\na\u271d b c : \u03b1\ninst\u271d : Subsingleton \u03b1\ns : Set \u03b1\nh : \u00acs = \u2205\na : \u03b1\nha : a \u2208 s\n\u22a2 \u22a5 \u2208 s\n[PROOFSTEP]\nrwa [Subsingleton.elim \u22a5 a]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d t : Set \u03b1\na b c : \u03b1\ninst\u271d : Subsingleton \u03b1\ns : Set \u03b1\n\u22a2 s = \u2205 \u2192 s \u2260 {\u22a5}\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\ninst\u271d : Subsingleton \u03b1\n\u22a2 \u2205 \u2260 {\u22a5}\n[PROOFSTEP]\nexact (Set.singleton_nonempty _).ne_empty.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\nhs : Intersecting s\nh : \u2200 (t : Set \u03b1), Intersecting t \u2192 s \u2286 t \u2192 s = t\n\u22a2 IsUpperSet s\n[PROOFSTEP]\nclassical\nrintro a b hab ha\nrw [h (Insert.insert b s) _ (subset_insert _ _)]\n\u00b7 exact mem_insert _ _\nexact hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na b c : \u03b1\nhs : Intersecting s\nh : \u2200 (t : Set \u03b1), Intersecting t \u2192 s \u2286 t \u2192 s = t\n\u22a2 IsUpperSet s\n[PROOFSTEP]\nrintro a b hab ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nhs : Intersecting s\nh : \u2200 (t : Set \u03b1), Intersecting t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 s\n\u22a2 b \u2208 s\n[PROOFSTEP]\nrw [h (Insert.insert b s) _ (subset_insert _ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nhs : Intersecting s\nh : \u2200 (t : Set \u03b1), Intersecting t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 s\n\u22a2 b \u2208 insert b s\n[PROOFSTEP]\nexact mem_insert _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns t : Set \u03b1\na\u271d b\u271d c : \u03b1\nhs : Intersecting s\nh : \u2200 (t : Set \u03b1), Intersecting t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 s\n\u22a2 Intersecting (insert b s)\n[PROOFSTEP]\nexact hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na b c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 IsUpperSet \u2191s\n[PROOFSTEP]\nclassical\nrintro a b hab ha\nrw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]\n\u00b7 exact mem_insert_self _ _\nrw [coe_insert]\nexact hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na b c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 IsUpperSet \u2191s\n[PROOFSTEP]\nrintro a b hab ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na\u271d b\u271d c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 \u2191s\n\u22a2 b \u2208 \u2191s\n[PROOFSTEP]\nrw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na\u271d b\u271d c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 \u2191s\n\u22a2 b \u2208 \u2191(insert b s)\n[PROOFSTEP]\nexact mem_insert_self _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na\u271d b\u271d c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 \u2191s\n\u22a2 Intersecting \u2191(insert b s)\n[PROOFSTEP]\nrw [coe_insert]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\ns\u271d t : Set \u03b1\na\u271d b\u271d c : \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na b : \u03b1\nhab : a \u2264 b\nha : a \u2208 \u2191s\n\u22a2 Intersecting (insert b \u2191s)\n[PROOFSTEP]\nexact hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n[PROOFSTEP]\nrw [Finset.disjoint_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 \u00aca \u2208 map { toFun := compl, inj' := (_ : Function.Injective compl) } s\n[PROOFSTEP]\nrintro x hx hxc\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nx : \u03b1\nhx : x \u2208 s\nhxc : x \u2208 map { toFun := compl, inj' := (_ : Function.Injective compl) } s\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, hx', rfl\u27e9 := mem_map.mp hxc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nx : \u03b1\nhx' : x \u2208 s\nhx : \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208 s\nhxc :\n \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208\n map { toFun := compl, inj' := (_ : Function.Injective compl) } s\n\u22a2 False\n[PROOFSTEP]\nexact hs.not_compl_mem hx' hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 2 * card s \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nclassical\nrefine' (s.disjUnion _ hs.disjoint_map_compl).card_le_univ.trans_eq' _\nrw [two_mul, card_disjUnion, card_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 2 * card s \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nrefine' (s.disjUnion _ hs.disjoint_map_compl).card_le_univ.trans_eq' _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 2 * card s =\n card\n (disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)))\n[PROOFSTEP]\nrw [two_mul, card_disjUnion, card_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 (\u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t) \u2194 2 * card s = Fintype.card \u03b1\n[PROOFSTEP]\nclassical\nrefine'\n \u27e8fun h => _, fun h t ht hst =>\n Finset.eq_of_subset_of_card_le hst <| le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) two_pos\u27e9\nsuffices s.disjUnion (s.map \u27e8compl, compl_injective\u27e9) hs.disjoint_map_compl = Finset.univ by\n rw [Fintype.card, \u2190 this, two_mul, card_disjUnion, card_map]\nrw [\u2190 coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,\n image_eq_preimage_of_inverse compl_compl compl_compl]\nrefine' eq_univ_of_forall fun a => _\nsimp_rw [mem_union, mem_preimage]\nby_contra' ha\nrefine' s.ne_insert_of_not_mem _ ha.1 (h _ _ <| s.subset_insert _)\nrw [coe_insert]\nrefine' hs.insert _ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb\nrintro rfl\nhave := h {\u22a4} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)\nrw [compl_bot] at ha \nrw [coe_eq_empty.1 ((hs.isUpperSet' h).not_top_mem.1 ha.2)] at this \nexact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 (\u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t) \u2194 2 * card s = Fintype.card \u03b1\n[PROOFSTEP]\nrefine'\n \u27e8fun h => _, fun h t ht hst =>\n Finset.eq_of_subset_of_card_le hst <| le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) two_pos\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 2 * card s = Fintype.card \u03b1\n[PROOFSTEP]\nsuffices s.disjUnion (s.map \u27e8compl, compl_injective\u27e9) hs.disjoint_map_compl = Finset.univ by\n rw [Fintype.card, \u2190 this, two_mul, card_disjUnion, card_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nthis :\n disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)) =\n Finset.univ\n\u22a2 2 * card s = Fintype.card \u03b1\n[PROOFSTEP]\nrw [Fintype.card, \u2190 this, two_mul, card_disjUnion, card_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)) =\n Finset.univ\n[PROOFSTEP]\nrw [\u2190 coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,\n image_eq_preimage_of_inverse compl_compl compl_compl]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2191s \u222a compl \u207b\u00b9' \u2191s = univ\n[PROOFSTEP]\nrefine' eq_univ_of_forall fun a => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\n\u22a2 a \u2208 \u2191s \u222a compl \u207b\u00b9' \u2191s\n[PROOFSTEP]\nsimp_rw [mem_union, mem_preimage]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\n\u22a2 a \u2208 \u2191s \u2228 a\u1d9c \u2208 \u2191s\n[PROOFSTEP]\nby_contra' ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\nha : \u00aca \u2208 \u2191s \u2227 \u00aca\u1d9c \u2208 \u2191s\n\u22a2 False\n[PROOFSTEP]\nrefine' s.ne_insert_of_not_mem _ ha.1 (h _ _ <| s.subset_insert _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\nha : \u00aca \u2208 \u2191s \u2227 \u00aca\u1d9c \u2208 \u2191s\n\u22a2 Intersecting \u2191(insert a s)\n[PROOFSTEP]\nrw [coe_insert]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\nha : \u00aca \u2208 \u2191s \u2227 \u00aca\u1d9c \u2208 \u2191s\n\u22a2 Intersecting (insert a \u2191s)\n[PROOFSTEP]\nrefine' hs.insert _ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\na : \u03b1\nha : \u00aca \u2208 \u2191s \u2227 \u00aca\u1d9c \u2208 \u2191s\n\u22a2 a \u2260 \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a5\u1d9c \u2208 \u2191s\n\u22a2 False\n[PROOFSTEP]\nhave := h {\u22a4} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a5\u1d9c \u2208 \u2191s\n\u22a2 Intersecting \u2191{\u22a4}\n[PROOFSTEP]\nrw [coe_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a5\u1d9c \u2208 \u2191s\n\u22a2 Intersecting {\u22a4}\n[PROOFSTEP]\nexact intersecting_singleton.2 top_ne_bot\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a5\u1d9c \u2208 \u2191s\nthis : s \u2286 {\u22a4} \u2192 s = {\u22a4}\n\u22a2 False\n[PROOFSTEP]\nrw [compl_bot] at ha \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a4 \u2208 \u2191s\nthis : s \u2286 {\u22a4} \u2192 s = {\u22a4}\n\u22a2 False\n[PROOFSTEP]\nrw [coe_eq_empty.1 ((hs.isUpperSet' h).not_top_mem.1 ha.2)] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\nha : \u00ac\u22a5 \u2208 \u2191s \u2227 \u00ac\u22a4 \u2208 \u2191s\nthis : \u2205 \u2286 {\u22a4} \u2192 \u2205 = {\u22a4}\n\u22a2 False\n[PROOFSTEP]\nexact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nhave := hs.card_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nthis : 2 * card s \u2264 Fintype.card \u03b1\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nrw [mul_comm, \u2190 Nat.le_div_iff_mul_le' two_pos] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nthis : card s \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nrevert hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : card s \u2264 Fintype.card \u03b1 / 2\n\u22a2 Intersecting \u2191s \u2192 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nrefine' s.strongDownwardInductionOn _ this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\nthis : card s \u2264 Fintype.card \u03b1 / 2\n\u22a2 \u2200 (t\u2081 : Finset \u03b1),\n (\u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n t\u2081 \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t) \u2192\n card t\u2081 \u2264 Fintype.card \u03b1 / 2 \u2192 Intersecting \u2191t\u2081 \u2192 \u2203 t, t\u2081 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nrintro s ih _hcard hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nby_cases h : \u2200 t : Finset \u03b1, (t : Set \u03b1).Intersecting \u2192 s \u2286 t \u2192 s = t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nh : \u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nexact \u27e8s, Subset.rfl, hs.is_max_iff_card_eq.1 h, hs\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nh : \u00ac\u2200 (t : Finset \u03b1), Intersecting \u2191t \u2192 s \u2286 t \u2192 s = t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nh : \u2203 t, Intersecting \u2191t \u2227 s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nobtain \u27e8t, ht, hst\u27e9 := h\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nt : Finset \u03b1\nht : Intersecting \u2191t\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 \u2203 t, s \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n[PROOFSTEP]\nrefine' (ih _ (_root_.ssubset_iff_subset_ne.2 hst) ht).imp fun u => And.imp_left hst.1.trans\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nt : Finset \u03b1\nht : Intersecting \u2191t\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 card t \u2264 Fintype.card \u03b1 / 2\n[PROOFSTEP]\nrw [Nat.le_div_iff_mul_le' two_pos, mul_comm]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : BooleanAlgebra \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Fintype \u03b1\ns\u271d : Finset \u03b1\nthis : card s\u271d \u2264 Fintype.card \u03b1 / 2\ns : Finset \u03b1\nih :\n \u2200 {t\u2082 : Finset \u03b1},\n card t\u2082 \u2264 Fintype.card \u03b1 / 2 \u2192\n s \u2282 t\u2082 \u2192 Intersecting \u2191t\u2082 \u2192 \u2203 t, t\u2082 \u2286 t \u2227 2 * card t = Fintype.card \u03b1 \u2227 Intersecting \u2191t\n_hcard : card s \u2264 Fintype.card \u03b1 / 2\nhs : Intersecting \u2191s\nt : Finset \u03b1\nht : Intersecting \u2191t\nhst : s \u2286 t \u2227 s \u2260 t\n\u22a2 2 * card t \u2264 Fintype.card \u03b1\n[PROOFSTEP]\nexact ht.card_le\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SetFamily.Intersecting", "llama_tokens": 10481, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.2631765570048666}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj X).obj X_1 \u27f6 (Functor.empty C).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 (Functor.empty C).obj X_1 \u27f6 ((Functor.const (Discrete PEmpty)).obj X).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 (X : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj Y).obj X \u27f6 F.obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 \u2200 \u2983X Y_1 : Discrete PEmpty\u2984 (f : X \u27f6 Y_1),\n ((Functor.const (Discrete PEmpty)).obj Y).map f \u226b id (Discrete.casesOn Y_1 fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) \u226b F.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nt : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }\nX : C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj X).obj X_1 \u27f6 F.obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nt : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }\nX : C\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const (Discrete PEmpty)).obj X).map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b F.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nt : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }\nX : C\nf : X \u27f6 Y\n\u22a2 (X_1 : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj X).obj X_1 \u27f6 F.obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nt : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }\nX : C\nf : X \u27f6 Y\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const (Discrete PEmpty)).obj X).map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b F.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nt : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }\nX : C\nf : X \u27f6 Y\n\u22a2 \u2200 (j : Discrete PEmpty),\n f \u226b\n NatTrans.app { pt := Y, \u03c0 := NatTrans.mk fun X => id (Discrete.casesOn X fun as => False.elim (_ : False)) }.\u03c0\n j =\n NatTrans.app { pt := X, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }.\u03c0\n j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 Function.LeftInverse (fun u => IsLimit.mk fun s => default) fun t X =>\n {\n toInhabited :=\n {\n default :=\n IsLimit.lift t\n { pt := X, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) } },\n uniq :=\n (_ :\n \u2200 (f : X \u27f6 Y),\n f =\n IsLimit.lift t\n { pt := X, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }) }\n[PROOFSTEP]\ndsimp [Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 \u2200 (x : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => False.elim (_ : False) }),\n (IsLimit.mk fun s => IsLimit.lift x { pt := s.pt, \u03c0 := NatTrans.mk fun X_1 => False.elim (_ : False) }) = x\n[PROOFSTEP]\nintro x\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nx : IsLimit { pt := Y, \u03c0 := NatTrans.mk fun X => False.elim (_ : False) }\n\u22a2 (IsLimit.mk fun s => IsLimit.lift x { pt := s.pt, \u03c0 := NatTrans.mk fun X_1 => False.elim (_ : False) }) = x\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 Function.RightInverse (fun u => IsLimit.mk fun s => default) fun t X =>\n {\n toInhabited :=\n {\n default :=\n IsLimit.lift t\n { pt := X, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) } },\n uniq :=\n (_ :\n \u2200 (f : X \u27f6 Y),\n f =\n IsLimit.lift t\n { pt := X, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }) }\n[PROOFSTEP]\ndsimp [Function.RightInverse, Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\n\u22a2 \u2200 (x : (X : C) \u2192 Unique (X \u27f6 Y)),\n (fun X => { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 Y), f = default) }) = x\n[PROOFSTEP]\nintro u\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nu : (X : C) \u2192 Unique (X \u27f6 Y)\n\u22a2 (fun X => { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 Y), f = default) }) = u\n[PROOFSTEP]\nfunext X\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nY : C\nu : (X : C) \u2192 Unique (X \u27f6 Y)\nX : C\n\u22a2 { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 Y), f = default) } = u X\n[PROOFSTEP]\nsimp only\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 F.obj X_1 \u27f6 ((Functor.const (Discrete PEmpty)).obj X).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n F.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b ((Functor.const (Discrete PEmpty)).obj X).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX\u271d : C\nt : IsColimit { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }\nX : C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 F.obj X_1 \u27f6 ((Functor.const (Discrete PEmpty)).obj X).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX\u271d : C\nt : IsColimit { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }\nX : C\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n F.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b ((Functor.const (Discrete PEmpty)).obj X).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX\u271d : C\nt : IsColimit { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }\nX : C\nf : X\u271d \u27f6 X\n\u22a2 (X_1 : Discrete PEmpty) \u2192 F.obj X_1 \u27f6 ((Functor.const (Discrete PEmpty)).obj X).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX\u271d : C\nt : IsColimit { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }\nX : C\nf : X\u271d \u27f6 X\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n F.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b ((Functor.const (Discrete PEmpty)).obj X).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX\u271d : C\nt : IsColimit { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }\nX : C\nf : X\u271d \u27f6 X\n\u22a2 \u2200 (j : Discrete PEmpty),\n NatTrans.app { pt := X\u271d, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) }.\u03b9\n j \u226b\n f =\n NatTrans.app { pt := X, \u03b9 := NatTrans.mk fun X_2 => id (Discrete.casesOn X_2 fun as => False.elim (_ : False)) }.\u03b9\n j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 Function.LeftInverse (fun u => IsColimit.mk fun s => default) fun t X_1 =>\n {\n toInhabited :=\n {\n default :=\n IsColimit.desc t\n { pt := X_1, \u03b9 := NatTrans.mk fun X_2 => id (Discrete.casesOn X_2 fun as => False.elim (_ : False)) } },\n uniq :=\n (_ :\n \u2200 (f : X \u27f6 X_1),\n f =\n IsColimit.desc t\n { pt := X_1, \u03b9 := NatTrans.mk fun X_2 => id (Discrete.casesOn X_2 fun as => False.elim (_ : False)) }) }\n[PROOFSTEP]\ndsimp [Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 \u2200 (x : IsColimit { pt := X, \u03b9 := NatTrans.mk fun X_1 => False.elim (_ : False) }),\n (IsColimit.mk fun s => IsColimit.desc x { pt := s.pt, \u03b9 := NatTrans.mk fun X_2 => False.elim (_ : False) }) = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\nx\u271d : IsColimit { pt := X, \u03b9 := NatTrans.mk fun X_1 => False.elim (_ : False) }\n\u22a2 (IsColimit.mk fun s => IsColimit.desc x\u271d { pt := s.pt, \u03b9 := NatTrans.mk fun X_2 => False.elim (_ : False) }) = x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 Function.RightInverse (fun u => IsColimit.mk fun s => default) fun t X_1 =>\n {\n toInhabited :=\n {\n default :=\n IsColimit.desc t\n { pt := X_1, \u03b9 := NatTrans.mk fun X_2 => id (Discrete.casesOn X_2 fun as => False.elim (_ : False)) } },\n uniq :=\n (_ :\n \u2200 (f : X \u27f6 X_1),\n f =\n IsColimit.desc t\n { pt := X_1, \u03b9 := NatTrans.mk fun X_2 => id (Discrete.casesOn X_2 fun as => False.elim (_ : False)) }) }\n[PROOFSTEP]\ndsimp [Function.RightInverse, Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\n\u22a2 \u2200 (x : (Y : C) \u2192 Unique (X \u27f6 Y)),\n (fun X_1 => { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 X_1), f = default) }) = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\nx\u271d : (Y : C) \u2192 Unique (X \u27f6 Y)\n\u22a2 (fun X_1 => { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 X_1), f = default) }) = x\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nF : Discrete PEmpty \u2964 C\nX : C\nx\u271d\u00b9 : (Y : C) \u2192 Unique (X \u27f6 Y)\nx\u271d : C\n\u22a2 { toInhabited := { default := default }, uniq := (_ : \u2200 (f : X \u27f6 x\u271d), f = default) } = x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsTerminal X\nf g : Y \u27f6 X\n\u22a2 \u2200 (j : Discrete PEmpty), f \u226b NatTrans.app (asEmptyCone X).\u03c0 j = g \u226b NatTrans.app (asEmptyCone X).\u03c0 j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsInitial X\nf g : X \u27f6 Y\n\u22a2 \u2200 (j : Discrete PEmpty), NatTrans.app (asEmptyCocone X).\u03b9 j \u226b f = NatTrans.app (asEmptyCocone X).\u03b9 j \u226b g\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsTerminal X\nf : X \u27f6 Y\n\u22a2 Mono f\n[PROOFSTEP]\nhaveI := t.isSplitMono_from f\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsTerminal X\nf : X \u27f6 Y\nthis : IsSplitMono f\n\u22a2 Mono f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsInitial X\nf : Y \u27f6 X\n\u22a2 Epi f\n[PROOFSTEP]\nhaveI := t.isSplitEpi_to f\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\nt : IsInitial X\nf : Y \u27f6 X\nthis : IsSplitEpi f\n\u22a2 Epi f\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\n\u22a2 (X : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj c.pt).obj X \u27f6 F\u2081.obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\n\u22a2 \u2200 \u2983X Y : Discrete PEmpty\u2984 (f : X \u27f6 Y),\n ((Functor.const (Discrete PEmpty)).obj c.pt).map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) \u226b F\u2081.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\nf : c.pt \u27f6 c\u2082.pt\nx\u271d : \u2200 (j : Discrete PEmpty), f \u226b NatTrans.app c\u2082.\u03c0 j = NatTrans.app c.\u03c0 j\n\u22a2 f =\n (fun c =>\n IsLimit.lift hl\n { pt := c.pt, \u03c0 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) } \u226b\n hi.hom)\n c\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\nf : c.pt \u27f6 c\u2082.pt\nx\u271d : \u2200 (j : Discrete PEmpty), f \u226b NatTrans.app c\u2082.\u03c0 j = NatTrans.app c.\u03c0 j\n\u22a2 f = IsLimit.lift hl { pt := c.pt, \u03c0 := NatTrans.mk fun X_1 => False.elim (_ : False) } \u226b hi.hom\n[PROOFSTEP]\nrw [\u2190 hl.uniq _ (f \u226b hi.inv) _]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\nf : c.pt \u27f6 c\u2082.pt\nx\u271d : \u2200 (j : Discrete PEmpty), f \u226b NatTrans.app c\u2082.\u03c0 j = NatTrans.app c.\u03c0 j\n\u22a2 f = (f \u226b hi.inv) \u226b hi.hom\n[PROOFSTEP]\nsimp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nhl : IsLimit c\u2081\nc\u2082 : Cone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cone F\u2082\nf : c.pt \u27f6 c\u2082.pt\nx\u271d : \u2200 (j : Discrete PEmpty), f \u226b NatTrans.app c\u2082.\u03c0 j = NatTrans.app c.\u03c0 j\n\u22a2 \u2200 (j : Discrete PEmpty),\n (f \u226b hi.inv) \u226b NatTrans.app c\u2081.\u03c0 j =\n NatTrans.app { pt := c.pt, \u03c0 := NatTrans.mk fun X_1 => False.elim (_ : False) }.\u03c0 j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 Function.LeftInverse (fun hl => isLimitChangeEmptyCone C hl c\u2081 h.symm) fun hl => isLimitChangeEmptyCone C hl c\u2082 h\n[PROOFSTEP]\ndsimp [Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 \u2200 (x : IsLimit c\u2081), isLimitChangeEmptyCone C (isLimitChangeEmptyCone C x c\u2082 h) c\u2081 h.symm = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsLimit c\u2081\n\u22a2 isLimitChangeEmptyCone C (isLimitChangeEmptyCone C x\u271d c\u2082 h) c\u2081 h.symm = x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 Function.RightInverse (fun hl => isLimitChangeEmptyCone C hl c\u2081 h.symm) fun hl => isLimitChangeEmptyCone C hl c\u2082 h\n[PROOFSTEP]\ndsimp [Function.LeftInverse, Function.RightInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 \u2200 (x : IsLimit c\u2082), isLimitChangeEmptyCone C (isLimitChangeEmptyCone C x c\u2081 h.symm) c\u2082 h = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsLimit c\u2082\n\u22a2 isLimitChangeEmptyCone C (isLimitChangeEmptyCone C x\u271d c\u2081 h.symm) c\u2082 h = x\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cone F\u2081\nc\u2082 : Cone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsLimit c\u2082\n\u22a2 isLimitChangeEmptyCone C (isLimitChangeEmptyCone C x\u271d c\u2081 h.symm) c\u2082 h = x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nh : HasLimit F\u2081\n\u22a2 (X : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj (limit F\u2081)).obj X \u27f6 F\u2082.obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nh : HasLimit F\u2081\n\u22a2 \u2200 \u2983X Y : Discrete PEmpty\u2984 (f : X \u27f6 Y),\n ((Functor.const (Discrete PEmpty)).obj (limit F\u2081)).map f \u226b\n id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) \u226b F\u2082.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\n\u22a2 (X : Discrete PEmpty) \u2192 F\u2081.obj X \u27f6 ((Functor.const (Discrete PEmpty)).obj c.pt).obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\n\u22a2 \u2200 \u2983X Y : Discrete PEmpty\u2984 (f : X \u27f6 Y),\n F\u2081.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) \u226b ((Functor.const (Discrete PEmpty)).obj c.pt).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\nf : c\u2082.pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app c\u2082.\u03b9 j \u226b f = NatTrans.app c.\u03b9 j\n\u22a2 f =\n (fun c =>\n hi.inv \u226b\n IsColimit.desc hl\n { pt := c.pt, \u03b9 := NatTrans.mk fun X_1 => id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) })\n c\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\nf : c\u2082.pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app c\u2082.\u03b9 j \u226b f = NatTrans.app c.\u03b9 j\n\u22a2 f = hi.inv \u226b IsColimit.desc hl { pt := c.pt, \u03b9 := NatTrans.mk fun X_1 => False.elim (_ : False) }\n[PROOFSTEP]\nrw [\u2190 hl.uniq _ (hi.hom \u226b f) _]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\nf : c\u2082.pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app c\u2082.\u03b9 j \u226b f = NatTrans.app c.\u03b9 j\n\u22a2 f = hi.inv \u226b hi.hom \u226b f\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nhl : IsColimit c\u2081\nc\u2082 : Cocone F\u2082\nhi : c\u2081.pt \u2245 c\u2082.pt\nc : Cocone F\u2082\nf : c\u2082.pt \u27f6 c.pt\nx\u271d : \u2200 (j : Discrete PEmpty), NatTrans.app c\u2082.\u03b9 j \u226b f = NatTrans.app c.\u03b9 j\n\u22a2 \u2200 (j : Discrete PEmpty),\n NatTrans.app c\u2081.\u03b9 j \u226b hi.hom \u226b f =\n NatTrans.app { pt := c.pt, \u03b9 := NatTrans.mk fun X_1 => False.elim (_ : False) }.\u03b9 j\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 Function.LeftInverse (fun hl => isColimitChangeEmptyCocone C hl c\u2081 h.symm) fun hl =>\n isColimitChangeEmptyCocone C hl c\u2082 h\n[PROOFSTEP]\ndsimp [Function.LeftInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 \u2200 (x : IsColimit c\u2081), isColimitChangeEmptyCocone C (isColimitChangeEmptyCocone C x c\u2082 h) c\u2081 h.symm = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsColimit c\u2081\n\u22a2 isColimitChangeEmptyCocone C (isColimitChangeEmptyCocone C x\u271d c\u2082 h) c\u2081 h.symm = x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 Function.RightInverse (fun hl => isColimitChangeEmptyCocone C hl c\u2081 h.symm) fun hl =>\n isColimitChangeEmptyCocone C hl c\u2082 h\n[PROOFSTEP]\ndsimp [Function.LeftInverse, Function.RightInverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\n\u22a2 \u2200 (x : IsColimit c\u2082), isColimitChangeEmptyCocone C (isColimitChangeEmptyCocone C x c\u2081 h.symm) c\u2082 h = x\n[PROOFSTEP]\nintro\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsColimit c\u2082\n\u22a2 isColimitChangeEmptyCocone C (isColimitChangeEmptyCocone C x\u271d c\u2081 h.symm) c\u2082 h = x\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nc\u2081 : Cocone F\u2081\nc\u2082 : Cocone F\u2082\nh : c\u2081.pt \u2245 c\u2082.pt\nx\u271d : IsColimit c\u2082\n\u22a2 isColimitChangeEmptyCocone C (isColimitChangeEmptyCocone C x\u271d c\u2081 h.symm) c\u2082 h = x\u271d\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nh : HasColimit F\u2081\n\u22a2 (X : Discrete PEmpty) \u2192 F\u2082.obj X \u27f6 ((Functor.const (Discrete PEmpty)).obj (colimit F\u2081)).obj X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty \u2964 C\nF\u2082 : Discrete PEmpty \u2964 C\nh : HasColimit F\u2081\n\u22a2 \u2200 \u2983X Y : Discrete PEmpty\u2984 (f : X \u27f6 Y),\n F\u2082.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X fun as => False.elim (_ : False)) \u226b\n ((Functor.const (Discrete PEmpty)).obj (colimit F\u2081)).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nh : IsTerminal X\nF : Discrete PEmpty \u2964 C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 ((Functor.const (Discrete PEmpty)).obj X).obj X_1 \u27f6 F.obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nh : IsTerminal X\nF : Discrete PEmpty \u2964 C\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const (Discrete PEmpty)).obj X).map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b F.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nh : IsInitial X\nF : Discrete PEmpty \u2964 C\n\u22a2 (X_1 : Discrete PEmpty) \u2192 F.obj X_1 \u27f6 ((Functor.const (Discrete PEmpty)).obj X).obj X_1\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nh : IsInitial X\nF : Discrete PEmpty \u2964 C\n\u22a2 \u2200 \u2983X_1 Y : Discrete PEmpty\u2984 (f : X_1 \u27f6 Y),\n F.map f \u226b id (Discrete.casesOn Y fun as => False.elim (_ : False)) =\n id (Discrete.casesOn X_1 fun as => False.elim (_ : False)) \u226b ((Functor.const (Discrete PEmpty)).obj X).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : HasTerminal C\nP Q : C\nf : P \u27f6 Q\n\u22a2 f \u226b from Q = from P\n[PROOFSTEP]\naesop\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : HasInitial C\nP Q : C\nf : P \u27f6 Q\n\u22a2 to P \u226b f = to Q\n[PROOFSTEP]\naesop\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u271d\nJ : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} J\nC : Type u_2\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : HasTerminal C\nj : J\n\u22a2 limitConstTerminal.inv \u226b limit.\u03c0 ((Functor.const J).obj (\u22a4_ C)) j = terminal.from (\u22a4_ C)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\u271d\nJ : Type u_1\ninst\u271d\u00b2 : Category.{u_3, u_1} J\nC : Type u_2\ninst\u271d\u00b9 : Category.{u_4, u_2} C\ninst\u271d : HasInitial C\nj : J\n\u22a2 colimit.\u03b9 ((Functor.const J).obj (\u22a5_ C)) j \u226b colimitConstInitial.hom = initial.to (\u22a5_ C)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : InitialMonoClass C\nI X : C\nhI : IsInitial I\nf : I \u27f6 X\n\u22a2 Mono f\n[PROOFSTEP]\nrw [hI.hom_ext f (hI.to X)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : InitialMonoClass C\nI X : C\nhI : IsInitial I\nf : I \u27f6 X\n\u22a2 Mono (to hI X)\n[PROOFSTEP]\napply InitialMonoClass.isInitial_mono_from\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nI : C\nhI : IsInitial I\nh : \u2200 (X : C), Mono (IsInitial.to hI X)\nI' X : C\nhI' : IsInitial I'\n\u22a2 Mono (IsInitial.to hI' X)\n[PROOFSTEP]\nrw [hI'.hom_ext (hI'.to X) ((hI'.uniqueUpToIso hI).hom \u226b hI.to X)]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nI : C\nhI : IsInitial I\nh : \u2200 (X : C), Mono (IsInitial.to hI X)\nI' X : C\nhI' : IsInitial I'\n\u22a2 Mono ((IsInitial.uniqueUpToIso hI' hI).hom \u226b IsInitial.to hI X)\n[PROOFSTEP]\napply mono_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\nj j' : J\nk : j \u27f6 j'\n\u22a2 ((Functor.const J).obj (F.obj X)).map k \u226b (fun j => F.map (IsInitial.to tX j)) j' =\n (fun j => F.map (IsInitial.to tX j)) j \u226b F.map k\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\nj j' : J\nk : j \u27f6 j'\n\u22a2 \ud835\udfd9 (F.obj X) \u226b F.map (IsInitial.to tX j') = F.map (IsInitial.to tX j) \u226b F.map k\n[PROOFSTEP]\nrw [\u2190 F.map_comp, Category.id_comp, tX.hom_ext (tX.to j \u226b k) (tX.to j')]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => NatTrans.app s.\u03c0 X) s\n[PROOFSTEP]\nconv_lhs => dsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n| m\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n| m\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n| m\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = (fun s => NatTrans.app s.\u03c0 X) s\n[PROOFSTEP]\nsimp_rw [\u2190 w X, coneOfDiagramInitial_\u03c0_app, tX.hom_ext (tX.to X) (\ud835\udfd9 _)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = m \u226b F.map (\ud835\udfd9 X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsInitial X\nF : J \u2964 C\ns : Cone F\nm : s.pt \u27f6 (coneOfDiagramInitial tX F).pt\nw : \u2200 (j : J), m \u226b NatTrans.app (coneOfDiagramInitial tX F).\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m = m \u226b F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsTerminal X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 \u2200 \u2983X_1 Y : J\u2984 (f : X_1 \u27f6 Y),\n ((Functor.const J).obj (F.obj X)).map f \u226b (fun i => inv (F.map (IsTerminal.from hX i))) Y =\n (fun i => inv (F.map (IsTerminal.from hX i))) X_1 \u226b F.map f\n[PROOFSTEP]\nintro i j f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsTerminal X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\ni j : J\nf : i \u27f6 j\n\u22a2 ((Functor.const J).obj (F.obj X)).map f \u226b (fun i => inv (F.map (IsTerminal.from hX i))) j =\n (fun i => inv (F.map (IsTerminal.from hX i))) i \u226b F.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsTerminal X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\ni j : J\nf : i \u27f6 j\n\u22a2 \ud835\udfd9 (F.obj X) \u226b inv (F.map (IsTerminal.from hX j)) = inv (F.map (IsTerminal.from hX i)) \u226b F.map f\n[PROOFSTEP]\nsimp only [IsIso.eq_inv_comp, IsIso.comp_inv_eq, Category.id_comp, \u2190 F.map_comp, hX.hom_ext (hX.from i) (f \u226b hX.from j)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\nj j' : J\nk : j \u27f6 j'\n\u22a2 F.map k \u226b (fun j => F.map (IsTerminal.from tX j)) j' =\n (fun j => F.map (IsTerminal.from tX j)) j \u226b ((Functor.const J).obj (F.obj X)).map k\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\nj j' : J\nk : j \u27f6 j'\n\u22a2 F.map k \u226b F.map (IsTerminal.from tX j') = F.map (IsTerminal.from tX j) \u226b \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, Category.comp_id, tX.hom_ext (k \u226b tX.from j') (tX.from j)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s => NatTrans.app s.\u03b9 X) s\n[PROOFSTEP]\nconv_rhs =>\n dsimp\n -- Porting note: why do I need this much firepower?\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n| (fun s => NatTrans.app s.\u03b9 X) s\n[PROOFSTEP]\ndsimp\n -- Porting note: why do I need this much firepower?\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n| (fun s => NatTrans.app s.\u03b9 X) s\n[PROOFSTEP]\ndsimp\n -- Porting note: why do I need this much firepower?\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n| (fun s => NatTrans.app s.\u03b9 X) s\n[PROOFSTEP]\ndsimp\n -- Porting note: why do I need this much firepower?\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = NatTrans.app s.\u03b9 X\n[PROOFSTEP]\nrw [\u2190 w X, coconeOfDiagramTerminal_\u03b9_app, tX.hom_ext (tX.from X) (\ud835\udfd9 _)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d : Category.{v, u} J\nX : J\ntX : IsTerminal X\nF : J \u2964 C\ns : Cocone F\nm : (coconeOfDiagramTerminal tX F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (coconeOfDiagramTerminal tX F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = F.map (\ud835\udfd9 X) \u226b m\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsInitial X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 \u2200 \u2983X_1 Y : J\u2984 (f : X_1 \u27f6 Y),\n F.map f \u226b (fun i => inv (F.map (IsInitial.to hX i))) Y =\n (fun i => inv (F.map (IsInitial.to hX i))) X_1 \u226b ((Functor.const J).obj (F.obj X)).map f\n[PROOFSTEP]\nintro i j f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsInitial X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\ni j : J\nf : i \u27f6 j\n\u22a2 F.map f \u226b (fun i => inv (F.map (IsInitial.to hX i))) j =\n (fun i => inv (F.map (IsInitial.to hX i))) i \u226b ((Functor.const J).obj (F.obj X)).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nX : J\nhX : IsInitial X\nF : J \u2964 C\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\ni j : J\nf : i \u27f6 j\n\u22a2 F.map f \u226b inv (F.map (IsInitial.to hX j)) = inv (F.map (IsInitial.to hX i)) \u226b \ud835\udfd9 (F.obj X)\n[PROOFSTEP]\nsimp only [IsIso.eq_inv_comp, IsIso.comp_inv_eq, Category.comp_id, \u2190 F.map_comp, hX.hom_ext (hX.to i \u226b f) (hX.to j)]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d : HasLimit F\n\u22a2 limit.\u03c0 F j \u226b limit.lift F (coneOfDiagramInitial I F) = \ud835\udfd9 (limit F)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d : HasLimit F\nj\u271d : J\n\u22a2 (limit.\u03c0 F j \u226b limit.lift F (coneOfDiagramInitial I F)) \u226b limit.\u03c0 F j\u271d = \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d : HasLimit F\n\u22a2 limit.lift F (coneOfDiagramInitial I F) \u226b limit.\u03c0 F j = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 limit.\u03c0 F j \u226b limit.lift F (coneOfDiagramTerminal I F) = \ud835\udfd9 (limit F)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\nj\u271d : J\n\u22a2 (limit.\u03c0 F j \u226b limit.lift F (coneOfDiagramTerminal I F)) \u226b limit.\u03c0 F j\u271d = \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 limit.lift F (coneOfDiagramTerminal I F) \u226b limit.\u03c0 F j = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d : HasColimit F\n\u22a2 colimit.\u03b9 F j \u226b colimit.desc F (coconeOfDiagramTerminal I F) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d : HasColimit F\n\u22a2 colimit.desc F (coconeOfDiagramTerminal I F) \u226b colimit.\u03b9 F j = \ud835\udfd9 (colimit F)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b9 : Category.{v, u} J\nj : J\nI : IsTerminal j\nF : J \u2964 C\ninst\u271d : HasColimit F\nj\u271d : J\n\u22a2 colimit.\u03b9 F j\u271d \u226b colimit.desc F (coconeOfDiagramTerminal I F) \u226b colimit.\u03b9 F j = colimit.\u03b9 F j\u271d \u226b \ud835\udfd9 (colimit F)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 colimit.\u03b9 F j \u226b colimit.desc F (coconeOfDiagramInitial I F) = \ud835\udfd9 (F.obj j) \u2227\n colimit.desc F (coconeOfDiagramInitial I F) \u226b colimit.\u03b9 F j = \ud835\udfd9 (colimit F)\n[PROOFSTEP]\nrefine \u27e8?_, by ext; simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 colimit.desc F (coconeOfDiagramInitial I F) \u226b colimit.\u03b9 F j = \ud835\udfd9 (colimit F)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\nj\u271d : J\n\u22a2 colimit.\u03b9 F j\u271d \u226b colimit.desc F (coconeOfDiagramInitial I F) \u226b colimit.\u03b9 F j = colimit.\u03b9 F j\u271d \u226b \ud835\udfd9 (colimit F)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 colimit.\u03b9 F j \u226b colimit.desc F (coconeOfDiagramInitial I F) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 colimit.\u03b9 F j \u226b colimit.desc F (coconeOfDiagramInitial I F) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\nsimp only [colimit.\u03b9_desc, coconeOfDiagramInitial_pt, coconeOfDiagramInitial_\u03b9_app, Functor.const_obj_obj,\n IsInitial.to_self, Functor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 inv (\ud835\udfd9 (F.obj j)) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\ndsimp [inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 Classical.choose (_ : \u2203 inv, \ud835\udfd9 (F.obj j) \u226b inv = \ud835\udfd9 (F.obj j) \u2227 inv \u226b \ud835\udfd9 (F.obj j) = \ud835\udfd9 (F.obj j)) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\nsimp only [Category.id_comp, Category.comp_id, and_self]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : Type u\ninst\u271d\u00b2 : Category.{v, u} J\nj : J\nI : IsInitial j\nF : J \u2964 C\ninst\u271d\u00b9 : HasColimit F\ninst\u271d : \u2200 (i j : J) (f : i \u27f6 j), IsIso (F.map f)\n\u22a2 Classical.choose (_ : \u2203 x, (fun x => x = \ud835\udfd9 (F.obj j)) x) = \ud835\udfd9 (F.obj j)\n[PROOFSTEP]\napply @Classical.choose_spec _ (fun x => x = \ud835\udfd9 F.obj j) _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "llama_tokens": 18448, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.26294014911576163}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP : PresheafOfModules R\nX : C\u1d52\u1d56\n\u22a2 map P (\ud835\udfd9 X) = id'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP : PresheafOfModules R\nX : C\u1d52\u1d56\nx\u271d : \u2191(obj P X)\n\u22a2 \u2191(map P (\ud835\udfd9 X)) x\u271d = \u2191id' x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP : PresheafOfModules R\nX Y Z : C\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 map P (f \u226b g) = comp (map P g) (map P f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP : PresheafOfModules R\nX Y Z : C\u1d52\u1d56\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx\u271d : \u2191(obj P X)\n\u22a2 \u2191(map P (f \u226b g)) x\u271d = \u2191(comp (map P g) (map P f)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR\u271d : C\u1d52\u1d56 \u2964 RingCat\nP Q R : PresheafOfModules R\u271d\nf : Hom P Q\ng : Hom Q R\nx\u271d\u00b2 : C\u1d52\u1d56\nx\u271d\u00b9 : \u2191(R\u271d.obj x\u271d\u00b2)\nx\u271d : \u2191(P.presheaf.obj x\u271d\u00b2)\n\u22a2 \u2191(NatTrans.app (f.hom \u226b g.hom) x\u271d\u00b2) (x\u271d\u00b9 \u2022 x\u271d) = x\u271d\u00b9 \u2022 \u2191(NatTrans.app (f.hom \u226b g.hom) x\u271d\u00b2) x\u271d\n[PROOFSTEP]\nsimp [Hom.map_smul]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP Q : PresheafOfModules R\nf g : P \u27f6 Q\nw : \u2200 (X : C\u1d52\u1d56), app f X = app g X\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP Q : PresheafOfModules R\ng : P \u27f6 Q\nhom\u271d : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)), \u2191(NatTrans.app hom\u271d X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d X) x\nw : \u2200 (X : C\u1d52\u1d56), app { hom := hom\u271d, map_smul := map_smul\u271d } X = app g X\n\u22a2 { hom := hom\u271d, map_smul := map_smul\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP Q : PresheafOfModules R\nhom\u271d\u00b9 : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d\u00b9 :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)),\n \u2191(NatTrans.app hom\u271d\u00b9 X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d\u00b9 X) x\nhom\u271d : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)), \u2191(NatTrans.app hom\u271d X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d X) x\nw : \u2200 (X : C\u1d52\u1d56), app { hom := hom\u271d\u00b9, map_smul := map_smul\u271d\u00b9 } X = app { hom := hom\u271d, map_smul := map_smul\u271d } X\n\u22a2 { hom := hom\u271d\u00b9, map_smul := map_smul\u271d\u00b9 } = { hom := hom\u271d, map_smul := map_smul\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_hom\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP Q : PresheafOfModules R\nhom\u271d\u00b9 : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d\u00b9 :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)),\n \u2191(NatTrans.app hom\u271d\u00b9 X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d\u00b9 X) x\nhom\u271d : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)), \u2191(NatTrans.app hom\u271d X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d X) x\nw : \u2200 (X : C\u1d52\u1d56), app { hom := hom\u271d\u00b9, map_smul := map_smul\u271d\u00b9 } X = app { hom := hom\u271d, map_smul := map_smul\u271d } X\n\u22a2 hom\u271d\u00b9 = hom\u271d\n[PROOFSTEP]\next X x\n[GOAL]\ncase mk.mk.e_hom.w.h.w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nR : C\u1d52\u1d56 \u2964 RingCat\nP Q : PresheafOfModules R\nhom\u271d\u00b9 : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d\u00b9 :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)),\n \u2191(NatTrans.app hom\u271d\u00b9 X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d\u00b9 X) x\nhom\u271d : P.presheaf \u27f6 Q.presheaf\nmap_smul\u271d :\n \u2200 (X : C\u1d52\u1d56) (r : \u2191(R.obj X)) (x : \u2191(P.presheaf.obj X)), \u2191(NatTrans.app hom\u271d X) (r \u2022 x) = r \u2022 \u2191(NatTrans.app hom\u271d X) x\nw : \u2200 (X : C\u1d52\u1d56), app { hom := hom\u271d\u00b9, map_smul := map_smul\u271d\u00b9 } X = app { hom := hom\u271d, map_smul := map_smul\u271d } X\nX : C\u1d52\u1d56\nx : \u2191(P.presheaf.obj X)\n\u22a2 \u2191(NatTrans.app hom\u271d\u00b9 X) x = \u2191(NatTrans.app hom\u271d X) x\n[PROOFSTEP]\nexact LinearMap.congr_fun (w X) x\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.Presheaf", "llama_tokens": 2184, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.4493926344647596, "lm_q1q2_score": 0.2629401491157616}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\na : R\nh : a \u2208 closure s\nb : R\nih : \u2203 L, (\u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2227 List.sum (List.map List.prod L) = b\nL1 : List (List R)\nh1 : \u2200 (l : List R), l \u2208 L1 \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nx\u271d : AddGroup.InClosure (Monoid.Closure s) (List.sum (List.map List.prod L1))\n\u22a2 List.sum (List.map List.prod (List.map (List.cons (-1)) L1)) = -List.sum (List.map List.prod L1)\n[PROOFSTEP]\nsimp only [List.map_map, (\u00b7 \u2218 \u00b7), List.prod_cons, neg_one_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\na : R\nh : a \u2208 closure s\nb : R\nih : \u2203 L, (\u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2227 List.sum (List.map List.prod L) = b\nL1 : List (List R)\nh1 : \u2200 (l : List R), l \u2208 L1 \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nx\u271d : AddGroup.InClosure (Monoid.Closure s) (List.sum (List.map List.prod L1))\n\u22a2 List.sum (List.map (fun x => -List.prod x) L1) = -List.sum (List.map List.prod L1)\n[PROOFSTEP]\nrefine' List.recOn L1 neg_zero.symm fun hd tl ih \u21a6 _\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\na : R\nh : a \u2208 closure s\nb : R\nih\u271d : \u2203 L, (\u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2227 List.sum (List.map List.prod L) = b\nL1 : List (List R)\nh1 : \u2200 (l : List R), l \u2208 L1 \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nx\u271d : AddGroup.InClosure (Monoid.Closure s) (List.sum (List.map List.prod L1))\nhd : List R\ntl : List (List R)\nih : List.sum (List.map (fun x => -List.prod x) tl) = -List.sum (List.map List.prod tl)\n\u22a2 List.sum (List.map (fun x => -List.prod x) (hd :: tl)) = -List.sum (List.map List.prod (hd :: tl))\n[PROOFSTEP]\nrw [List.map_cons, List.sum_cons, ih, List.map_cons, List.sum_cons, neg_add]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\na : R\nh : a \u2208 closure s\nr1 r2 : R\nih1 : \u2203 L, (\u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2227 List.sum (List.map List.prod L) = r1\nih2 : \u2203 L, (\u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2227 List.sum (List.map List.prod L) = r2\nL1 : List (List R)\nh1 : \u2200 (l : List R), l \u2208 L1 \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nL2 : List (List R)\nh2 : \u2200 (l : List R), l \u2208 L2 \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nx\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) (List.sum (List.map List.prod L1))\nx\u271d : AddGroup.InClosure (Monoid.Closure s) (List.sum (List.map List.prod L2))\n\u22a2 List.sum (List.map List.prod (L1 ++ L2)) = List.sum (List.map List.prod L1) + List.sum (List.map List.prod L2)\n[PROOFSTEP]\nrw [List.map_append, List.sum_append]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nx : R\nhx : x \u2208 closure s\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\n\u22a2 C x\n[PROOFSTEP]\nhave h0 : C 0 := add_neg_self (1 : R) \u25b8 ha h1 hneg1\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nx : R\nhx : x \u2208 closure s\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\n\u22a2 C x\n[PROOFSTEP]\nrcases exists_list_of_mem_closure hx with \u27e8L, HL, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhx : List.sum (List.map List.prod L) \u2208 closure s\n\u22a2 C (List.sum (List.map List.prod L))\n[PROOFSTEP]\nclear hx\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\n\u22a2 C (List.sum (List.map List.prod L))\n[PROOFSTEP]\ninduction' L with hd tl ih\n[GOAL]\ncase intro.intro.nil\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nHL : \u2200 (l : List R), l \u2208 [] \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\n\u22a2 C (List.sum (List.map List.prod []))\n[PROOFSTEP]\nexact h0\n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : \u2200 (l : List R), l \u2208 hd :: tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\n\u22a2 C (List.sum (List.map List.prod (hd :: tl)))\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL \n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : (\u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1) \u2227 \u2200 (x : List R), x \u2208 tl \u2192 \u2200 (x_1 : R), x_1 \u2208 x \u2192 x_1 \u2208 s \u2228 x_1 = -1\n\u22a2 C (List.sum (List.map List.prod (hd :: tl)))\n[PROOFSTEP]\nsuffices C (List.prod hd) by\n rw [List.map_cons, List.sum_cons]\n exact ha this (ih HL.2)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : (\u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1) \u2227 \u2200 (x : List R), x \u2208 tl \u2192 \u2200 (x_1 : R), x_1 \u2208 x \u2192 x_1 \u2208 s \u2228 x_1 = -1\nthis : C (List.prod hd)\n\u22a2 C (List.sum (List.map List.prod (hd :: tl)))\n[PROOFSTEP]\nrw [List.map_cons, List.sum_cons]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : (\u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1) \u2227 \u2200 (x : List R), x \u2208 tl \u2192 \u2200 (x_1 : R), x_1 \u2208 x \u2192 x_1 \u2208 s \u2228 x_1 = -1\nthis : C (List.prod hd)\n\u22a2 C (List.prod hd + List.sum (List.map List.prod tl))\n[PROOFSTEP]\nexact ha this (ih HL.2)\n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : (\u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1) \u2227 \u2200 (x : List R), x \u2208 tl \u2192 \u2200 (x_1 : R), x_1 \u2208 x \u2192 x_1 \u2208 s \u2228 x_1 = -1\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nreplace HL := HL.1\n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\ntl : List (List R)\nih : (\u2200 (l : List R), l \u2208 tl \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1) \u2192 C (List.sum (List.map List.prod tl))\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nclear ih tl\n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nsuffices \u2203 L, (\u2200 x \u2208 L, x \u2208 s) \u2227 (List.prod hd = List.prod L \u2228 List.prod hd = -List.prod L)\n by\n rcases this with \u27e8L, HL', HP | HP\u27e9 <;> rw [HP] <;> clear HP HL\n \u00b7 induction' L with hd tl ih\n \u00b7 exact h1\n rw [List.forall_mem_cons] at HL' \n rw [List.prod_cons]\n exact hs _ HL'.1 _ (ih HL'.2)\n \u00b7 induction' L with hd tl ih\n \u00b7 exact hneg1\n rw [List.prod_cons, neg_mul_eq_mul_neg]\n rw [List.forall_mem_cons] at HL' \n exact hs _ HL'.1 _ (ih HL'.2)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nthis : \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod hd = List.prod L \u2228 List.prod hd = -List.prod L)\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nrcases this with \u27e8L, HL', HP | HP\u27e9\n[GOAL]\ncase intro.intro.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod hd = List.prod L\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nrw [HP]\n[GOAL]\ncase intro.intro.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod hd = -List.prod L\n\u22a2 C (List.prod hd)\n[PROOFSTEP]\nrw [HP]\n[GOAL]\ncase intro.intro.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod hd = List.prod L\n\u22a2 C (List.prod L)\n[PROOFSTEP]\nclear HP HL\n[GOAL]\ncase intro.intro.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod hd = -List.prod L\n\u22a2 C (-List.prod L)\n[PROOFSTEP]\nclear HP HL\n[GOAL]\ncase intro.intro.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd L : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\n\u22a2 C (List.prod L)\n[PROOFSTEP]\ninduction' L with hd tl ih\n[GOAL]\ncase intro.intro.inl.nil\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHL' : \u2200 (x : R), x \u2208 [] \u2192 x \u2208 s\n\u22a2 C (List.prod [])\n[PROOFSTEP]\nexact h1\n[GOAL]\ncase intro.intro.inl.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (List.prod tl)\nHL' : \u2200 (x : R), x \u2208 hd :: tl \u2192 x \u2208 s\n\u22a2 C (List.prod (hd :: tl))\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL' \n[GOAL]\ncase intro.intro.inl.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (List.prod tl)\nHL' : hd \u2208 s \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s\n\u22a2 C (List.prod (hd :: tl))\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\ncase intro.intro.inl.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (List.prod tl)\nHL' : hd \u2208 s \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s\n\u22a2 C (hd * List.prod tl)\n[PROOFSTEP]\nexact hs _ HL'.1 _ (ih HL'.2)\n[GOAL]\ncase intro.intro.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd L : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\n\u22a2 C (-List.prod L)\n[PROOFSTEP]\ninduction' L with hd tl ih\n[GOAL]\ncase intro.intro.inr.nil\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHL' : \u2200 (x : R), x \u2208 [] \u2192 x \u2208 s\n\u22a2 C (-List.prod [])\n[PROOFSTEP]\nexact hneg1\n[GOAL]\ncase intro.intro.inr.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (-List.prod tl)\nHL' : \u2200 (x : R), x \u2208 hd :: tl \u2192 x \u2208 s\n\u22a2 C (-List.prod (hd :: tl))\n[PROOFSTEP]\nrw [List.prod_cons, neg_mul_eq_mul_neg]\n[GOAL]\ncase intro.intro.inr.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (-List.prod tl)\nHL' : \u2200 (x : R), x \u2208 hd :: tl \u2192 x \u2208 s\n\u22a2 C (hd * -List.prod tl)\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL' \n[GOAL]\ncase intro.intro.inr.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d L : List R\nHL'\u271d : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nhd : R\ntl : List R\nih : (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s) \u2192 C (-List.prod tl)\nHL' : hd \u2208 s \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s\n\u22a2 C (hd * -List.prod tl)\n[PROOFSTEP]\nexact hs _ HL'.1 _ (ih HL'.2)\n[GOAL]\ncase intro.intro.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod hd = List.prod L \u2228 List.prod hd = -List.prod L)\n[PROOFSTEP]\ninduction' hd with hd tl ih\n[GOAL]\ncase intro.intro.cons.nil\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd \u2192 x \u2208 s \u2228 x = -1\nHL : \u2200 (x : R), x \u2208 [] \u2192 x \u2208 s \u2228 x = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod [] = List.prod L \u2228 List.prod [] = -List.prod L)\n[PROOFSTEP]\nexact \u27e8[], List.forall_mem_nil _, Or.inl rfl\u27e9\n[GOAL]\ncase intro.intro.cons.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : \u2200 (x : R), x \u2208 hd :: tl \u2192 x \u2208 s \u2228 x = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nrw [List.forall_mem_cons] at HL \n[GOAL]\ncase intro.intro.cons.cons\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nrcases ih HL.2 with \u27e8L, HL', HP | HP\u27e9\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = List.prod L\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\ncases' HL.1 with hhd hhd\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = -List.prod L\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\ncases' HL.1 with hhd hhd\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inl.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = List.prod L\nhhd : hd \u2208 s\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nexact \u27e8hd :: L, List.forall_mem_cons.2 \u27e8hhd, HL'\u27e9, Or.inl <| by rw [List.prod_cons, List.prod_cons, HP]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = List.prod L\nhhd : hd \u2208 s\n\u22a2 List.prod (hd :: tl) = List.prod (hd :: L)\n[PROOFSTEP]\nrw [List.prod_cons, List.prod_cons, HP]\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inl.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = List.prod L\nhhd : hd = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nexact \u27e8L, HL', Or.inr <| by rw [List.prod_cons, hhd, neg_one_mul, HP]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = List.prod L\nhhd : hd = -1\n\u22a2 List.prod (hd :: tl) = -List.prod L\n[PROOFSTEP]\nrw [List.prod_cons, hhd, neg_one_mul, HP]\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inr.inl\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = -List.prod L\nhhd : hd \u2208 s\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nexact\n \u27e8hd :: L, List.forall_mem_cons.2 \u27e8hhd, HL'\u27e9, Or.inr <| by rw [List.prod_cons, List.prod_cons, HP, neg_mul_eq_mul_neg]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = -List.prod L\nhhd : hd \u2208 s\n\u22a2 List.prod (hd :: tl) = -List.prod (hd :: L)\n[PROOFSTEP]\nrw [List.prod_cons, List.prod_cons, HP, neg_mul_eq_mul_neg]\n[GOAL]\ncase intro.intro.cons.cons.intro.intro.inr.inr\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = -List.prod L\nhhd : hd = -1\n\u22a2 \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod (hd :: tl) = List.prod L \u2228 List.prod (hd :: tl) = -List.prod L)\n[PROOFSTEP]\nexact \u27e8L, HL', Or.inl <| by rw [List.prod_cons, hhd, HP, neg_one_mul, neg_neg]\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : Ring R\ncR : Type u\ninst\u271d : CommRing cR\ns : Set R\nC : R \u2192 Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : \u2200 (z : R), z \u2208 s \u2192 \u2200 (n : R), C n \u2192 C (z * n)\nha : \u2200 {x y : R}, C x \u2192 C y \u2192 C (x + y)\nh0 : C 0\nL\u271d : List (List R)\nHL\u271d\u00b9 : \u2200 (l : List R), l \u2208 L\u271d \u2192 \u2200 (x : R), x \u2208 l \u2192 x \u2208 s \u2228 x = -1\nhd\u271d : List R\nHL\u271d : \u2200 (x : R), x \u2208 hd\u271d \u2192 x \u2208 s \u2228 x = -1\nhd : R\ntl : List R\nih :\n (\u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1) \u2192\n \u2203 L, (\u2200 (x : R), x \u2208 L \u2192 x \u2208 s) \u2227 (List.prod tl = List.prod L \u2228 List.prod tl = -List.prod L)\nHL : (hd \u2208 s \u2228 hd = -1) \u2227 \u2200 (x : R), x \u2208 tl \u2192 x \u2208 s \u2228 x = -1\nL : List R\nHL' : \u2200 (x : R), x \u2208 L \u2192 x \u2208 s\nHP : List.prod tl = -List.prod L\nhhd : hd = -1\n\u22a2 List.prod (hd :: tl) = List.prod L\n[PROOFSTEP]\nrw [List.prod_cons, hhd, HP, neg_one_mul, neg_neg]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\n\u22a2 \u2191f '' closure s = closure (\u2191f '' s)\n[PROOFSTEP]\nrefine'\n le_antisymm _ (closure_subset (RingHom.isSubring_image _ closure.isSubring) <| Set.image_subset _ subset_closure)\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\n\u22a2 \u2191f '' closure s \u2264 closure (\u2191f '' s)\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2191f x \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\napply AddGroup.InClosure.recOn (motive := fun {x} _ \u21a6 f x \u2208 closure (f '' s)) hx _\n[GOAL]\ncase intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2191f 0 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nintros\n[GOAL]\ncase intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2200 {a : R}, AddGroup.InClosure (Monoid.Closure s) a \u2192 \u2191f a \u2208 closure (\u2191f '' s) \u2192 \u2191f (-a) \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nintros\n[GOAL]\ncase intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2200 {a b : R},\n AddGroup.InClosure (Monoid.Closure s) a \u2192\n AddGroup.InClosure (Monoid.Closure s) b \u2192\n \u2191f a \u2208 closure (\u2191f '' s) \u2192 \u2191f b \u2208 closure (\u2191f '' s) \u2192 \u2191f (a + b) \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2200 {a : R} (a_1 : a \u2208 Monoid.Closure s),\n (fun {x} x_1 => \u2191f x \u2208 closure (\u2191f '' s)) (_ : AddGroup.InClosure (Monoid.Closure s) a)\n[PROOFSTEP]\nintros\n[GOAL]\ncase intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2191f 0 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nrw [f.map_zero]\n[GOAL]\ncase intro.intro.zero\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 0 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\napply closure.isSubring.zero_mem\n[GOAL]\ncase intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : \u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f (-a\u271d\u00b9) \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nrw [f.map_neg]\n[GOAL]\ncase intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : \u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n\u22a2 -\u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\napply closure.isSubring.neg_mem\n[GOAL]\ncase intro.intro.neg\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b9\na_ih\u271d : \u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : \u2191f a\u271d\u00b2 \u2208 closure (\u2191f '' s)\na_ih\u271d : \u2191f b\u271d \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f (a\u271d\u00b2 + b\u271d) \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nrw [f.map_add]\n[GOAL]\ncase intro.intro.add\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : \u2191f a\u271d\u00b2 \u2208 closure (\u2191f '' s)\na_ih\u271d : \u2191f b\u271d \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f a\u271d\u00b2 + \u2191f b\u271d \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\napply closure.isSubring.add_mem\n[GOAL]\ncase intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : \u2191f a\u271d\u00b2 \u2208 closure (\u2191f '' s)\na_ih\u271d : \u2191f b\u271d \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f a\u271d\u00b2 \u2208 closure (\u2191f '' s)\ncase intro.intro.add.a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : R\na\u271d\u00b9 : AddGroup.InClosure (Monoid.Closure s) a\u271d\u00b2\na\u271d : AddGroup.InClosure (Monoid.Closure s) b\u271d\na_ih\u271d\u00b9 : \u2191f a\u271d\u00b2 \u2208 closure (\u2191f '' s)\na_ih\u271d : \u2191f b\u271d \u2208 closure (\u2191f '' s)\n\u22a2 \u2191f b\u271d \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\nassumption'\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 \u2191f a\u271d\u00b9 \u2208 closure (\u2191f '' s)\n[PROOFSTEP]\napply AddGroup.mem_closure\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 \u2191f a\u271d\u00b9 \u2208 Monoid.Closure (\u2191f '' s)\n[PROOFSTEP]\nrw [\u2190 Monoid.image_closure f.to_isMonoidHom]\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 \u2191f a\u271d\u00b9 \u2208 \u2191f '' Monoid.Closure s\n[PROOFSTEP]\napply Set.mem_image_of_mem\n[GOAL]\ncase a.h\nR : Type u\ninst\u271d\u00b2 : Ring R\ncR : Type u\ninst\u271d\u00b9 : CommRing cR\ns\u271d : Set R\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\ns : Set R\nx : R\nhx : x \u2208 closure s\na\u271d\u00b9 : R\na\u271d : a\u271d\u00b9 \u2208 Monoid.Closure s\n\u22a2 a\u271d\u00b9 \u2208 Monoid.Closure s\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Deprecated.Subring", "llama_tokens": 18978, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.26277656757891465}} {"text": "[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift #{ x // IsAlgebraic R x } \u2264 lift #R[X] * \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 mk_uLift, \u2190 mk_uLift]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 #(ULift { x // IsAlgebraic R x }) \u2264 #(ULift R[X]) * \u2135\u2080\n[PROOFSTEP]\nchoose g hg\u2081 hg\u2082 using fun x : {x : A | IsAlgebraic R x} => x.coe_prop\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\n\u22a2 #(ULift { x // IsAlgebraic R x }) \u2264 #(ULift R[X]) * \u2135\u2080\n[PROOFSTEP]\nrefine' lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => _\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nf : R[X]\n\u22a2 lift #\u2191(g \u207b\u00b9' {f}) \u2264 \u2135\u2080\n[PROOFSTEP]\nrw [lift_le_aleph0, le_aleph0_iff_set_countable]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nf : R[X]\n\u22a2 Set.Countable (g \u207b\u00b9' {f})\n[PROOFSTEP]\nsuffices : MapsTo (\u2191) (g \u207b\u00b9' { f }) (f.rootSet A)\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nf : R[X]\nthis : MapsTo Subtype.val (g \u207b\u00b9' {f}) (rootSet f A)\n\u22a2 Set.Countable (g \u207b\u00b9' {f})\ncase this\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nf : R[X]\n\u22a2 MapsTo Subtype.val (g \u207b\u00b9' {f}) (rootSet f A)\n[PROOFSTEP]\nexact this.countable_of_injOn (Subtype.coe_injective.injOn _) (f.rootSet_finite A).countable\n[GOAL]\ncase this\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nf : R[X]\n\u22a2 MapsTo Subtype.val (g \u207b\u00b9' {f}) (rootSet f A)\n[PROOFSTEP]\nrintro x (rfl : g x = f)\n[GOAL]\ncase this\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\ng : \u2191{x | IsAlgebraic R x} \u2192 R[X]\nhg\u2081 : \u2200 (x : \u2191{x | IsAlgebraic R x}), g x \u2260 0\nhg\u2082 : \u2200 (x : \u2191{x | IsAlgebraic R x}), \u2191(aeval \u2191x) (g x) = 0\nx : \u2191{x | IsAlgebraic R x}\n\u22a2 \u2191x \u2208 rootSet (g x) A\n[PROOFSTEP]\nexact mem_rootSet.2 \u27e8hg\u2081 x, hg\u2082 x\u27e9\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift (max #R \u2135\u2080) * \u2135\u2080 \u2264 max (lift #R) \u2135\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : Countable R\n\u22a2 Set.Countable {x | IsAlgebraic R x}\n[PROOFSTEP]\nrw [\u2190 le_aleph0_iff_set_countable, \u2190 lift_le]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : Countable R\n\u22a2 lift #\u2191{x | IsAlgebraic R x} \u2264 lift \u2135\u2080\n[PROOFSTEP]\napply (cardinal_mk_lift_le_max R A).trans\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : NoZeroSMulDivisors R A\ninst\u271d : Countable R\n\u22a2 max (lift #R) \u2135\u2080 \u2264 lift \u2135\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\nR A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 #{ x // IsAlgebraic R x } \u2264 #R[X] * \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 lift_id #_, \u2190 lift_id #(R[X])]\n[GOAL]\nR A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift #{ x // IsAlgebraic R x } \u2264 lift #R[X] * \u2135\u2080\n[PROOFSTEP]\nexact cardinal_mk_lift_le_mul R A\n[GOAL]\nR A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 #{ x // IsAlgebraic R x } \u2264 max #R \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 lift_id #_, \u2190 lift_id #R]\n[GOAL]\nR A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift #{ x // IsAlgebraic R x } \u2264 max (lift #R) \u2135\u2080\n[PROOFSTEP]\nexact cardinal_mk_lift_le_max R A\n", "meta": {"mathlib_filename": "Mathlib.Algebra.AlgebraicCard", "llama_tokens": 2758, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.2623675006208503}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\ny : \u03b1\nys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts r (y :: ys) f).fst = y :: ys ++ ts\n[PROOFSTEP]\nsimp [permutationsAux2, permutationsAux2_fst t _ _ ys]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\ny : \u03b1\nys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts r (y :: ys) f).snd =\n f (t :: y :: ys ++ ts) :: (permutationsAux2 t ts r ys fun x => f (y :: x)).snd\n[PROOFSTEP]\nsimp [permutationsAux2, permutationsAux2_fst t _ _ ys]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts [] ys f).snd ++ r = (permutationsAux2 t ts r ys f).snd\n[PROOFSTEP]\ninduction ys generalizing f\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts [] [] f).snd ++ r = (permutationsAux2 t ts r [] f).snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (f : List \u03b1 \u2192 \u03b2), (permutationsAux2 t ts [] tail\u271d f).snd ++ r = (permutationsAux2 t ts r tail\u271d f).snd\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts [] (head\u271d :: tail\u271d) f).snd ++ r = (permutationsAux2 t ts r (head\u271d :: tail\u271d) f).snd\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t [] r ys fun x => f (x ++ ts)).snd = (permutationsAux2 t ts r ys f).snd\n[PROOFSTEP]\ninduction' ys with ys_hd _ ys_ih generalizing f\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d f : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t [] r [] fun x => f (x ++ ts)).snd = (permutationsAux2 t ts r [] f).snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2), (permutationsAux2 t [] r tail\u271d fun x => f (x ++ ts)).snd = (permutationsAux2 t ts r tail\u271d f).snd\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t [] r (ys_hd :: tail\u271d) fun x => f (x ++ ts)).snd = (permutationsAux2 t ts r (ys_hd :: tail\u271d) f).snd\n[PROOFSTEP]\nsimp [ys_ih fun xs => f (ys_hd :: xs)]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts ys : List \u03b1\nr : List \u03b2\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r ys f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g ys) f').snd\n[PROOFSTEP]\ninduction' ys with ys_hd _ ys_ih generalizing f f'\n[GOAL]\ncase nil\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r [] f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g []) f').snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 map g' (permutationsAux2 t ts r (ys_hd :: tail\u271d) f).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g (ys_hd :: tail\u271d)) f').snd\n[PROOFSTEP]\nsimp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).fst)) =\n f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n map g' (permutationsAux2 t ts r tail\u271d fun x => f (ys_hd :: x)).snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).snd\n[PROOFSTEP]\nrw [ys_ih, permutationsAux2_fst]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts)) \u2227\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) ?cons.f').snd =\n (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) fun x => f' (g ys_hd :: x)).snd\ncase cons.f'\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 List \u03b1' \u2192 \u03b2'\ncase cons.f'\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 List \u03b1' \u2192 \u03b2'\ncase cons.H\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 \u2200 (a : List \u03b1), g' (f (ys_hd :: a)) = ?cons.f' (map g a)\n[PROOFSTEP]\nrefine' \u27e8_, rfl\u27e9\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (g t :: g ys_hd :: (map g tail\u271d ++ map g ts))\n[PROOFSTEP]\nsimp only [\u2190 map_cons, \u2190 map_append]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 g' (f (t :: ys_hd :: (tail\u271d ++ ts))) = f' (map g (t :: ys_hd :: (tail\u271d ++ ts)))\n[PROOFSTEP]\napply H\n[GOAL]\ncase cons.H\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\n\u22a2 \u2200 (a : List \u03b1), g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase cons.H\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b1' : Type u_5\n\u03b2' : Type u_6\ng : \u03b1 \u2192 \u03b1'\ng' : \u03b2 \u2192 \u03b2'\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nf\u271d : List \u03b1 \u2192 \u03b2\nf'\u271d : List \u03b1' \u2192 \u03b2'\nH\u271d : \u2200 (a : List \u03b1), g' (f\u271d a) = f'\u271d (map g a)\nys_hd : \u03b1\ntail\u271d : List \u03b1\nys_ih :\n \u2200 (f : List \u03b1 \u2192 \u03b2) (f' : List \u03b1' \u2192 \u03b2'),\n (\u2200 (a : List \u03b1), g' (f a) = f' (map g a)) \u2192\n map g' (permutationsAux2 t ts r tail\u271d f).snd = (permutationsAux2 (g t) (map g ts) (map g' r) (map g tail\u271d) f').snd\nf : List \u03b1 \u2192 \u03b2\nf' : List \u03b1' \u2192 \u03b2'\nH : \u2200 (a : List \u03b1), g' (f a) = f' (map g a)\na : List \u03b1\n\u22a2 g' (f (ys_hd :: a)) = f' (g ys_hd :: map g a)\n[PROOFSTEP]\napply H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 map f (permutationsAux2 t ts [] ys id).snd = (permutationsAux2 t ts [] ys f).snd\n[PROOFSTEP]\nrw [map_permutationsAux2' id, map_id, map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 (id t) ts (map f []) ys ?f').snd = (permutationsAux2 t ts [] ys f).snd\ncase f'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 List \u03b1 \u2192 \u03b2\ncase f'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 List \u03b1 \u2192 \u03b2\ncase f'\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 List \u03b1 \u2192 \u03b2\ncase H \u03b1 : Type u_1 \u03b2 : Type u_2 t : \u03b1 ts ys : List \u03b1 f : List \u03b1 \u2192 \u03b2 \u22a2 \u2200 (a : List \u03b1), f (id a) = ?f' (map id a)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (a : List \u03b1), f (id a) = f (map id a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List \u03b2\nys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 (permutationsAux2 t ts r ys f).snd = map (fun x => f (x ++ ts)) (permutationsAux2 t [] [] ys id).snd ++ r\n[PROOFSTEP]\nrw [\u2190 permutationsAux2_append, map_permutationsAux2, permutationsAux2_comp_append]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts : List \u03b1\n\u22a2 map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n[PROOFSTEP]\ninduction' ts with a ts ih <;> [rfl; (simp [\u2190 ih]; rfl)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts : List \u03b1\n\u22a2 map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n[PROOFSTEP]\ninduction' ts with a ts ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\n\u22a2 map (map f) (permutations'Aux t []) = permutations'Aux (f t) (map f [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt a : \u03b1\nts : List \u03b1\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n\u22a2 map (map f) (permutations'Aux t (a :: ts)) = permutations'Aux (f t) (map f (a :: ts))\n[PROOFSTEP]\nsimp [\u2190 ih]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt a : \u03b1\nts : List \u03b1\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n\u22a2 map (map f \u2218 cons a) (permutations'Aux t ts) = map (cons (f a) \u2218 map f) (permutations'Aux t ts)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\n\u22a2 permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).snd\n[PROOFSTEP]\ninduction' ts with a ts ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\n\u22a2 permutations'Aux t [] = (permutationsAux2 t [] [[] ++ [t]] [] id).snd\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt a : \u03b1\nts : List \u03b1\nih : permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).snd\n\u22a2 permutations'Aux t (a :: ts) = (permutationsAux2 t [] [a :: ts ++ [t]] (a :: ts) id).snd\n[PROOFSTEP]\nsimp [permutations'Aux, permutationsAux2_snd_cons, ih]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt a : \u03b1\nts : List \u03b1\nih : permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).snd\n\u22a2 map (cons a) (permutationsAux2 t [] [ts ++ [t]] ts id).snd =\n (permutationsAux2 t [] [a :: (ts ++ [t])] ts fun x => a :: x).snd\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 permutationsAux2_append]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt a : \u03b1\nts : List \u03b1\nih : permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).snd\n\u22a2 map (cons a) ((permutationsAux2 t [] [] ts id).snd ++ [ts ++ [t]]) =\n (permutationsAux2 t [] [] ts fun x => a :: x).snd ++ [a :: (ts ++ [t])]\n[PROOFSTEP]\nsimp [map_permutationsAux2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys l l' : List \u03b1\n\u22a2 l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\ninduction' ys with y ys ih generalizing l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' l : List \u03b1\n\u22a2 l' \u2208 (permutationsAux2 t ts [] [] fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 [] = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl : List \u03b1\n\u22a2 l' \u2208 (permutationsAux2 t ts [] (y :: ys) fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 y :: ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrw [permutationsAux2_snd_cons, show (fun x : List \u03b1 => l ++ y :: x) = (l ++ [y] ++ \u00b7) by funext _; simp, mem_cons, ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl : List \u03b1\n\u22a2 (fun x => l ++ y :: x) = fun x => l ++ [y] ++ x\n[PROOFSTEP]\nfunext _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl x\u271d : List \u03b1\n\u22a2 l ++ y :: x\u271d = l ++ [y] ++ x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl : List \u03b1\n\u22a2 (l' = l ++ (t :: y :: ys ++ ts) \u2228 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts) \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 y :: ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl : List \u03b1\n\u22a2 (l' = l ++ (t :: y :: ys ++ ts) \u2228 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts) \u2192\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 y :: ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrintro (rfl | \u27e8l\u2081, l\u2082, l0, rfl, rfl\u27e9)\n[GOAL]\ncase cons.mp.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nys l : List \u03b1\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ (t :: y :: ys ++ ts) \u2208 (permutationsAux2 t ts [] ys fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l ++ (t :: y :: ys ++ ts) = l_1 ++ l\u2081 ++ t :: l\u2082 ++ ts\n\u22a2 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 y :: ys = l\u2081 ++ l\u2082 \u2227 l ++ (t :: y :: ys ++ ts) = l ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nexact \u27e8[], y :: ys, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nys l : List \u03b1\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ (t :: y :: ys ++ ts) \u2208 (permutationsAux2 t ts [] ys fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l ++ (t :: y :: ys ++ ts) = l_1 ++ l\u2081 ++ t :: l\u2082 ++ ts\n\u22a2 y :: ys \u2260 [] \u2227 y :: ys = [] ++ y :: ys \u2227 l ++ (t :: y :: ys ++ ts) = l ++ [] ++ t :: y :: ys ++ ts\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.mp.inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] (l\u2081 ++ l\u2082) fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1,\n l\u2082_1 \u2260 [] \u2227 l\u2081 ++ l\u2082 = l\u2081_1 ++ l\u2082_1 \u2227 l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 \u2203 l\u2081_1 l\u2082_1,\n l\u2082_1 \u2260 [] \u2227 y :: (l\u2081 ++ l\u2082) = l\u2081_1 ++ l\u2082_1 \u2227 l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts = l ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nexact \u27e8y :: l\u2081, l\u2082, l0, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] (l\u2081 ++ l\u2082) fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1,\n l\u2082_1 \u2260 [] \u2227 l\u2081 ++ l\u2082 = l\u2081_1 ++ l\u2082_1 \u2227 l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 y :: (l\u2081 ++ l\u2082) = y :: l\u2081 ++ l\u2082 \u2227 l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts = l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d l' : List \u03b1\ny : \u03b1\nys : List \u03b1\nih :\n \u2200 {l : List \u03b1},\n l' \u2208 (permutationsAux2 t ts [] ys fun x => l ++ x).snd \u2194\n \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts\nl : List \u03b1\n\u22a2 (\u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 y :: ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ l\u2081 ++ t :: l\u2082 ++ ts) \u2192\n l' = l ++ (t :: y :: ys ++ ts) \u2228 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l' = l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrintro \u27e8_ | \u27e8y', l\u2081\u27e9, l\u2082, l0, ye, rfl\u27e9\n[GOAL]\ncase cons.mpr.intro.nil.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nys l l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nye : y :: ys = [] ++ l\u2082\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ [] ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] ys fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082_1 \u2227 l ++ [] ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081 ++ t :: l\u2082_1 ++ ts\n\u22a2 l ++ [] ++ t :: l\u2082 ++ ts = l ++ (t :: y :: ys ++ ts) \u2228\n \u2203 l\u2081 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082_1 \u2227 l ++ [] ++ t :: l\u2082 ++ ts = l ++ [y] ++ l\u2081 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nsimp [ye]\n[GOAL]\ncase cons.mpr.intro.cons.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nys l : List \u03b1\ny' : \u03b1\nl\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nye : y :: ys = y' :: l\u2081 ++ l\u2082\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] ys fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ (t :: y :: ys ++ ts) \u2228\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ [y] ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nsimp only [cons_append] at ye \n[GOAL]\ncase cons.mpr.intro.cons.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nys l : List \u03b1\ny' : \u03b1\nl\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nye : y :: ys = y' :: (l\u2081 ++ l\u2082)\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] ys fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ (t :: y :: ys ++ ts) \u2228\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 ys = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y' :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ [y] ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nrcases ye with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase cons.mpr.intro.cons.intro.intro.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] (l\u2081 ++ l\u2082) fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 l\u2081 ++ l\u2082 = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ (t :: y :: (l\u2081 ++ l\u2082) ++ ts) \u2228\n \u2203 l\u2081_1 l\u2082_1,\n l\u2082_1 \u2260 [] \u2227 l\u2081 ++ l\u2082 = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ [y] ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n[PROOFSTEP]\nexact Or.inr \u27e8l\u2081, l\u2082, l0, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts l\u271d : List \u03b1\ny : \u03b1\nl l\u2081 l\u2082 : List \u03b1\nl0 : l\u2082 \u2260 []\nih :\n \u2200 {l_1 : List \u03b1},\n l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts \u2208 (permutationsAux2 t ts [] (l\u2081 ++ l\u2082) fun x => l_1 ++ x).snd \u2194\n \u2203 l\u2081_1 l\u2082_1, l\u2082_1 \u2260 [] \u2227 l\u2081 ++ l\u2082 = l\u2081_1 ++ l\u2082_1 \u2227 l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts = l_1 ++ l\u2081_1 ++ t :: l\u2082_1 ++ ts\n\u22a2 l\u2081 ++ l\u2082 = l\u2081 ++ l\u2082 \u2227 l ++ y :: l\u2081 ++ t :: l\u2082 ++ ts = l ++ [y] ++ l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys l : List \u03b1\n\u22a2 l \u2208 (permutationsAux2 t ts [] ys id).snd \u2194 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrw [show @id (List \u03b1) = ([] ++ \u00b7) by funext _; rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys l : List \u03b1\n\u22a2 id = fun x => [] ++ x\n[PROOFSTEP]\nfunext _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys l x\u271d : List \u03b1\n\u22a2 id x\u271d = [] ++ x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys l : List \u03b1\n\u22a2 l \u2208 (permutationsAux2 t ts [] ys fun x => [] ++ x).snd \u2194 \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\napply mem_permutationsAux2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts ys : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 length (permutationsAux2 t ts [] ys f).snd = length ys\n[PROOFSTEP]\ninduction ys generalizing f\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nf : List \u03b1 \u2192 \u03b2\n\u22a2 length (permutationsAux2 t ts [] [] f).snd = length []\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : \u2200 (f : List \u03b1 \u2192 \u03b2), length (permutationsAux2 t ts [] tail\u271d f).snd = length tail\u271d\nf : List \u03b1 \u2192 \u03b2\n\u22a2 length (permutationsAux2 t ts [] (head\u271d :: tail\u271d) f).snd = length (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L =\n (List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r\n[PROOFSTEP]\ninduction' L with l L ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List (List \u03b1)\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).snd) r [] =\n (List.bind [] fun y => (permutationsAux2 t ts [] y id).snd) ++ r\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List (List \u03b1)\nl : List \u03b1\nL : List (List \u03b1)\nih :\n foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L =\n (List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).snd) r (l :: L) =\n (List.bind (l :: L) fun y => (permutationsAux2 t ts [] y id).snd) ++ r\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr : List (List \u03b1)\nl : List \u03b1\nL : List (List \u03b1)\nih :\n foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L =\n (List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r\n\u22a2 (permutationsAux2 t ts ((List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r) l id).snd =\n (permutationsAux2 t ts [] l id).snd ++ ((List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r)\n[PROOFSTEP]\nrw [\u2190 permutationsAux2_append]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nl' : List \u03b1\n\u22a2 l' \u2208 foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L \u2194\n l' \u2208 r \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 L \u2227 l\u2082 \u2260 [] \u2227 l' = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nhave :\n (\u2203 a : List \u03b1, a \u2208 L \u2227 \u2203 l\u2081 l\u2082 : List \u03b1, \u00acl\u2082 = nil \u2227 a = l\u2081 ++ l\u2082 \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts)) \u2194\n \u2203 l\u2081 l\u2082 : List \u03b1, \u00acl\u2082 = nil \u2227 l\u2081 ++ l\u2082 \u2208 L \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts) :=\n \u27e8fun \u27e8_, aL, l\u2081, l\u2082, l0, e, h\u27e9 => \u27e8l\u2081, l\u2082, l0, e \u25b8 aL, h\u27e9, fun \u27e8l\u2081, l\u2082, l0, aL, h\u27e9 => \u27e8_, aL, l\u2081, l\u2082, l0, rfl, h\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nl' : List \u03b1\nthis :\n (\u2203 a, a \u2208 L \u2227 \u2203 l\u2081 l\u2082, \u00acl\u2082 = [] \u2227 a = l\u2081 ++ l\u2082 \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts)) \u2194\n \u2203 l\u2081 l\u2082, \u00acl\u2082 = [] \u2227 l\u2081 ++ l\u2082 \u2208 L \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts)\n\u22a2 l' \u2208 foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L \u2194\n l' \u2208 r \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 L \u2227 l\u2082 \u2260 [] \u2227 l' = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nrw [foldr_permutationsAux2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nl' : List \u03b1\nthis :\n (\u2203 a, a \u2208 L \u2227 \u2203 l\u2081 l\u2082, \u00acl\u2082 = [] \u2227 a = l\u2081 ++ l\u2082 \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts)) \u2194\n \u2203 l\u2081 l\u2082, \u00acl\u2082 = [] \u2227 l\u2081 ++ l\u2082 \u2208 L \u2227 l' = l\u2081 ++ t :: (l\u2082 ++ ts)\n\u22a2 l' \u2208 (List.bind L fun y => (permutationsAux2 t ts [] y id).snd) ++ r \u2194\n l' \u2208 r \u2228 \u2203 l\u2081 l\u2082, l\u2081 ++ l\u2082 \u2208 L \u2227 l\u2082 \u2260 [] \u2227 l' = l\u2081 ++ t :: l\u2082 ++ ts\n[PROOFSTEP]\nsimp only [mem_permutationsAux2', \u2190 this, or_comm, and_left_comm, mem_append, mem_bind, append_assoc, cons_append,\n exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\n\u22a2 length (foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L) = sum (map length L) + length r\n[PROOFSTEP]\nsimp [foldr_permutationsAux2, (\u00b7 \u2218 \u00b7), length_permutationsAux2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nn : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n\n\u22a2 length (foldr (fun y r => (permutationsAux2 t ts r y id).snd) r L) = n * length L + length r\n[PROOFSTEP]\nrw [length_foldr_permutationsAux2, (_ : List.sum (map length L) = n * length L)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nn : \u2115\nH : \u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n\n\u22a2 sum (map length L) = n * length L\n[PROOFSTEP]\ninduction' L with l L ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L : List (List \u03b1)\nn : \u2115\nH\u271d : \u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n\nH : \u2200 (l : List \u03b1), l \u2208 [] \u2192 length l = n\n\u22a2 sum (map length []) = n * length []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L\u271d : List (List \u03b1)\nn : \u2115\nH\u271d : \u2200 (l : List \u03b1), l \u2208 L\u271d \u2192 length l = n\nl : List \u03b1\nL : List (List \u03b1)\nih : (\u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n) \u2192 sum (map length L) = n * length L\nH : \u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 length l_1 = n\n\u22a2 sum (map length (l :: L)) = n * length (l :: L)\n[PROOFSTEP]\nhave sum_map : sum (map length L) = n * length L := ih fun l m => H l (mem_cons_of_mem _ m)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L\u271d : List (List \u03b1)\nn : \u2115\nH\u271d : \u2200 (l : List \u03b1), l \u2208 L\u271d \u2192 length l = n\nl : List \u03b1\nL : List (List \u03b1)\nih : (\u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n) \u2192 sum (map length L) = n * length L\nH : \u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 length l_1 = n\nsum_map : sum (map length L) = n * length L\n\u22a2 sum (map length (l :: L)) = n * length (l :: L)\n[PROOFSTEP]\nhave length_l : length l = n := H _ (mem_cons_self _ _)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts : List \u03b1\nr L\u271d : List (List \u03b1)\nn : \u2115\nH\u271d : \u2200 (l : List \u03b1), l \u2208 L\u271d \u2192 length l = n\nl : List \u03b1\nL : List (List \u03b1)\nih : (\u2200 (l : List \u03b1), l \u2208 L \u2192 length l = n) \u2192 sum (map length L) = n * length L\nH : \u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 length l_1 = n\nsum_map : sum (map length L) = n * length L\nlength_l : length l = n\n\u22a2 sum (map length (l :: L)) = n * length (l :: L)\n[PROOFSTEP]\nsimp [sum_map, length_l, mul_add, add_comm, mul_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis : List \u03b1\n\u22a2 permutationsAux [] is = []\n[PROOFSTEP]\nrw [permutationsAux, permutationsAux.rec]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts is : List \u03b1\n\u22a2 permutationsAux (t :: ts) is =\n foldr (fun y r => (permutationsAux2 t ts r y id).snd) (permutationsAux ts (t :: is)) (permutations is)\n[PROOFSTEP]\nrw [permutationsAux, permutationsAux.rec]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nt : \u03b1\nts is : List \u03b1\n\u22a2 foldr (fun y r => (permutationsAux2 t ts r y id).snd)\n (permutationsAux.rec (fun x => [])\n (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).snd) IH1 (is :: IH2)) ts (t :: is))\n (is ::\n permutationsAux.rec (fun x => [])\n (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).snd) IH1 (is :: IH2)) is []) =\n foldr (fun y r => (permutationsAux2 t ts r y id).snd)\n (permutationsAux.rec (fun x => [])\n (fun t ts is IH1 IH2 => foldr (fun y r => (permutationsAux2 t ts r y id).snd) IH1 (is :: IH2)) ts (t :: is))\n (permutations is)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 permutations [] = [[]]\n[PROOFSTEP]\nrw [permutations, permutationsAux_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (ts is : List \u03b1), map (map f) (permutationsAux ts is) = permutationsAux (map f ts) (map f is)\n[PROOFSTEP]\nrefine' permutationsAux.rec (by simp) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (is : List \u03b1), map (map f) (permutationsAux [] is) = permutationsAux (map f []) (map f is)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (t : \u03b1) (ts is : List \u03b1),\n map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is)) \u2192\n map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f []) \u2192\n map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)\n[PROOFSTEP]\nintrov IH1 IH2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts is : List \u03b1\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f [])\n\u22a2 map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)\n[PROOFSTEP]\nrw [map] at IH2 \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts is : List \u03b1\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) []\n\u22a2 map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)\n[PROOFSTEP]\nsimp only [foldr_permutationsAux2, map_append, map, map_map_permutationsAux2, permutations, bind_map, IH1, append_assoc,\n permutationsAux_cons, cons_bind, \u2190 IH2, map_bind]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nts : List \u03b1\n\u22a2 map (map f) (permutations ts) = permutations (map f ts)\n[PROOFSTEP]\nrw [permutations, permutations, map, map_permutationsAux, map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nts : List \u03b1\n\u22a2 map (map f) (permutations' ts) = permutations' (map f ts)\n[PROOFSTEP]\ninduction' ts with t ts ih <;> [rfl; simp [\u2190 ih, map_bind, \u2190 map_map_permutations'Aux, bind_map]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nts : List \u03b1\n\u22a2 map (map f) (permutations' ts) = permutations' (map f ts)\n[PROOFSTEP]\ninduction' ts with t ts ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (map f) (permutations' []) = permutations' (map f [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts : List \u03b1\nih : map (map f) (permutations' ts) = permutations' (map f ts)\n\u22a2 map (map f) (permutations' (t :: ts)) = permutations' (map f (t :: ts))\n[PROOFSTEP]\nsimp [\u2190 ih, map_bind, \u2190 map_map_permutations'Aux, bind_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis is' ts : List \u03b1\n\u22a2 permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\n[PROOFSTEP]\ninduction' is with t is ih generalizing is'\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts is' : List \u03b1\n\u22a2 permutationsAux ([] ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux [] is') ++ permutationsAux ts (reverse [] ++ is')\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' : List \u03b1\n\u22a2 permutationsAux (t :: is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux (t :: is) is') ++ permutationsAux ts (reverse (t :: is) ++ is')\n[PROOFSTEP]\nsimp only [foldr_permutationsAux2, ih, bind_map, cons_append, permutationsAux_cons, map_append, reverse_cons,\n append_assoc, singleton_append]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' : List \u03b1\n\u22a2 (List.bind (permutations is') fun y => (permutationsAux2 t (is ++ ts) [] y id).snd) ++\n (map (fun x => x ++ ts) (permutationsAux is (t :: is')) ++ permutationsAux ts (reverse is ++ t :: is')) =\n (List.bind (permutations is') fun a => map (fun x => x ++ ts) (permutationsAux2 t is [] a id).snd) ++\n (map (fun x => x ++ ts) (permutationsAux is (t :: is')) ++ permutationsAux ts (reverse is ++ t :: ([] ++ is')))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase cons.e_a.e_b\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' : List \u03b1\n\u22a2 (fun y => (permutationsAux2 t (is ++ ts) [] y id).snd) = fun a =>\n map (fun x => x ++ ts) (permutationsAux2 t is [] a id).snd\n[PROOFSTEP]\nfunext _\n[GOAL]\ncase cons.e_a.e_b.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' x\u271d : List \u03b1\n\u22a2 (permutationsAux2 t (is ++ ts) [] x\u271d id).snd = map (fun x => x ++ ts) (permutationsAux2 t is [] x\u271d id).snd\n[PROOFSTEP]\nrw [map_permutationsAux2]\n[GOAL]\ncase cons.e_a.e_b.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' x\u271d : List \u03b1\n\u22a2 (permutationsAux2 t (is ++ ts) [] x\u271d id).snd = (permutationsAux2 t is [] x\u271d fun x => x ++ ts).snd\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 permutationsAux2_comp_append]\n[GOAL]\ncase cons.e_a.e_b.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis'\u271d ts : List \u03b1\nt : \u03b1\nis : List \u03b1\nih :\n \u2200 (is' : List \u03b1),\n permutationsAux (is ++ ts) is' =\n map (fun x => x ++ ts) (permutationsAux is is') ++ permutationsAux ts (reverse is ++ is')\nis' x\u271d : List \u03b1\n\u22a2 (permutationsAux2 t [] [] x\u271d fun x => id (x ++ (is ++ ts))).snd =\n (permutationsAux2 t [] [] x\u271d fun x => x ++ is ++ ts).snd\n[PROOFSTEP]\nsimp only [id, append_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nis ts : List \u03b1\n\u22a2 permutations (is ++ ts) = map (fun x => x ++ ts) (permutations is) ++ permutationsAux ts (reverse is)\n[PROOFSTEP]\nsimp [permutations, permutationsAux_append]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Permutation", "llama_tokens": 17600, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.26183791066279083}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 \u2200 (i j : \u2115),\n ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n ((fun n => Iso.refl (HomologicalComplex.X (Splitting.nondegComplex (\u0393\u2080.splitting K)) n)) i).hom \u226b\n HomologicalComplex.d K i j =\n HomologicalComplex.d (Splitting.nondegComplex (\u0393\u2080.splitting K)) i j \u226b\n ((fun n => Iso.refl (HomologicalComplex.X (Splitting.nondegComplex (\u0393\u2080.splitting K)) n)) j).hom\n[PROOFSTEP]\nrintro _ n (rfl : n + 1 = _)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 ((fun n => Iso.refl (HomologicalComplex.X (Splitting.nondegComplex (\u0393\u2080.splitting K)) n)) (n + 1)).hom \u226b\n HomologicalComplex.d K (n + 1) n =\n HomologicalComplex.d (Splitting.nondegComplex (\u0393\u2080.splitting K)) (n + 1) n \u226b\n ((fun n => Iso.refl (HomologicalComplex.X (Splitting.nondegComplex (\u0393\u2080.splitting K)) n)) n).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 \ud835\udfd9 (Splitting.N (\u0393\u2080.splitting K) (n + 1)) \u226b HomologicalComplex.d K (n + 1) n =\n (Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n HomologicalComplex.d (AlternatingFaceMapComplex.obj (\u0393\u2080.obj K)) (n + 1) n \u226b\n Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n]))) \u226b\n \ud835\udfd9 (Splitting.N (\u0393\u2080.splitting K) n)\n[PROOFSTEP]\nsimp only [id_comp, comp_id, AlternatingFaceMapComplex.obj_d_eq, Preadditive.sum_comp, Preadditive.comp_sum]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.d K (n + 1) n =\n Finset.sum Finset.univ fun j =>\n Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n ((-1) ^ \u2191j \u2022 \u03b4 (\u0393\u2080.obj K) j) \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n]))\n[PROOFSTEP]\nrw [Fintype.sum_eq_single (0 : Fin (n + 2))]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.d K (n + 1) n =\n Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n ((-1) ^ \u21910 \u2022 \u03b4 (\u0393\u2080.obj K) 0) \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n]))\n[PROOFSTEP]\nsimp only [Fin.val_zero, pow_zero, one_zsmul]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.d K (n + 1) n =\n Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n \u03b4 (\u0393\u2080.obj K) 0 \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n]))\n[PROOFSTEP]\nerw [\u0393\u2080.Obj.mapMono_on_summand_id_assoc, \u0393\u2080.Obj.Termwise.mapMono_\u03b4\u2080, Splitting.\u03b9_\u03c0Summand_eq_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 \u2200 (x : Fin (n + 2)),\n x \u2260 0 \u2192\n Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n ((-1) ^ \u2191x \u2022 \u03b4 (\u0393\u2080.obj K) x) \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n])) =\n 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n ((-1) ^ \u2191i \u2022 \u03b4 (\u0393\u2080.obj K) i) \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n])) =\n 0\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n ((-1) ^ \u2191i \u2022 \u03b4 (\u0393\u2080.obj K) i) \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n])) =\n 0\n[PROOFSTEP]\nsimp only [Preadditive.zsmul_comp, Preadditive.comp_zsmul, assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\n\u22a2 (-1) ^ \u2191i \u2022\n Splitting.\u03b9Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n + 1])) \u226b\n \u03b4 (\u0393\u2080.obj K) i \u226b Splitting.\u03c0Summand (\u0393\u2080.splitting K) (Splitting.IndexSet.id (op [n])) =\n 0\n[PROOFSTEP]\nerw [\u0393\u2080.Obj.mapMono_on_summand_id_assoc, \u0393\u2080.Obj.Termwise.mapMono_eq_zero, zero_comp, zsmul_zero]\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\n\u22a2 [n + 1] \u2260 [n]\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\nh : [n + 1] = [n]\n\u22a2 False\n[PROOFSTEP]\nreplace h := congr_arg SimplexCategory.len h\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\nh : SimplexCategory.len [n + 1] = SimplexCategory.len [n]\n\u22a2 False\n[PROOFSTEP]\nchange n + 1 = n at h \n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\nh : n + 1 = n\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\u2082\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.42, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\ni : Fin (n + 2)\nhi : i \u2260 0\n\u22a2 \u00acIs\u03b4\u2080 (SimplexCategory.\u03b4 i)\n[PROOFSTEP]\nsimpa only [Is\u03b4\u2080.iff] using hi\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 N\u2081\u0393\u2080.app K =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).symm \u226a\u226b\n (toKaroubi (ChainComplex C \u2115)).mapIso (\u0393\u2080NondegComplexIso K)\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 (N\u2081\u0393\u2080.app K).hom =\n ((Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).symm \u226a\u226b\n (toKaroubi (ChainComplex C \u2115)).mapIso (\u0393\u2080NondegComplexIso K)).hom\n[PROOFSTEP]\ndsimp [N\u2081\u0393\u2080]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 (((Karoubi.Hom.mk PInfty \u226b NatTrans.app Split.toKaroubiNondegComplexFunctorIsoN\u2081.inv (Split.mk' (\u0393\u2080.splitting K))) \u226b\n Karoubi.Hom.mk (\ud835\udfd9 (Splitting.nondegComplex (\u0393\u2080.splitting K)))) \u226b\n (toKaroubi (ChainComplex C \u2115)).map (NatTrans.app \u0393\u2080'CompNondegComplexFunctor.hom K)) \u226b\n Karoubi.Hom.mk (\ud835\udfd9 K) =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom\n[PROOFSTEP]\nerw [id_comp, comp_id, comp_id]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app Split.toKaroubiNondegComplexFunctorIsoN\u2081.inv (Split.mk' (\u0393\u2080.splitting K)) \u226b\n (toKaroubi (ChainComplex C \u2115)).map (NatTrans.app \u0393\u2080'CompNondegComplexFunctor.hom K) =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app N\u2081\u0393\u2080.hom K =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom\n[PROOFSTEP]\nchange (N\u2081\u0393\u2080.app K).hom = _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 (N\u2081\u0393\u2080.app K).hom =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom\n[PROOFSTEP]\nsimp only [N\u2081\u0393\u2080_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 ((Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).symm \u226a\u226b\n (toKaroubi (ChainComplex C \u2115)).mapIso (\u0393\u2080NondegComplexIso K)).hom =\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app N\u2081\u0393\u2080.inv K =\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).inv \u226b\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom\n[PROOFSTEP]\nchange (N\u2081\u0393\u2080.app K).inv = _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 (N\u2081\u0393\u2080.app K).inv =\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).inv \u226b\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom\n[PROOFSTEP]\nsimp only [N\u2081\u0393\u2080_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 ((Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).symm \u226a\u226b\n (toKaroubi (ChainComplex C \u2115)).mapIso (\u0393\u2080NondegComplexIso K)).inv =\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).inv \u226b\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (NatTrans.app N\u2081\u0393\u2080.hom K).f n =\n HomologicalComplex.Hom.f (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv.f n\n[PROOFSTEP]\nrw [N\u2081\u0393\u2080_hom_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n ((Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv \u226b\n (toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).hom).f\n n =\n HomologicalComplex.Hom.f (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).inv.f n\n[PROOFSTEP]\napply comp_id\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (NatTrans.app N\u2081\u0393\u2080.inv K).f n =\n HomologicalComplex.Hom.f (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom.f n\n[PROOFSTEP]\nrw [N\u2081\u0393\u2080_inv_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n ((toKaroubi (ChainComplex C \u2115)).map (\u0393\u2080NondegComplexIso K).inv \u226b\n (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom).f\n n =\n HomologicalComplex.Hom.f (Splitting.toKaroubiNondegComplexIsoN\u2081 (\u0393\u2080.splitting K)).hom.f n\n[PROOFSTEP]\napply id_comp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081\n[PROOFSTEP]\nhave h := Functor.congr_obj (functorExtension\u2082_comp_whiskeringLeft_toKaroubi (ChainComplex C \u2115) (SimplicialObject C)) \u0393\u2080\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nh :\n (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C) \u22d9\n (whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (SimplicialObject C))).obj\n (toKaroubi (ChainComplex C \u2115))).obj\n \u0393\u2080 =\n ((whiskeringRight (ChainComplex C \u2115) (SimplicialObject C) (Karoubi (SimplicialObject C))).obj\n (toKaroubi (SimplicialObject C))).obj\n \u0393\u2080\n\u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081\n[PROOFSTEP]\nhave h' :=\n Functor.congr_obj (functorExtension\u2081_comp_whiskeringLeft_toKaroubi (SimplicialObject C) (ChainComplex C \u2115)) N\u2081\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nh :\n (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C) \u22d9\n (whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (SimplicialObject C))).obj\n (toKaroubi (ChainComplex C \u2115))).obj\n \u0393\u2080 =\n ((whiskeringRight (ChainComplex C \u2115) (SimplicialObject C) (Karoubi (SimplicialObject C))).obj\n (toKaroubi (SimplicialObject C))).obj\n \u0393\u2080\nh' :\n (functorExtension\u2081 (SimplicialObject C) (ChainComplex C \u2115) \u22d9\n (whiskeringLeft (SimplicialObject C) (Karoubi (SimplicialObject C)) (Karoubi (ChainComplex C \u2115))).obj\n (toKaroubi (SimplicialObject C))).obj\n N\u2081 =\n (\ud835\udfed (SimplicialObject C \u2964 Karoubi (ChainComplex C \u2115))).obj N\u2081\n\u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081\n[PROOFSTEP]\ndsimp [N\u2082, \u0393\u2082, functorExtension\u2081] at h h' \u22a2\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nh :\n toKaroubi (ChainComplex C \u2115) \u22d9 (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C)).obj \u0393\u2080 =\n \u0393\u2080 \u22d9 toKaroubi (SimplicialObject C)\nh' : toKaroubi (SimplicialObject C) \u22d9 FunctorExtension\u2081.obj N\u2081 = N\u2081\n\u22a2 toKaroubi (ChainComplex C \u2115) \u22d9\n (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C)).obj \u0393\u2080 \u22d9 FunctorExtension\u2081.obj N\u2081 =\n \u0393\u2080 \u22d9 N\u2081\n[PROOFSTEP]\nrw [\u2190 Functor.assoc, h, Functor.assoc, h']\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX : Karoubi (ChainComplex C \u2115)\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (NatTrans.app N\u2082\u0393\u2082.inv X).f n =\n HomologicalComplex.Hom.f X.p n \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting X.X) (Splitting.IndexSet.id (op [n]))\n[PROOFSTEP]\nsimp only [N\u2082\u0393\u2082, Functor.preimageIso, Iso.trans, whiskeringLeft_obj_preimage_app, N\u2082\u0393\u2082ToKaroubiIso_inv, assoc,\n Functor.id_map, NatTrans.comp_app, eqToHom_app, Karoubi.comp_f, Karoubi.eqToHom_f, Karoubi.decompId_p_f,\n HomologicalComplex.comp_f, N\u2081\u0393\u2080_inv_app_f_f, Splitting.toKaroubiNondegComplexIsoN\u2081_hom_f_f, Functor.comp_map,\n Functor.comp_obj, Karoubi.decompId_i_f, eqToHom_refl, comp_id, N\u2082_map_f_f, \u0393\u2082_map_f_app, N\u2081_obj_p,\n PInfty_on_\u0393\u2080_splitting_summand_eq_self_assoc, toKaroubi_obj_X, Splitting.\u03b9_desc, Splitting.IndexSet.id_fst,\n SimplexCategory.len_mk, unop_op, Karoubi.HomologicalComplex.p_idem_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 whiskerLeft (toKaroubi (ChainComplex C \u2115)) N\u2082\u0393\u2082.hom = N\u2082\u0393\u2082ToKaroubiIso.hom \u226b N\u2081\u0393\u2080.hom\n[PROOFSTEP]\nlet e : _ \u2245 toKaroubi (ChainComplex C \u2115) \u22d9 \ud835\udfed _ := N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\ne : toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 \u2245 toKaroubi (ChainComplex C \u2115) \u22d9 \ud835\udfed (Karoubi (ChainComplex C \u2115)) :=\n N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080\n\u22a2 whiskerLeft (toKaroubi (ChainComplex C \u2115)) N\u2082\u0393\u2082.hom = N\u2082\u0393\u2082ToKaroubiIso.hom \u226b N\u2081\u0393\u2080.hom\n[PROOFSTEP]\nhave h := ((whiskeringLeft _ _ (Karoubi (ChainComplex C \u2115))).obj (toKaroubi (ChainComplex C \u2115))).image_preimage e.hom\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\ne : toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 \u2245 toKaroubi (ChainComplex C \u2115) \u22d9 \ud835\udfed (Karoubi (ChainComplex C \u2115)) :=\n N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080\nh :\n ((whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (ChainComplex C \u2115))).obj\n (toKaroubi (ChainComplex C \u2115))).map\n (((whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (ChainComplex C \u2115))).obj\n (toKaroubi (ChainComplex C \u2115))).preimage\n e.hom) =\n e.hom\n\u22a2 whiskerLeft (toKaroubi (ChainComplex C \u2115)) N\u2082\u0393\u2082.hom = N\u2082\u0393\u2082ToKaroubiIso.hom \u226b N\u2081\u0393\u2080.hom\n[PROOFSTEP]\ndsimp only [whiskeringLeft, N\u2082\u0393\u2082, Functor.preimageIso] at h \u22a2\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\ne : toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 \u2245 toKaroubi (ChainComplex C \u2115) \u22d9 \ud835\udfed (Karoubi (ChainComplex C \u2115)) :=\n N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080\nh :\n whiskerLeft (toKaroubi (ChainComplex C \u2115))\n ((CategoryTheory.Functor.mk\n { obj := fun G => toKaroubi (ChainComplex C \u2115) \u22d9 G,\n map := fun {X Y} \u03b1 => whiskerLeft (toKaroubi (ChainComplex C \u2115)) \u03b1 }).preimage\n (N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080).hom) =\n (N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080).hom\n\u22a2 whiskerLeft (toKaroubi (ChainComplex C \u2115))\n ((CategoryTheory.Functor.mk\n { obj := fun G => toKaroubi (ChainComplex C \u2115) \u22d9 G,\n map := fun {X Y} \u03b1 => whiskerLeft (toKaroubi (ChainComplex C \u2115)) \u03b1 }).preimage\n (N\u2082\u0393\u2082ToKaroubiIso \u226a\u226b N\u2081\u0393\u2080).hom) =\n N\u2082\u0393\u2082ToKaroubiIso.hom \u226b N\u2081\u0393\u2080.hom\n[PROOFSTEP]\nexact h\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.GammaCompN", "llama_tokens": 8245, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.41489884579676883, "lm_q1q2_score": 0.26129276349019154}} {"text": "[GOAL]\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj : J\nU : Opens \u2191\u2191(F.obj j)\n\u22a2 \ud835\udfd9 \u2191(F.obj j) = (F.map (\ud835\udfd9 j)).base\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj : J\nU : Opens \u2191\u2191(F.obj j)\n\u22a2 NatTrans.app (F.map (\ud835\udfd9 j)).c (op U) =\n NatTrans.app (Pushforward.id (F.obj j).presheaf).inv (op U) \u226b\n NatTrans.app (pushforwardEq (_ : \ud835\udfd9 \u2191(F.obj j) = (F.map (\ud835\udfd9 j)).base) (F.obj j).presheaf).hom (op U)\n[PROOFSTEP]\ncases U\n[GOAL]\ncase mk\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj : J\ncarrier\u271d : Set \u2191\u2191(F.obj j)\nis_open'\u271d : IsOpen carrier\u271d\n\u22a2 NatTrans.app (F.map (\ud835\udfd9 j)).c (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) =\n NatTrans.app (Pushforward.id (F.obj j).presheaf).inv (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) \u226b\n NatTrans.app (pushforwardEq (_ : \ud835\udfd9 \u2191(F.obj j) = (F.map (\ud835\udfd9 j)).base) (F.obj j).presheaf).hom\n (op { carrier := carrier\u271d, is_open' := is_open'\u271d })\n[PROOFSTEP]\nsimp [PresheafedSpace.congr_app (F.map_id j)]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : Opens \u2191\u2191(F.obj j\u2083)\n\u22a2 (F.map f).base \u226b (F.map g).base = (F.map (f \u226b g)).base\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : Opens \u2191\u2191(F.obj j\u2083)\n\u22a2 (F.map f).base \u226b (F.map g).base = (F.map f \u226b F.map g).base\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : Opens \u2191\u2191(F.obj j\u2083)\n\u22a2 NatTrans.app (F.map (f \u226b g)).c (op U) =\n NatTrans.app (F.map g).c (op U) \u226b\n NatTrans.app (pushforwardMap (F.map g).base (F.map f).c) (op U) \u226b\n NatTrans.app (Pushforward.comp (F.obj j\u2081).presheaf (F.map f).base (F.map g).base).inv (op U) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F.map f).base \u226b (F.map g).base = (F.map (f \u226b g)).base) (F.obj j\u2081).presheaf).hom (op U)\n[PROOFSTEP]\ncases U\n[GOAL]\ncase mk\nJ : Type u'\ninst\u271d\u00b9 : Category.{v', u'} J\nC : Type u\ninst\u271d : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\ncarrier\u271d : Set \u2191\u2191(F.obj j\u2083)\nis_open'\u271d : IsOpen carrier\u271d\n\u22a2 NatTrans.app (F.map (f \u226b g)).c (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) =\n NatTrans.app (F.map g).c (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) \u226b\n NatTrans.app (pushforwardMap (F.map g).base (F.map f).c) (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) \u226b\n NatTrans.app (Pushforward.comp (F.obj j\u2081).presheaf (F.map f).base (F.map g).base).inv\n (op { carrier := carrier\u271d, is_open' := is_open'\u271d }) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F.map f).base \u226b (F.map g).base = (F.map (f \u226b g)).base) (F.obj j\u2081).presheaf).hom\n (op { carrier := carrier\u271d, is_open' := is_open'\u271d })\n[PROOFSTEP]\nsimp [PresheafedSpace.congr_app (F.map_comp f g)]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nj k : J\u1d52\u1d56\nf : j \u27f6 k\n\u22a2 (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)\n[PROOFSTEP]\nrw [\u2190 colimit.w F f.unop, comp_base]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nj k : J\u1d52\u1d56\nf : j \u27f6 k\n\u22a2 (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map ((F.map f.unop).base \u226b (colimit.\u03b9 F j.unop).base)).obj U)\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nx : J\u1d52\u1d56\n\u22a2 { obj := fun j => (F.obj j.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)),\n map := fun {j k} f =>\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) }.map\n (\ud835\udfd9 x) =\n \ud835\udfd9\n ({ obj := fun j => (F.obj j.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)),\n map := fun {j k} f =>\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) }.obj\n x)\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nx : J\u1d52\u1d56\n\u22a2 NatTrans.app (F.map (\ud835\udfd9 x.unop)).c (op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)) \u226b\n (F.obj x.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (\ud835\udfd9 x.unop)).base).obj ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U))) =\n \ud835\udfd9 ((F.obj x.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)))\n[PROOFSTEP]\nsimp [map_id_c_app, pushforwardObj_obj, op_obj, unop_op, pushforwardEq_hom_app, eqToHom_op, id_eq, eqToHom_map, assoc,\n eqToHom_trans, eqToHom_refl, comp_id, TopCat.Presheaf.Pushforward.id_inv_app']\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nx : J\u1d52\u1d56\n\u22a2 NatTrans.app (Pushforward.id (F.obj x.unop).presheaf).inv (op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)) =\n \ud835\udfd9 ((F.obj x.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)))\n[PROOFSTEP]\nrw [TopCat.Presheaf.Pushforward.id_inv_app']\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\nx : J\u1d52\u1d56\n\u22a2 (F.obj x.unop).presheaf.map\n (\ud835\udfd9\n (op\n { carrier := ((Opens.map (colimit.\u03b9 F x.unop).base).toPrefunctor.1 U).carrier,\n is_open' := (_ : IsOpen ((Opens.map (colimit.\u03b9 F x.unop).base).toPrefunctor.1 U).carrier) })) =\n \ud835\udfd9 ((F.obj x.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F x.unop).base).obj U)))\n[PROOFSTEP]\nsimp only [Opens.carrier_eq_coe, Opens.mk_coe, map_id]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 { obj := fun j => (F.obj j.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)),\n map := fun {j k} f =>\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) }.map\n (f \u226b g) =\n { obj := fun j => (F.obj j.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)),\n map := fun {j k} f =>\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) }.map\n f \u226b\n { obj := fun j => (F.obj j.unop).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)),\n map := fun {j k} f =>\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) }.map\n g\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 NatTrans.app (F.map (g.unop \u226b f.unop)).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n (NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)))) \u226b\n NatTrans.app (F.map g.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\nsimp_rw [map_comp_c_app]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 (NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app (pushforwardMap (F.map f.unop).base (F.map g.unop).c)\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app (Pushforward.comp (F.obj k.unop).presheaf (F.map g.unop).base (F.map f.unop).base).inv\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F.map g.unop).base \u226b (F.map f.unop).base = (F.map (g.unop \u226b f.unop)).base)\n (F.obj k.unop).presheaf).hom\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U))) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n (NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)))) \u226b\n NatTrans.app (F.map g.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\nsimp only [op_obj, unop_op, eqToHom_op, id_eq, id_comp, assoc, eqToHom_trans]\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app (pushforwardMap (F.map f.unop).base (F.map g.unop).c)\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app (Pushforward.comp (F.obj k.unop).presheaf (F.map g.unop).base (F.map f.unop).base).inv\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F.map g.unop).base \u226b (F.map f.unop).base = (F.map (g.unop \u226b f.unop)).base)\n (F.obj k.unop).presheaf).hom\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n NatTrans.app (F.map f.unop).c (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U))) \u226b\n NatTrans.app (F.map g.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 NatTrans.app (pushforwardMap (F.map f.unop).base (F.map g.unop).c)\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app (Pushforward.comp (F.obj k.unop).presheaf (F.map g.unop).base (F.map f.unop).base).inv\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F.map g.unop).base \u226b (F.map f.unop).base = (F.map (g.unop \u226b f.unop)).base)\n (F.obj k.unop).presheaf).hom\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U))) \u226b\n NatTrans.app (F.map g.unop).c (op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\nrw [TopCat.Presheaf.Pushforward.comp_inv_app, TopCat.Presheaf.pushforwardEq_hom_app,\n CategoryTheory.NatTrans.naturality_assoc, TopCat.Presheaf.pushforwardMap_app]\n[GOAL]\ncase e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 NatTrans.app (F.map g.unop).c\n ((Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U))) \u226b\n \ud835\udfd9\n (((F.map f.unop).base _* ((F.map g.unop).base _* (F.obj k.unop).presheaf)).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U))) \u226b\n (F.obj k.unop).presheaf.map\n (_root_.id\n (eqToHom\n (_ :\n (Opens.map (F.map (g.unop \u226b f.unop)).base).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)).unop =\n (Opens.map ((F.map g.unop).base \u226b (F.map f.unop).base)).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)).unop)).op) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n NatTrans.app (F.map g.unop).c\n (op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U))) \u226b\n ((F.map g.unop).base _* (F.obj k.unop).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U))) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a.e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(colimit F)\ni j k : J\u1d52\u1d56\nf : i \u27f6 j\ng : j \u27f6 k\n\u22a2 \ud835\udfd9\n (((F.map f.unop).base _* ((F.map g.unop).base _* (F.obj k.unop).presheaf)).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U))) \u226b\n (F.obj k.unop).presheaf.map\n (_root_.id\n (eqToHom\n (_ :\n (Opens.map (F.map (g.unop \u226b f.unop)).base).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)).unop =\n (Opens.map ((F.map g.unop).base \u226b (F.map f.unop).base)).obj\n (op ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)).unop)).op) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map (g.unop \u226b f.unop)).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U))) =\n ((F.map g.unop).base _* (F.obj k.unop).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (F.map f.unop).base).obj ((Opens.map (colimit.\u03b9 F i.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F j.unop).base).obj U))) \u226b\n (F.obj k.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g.unop).base).obj ((Opens.map (colimit.\u03b9 F j.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F k.unop).base).obj U)))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\n\u22a2 { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j) =\n \ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n[PROOFSTEP]\napply (opEquiv _ _).injective\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\n\u22a2 \u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)) =\n \u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n[PROOFSTEP]\nrefine NatTrans.ext _ _ (funext fun U => ?_)\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)))\n U =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)))\n U\n[PROOFSTEP]\ninduction U with\n| h U =>\n rcases U with \u27e8U, hU\u27e9\n dsimp [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op, pushforwardObj_obj, op_obj,\n Opens.map_obj, opEquiv, Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op, unop_id]\n -- Porting note : some `simp` lemmas are not picked up\n rw [NatTrans.comp_app, pushforwardMap_app, NatTrans.id_app]\n simp only [op_obj, unop_op, Opens.map_obj, map_id_c_app, Opens.map_id_obj', map_id, pushforwardEq_hom_app, eqToHom_op,\n id_eq, eqToHom_map, id_comp, TopCat.Presheaf.Pushforward.id_inv_app']\n rw [NatTrans.comp_app, Pushforward.comp_inv_app, id_comp]\n dsimp\n simp\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)))\n U =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)))\n U\n[PROOFSTEP]\ninduction U with\n| h U =>\n rcases U with \u27e8U, hU\u27e9\n dsimp [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op, pushforwardObj_obj, op_obj,\n Opens.map_obj, opEquiv, Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op, unop_id]\n -- Porting note : some `simp` lemmas are not picked up\n rw [NatTrans.comp_app, pushforwardMap_app, NatTrans.id_app]\n simp only [op_obj, unop_op, Opens.map_obj, map_id_c_app, Opens.map_id_obj', map_id, pushforwardEq_hom_app, eqToHom_op,\n id_eq, eqToHom_map, id_comp, TopCat.Presheaf.Pushforward.id_inv_app']\n rw [NatTrans.comp_app, Pushforward.comp_inv_app, id_comp]\n dsimp\n simp\n[GOAL]\ncase a.h\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Opens \u2191(colimit (F \u22d9 forget C))\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)))\n (op U) =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)))\n (op U)\n[PROOFSTEP]\n\n| h U =>\n rcases U with \u27e8U, hU\u27e9\n dsimp [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op, pushforwardObj_obj, op_obj,\n Opens.map_obj, opEquiv, Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op, unop_id]\n -- Porting note : some `simp` lemmas are not picked up\n rw [NatTrans.comp_app, pushforwardMap_app, NatTrans.id_app]\n simp only [op_obj, unop_op, Opens.map_obj, map_id_c_app, Opens.map_id_obj', map_id, pushforwardEq_hom_app, eqToHom_op,\n id_eq, eqToHom_map, id_comp, TopCat.Presheaf.Pushforward.id_inv_app']\n rw [NatTrans.comp_app, Pushforward.comp_inv_app, id_comp]\n dsimp\n simp\n[GOAL]\ncase a.h\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Opens \u2191(colimit (F \u22d9 forget C))\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)))\n (op U) =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)))\n (op U)\n[PROOFSTEP]\nrcases U with \u27e8U, hU\u27e9\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (\ud835\udfd9 j)))\n (op { carrier := U, is_open' := hU }) =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j))\n (\ud835\udfd9\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j)))\n (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\ndsimp [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op, pushforwardObj_obj, op_obj, Opens.map_obj,\n opEquiv, Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op, unop_id]\n -- Porting note : some `simp` lemmas are not picked up\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 NatTrans.app\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j) (F.map (\ud835\udfd9 j)).c \u226b\n (Pushforward.comp (F.obj j).presheaf (F.map (\ud835\udfd9 j)).base (colimit.\u03b9 (F \u22d9 forget C) j)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map (\ud835\udfd9 j) \u226b colimit.\u03b9 (F \u22d9 forget C) j = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom)\n (op { carrier := U, is_open' := hU }) =\n NatTrans.app (\ud835\udfd9 (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf)) (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\nrw [NatTrans.comp_app, pushforwardMap_app, NatTrans.id_app]\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 NatTrans.app (F.map (\ud835\udfd9 j)).c ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j)).op.obj (op { carrier := U, is_open' := hU })) \u226b\n NatTrans.app\n ((Pushforward.comp (F.obj j).presheaf (F.map (\ud835\udfd9 j)).base (colimit.\u03b9 (F \u22d9 forget C) j)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map (\ud835\udfd9 j) \u226b colimit.\u03b9 (F \u22d9 forget C) j = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom)\n (op { carrier := U, is_open' := hU }) =\n \ud835\udfd9 ((colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf).obj (op { carrier := U, is_open' := hU }))\n[PROOFSTEP]\nsimp only [op_obj, unop_op, Opens.map_obj, map_id_c_app, Opens.map_id_obj', map_id, pushforwardEq_hom_app, eqToHom_op,\n id_eq, eqToHom_map, id_comp, TopCat.Presheaf.Pushforward.id_inv_app']\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 eqToHom\n (_ :\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) }) =\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) })) \u226b\n NatTrans.app\n ((Pushforward.comp (F.obj j).presheaf (F.map (\ud835\udfd9 j)).base (colimit.\u03b9 (F \u22d9 forget C) j)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map (\ud835\udfd9 j) \u226b colimit.\u03b9 (F \u22d9 forget C) j = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom)\n (op { carrier := U, is_open' := hU }) =\n \ud835\udfd9 ((colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf).obj (op { carrier := U, is_open' := hU }))\n[PROOFSTEP]\nrw [NatTrans.comp_app, Pushforward.comp_inv_app, id_comp]\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 eqToHom\n (_ :\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) }) =\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) })) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F \u22d9 forget C).map (\ud835\udfd9 j) \u226b colimit.\u03b9 (F \u22d9 forget C) j = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom\n (op { carrier := U, is_open' := hU }) =\n \ud835\udfd9 ((colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf).obj (op { carrier := U, is_open' := hU }))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a.h.mk\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj : J\nU : Set \u2191(colimit (F \u22d9 forget C))\nhU : IsOpen U\n\u22a2 eqToHom\n (_ :\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(\ud835\udfd9 \u2191(F.obj j)) \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) }) =\n (F.obj j).presheaf.obj\n (op\n { carrier := \u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U),\n is_open' := (_ : IsOpen (\u2191(F.map (\ud835\udfd9 j)).base \u207b\u00b9' (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U))) })) \u226b\n NatTrans.app\n (pushforwardEq (_ : (F \u22d9 forget C).map (\ud835\udfd9 j) \u226b colimit.\u03b9 (F \u22d9 forget C) j = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom\n (op { carrier := U, is_open' := hU }) =\n \ud835\udfd9\n ((F.obj j).presheaf.obj\n (op\n { carrier := \u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U,\n is_open' := (_ : IsOpen (\u2191(colimit.\u03b9 (F \u22d9 forget C) j) \u207b\u00b9' U)) }))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\n\u22a2 { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (f \u226b g) =\n { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n f \u226b\n { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n g\n[PROOFSTEP]\napply (opEquiv _ _).injective\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\n\u22a2 \u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2081)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2083))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (f \u226b g)) =\n \u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2081)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2083))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n f \u226b\n { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n g)\n[PROOFSTEP]\nrefine NatTrans.ext _ _ (funext fun U => ?_)\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2081)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2083))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n (f \u226b g)))\n U =\n NatTrans.app\n (\u2191(opEquiv\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2081)\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.obj\n j\u2083))\n ({ obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n f \u226b\n { obj := fun j => op (colimit.\u03b9 (F \u22d9 forget C) j _* (F.obj j).presheaf),\n map := fun {j j'} f =>\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf ((F \u22d9 forget C).map f) (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom).op }.map\n g))\n U\n[PROOFSTEP]\ndsimp only [comp_obj, forget_obj, Functor.comp_map, forget_map, op_comp, unop_op, pushforwardObj_obj, op_obj, opEquiv,\n Equiv.coe_fn_mk, unop_comp, Quiver.Hom.unop_op]\n -- Porting note : some `simp` lemmas are not picked up\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j\u2083) (F.map (f \u226b g)).c \u226b\n (Pushforward.comp (F.obj j\u2081).presheaf (F.map (f \u226b g)).base (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map (f \u226b g) \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083 = colimit.\u03b9 (F \u22d9 forget C) j\u2081)\n (F.obj j\u2081).presheaf).hom)\n U =\n NatTrans.app\n ((pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j\u2083) (F.map g).c \u226b\n (Pushforward.comp (F.obj j\u2082).presheaf (F.map g).base (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map g \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083 = colimit.\u03b9 (F \u22d9 forget C) j\u2082)\n (F.obj j\u2082).presheaf).hom) \u226b\n pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j\u2082) (F.map f).c \u226b\n (Pushforward.comp (F.obj j\u2081).presheaf (F.map f).base (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2082 = colimit.\u03b9 (F \u22d9 forget C) j\u2081)\n (F.obj j\u2081).presheaf).hom)\n U\n[PROOFSTEP]\nrw [NatTrans.comp_app, pushforwardMap_app, NatTrans.comp_app, Pushforward.comp_inv_app, id_comp, pushforwardEq_hom_app,\n NatTrans.comp_app, NatTrans.comp_app, NatTrans.comp_app, pushforwardMap_app, Pushforward.comp_inv_app, id_comp,\n pushforwardEq_hom_app, NatTrans.comp_app, NatTrans.comp_app, pushforwardEq_hom_app, Pushforward.comp_inv_app, id_comp,\n pushforwardMap_app]\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app (F.map (f \u226b g)).c ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).op.obj U) \u226b\n (F.obj j\u2081).presheaf.map\n (_root_.id\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop =\n (Opens.map ((F.map (f \u226b g)).base \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)).op) =\n (NatTrans.app (F.map g).c ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).op.obj U) \u226b\n (F.obj j\u2082).presheaf.map\n (_root_.id\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop =\n (Opens.map ((F.map g).base \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)).op)) \u226b\n NatTrans.app (F.map f).c ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).op.obj U) \u226b\n (F.obj j\u2081).presheaf.map\n (_root_.id\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop =\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)).op)\n[PROOFSTEP]\nsimp only [pushforwardObj_obj, op_obj, unop_op, map_comp_c_app, pushforwardMap_app, Opens.map_comp_obj,\n Pushforward.comp_inv_app, pushforwardEq_hom_app, eqToHom_op, id_eq, eqToHom_map, id_comp, assoc, eqToHom_trans]\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app (F.map g).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) \u226b\n NatTrans.app (F.map f).c\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op\n ((Opens.map (F.map f).base).obj\n ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop))) =\n NatTrans.app (F.map g).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2082).presheaf.obj\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) =\n (F.obj j\u2082).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app (F.map g).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) \u226b\n NatTrans.app (F.map f).c\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op\n ((Opens.map (F.map f).base).obj\n ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop))) =\n NatTrans.app (F.map g).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2082).presheaf.obj\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) =\n (F.obj j\u2082).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop)))\n[PROOFSTEP]\ncongr 1\n -- The key fact is `(F.map f).c.congr`,\n -- which allows us in rewrite in the argument of `(F.map f).c.app`.\n[GOAL]\ncase a.e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 NatTrans.app (F.map f).c\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op\n ((Opens.map (F.map f).base).obj\n ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop))) =\n eqToHom\n (_ :\n (F.obj j\u2082).presheaf.obj\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) =\n (F.obj j\u2082).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop)))\n[PROOFSTEP]\nrw [@NatTrans.congr (\u03b1 := (F.map f).c)\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))\n (op ((Opens.map (colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j\u2082)).obj (unop U))) _]\n[GOAL]\ncase a.e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 ((F.obj j\u2082).presheaf.map (eqToHom ?m.93087) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n ((F.map f).base _* (F.obj j\u2081).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop) =\n op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))))) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op\n ((Opens.map (F.map f).base).obj\n ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop))) =\n eqToHom\n (_ :\n (F.obj j\u2082).presheaf.obj\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) =\n (F.obj j\u2082).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop)))\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) =\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)\n[PROOFSTEP]\nswap\n -- Now we show the open sets are equal.\n[GOAL]\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) =\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)\n[PROOFSTEP]\napply unop_injective\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))).unop =\n (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)).unop\n[PROOFSTEP]\nrw [\u2190 Opens.map_comp_obj]\n[GOAL]\ncase a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 (op ((Opens.map ((F.map g).base \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)).unop =\n (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)).unop\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.e_self.e_x.e_self.e_self.e_f\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 (F.map g).base \u226b colimit.\u03b9 (F \u22d9 forget C) j\u2083 = colimit.\u03b9 (F \u22d9 forget C) j\u2082\n[PROOFSTEP]\nexact colimit.w (F \u22d9 PresheafedSpace.forget C) g\n[GOAL]\ncase a.e_a\nJ : Type u'\ninst\u271d\u00b2 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimitsOfShape J TopCat\nF : J \u2964 PresheafedSpace C\nj\u2081 j\u2082 j\u2083 : J\nf : j\u2081 \u27f6 j\u2082\ng : j\u2082 \u27f6 j\u2083\nU : (Opens \u2191(colimit (F \u22d9 forget C)))\u1d52\u1d56\n\u22a2 ((F.obj j\u2082).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)) =\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n ((F.map f).base _* (F.obj j\u2081).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop) =\n op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))))) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op\n ((Opens.map (F.map f).base).obj\n ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop)))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop))) =\n eqToHom\n (_ :\n (F.obj j\u2082).presheaf.obj\n (op ((Opens.map (F.map g).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2083)).obj U.unop))) =\n (F.obj j\u2082).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) \u226b\n NatTrans.app (F.map f).c (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop)) \u226b\n eqToHom\n (_ :\n (F.obj j\u2081).presheaf.obj\n (op ((Opens.map (F.map f).base).obj ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2082)).obj U.unop))) =\n (F.obj j\u2081).presheaf.obj (op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j\u2081)).obj U.unop)))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f \u226b\n (fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j' =\n (fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j \u226b\n ((Functor.const J).obj (colimit F)).map f\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\n\u22a2 (F.map f \u226b\n (fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').base =\n ((fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j \u226b\n ((Functor.const J).obj (colimit F)).map f).base\n[PROOFSTEP]\next x\n[GOAL]\ncase w.w\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nx : (CategoryTheory.forget TopCat).obj \u2191(F.obj j)\n\u22a2 \u2191(F.map f \u226b\n (fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').base\n x =\n \u2191((fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j \u226b\n ((Functor.const J).obj (colimit F)).map f).base\n x\n[PROOFSTEP]\nexact colimit.w_apply (F \u22d9 PresheafedSpace.forget C) f x\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\n\u22a2 (F.map f \u226b\n (fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n (F.map f \u226b\n (fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j,\n c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').base).op =\n (Opens.map\n ((fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j,\n c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j \u226b\n ((Functor.const J).obj (colimit F)).map f).base).op))\n (F.obj j).presheaf =\n ((fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j \u226b\n ((Functor.const J).obj (colimit F)).map f).c\n[PROOFSTEP]\next \u27e8U, hU\u27e9\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 NatTrans.app\n ((F.map f \u226b\n (fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n (F.map f \u226b\n (fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j,\n c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j').base).op =\n (Opens.map\n ((fun j =>\n { base := colimit.\u03b9 (F \u22d9 forget C) j,\n c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) })\n j \u226b\n ((Functor.const J).obj (colimit F)).map f).base).op))\n (F.obj j).presheaf)\n (op { carrier := U, is_open' := hU }) =\n NatTrans.app\n ((fun j => { base := colimit.\u03b9 (F \u22d9 forget C) j, c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j) }) j \u226b\n ((Functor.const J).obj (colimit F)).map f).c\n (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 NatTrans.app\n ((F.map f \u226b\n { base := colimit.\u03b9 (F \u22d9 forget C) j', c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j') }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op))\n (F.obj j).presheaf)\n (op { carrier := U, is_open' := hU }) =\n NatTrans.app (\ud835\udfd9 (colimit F)).c (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j)) (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\nrw [PresheafedSpace.id_c_app, map_id]\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 NatTrans.app\n ((F.map f \u226b\n { base := colimit.\u03b9 (F \u22d9 forget C) j', c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j') }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op))\n (F.obj j).presheaf)\n (op { carrier := U, is_open' := hU }) =\n \ud835\udfd9 ((colimit F).presheaf.obj (op { carrier := U, is_open' := hU })) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j)) (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\nerw [id_comp]\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 NatTrans.app\n ((F.map f \u226b\n { base := colimit.\u03b9 (F \u22d9 forget C) j', c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j') }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op))\n (F.obj j).presheaf)\n (op { carrier := U, is_open' := hU }) =\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j)) (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\nrw [NatTrans.comp_app, PresheafedSpace.comp_c_app, whiskerRight_app, eqToHom_app, \u2190\n congr_arg NatTrans.app (limit.w (pushforwardDiagramToColimit F).leftOp f.op), NatTrans.comp_app, Functor.leftOp_map,\n pushforwardDiagramToColimit_map]\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 (NatTrans.app { base := colimit.\u03b9 (F \u22d9 forget C) j', c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j') }.c\n (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app (F.map f).c\n (op\n ((Opens.map\n { base := colimit.\u03b9 (F \u22d9 forget C) j',\n c := limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j') }.base).obj\n (op { carrier := U, is_open' := hU }).unop))) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op.obj (op { carrier := U, is_open' := hU }) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op.obj\n (op { carrier := U, is_open' := hU }))) =\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j')) (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) (op j').unop) (F.map f.op.unop).c \u226b\n (Pushforward.comp (F.obj (op j).unop).presheaf ((F \u22d9 forget C).map f.op.unop)\n (colimit.\u03b9 (F \u22d9 forget C) (op j').unop)).inv \u226b\n (pushforwardEq\n (_ :\n (F \u22d9 forget C).map f.op.unop \u226b colimit.\u03b9 (F \u22d9 forget C) (op j').unop =\n colimit.\u03b9 (F \u22d9 forget C) (op j).unop)\n (F.obj (op j).unop).presheaf).hom).op.unop\n (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 (NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j')) (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app (F.map f).c\n (op\n { carrier := \u2191(colimit.\u03b9 (F \u22d9 forget C) j') \u207b\u00b9' U,\n is_open' := (_ : IsOpen (\u2191(colimit.\u03b9 (F \u22d9 forget C) j') \u207b\u00b9' U)) })) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op.obj (op { carrier := U, is_open' := hU }) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op.obj\n (op { carrier := U, is_open' := hU }))) =\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j')) (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app\n (pushforwardMap (colimit.\u03b9 (F \u22d9 forget C) j') (F.map f).c \u226b\n (Pushforward.comp (F.obj j).presheaf (F.map f).base (colimit.\u03b9 (F \u22d9 forget C) j')).inv \u226b\n (pushforwardEq (_ : (F \u22d9 forget C).map f \u226b colimit.\u03b9 (F \u22d9 forget C) j' = colimit.\u03b9 (F \u22d9 forget C) j)\n (F.obj j).presheaf).hom)\n (op { carrier := U, is_open' := hU })\n[PROOFSTEP]\nrw [NatTrans.comp_app, NatTrans.comp_app, pushforwardEq_hom_app, id.def, eqToHom_op, Pushforward.comp_inv_app, id_comp,\n pushforwardMap_app, \u2190 assoc]\n[GOAL]\ncase h.w.mk\nJ : Type u'\ninst\u271d\u00b3 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasColimitsOfShape J TopCat\ninst\u271d : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\nF : J \u2964 PresheafedSpace C\nj j' : J\nf : j \u27f6 j'\nU : Set \u2191\u2191(((Functor.const J).obj (colimit F)).obj j')\nhU : IsOpen U\n\u22a2 (NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j')) (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app (F.map f).c\n (op\n { carrier := \u2191(colimit.\u03b9 (F \u22d9 forget C) j') \u207b\u00b9' U,\n is_open' := (_ : IsOpen (\u2191(colimit.\u03b9 (F \u22d9 forget C) j') \u207b\u00b9' U)) })) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).op.obj (op { carrier := U, is_open' := hU }) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b \ud835\udfd9 (Limits.colimit (F \u22d9 forget C)))).op.obj\n (op { carrier := U, is_open' := hU }))) =\n (NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j')) (op { carrier := U, is_open' := hU }) \u226b\n NatTrans.app (F.map f).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j')).op.obj (op { carrier := U, is_open' := hU }))) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n op\n ((Opens.map ((F.map f).base \u226b colimit.\u03b9 (F \u22d9 forget C) j')).obj\n (op { carrier := U, is_open' := hU }).unop) =\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j)).obj (op { carrier := U, is_open' := hU }).unop)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\n\u22a2 s.pt.presheaf.obj U \u27f6\n (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s) _* limit (pushforwardDiagramToColimit F).leftOp).obj U\n[PROOFSTEP]\nrefine'\n limit.lift _\n { pt := s.pt.presheaf.obj U\n \u03c0 :=\n { app := fun j => _\n naturality := fun j j' f => _ } } \u226b\n (limitObjIsoLimitCompEvaluation _ _).inv\n[GOAL]\ncase refine'_1\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 ((Functor.const J\u1d52\u1d56).obj (s.pt.presheaf.obj U)).obj j \u27f6\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U)).obj\n j\n[PROOFSTEP]\nrefine' (s.\u03b9.app (unop j)).c.app U \u226b (F.obj (unop j)).presheaf.map (eqToHom _)\n[GOAL]\ncase refine'_1\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nrw [\u2190 Opens.map_comp_obj]\n[GOAL]\ncase refine'_1\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\n\u22a2 ((Functor.const J\u1d52\u1d56).obj (s.pt.presheaf.obj U)).map f \u226b\n (fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))))\n j' =\n (fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))))\n j \u226b\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U)).map\n f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n NatTrans.app (NatTrans.app s.\u03b9 j'.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nrw [PresheafedSpace.congr_app (s.w f.unop).symm U]\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nhave w :=\n Functor.congr_obj (congr_arg Opens.map (colimit.\u03b9_desc ((PresheafedSpace.forget C).mapCocone s) (unop j))) (unop U)\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\nw :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nsimp only [Opens.map_comp_obj_unop] at w \n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\nw :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop) =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nreplace w := congr_arg op w\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\nw :\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nhave w' := NatTrans.congr (F.map f.unop).c w\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\nw :\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)\nw' :\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) =\n (F.obj j.unop).presheaf.map (eqToHom w) \u226b\n NatTrans.app (F.map f.unop).c (op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)) \u226b\n ((F.map f.unop).base _* (F.obj j'.unop).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nrw [w']\n[GOAL]\ncase refine'_2\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj j' : J\u1d52\u1d56\nf : j \u27f6 j'\nw :\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)\nw' :\n NatTrans.app (F.map f.unop).c\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) =\n (F.obj j.unop).presheaf.map (eqToHom w) \u226b\n NatTrans.app (F.map f.unop).c (op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)) \u226b\n ((F.map f.unop).base _* (F.obj j'.unop).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n\u22a2 \ud835\udfd9 (s.pt.presheaf.obj U) \u226b\n (NatTrans.app (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (F.map f.unop \u226b NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j'.unop).base).op.obj U))) \u226b\n (F.obj j'.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j'.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j'.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) =\n (NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n ((F.obj j.unop).presheaf.map (eqToHom w) \u226b\n NatTrans.app (F.map f.unop).c (op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop)) \u226b\n ((F.map f.unop).base _* (F.obj j'.unop).presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop) =\n op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj j'.unop).presheaf (F.map f.unop).base (colimit.\u03b9 (F \u22d9 forget C) j.unop)).inv\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) j.unop = colimit.\u03b9 (F \u22d9 forget C) j'.unop)\n (F.obj j'.unop).presheaf).hom\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 s.pt.presheaf.map i \u226b descCApp F s V =\n descCApp F s U \u226b (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s) _* (colimitCocone F).pt.presheaf).map i\n[PROOFSTEP]\ndsimp [descCApp]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 s.pt.presheaf.map i \u226b\n limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))\n { pt := s.pt.presheaf.obj V,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c V \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj V =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj V))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop))).inv =\n (limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))\n { pt := s.pt.presheaf.obj U,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))).inv) \u226b\n (limit (pushforwardDiagramToColimit F).leftOp).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop).op\n[PROOFSTEP]\nrefine limit_obj_ext (fun j => ?_)\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\n\u22a2 (s.pt.presheaf.map i \u226b\n limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))\n { pt := s.pt.presheaf.obj V,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c V \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj V =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj V))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop))).inv) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)) =\n ((limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))\n { pt := s.pt.presheaf.obj U,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))).inv) \u226b\n (limit (pushforwardDiagramToColimit F).leftOp).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop).op) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop))\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, NatTrans.naturality, limit.lift_\u03c0_assoc, eqToHom_map, assoc, pushforwardObj_map,\n NatTrans.naturality_assoc, op_map, limitObjIsoLimitCompEvaluation_inv_\u03c0_app_assoc,\n limitObjIsoLimitCompEvaluation_inv_\u03c0_app]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n ((pushforwardDiagramToColimit F).leftOp.obj j).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop).op\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op\n[PROOFSTEP]\nhave w :=\n Functor.congr_hom (congr_arg Opens.map (colimit.\u03b9_desc ((PresheafedSpace.forget C).mapCocone s) (unop j))) i.unop\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\nw :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop =\n eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n V.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj V.unop) \u226b\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).map i.unop \u226b\n eqToHom\n (_ :\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n U.unop)\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op\n[PROOFSTEP]\nsimp only [Opens.map_comp_map] at w \n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\nw :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop) =\n eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n V.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj V.unop) \u226b\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).map i.unop \u226b\n eqToHom\n (_ :\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n U.unop)\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op\n[PROOFSTEP]\nreplace w := congr_arg Quiver.Hom.op w\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\nw :\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op =\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n V.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj V.unop) \u226b\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).map i.unop \u226b\n eqToHom\n (_ :\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n U.unop)).op\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op\n[PROOFSTEP]\nrw [w]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nU V : (Opens \u2191\u2191s.pt)\u1d52\u1d56\ni : U \u27f6 V\nj : J\u1d52\u1d56\nw :\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).map\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).map i.unop)).op =\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n V.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj V.unop) \u226b\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).map i.unop \u226b\n eqToHom\n (_ :\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n U.unop)).op\n\u22a2 NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).map i.unop).op \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj V.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj V.unop)))) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n V.unop =\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj V.unop) \u226b\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).map i.unop \u226b\n eqToHom\n (_ :\n (Opens.map (NatTrans.app ((forget C).mapCocone s).\u03b9 j.unop)).obj U.unop =\n (Opens.map\n (colimit.\u03b9 (F \u22d9 forget C) j.unop \u226b\n colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj\n U.unop)).op\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\n\u22a2 NatTrans.app (colimitCocone F).\u03b9 j \u226b desc F s = NatTrans.app s.\u03b9 j\n[PROOFSTEP]\next U\n[GOAL]\ncase w.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : (CategoryTheory.forget TopCat).obj \u2191(F.obj j)\n\u22a2 \u2191(NatTrans.app (colimitCocone F).\u03b9 j \u226b desc F s).base U = \u2191(NatTrans.app s.\u03b9 j).base U\n[PROOFSTEP]\nsimp [desc]\n[GOAL]\ncase h.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : Opens \u2191\u2191s.pt\n\u22a2 NatTrans.app\n ((NatTrans.app (colimitCocone F).\u03b9 j \u226b desc F s).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app (colimitCocone F).\u03b9 j \u226b desc F s).base).op =\n (Opens.map (NatTrans.app s.\u03b9 j).base).op))\n (F.obj j).presheaf)\n (op U) =\n NatTrans.app (NatTrans.app s.\u03b9 j).c (op U)\n[PROOFSTEP]\nrw [NatTrans.comp_app, PresheafedSpace.comp_c_app, whiskerRight_app]\n[GOAL]\ncase h.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : Opens \u2191\u2191s.pt\n\u22a2 (NatTrans.app (desc F s).c (op U) \u226b\n NatTrans.app (NatTrans.app (colimitCocone F).\u03b9 j).c (op ((Opens.map (desc F s).base).obj (op U).unop))) \u226b\n (F.obj j).presheaf.map\n (NatTrans.app\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app (colimitCocone F).\u03b9 j \u226b desc F s).base).op =\n (Opens.map (NatTrans.app s.\u03b9 j).base).op))\n (op U)) =\n NatTrans.app (NatTrans.app s.\u03b9 j).c (op U)\n[PROOFSTEP]\ndsimp [desc, descCApp]\n[GOAL]\ncase h.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : Opens \u2191\u2191s.pt\n\u22a2 ((limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U)))\n { pt := s.pt.presheaf.obj (op U),\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c (op U) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj (op U) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj (op U)))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U))).inv) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp (op j))\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U))) \u226b\n (F.obj j).presheaf.map\n (NatTrans.app\n (eqToHom\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op =\n (Opens.map (NatTrans.app s.\u03b9 j).base).op))\n (op U)) =\n NatTrans.app (NatTrans.app s.\u03b9 j).c (op U)\n[PROOFSTEP]\nsimp only [eqToHom_app, op_obj, Opens.map_comp_obj, eqToHom_map, Functor.leftOp, assoc]\n[GOAL]\ncase h.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : Opens \u2191\u2191s.pt\n\u22a2 limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun {X Y} f => ((pushforwardDiagramToColimit F).map f.unop).unop } \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U)))\n { pt := s.pt.presheaf.obj (op U),\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c (op U) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj (op U) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj (op U)))) } \u226b\n (limitObjIsoLimitCompEvaluation\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun {X Y} f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U))).inv \u226b\n NatTrans.app\n (limit.\u03c0\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun {X Y} f => ((pushforwardDiagramToColimit F).map f.unop).unop })\n (op j))\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U)) \u226b\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U))) =\n (F.obj j).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j).base).obj U))) =\n NatTrans.app (NatTrans.app s.\u03b9 j).c (op U)\n[PROOFSTEP]\nrw [limitObjIsoLimitCompEvaluation_inv_\u03c0_app_assoc]\n[GOAL]\ncase h.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nj : J\nU : Opens \u2191\u2191s.pt\n\u22a2 limit.lift\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun {X Y} f => ((pushforwardDiagramToColimit F).map f.unop).unop } \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U)))\n { pt := s.pt.presheaf.obj (op U),\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c (op U) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj (op U) =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj (op U)))) } \u226b\n limit.\u03c0\n (CategoryTheory.Functor.mk\n { obj := fun X => ((pushforwardDiagramToColimit F).obj X.unop).unop,\n map := fun {X Y} f => ((pushforwardDiagramToColimit F).map f.unop).unop } \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U)))\n (op j) \u226b\n eqToHom\n (_ :\n (F.obj j).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U))) =\n (F.obj j).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j).base).obj U))) =\n NatTrans.app (NatTrans.app s.\u03b9 j).c (op U)\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m = (fun s => desc F s) s\n[PROOFSTEP]\nhave t : m.base = colimit.desc (F \u22d9 PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) :=\n by\n dsimp\n ext j\n rw [colimit.\u03b9_desc, mapCocone_\u03b9_app, \u2190 w j]\n simp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\n[PROOFSTEP]\next j\n[GOAL]\ncase w.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nj : J\nx\u271d : (CategoryTheory.forget TopCat).obj ((F \u22d9 forget C).obj j)\n\u22a2 \u2191(colimit.\u03b9 (F \u22d9 forget C) j \u226b m.base) x\u271d =\n \u2191(colimit.\u03b9 (F \u22d9 forget C) j \u226b colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)) x\u271d\n[PROOFSTEP]\nrw [colimit.\u03b9_desc, mapCocone_\u03b9_app, \u2190 w j]\n[GOAL]\ncase w.w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nj : J\nx\u271d : (CategoryTheory.forget TopCat).obj ((F \u22d9 forget C).obj j)\n\u22a2 \u2191(colimit.\u03b9 (F \u22d9 forget C) j \u226b m.base) x\u271d = \u2191((forget C).map (NatTrans.app (colimitCocone F).\u03b9 j \u226b m)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\n\u22a2 m = (fun s => desc F s) s\n[PROOFSTEP]\next : 1\n[GOAL]\ncase w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\n\u22a2 m.base = ((fun s => desc F s) s).base\n[PROOFSTEP]\nexact t\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\n\u22a2 m.c \u226b\n whiskerRight (eqToHom (_ : (Opens.map m.base).op = (Opens.map ((fun s => desc F s) s).base).op))\n (colimitCocone F).pt.presheaf =\n ((fun s => desc F s) s).c\n[PROOFSTEP]\nrefine NatTrans.ext _ _ (funext fun U => limit_obj_ext fun j => ?_)\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app\n (m.c \u226b\n whiskerRight (eqToHom (_ : (Opens.map m.base).op = (Opens.map ((fun s => desc F s) s).base).op))\n (colimitCocone F).pt.presheaf)\n U \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n ((Opens.map ((fun s => desc F s) s).base).op.obj U) =\n NatTrans.app ((fun s => desc F s) s).c U \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n ((Opens.map ((fun s => desc F s) s).base).op.obj U)\n[PROOFSTEP]\ndsimp only [colimitCocone_pt, colimit_carrier, leftOp_obj, pushforwardDiagramToColimit_obj, comp_obj, forget_obj,\n unop_op, op_obj, desc, colimit_presheaf, descCApp, mapCocone_pt, pushforwardObj_obj, const_obj_obj, id_eq,\n evaluation_obj_obj, Eq.ndrec, eq_mpr_eq_cast]\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app\n (m.c \u226b\n whiskerRight\n (eqToHom\n (_ : (Opens.map m.base).op = (Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op))\n (limit (pushforwardDiagramToColimit F).leftOp))\n U \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n (limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))\n { pt := s.pt.presheaf.obj U,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))).inv) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nrw [NatTrans.comp_app, whiskerRight_app]\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 (NatTrans.app m.c U \u226b\n (limit (pushforwardDiagramToColimit F).leftOp).map\n (NatTrans.app\n (eqToHom\n (_ : (Opens.map m.base).op = (Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op))\n U)) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n (limit.lift\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191(Limits.colimit (F \u22d9 forget C)))\u1d52\u1d56 C).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)))\n { pt := s.pt.presheaf.obj U,\n \u03c0 :=\n NatTrans.mk fun j =>\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U =\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).op.obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).op.obj U))) } \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))).inv) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))\n[PROOFSTEP]\nsimp only [pushforwardObj_obj, op_obj, comp_obj, eqToHom_app, eqToHom_map, assoc,\n limitObjIsoLimitCompEvaluation_inv_\u03c0_app, limit.lift_\u03c0]\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app m.c U \u226b\n eqToHom\n (_ :\n (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =\n (limit (pushforwardDiagramToColimit F).leftOp).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n NatTrans.app (NatTrans.app s.\u03b9 j.unop).c U \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n[PROOFSTEP]\nrw [PresheafedSpace.congr_app (w (unop j)).symm U]\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app m.c U \u226b\n eqToHom\n (_ :\n (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =\n (limit (pushforwardDiagramToColimit F).leftOp).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n (NatTrans.app (NatTrans.app (colimitCocone F).\u03b9 j.unop \u226b m).c U \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n (Opens.map (NatTrans.app (colimitCocone F).\u03b9 j.unop \u226b m).base).op.obj U =\n (Opens.map (NatTrans.app s.\u03b9 j.unop).base).op.obj U))) \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\n\u22a2 NatTrans.app m.c U \u226b\n eqToHom\n (_ :\n (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =\n (limit (pushforwardDiagramToColimit F).leftOp).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n ((NatTrans.app m.c U \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j) (op ((Opens.map m.base).obj U.unop))) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app (colimitCocone F).\u03b9 j.unop \u226b m).base).obj U.unop) =\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)))) \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n[PROOFSTEP]\nhave w := congr_arg op (Functor.congr_obj (congr_arg Opens.map t) (unop U))\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw\u271d : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\nw :\n op ((Opens.map m.base).obj U.unop) =\n op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)\n\u22a2 NatTrans.app m.c U \u226b\n eqToHom\n (_ :\n (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =\n (limit (pushforwardDiagramToColimit F).leftOp).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n ((NatTrans.app m.c U \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j) (op ((Opens.map m.base).obj U.unop))) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app (colimitCocone F).\u03b9 j.unop \u226b m).base).obj U.unop) =\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)))) \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n[PROOFSTEP]\nrw [NatTrans.congr (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j) w]\n[GOAL]\ncase h\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ns : Cocone F\nm : (colimitCocone F).pt \u27f6 s.pt\nw\u271d : \u2200 (j : J), NatTrans.app (colimitCocone F).\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nt : m.base = colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s)\nU : (Opens \u2191\u2191s.pt)\u1d52\u1d56\nj : J\u1d52\u1d56\nw :\n op ((Opens.map m.base).obj U.unop) =\n op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)\n\u22a2 NatTrans.app m.c U \u226b\n eqToHom\n (_ :\n (limit (pushforwardDiagramToColimit F).leftOp).obj (op ((Opens.map m.base).obj U.unop)) =\n (limit (pushforwardDiagramToColimit F).leftOp).obj\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) =\n ((NatTrans.app m.c U \u226b\n (limit (pushforwardDiagramToColimit F).leftOp).map (eqToHom w) \u226b\n NatTrans.app (limit.\u03c0 (pushforwardDiagramToColimit F).leftOp j)\n (op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop)) \u226b\n ((pushforwardDiagramToColimit F).leftOp.obj j).map\n (eqToHom\n (_ :\n op ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop) =\n op ((Opens.map m.base).obj U.unop)))) \u226b\n (F.obj j.unop).presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map (NatTrans.app (colimitCocone F).\u03b9 j.unop \u226b m).base).obj U.unop) =\n op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)))) \u226b\n eqToHom\n (_ :\n (F.obj j.unop).presheaf.obj (op ((Opens.map (NatTrans.app s.\u03b9 j.unop).base).obj U.unop)) =\n (F.obj j.unop).presheaf.obj\n (op\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) j.unop)).obj\n ((Opens.map (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s))).obj U.unop))))\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\n\u22a2 IsColimit ((forget C).mapCocone (colimitCocone F))\n[PROOFSTEP]\napply IsColimit.ofIsoColimit (colimit.isColimit _)\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\n\u22a2 colimit.cocone (F \u22d9 forget C) \u2245 (forget C).mapCocone (colimitCocone F)\n[PROOFSTEP]\nfapply Cocones.ext\n[GOAL]\ncase \u03c6\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\n\u22a2 (colimit.cocone (F \u22d9 forget C)).pt \u2245 ((forget C).mapCocone (colimitCocone F)).pt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\n\u22a2 autoParam\n (\u2200 (j : J),\n NatTrans.app (colimit.cocone (F \u22d9 forget C)).\u03b9 j \u226b (Iso.refl (colimit.cocone (F \u22d9 forget C)).pt).hom =\n NatTrans.app ((forget C).mapCocone (colimitCocone F)).\u03b9 j)\n _auto\u271d\n[PROOFSTEP]\nintro j\n[GOAL]\ncase w\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nj : J\n\u22a2 NatTrans.app (colimit.cocone (F \u22d9 forget C)).\u03b9 j \u226b (Iso.refl (colimit.cocone (F \u22d9 forget C)).pt).hom =\n NatTrans.app ((forget C).mapCocone (colimitCocone F)).\u03b9 j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ\u271d : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\u271d\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J\u271d TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u271d\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u271d\u1d52\u1d56 C\ninst\u271d : HasLimits C\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 PresheafedSpace C\n\u22a2 IsColimit ((forget C).mapCocone (colimitCocone F))\n[PROOFSTEP]\napply IsColimit.ofIsoColimit (colimit.isColimit _)\n[GOAL]\nJ\u271d : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\u271d\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J\u271d TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u271d\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u271d\u1d52\u1d56 C\ninst\u271d : HasLimits C\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 PresheafedSpace C\n\u22a2 colimit.cocone (F \u22d9 forget C) \u2245 (forget C).mapCocone (colimitCocone F)\n[PROOFSTEP]\nfapply Cocones.ext\n[GOAL]\ncase \u03c6\nJ\u271d : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\u271d\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J\u271d TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u271d\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u271d\u1d52\u1d56 C\ninst\u271d : HasLimits C\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 PresheafedSpace C\n\u22a2 (colimit.cocone (F \u22d9 forget C)).pt \u2245 ((forget C).mapCocone (colimitCocone F)).pt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase w\nJ\u271d : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\u271d\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J\u271d TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u271d\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u271d\u1d52\u1d56 C\ninst\u271d : HasLimits C\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 PresheafedSpace C\n\u22a2 autoParam\n (\u2200 (j : J),\n NatTrans.app (colimit.cocone (F \u22d9 forget C)).\u03b9 j \u226b (Iso.refl (colimit.cocone (F \u22d9 forget C)).pt).hom =\n NatTrans.app ((forget C).mapCocone (colimitCocone F)).\u03b9 j)\n _auto\u271d\n[PROOFSTEP]\nintro j\n[GOAL]\ncase w\nJ\u271d : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\u271d\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J\u271d TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u271d\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u271d\u1d52\u1d56 C\ninst\u271d : HasLimits C\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\nF : J \u2964 PresheafedSpace C\nj : J\n\u22a2 NatTrans.app (colimit.cocone (F \u22d9 forget C)).\u03b9 j \u226b (Iso.refl (colimit.cocone (F \u22d9 forget C)).pt).hom =\n NatTrans.app ((forget C).mapCocone (colimitCocone F)).\u03b9 j\n[PROOFSTEP]\nsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\n\u22a2 (Limits.colimit F).presheaf.obj (op U) \u2245 limit (componentwiseDiagram F U)\n[PROOFSTEP]\nrefine' ((sheafIsoOfIso (colimit.isoColimitCocone \u27e8_, colimitCoconeIsColimit F\u27e9).symm).app (op U)).trans _\n[GOAL]\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\n\u22a2 ((colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base _*\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt.presheaf).obj\n (op U) \u2245\n limit (componentwiseDiagram F U)\n[PROOFSTEP]\nrefine' (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso _)\n[GOAL]\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\n\u22a2 (pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191\u2191{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)\u1d52\u1d56 C).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) \u2245\n componentwiseDiagram F U\n[PROOFSTEP]\nfapply NatIso.ofComponents\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\n\u22a2 (X : J\u1d52\u1d56) \u2192\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191\u2191{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)\u1d52\u1d56\n C).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))).obj\n X \u2245\n (componentwiseDiagram F U).obj X\n[PROOFSTEP]\nintro X\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\n\u22a2 ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191\u2191{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)\u1d52\u1d56 C).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))).obj\n X \u2245\n (componentwiseDiagram F U).obj X\n[PROOFSTEP]\nrefine' (F.obj (unop X)).presheaf.mapIso (eqToIso _)\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\n\u22a2 (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)\n[PROOFSTEP]\nsimp only [Functor.op_obj, unop_op, op_inj_iff, Opens.map_coe, SetLike.ext'_iff, Set.preimage_preimage]\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\n\u22a2 (fun x =>\n \u2191(colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base\n (\u2191(colimit.\u03b9 (F \u22d9 forget C) X.unop) x)) \u207b\u00b9'\n \u2191U =\n \u2191(colimit.\u03b9 F X.unop).base \u207b\u00b9' \u2191U\n[PROOFSTEP]\nrefine congr_arg (Set.preimage . U.1) (funext fun x => ?_)\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\nx : (CategoryTheory.forget TopCat).obj ((F \u22d9 forget C).obj X.unop)\n\u22a2 \u2191(colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base\n (\u2191(colimit.\u03b9 (F \u22d9 forget C) X.unop) x) =\n \u2191(colimit.\u03b9 F X.unop).base x\n[PROOFSTEP]\nerw [\u2190 comp_app]\n[GOAL]\ncase app\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\nx : (CategoryTheory.forget TopCat).obj ((F \u22d9 forget C).obj X.unop)\n\u22a2 \u2191(colimit.\u03b9 (F \u22d9 forget C) X.unop \u226b\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base)\n x =\n \u2191(colimit.\u03b9 F X.unop).base x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase app.e_a\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX : J\u1d52\u1d56\nx : (CategoryTheory.forget TopCat).obj ((F \u22d9 forget C).obj X.unop)\n\u22a2 colimit.\u03b9 (F \u22d9 forget C) X.unop \u226b\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base =\n (colimit.\u03b9 F X.unop).base\n[PROOFSTEP]\nexact \u03b9_preservesColimitsIso_inv (forget C) F (unop X)\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\n\u22a2 autoParam\n (\u2200 {X Y : J\u1d52\u1d56} (f : X \u27f6 Y),\n ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191\u2191{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)\u1d52\u1d56\n C).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))).map\n f \u226b\n ((F.obj Y.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U)))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f)\n _auto\u271d\n[PROOFSTEP]\nintro X Y f\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 ((pushforwardDiagramToColimit F).leftOp \u22d9\n (evaluation (Opens \u2191\u2191{ cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.pt)\u1d52\u1d56\n C).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))).map\n f \u226b\n ((F.obj Y.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U)))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f\n[PROOFSTEP]\nchange ((F.map f.unop).c.app _ \u226b _ \u226b _) \u226b (F.obj (unop Y)).presheaf.map _ = _ \u226b _\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n NatTrans.app\n (Pushforward.comp (F.obj Y.unop).presheaf ((F \u22d9 forget C).map f.unop) (colimit.\u03b9 (F \u22d9 forget C) X.unop)).inv\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop = colimit.\u03b9 (F \u22d9 forget C) Y.unop)\n (F.obj Y.unop).presheaf).hom\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n (F.obj Y.unop).presheaf.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f\n[PROOFSTEP]\nrw [TopCat.Presheaf.Pushforward.comp_inv_app]\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n \ud835\udfd9\n ((colimit.\u03b9 (F \u22d9 forget C) X.unop _* ((F \u22d9 forget C).map f.unop _* (F.obj Y.unop).presheaf)).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop = colimit.\u03b9 (F \u22d9 forget C) Y.unop)\n (F.obj Y.unop).presheaf).hom\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n (F.obj Y.unop).presheaf.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f\n[PROOFSTEP]\nerw [Category.id_comp]\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop = colimit.\u03b9 (F \u22d9 forget C) Y.unop)\n (F.obj Y.unop).presheaf).hom\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n (F.obj Y.unop).presheaf.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n NatTrans.app\n (pushforwardEq\n (_ : (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop = colimit.\u03b9 (F \u22d9 forget C) Y.unop)\n (F.obj Y.unop).presheaf).hom\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) \u226b\n (F.obj Y.unop).presheaf.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U))).hom =\n ((F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).hom \u226b\n (componentwiseDiagram F U).map f\n[PROOFSTEP]\nerw [\u2190 (F.obj (unop Y)).presheaf.map_comp, (F.map f.unop).c.naturality_assoc, \u2190 (F.obj (unop Y)).presheaf.map_comp]\n[GOAL]\ncase naturality\nJ : Type u'\ninst\u271d\u2075 : Category.{v', u'} J\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasColimitsOfShape J TopCat\ninst\u271d\u00b2 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d\u00b9 : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\ninst\u271d : HasColimit F\nU : Opens \u2191\u2191(Limits.colimit F)\nX Y : J\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n (F.obj Y.unop).presheaf.map\n (NatTrans.app\n (NatIso.op\n (Opens.mapIso ((F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop)\n (colimit.\u03b9 (F \u22d9 forget C) Y.unop)\n (_ :\n (F \u22d9 forget C).map f.unop \u226b colimit.\u03b9 (F \u22d9 forget C) X.unop =\n colimit.\u03b9 (F \u22d9 forget C) Y.unop)).symm).hom\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) \u226b\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) Y.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U))).hom) =\n NatTrans.app (F.map f.unop).c\n ((Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))) \u226b\n (F.obj Y.unop).presheaf.map\n ((Opens.map (F.map f.unop).base).op.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U))).hom \u226b\n eqToHom\n (_ :\n (Opens.map (F.map f.unop).base).op.obj (op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)) =\n op ((Opens.map (colimit.\u03b9 F Y.unop).base).obj U)))\n[PROOFSTEP]\nrfl\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 (colimitPresheafObjIsoComponentwiseLimit F U).inv \u226b NatTrans.app (colimit.\u03b9 F j).c (op U) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\ndelta colimitPresheafObjIsoComponentwiseLimit\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 ((sheafIsoOfIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).symm).app\n (op U) \u226a\u226b\n limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) \u226a\u226b\n Limits.lim.mapIso\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U))))).inv \u226b\n NatTrans.app (colimit.\u03b9 F j).c (op U) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nrw [Iso.trans_inv, Iso.trans_inv, Iso.app_inv, sheafIsoOfIso_inv, pushforwardToOfIso_app, congr_app (Iso.symm_inv _)]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 (((Limits.lim.mapIso\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U))))).inv \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U))).inv) \u226b\n (NatTrans.app\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c\n (op\n ((Opens.map\n ((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv).obj\n (op U).unop)) \u226b\n (Limits.colimit F).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).op.obj\n (op\n ((Opens.map\n ((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv).obj\n (op U).unop)) =\n (Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.inv.base).op.obj\n (op\n ((Opens.map\n ((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv).obj\n (op U).unop))))) \u226b\n (Limits.colimit F).presheaf.map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.hom \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv \u207b\u00b9'\n \u2191(op U).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.hom \u207b\u00b9'\n \u2191(op\n {\n carrier :=\n \u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv \u207b\u00b9'\n \u2191(op U).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191((forget C).mapIso\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm).symm.inv \u207b\u00b9'\n \u2191(op U).unop)) }).unop)) } =\n op U))) \u226b\n NatTrans.app (colimit.\u03b9 F j).c (op U) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\ndsimp\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 ((limMap\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))).inv) \u226b\n (NatTrans.app\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) \u226b\n (Limits.colimit F).presheaf.map\n (\ud835\udfd9\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))))) \u226b\n (Limits.colimit F).presheaf.map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))) \u226b\n NatTrans.app (colimit.\u03b9 F j).c (op U) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nrw [map_id, comp_id, assoc, assoc, assoc, NatTrans.naturality]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 limMap\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))).inv \u226b\n NatTrans.app\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.c\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) \u226b\n NatTrans.app (colimit.\u03b9 F j).c\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base).obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nerw [\u2190 comp_c_app_assoc]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 limMap\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))).inv \u226b\n NatTrans.app\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).c\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nrw [congr_app (colimit.isoColimitCocone_\u03b9_hom _ _), assoc]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 limMap\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv \u226b\n (limitObjIsoLimitCompEvaluation (pushforwardDiagramToColimit F).leftOp\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))).inv \u226b\n NatTrans.app (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9 j).c\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nerw [limitObjIsoLimitCompEvaluation_inv_\u03c0_app_assoc, limMap_\u03c0_assoc]\n -- Porting note : `convert` doesn't work due to meta variable, so change to a `suffices` block\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 limit.\u03c0 (componentwiseDiagram F U) (op j) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nset f := _\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : ?m.311389 := ?m.311390\n\u22a2 limit.\u03c0 (componentwiseDiagram F U) (op j) \u226b\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nchange _ \u226b f = _\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\n\u22a2 limit.\u03c0 (componentwiseDiagram F U) (op j) \u226b f = limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nsuffices f_eq : f = \ud835\udfd9 _ by rw [f_eq, comp_id]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\nf_eq : f = \ud835\udfd9 ((componentwiseDiagram F U).obj (op j))\n\u22a2 limit.\u03c0 (componentwiseDiagram F U) (op j) \u226b f = limit.\u03c0 (componentwiseDiagram F U) (op j)\n[PROOFSTEP]\nrw [f_eq, comp_id]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\n\u22a2 f = \ud835\udfd9 ((componentwiseDiagram F U).obj (op j))\n[PROOFSTEP]\nerw [\u2190 (F.obj j).presheaf.map_id]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\n\u22a2 f = (F.obj j).presheaf.map (\ud835\udfd9 (op ((Opens.map (colimit.\u03b9 F (op j).unop).base).obj U)))\n[PROOFSTEP]\nchange (F.obj j).presheaf.map _ \u226b _ = _\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\n\u22a2 (F.obj j).presheaf.map\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) (op j).unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F (op j).unop).base).obj U))).inv \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U)) =\n (F.obj j).presheaf.map (\ud835\udfd9 (op ((Opens.map (colimit.\u03b9 F (op j).unop).base).obj U)))\n[PROOFSTEP]\nerw [\u2190 (F.obj j).presheaf.map_comp, \u2190 (F.obj j).presheaf.map_comp]\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\nf : (componentwiseDiagram F U).obj (op j) \u27f6 (F.obj j).presheaf.obj (op ((Opens.map (colimit.\u03b9 F j).base).obj U)) :=\n NatTrans.app\n (NatIso.ofComponents fun X =>\n (F.obj X.unop).presheaf.mapIso\n (eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) X.unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F X.unop).base).obj U)))).inv\n (op j) \u226b\n (F.obj j).presheaf.map\n (eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)))) \u226b\n ((colimit.\u03b9 F j).base _* (F.obj j).presheaf).map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))\n\u22a2 (F.obj j).presheaf.map\n ((eqToIso\n (_ :\n (Opens.map (colimit.\u03b9 (F \u22d9 forget C) (op j).unop)).op.obj\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).symm.hom.base).op.obj\n (op U)) =\n op ((Opens.map (colimit.\u03b9 F (op j).unop).base).obj U))).inv \u226b\n eqToHom\n (_ :\n (Opens.map\n (NatTrans.app { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }.cocone.\u03b9\n j).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U)) =\n (Opens.map\n (colimit.\u03b9 F j \u226b\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom).base).op.obj\n (op\n ((Opens.map\n (colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base).obj\n U))) \u226b\n (Opens.map (colimit.\u03b9 F j).base).op.map\n (eqToHom\n (_ :\n op\n {\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U),\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F }).hom.base \u207b\u00b9'\n \u2191{\n carrier :=\n \u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U,\n is_open' :=\n (_ :\n IsOpen\n (\u2191(colimit.isoColimitCocone\n { cocone := colimitCocone F,\n isColimit := colimitCoconeIsColimit F }).inv.base \u207b\u00b9'\n \u2191U)) })) } =\n op U))) =\n (F.obj j).presheaf.map (\ud835\udfd9 (op ((Opens.map (colimit.\u03b9 F (op j).unop).base).obj U)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nJ : Type u'\ninst\u271d\u2074 : Category.{v', u'} J\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasColimitsOfShape J TopCat\ninst\u271d\u00b9 : \u2200 (X : TopCat), HasLimitsOfShape J\u1d52\u1d56 (Presheaf C X)\ninst\u271d : HasLimitsOfShape J\u1d52\u1d56 C\nF : J \u2964 PresheafedSpace C\nU : Opens \u2191\u2191(Limits.colimit F)\nj : J\n\u22a2 (colimitPresheafObjIsoComponentwiseLimit F U).hom \u226b limit.\u03c0 (componentwiseDiagram F U) (op j) =\n NatTrans.app (colimit.\u03b9 F j).c (op U)\n[PROOFSTEP]\nrw [\u2190 Iso.eq_inv_comp, colimitPresheafObjIsoComponentwiseLimit_inv_\u03b9_app]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.PresheafedSpace.HasColimits", "llama_tokens": 95293, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.2609440421271814}} {"text": "[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun n => (-1) ^ n * \u2016q\u2016 ^ (2 * n) / \u2191(2 * n)!) c\nhs : HasSum (fun n => (-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / \u2191(2 * n + 1)!) s\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => q) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nreplace hc := hasSum_coe.mpr hc\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhs : HasSum (fun n => (-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / \u2191(2 * n + 1)!) s\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => q) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nreplace hs := (hs.div_const \u2016q\u2016).smul_const q\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => q) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nobtain rfl | hq0 := eq_or_ne q 0\n[GOAL]\ncase inl\nc s : \u211d\nhq : 0.re = 0\nhc : HasSum (fun a => \u2191((-1) ^ a * \u20160\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => 0) (\u2191c + (s / \u20160\u2016) \u2022 0)\n[PROOFSTEP]\nsimp_rw [expSeries_apply_zero, norm_zero, div_zero, zero_smul, add_zero]\n[GOAL]\ncase inl\nc s : \u211d\nhq : 0.re = 0\nhc : HasSum (fun a => \u2191((-1) ^ a * \u20160\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\n\u22a2 HasSum (fun n => Pi.single 0 1 n) \u2191c\n[PROOFSTEP]\nsimp_rw [norm_zero] at hc \n[GOAL]\ncase inl\nc s : \u211d\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\nhc : HasSum (fun a => \u2191((-1) ^ a * 0 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\n\u22a2 HasSum (fun n => Pi.single 0 1 n) \u2191c\n[PROOFSTEP]\nconvert hc using 1\n[GOAL]\ncase h.e'_5\nc s : \u211d\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\nhc : HasSum (fun a => \u2191((-1) ^ a * 0 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\n\u22a2 (fun n => Pi.single 0 1 n) = fun a => \u2191((-1) ^ a * 0 ^ (2 * a) / \u2191(2 * a)!)\n[PROOFSTEP]\next (_ | n) : 1\n[GOAL]\ncase h.e'_5.h.zero\nc s : \u211d\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\nhc : HasSum (fun a => \u2191((-1) ^ a * 0 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\n\u22a2 Pi.single 0 1 Nat.zero = \u2191((-1) ^ Nat.zero * 0 ^ (2 * Nat.zero) / \u2191(2 * Nat.zero)!)\n[PROOFSTEP]\nrw [pow_zero, Nat.zero_eq, mul_zero, pow_zero, Nat.factorial_zero, Nat.cast_one, div_one, one_mul, Pi.single_eq_same,\n coe_one]\n[GOAL]\ncase h.e'_5.h.succ\nc s : \u211d\nhq : 0.re = 0\nhs : HasSum (fun z => ((-1) ^ z * \u20160\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u20160\u2016) \u2022 0) ((s / \u20160\u2016) \u2022 0)\nhc : HasSum (fun a => \u2191((-1) ^ a * 0 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nn : \u2115\n\u22a2 Pi.single 0 1 (Nat.succ n) = \u2191((-1) ^ Nat.succ n * 0 ^ (2 * Nat.succ n) / \u2191(2 * Nat.succ n)!)\n[PROOFSTEP]\nrw [zero_pow (mul_pos two_pos (Nat.succ_pos _)), mul_zero, zero_div, Pi.single_eq_of_ne n.succ_ne_zero, coe_zero]\n[GOAL]\ncase inr\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => q) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nsimp_rw [expSeries_apply_eq]\n[GOAL]\ncase inr\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\n\u22a2 HasSum (fun n => (\u2191n !)\u207b\u00b9 \u2022 q ^ n) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nhave hq2 : q ^ 2 = -normSq q := sq_eq_neg_normSq.mpr hq\n[GOAL]\ncase inr\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\n\u22a2 HasSum (fun n => (\u2191n !)\u207b\u00b9 \u2022 q ^ n) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nhave hqn := norm_ne_zero_iff.mpr hq0\n[GOAL]\ncase inr\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\n\u22a2 HasSum (fun n => (\u2191n !)\u207b\u00b9 \u2022 q ^ n) (\u2191c + (s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nrefine' HasSum.even_add_odd _ _\n[GOAL]\ncase inr.refine'_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\n\u22a2 HasSum (fun k => (\u2191(2 * k)!)\u207b\u00b9 \u2022 q ^ (2 * k)) \u2191c\n[PROOFSTEP]\nconvert hc using 1\n[GOAL]\ncase h.e'_5\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\n\u22a2 (fun k => (\u2191(2 * k)!)\u207b\u00b9 \u2022 q ^ (2 * k)) = fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)\n[PROOFSTEP]\next n : 1\n[GOAL]\ncase h.e'_5.h\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\n\u22a2 (\u2191(2 * n)!)\u207b\u00b9 \u2022 q ^ (2 * n) = \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n) / \u2191(2 * n)!)\n[PROOFSTEP]\nletI k : \u211d := \u2191(2 * n)!\n[GOAL]\ncase h.e'_5.h\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 (\u2191(2 * n)!)\u207b\u00b9 \u2022 q ^ (2 * n) = \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n) / \u2191(2 * n)!)\n[PROOFSTEP]\ncalc\n k\u207b\u00b9 \u2022 q ^ (2 * n) = k\u207b\u00b9 \u2022 (-normSq q) ^ n := by rw [pow_mul, hq2]; norm_cast\n _ = k\u207b\u00b9 \u2022 \u2191((-1 : \u211d) ^ n * \u2016q\u2016 ^ (2 * n)) := ?_\n _ = \u2191((-1 : \u211d) ^ n * \u2016q\u2016 ^ (2 * n) / k) := ?_\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 k\u207b\u00b9 \u2022 q ^ (2 * n) = \u2191(k\u207b\u00b9 \u2022 (-\u2191normSq q) ^ n)\n[PROOFSTEP]\nrw [pow_mul, hq2]\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 k\u207b\u00b9 \u2022 (-\u2191(\u2191normSq q)) ^ n = \u2191(k\u207b\u00b9 \u2022 (-\u2191normSq q) ^ n)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_5.h.calc_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 \u2191(k\u207b\u00b9 \u2022 (-\u2191normSq q) ^ n) = k\u207b\u00b9 \u2022 \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_5.h.calc_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 \u2191(k\u207b\u00b9 \u2022 (-\u2191normSq q) ^ n) = k\u207b\u00b9 \u2022 \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n))\n[PROOFSTEP]\nrw [neg_pow, normSq_eq_norm_mul_self, pow_mul, sq]\n[GOAL]\ncase h.e'_5.h.calc_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 \u2191(k\u207b\u00b9 \u2022 ((-1) ^ n * (\u2016q\u2016 * \u2016q\u2016) ^ n)) = k\u207b\u00b9 \u2022 \u2191((-1) ^ n * (\u2016q\u2016 * \u2016q\u2016) ^ n)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_5.h.calc_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 (\u2191(2 * n)!)\u207b\u00b9 \u2022 ((-1) ^ n * (\u2191\u2016q\u2016 * \u2191\u2016q\u2016) ^ n) = (\u2191(2 * n)!)\u207b\u00b9 \u2022 ((-1) ^ n * (\u2191\u2016q\u2016 * \u2191\u2016q\u2016) ^ n)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_5.h.calc_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 k\u207b\u00b9 \u2022 \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n)) = \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n) / k)\n[PROOFSTEP]\nrw [\u2190 coe_mul_eq_smul, div_eq_mul_inv]\n[GOAL]\ncase h.e'_5.h.calc_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 \u2191k\u207b\u00b9 * \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n)) = \u2191((-1) ^ n * \u2016q\u2016 ^ (2 * n) * k\u207b\u00b9)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_5.h.calc_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n)!\n\u22a2 \u2191(k\u207b\u00b9 * (\u2191(Int.negSucc 0 ^ n) * \u2016q\u2016 ^ (2 * n))) = \u2191(\u2191(Int.negSucc 0 ^ n) * \u2016q\u2016 ^ (2 * n) * k\u207b\u00b9)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase inr.refine'_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\n\u22a2 HasSum (fun k => (\u2191(2 * k + 1)!)\u207b\u00b9 \u2022 q ^ (2 * k + 1)) ((s / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nconvert hs using 1\n[GOAL]\ncase h.e'_5\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\n\u22a2 (fun k => (\u2191(2 * k + 1)!)\u207b\u00b9 \u2022 q ^ (2 * k + 1)) = fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\next n : 1\n[GOAL]\ncase h.e'_5.h\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\n\u22a2 (\u2191(2 * n + 1)!)\u207b\u00b9 \u2022 q ^ (2 * n + 1) = ((-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / \u2191(2 * n + 1)! / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\nlet k : \u211d := \u2191(2 * n + 1)!\n[GOAL]\ncase h.e'_5.h\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 (\u2191(2 * n + 1)!)\u207b\u00b9 \u2022 q ^ (2 * n + 1) = ((-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / \u2191(2 * n + 1)! / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\ncalc\n k\u207b\u00b9 \u2022 q ^ (2 * n + 1) = k\u207b\u00b9 \u2022 ((-normSq q) ^ n * q) :=\n by\n rw [pow_succ', pow_mul, hq2]\n norm_cast\n _ = k\u207b\u00b9 \u2022 ((-1 : \u211d) ^ n * \u2016q\u2016 ^ (2 * n)) \u2022 q := ?_\n _ = ((-1 : \u211d) ^ n * \u2016q\u2016 ^ (2 * n + 1) / k / \u2016q\u2016) \u2022 q := ?_\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 k\u207b\u00b9 \u2022 q ^ (2 * n + 1) = k\u207b\u00b9 \u2022 (\u2191((-\u2191normSq q) ^ n) * q)\n[PROOFSTEP]\nrw [pow_succ', pow_mul, hq2]\n[GOAL]\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 k\u207b\u00b9 \u2022 ((-\u2191(\u2191normSq q)) ^ n * q) = k\u207b\u00b9 \u2022 (\u2191((-\u2191normSq q) ^ n) * q)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_5.h.calc_1\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 k\u207b\u00b9 \u2022 (\u2191((-\u2191normSq q) ^ n) * q) = k\u207b\u00b9 \u2022 ((-1) ^ n * \u2016q\u2016 ^ (2 * n)) \u2022 q\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_5.h.calc_1.e_a\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 \u2191((-\u2191normSq q) ^ n) * q = ((-1) ^ n * \u2016q\u2016 ^ (2 * n)) \u2022 q\n[PROOFSTEP]\nrw [neg_pow, normSq_eq_norm_mul_self, pow_mul, sq, \u2190 coe_mul_eq_smul]\n[GOAL]\ncase h.e'_5.h.calc_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 k\u207b\u00b9 \u2022 ((-1) ^ n * \u2016q\u2016 ^ (2 * n)) \u2022 q = ((-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / k / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\nrw [smul_smul]\n[GOAL]\ncase h.e'_5.h.calc_2\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 (k\u207b\u00b9 * ((-1) ^ n * \u2016q\u2016 ^ (2 * n))) \u2022 q = ((-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / k / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_5.h.calc_2.e_a\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 k\u207b\u00b9 * ((-1) ^ n * \u2016q\u2016 ^ (2 * n)) = (-1) ^ n * \u2016q\u2016 ^ (2 * n + 1) / k / \u2016q\u2016\n[PROOFSTEP]\nsimp_rw [pow_succ', mul_div_assoc, div_div_cancel_left' hqn]\n[GOAL]\ncase h.e'_5.h.calc_2.e_a\nq : \u210d\nhq : q.re = 0\nc s : \u211d\nhc : HasSum (fun a => \u2191((-1) ^ a * \u2016q\u2016 ^ (2 * a) / \u2191(2 * a)!)) \u2191c\nhs : HasSum (fun z => ((-1) ^ z * \u2016q\u2016 ^ (2 * z + 1) / \u2191(2 * z + 1)! / \u2016q\u2016) \u2022 q) ((s / \u2016q\u2016) \u2022 q)\nhq0 : q \u2260 0\nhq2 : q ^ 2 = -\u2191(\u2191normSq q)\nhqn : \u2016q\u2016 \u2260 0\nn : \u2115\nk : \u211d := \u2191(2 * n + 1)!\n\u22a2 (\u2191(2 * n + 1)!)\u207b\u00b9 * ((-1) ^ n * \u2016q\u2016 ^ (2 * n)) = (-1) ^ n * (\u2016q\u2016 ^ (2 * n) * (\u2191(2 * n + 1)!)\u207b\u00b9)\n[PROOFSTEP]\nring\n[GOAL]\nq : \u210d\nhq : q.re = 0\n\u22a2 exp \u211d q = \u2191(Real.cos \u2016q\u2016) + (Real.sin \u2016q\u2016 / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\nrw [exp_eq_tsum]\n[GOAL]\nq : \u210d\nhq : q.re = 0\n\u22a2 (fun x => \u2211' (n : \u2115), (\u2191n !)\u207b\u00b9 \u2022 x ^ n) q = \u2191(Real.cos \u2016q\u2016) + (Real.sin \u2016q\u2016 / \u2016q\u2016) \u2022 q\n[PROOFSTEP]\nrefine' HasSum.tsum_eq _\n[GOAL]\nq : \u210d\nhq : q.re = 0\n\u22a2 HasSum (fun n => (\u2191n !)\u207b\u00b9 \u2022 q ^ n) (\u2191(Real.cos \u2016q\u2016) + (Real.sin \u2016q\u2016 / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nsimp_rw [\u2190 expSeries_apply_eq]\n[GOAL]\nq : \u210d\nhq : q.re = 0\n\u22a2 HasSum (fun n => \u2191(expSeries \u211d \u210d n) fun x => q) (\u2191(Real.cos \u2016q\u2016) + (Real.sin \u2016q\u2016 / \u2016q\u2016) \u2022 q)\n[PROOFSTEP]\nexact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)\n[GOAL]\nq : \u210d\n\u22a2 exp \u211d q = exp \u211d q.re \u2022 (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q)\n[PROOFSTEP]\nrw [\u2190 exp_of_re_eq_zero q.im q.im_re, \u2190 coe_mul_eq_smul, \u2190 exp_coe, \u2190 exp_add_of_commute, re_add_im]\n[GOAL]\nq : \u210d\n\u22a2 Commute (\u2191q.re) (im q)\n[PROOFSTEP]\nexact Algebra.commutes q.re (_ : \u210d[\u211d])\n[GOAL]\nq : \u210d\n\u22a2 (exp \u211d q).re = exp \u211d q.re * Real.cos \u2016q - \u2191q.re\u2016\n[PROOFSTEP]\nsimp [exp_eq]\n[GOAL]\nq : \u210d\n\u22a2 im (exp \u211d q) = (exp \u211d q.re * (Real.sin \u2016im q\u2016 / \u2016im q\u2016)) \u2022 im q\n[PROOFSTEP]\nsimp [exp_eq, smul_smul]\n[GOAL]\nq : \u210d\n\u22a2 \u2191normSq (exp \u211d q) = \u2191normSq (exp \u211d q.re \u2022 (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q))\n[PROOFSTEP]\nrw [exp_eq]\n[GOAL]\nq : \u210d\n\u22a2 \u2191normSq (exp \u211d q.re \u2022 (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q)) =\n exp \u211d q.re ^ 2 * \u2191normSq (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q)\n[PROOFSTEP]\nrw [normSq_smul]\n[GOAL]\nq : \u210d\n\u22a2 exp \u211d q.re ^ 2 * \u2191normSq (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q) =\n exp \u211d q.re ^ 2 * (Real.cos \u2016im q\u2016 ^ 2 + Real.sin \u2016im q\u2016 ^ 2)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nq : \u210d\n\u22a2 \u2191normSq (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q) = Real.cos \u2016im q\u2016 ^ 2 + Real.sin \u2016im q\u2016 ^ 2\n[PROOFSTEP]\nobtain hv | hv := eq_or_ne \u2016q.im\u2016 0\n[GOAL]\ncase e_a.inl\nq : \u210d\nhv : \u2016im q\u2016 = 0\n\u22a2 \u2191normSq (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q) = Real.cos \u2016im q\u2016 ^ 2 + Real.sin \u2016im q\u2016 ^ 2\n[PROOFSTEP]\nsimp [hv]\n[GOAL]\ncase e_a.inr\nq : \u210d\nhv : \u2016im q\u2016 \u2260 0\n\u22a2 \u2191normSq (\u2191(Real.cos \u2016im q\u2016) + (Real.sin \u2016im q\u2016 / \u2016im q\u2016) \u2022 im q) = Real.cos \u2016im q\u2016 ^ 2 + Real.sin \u2016im q\u2016 ^ 2\n[PROOFSTEP]\nrw [normSq_add, normSq_smul, star_smul, coe_mul_eq_smul, smul_re, smul_re, star_re, im_re, smul_zero, smul_zero,\n mul_zero, add_zero, div_pow, normSq_coe, normSq_eq_norm_mul_self, \u2190 sq, div_mul_cancel _ (pow_ne_zero _ hv)]\n[GOAL]\nq : \u210d\n\u22a2 exp \u211d q.re ^ 2 * (Real.cos \u2016im q\u2016 ^ 2 + Real.sin \u2016im q\u2016 ^ 2) = exp \u211d q.re ^ 2\n[PROOFSTEP]\nrw [Real.cos_sq_add_sin_sq, mul_one]\n[GOAL]\nq : \u210d\n\u22a2 \u2016exp \u211d q\u2016 = \u2016exp \u211d q.re\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_real_inner (exp \u211d q), inner_self, normSq_exp, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.QuaternionExponential", "llama_tokens": 11087, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5467381372136563, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.26056427065648846}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\ninst\u271d : ConcreteCategory D\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : GrothendieckTopology.Cover J X\nx : Meq P S\nf : Y \u27f6 X\nI : GrothendieckTopology.Cover.Relation ((GrothendieckTopology.pullback J f).obj S)\n\u22a2 I.g\u2081 \u226b I.f\u2081 \u226b f = I.g\u2082 \u226b I.f\u2082 \u226b f\n[PROOFSTEP]\nsimp [I.w_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\ninst\u271d : ConcreteCategory D\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : GrothendieckTopology.Cover J X\nx : (forget D).obj (P.obj (op X))\nI : GrothendieckTopology.Cover.Relation S\n\u22a2 \u2191(P.map I.g\u2081.op) ((fun I => \u2191(P.map I.f.op) x) (GrothendieckTopology.Cover.Relation.fst I)) =\n \u2191(P.map I.g\u2082.op) ((fun I => \u2191(P.map I.f.op) x) (GrothendieckTopology.Cover.Relation.snd I))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b9 : Category.{max v u, w} D\ninst\u271d : ConcreteCategory D\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : GrothendieckTopology.Cover J X\nx : (forget D).obj (P.obj (op X))\nI : GrothendieckTopology.Cover.Relation S\n\u22a2 \u2191(P.map I.g\u2081.op) (\u2191(P.map I.f\u2081.op) x) = \u2191(P.map I.g\u2082.op) (\u2191(P.map I.f\u2082.op) x)\n[PROOFSTEP]\nsimp only [\u2190 comp_apply, \u2190 P.map_comp, \u2190 op_comp, I.w]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : PreservesLimits (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : GrothendieckTopology.Cover J X\ninst\u271d : HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nx : Meq P S\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 \u2191(Multiequalizer.\u03b9 (GrothendieckTopology.Cover.index S P) I) (\u2191(equiv P S).symm x) = \u2191x I\n[PROOFSTEP]\nrw [\u2190 equiv_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b3 : Category.{max v u, w} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : PreservesLimits (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : GrothendieckTopology.Cover J X\ninst\u271d : HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nx : Meq P S\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 \u2191(\u2191(equiv P S) (\u2191(equiv P S).symm x)) I = \u2191x I\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191((plusObj J P).map f.op) (mk x) = mk (Meq.pullback x f)\n[PROOFSTEP]\ndsimp [mk, plusObj]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191(colimMap (diagramPullback J P f) \u226b colimit.pre (diagram J P Y) (pullback J f).op)\n (\u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(colimit.\u03b9 (diagram J P Y) (op (Cover.pullback S f))) (\u2191(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))\n[PROOFSTEP]\nrw [\u2190 comp_apply (x := (Meq.equiv P S).symm x), \u03b9_colimMap_assoc, colimit.\u03b9_pre,\n comp_apply (x := (Meq.equiv P S).symm x)]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191(colimit.\u03b9 (diagram J P Y) ((pullback J f).op.obj (op S)))\n (\u2191(NatTrans.app (diagramPullback J P f) (op S)) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(colimit.\u03b9 (diagram J P Y) (op (Cover.pullback S f))) (\u2191(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191(NatTrans.app (diagramPullback J P f) (op S)) (\u2191(Meq.equiv P S).symm x) =\n \u2191(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f)\n[PROOFSTEP]\napply (Meq.equiv P _).injective\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191(Meq.equiv P ((pullback J f).op.obj (op S)).unop)\n (\u2191(NatTrans.app (diagramPullback J P f) (op S)) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Meq.equiv P ((pullback J f).op.obj (op S)).unop) (\u2191(Meq.equiv P (Cover.pullback S f)).symm (Meq.pullback x f))\n[PROOFSTEP]\nerw [Equiv.apply_symm_apply]\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\n\u22a2 \u2191(Meq.equiv P ((pullback J f).op.obj (op S)).unop)\n (\u2191(NatTrans.app (diagramPullback J P f) (op S)) (\u2191(Meq.equiv P S).symm x)) =\n Meq.pullback x f\n[PROOFSTEP]\next i\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\ni : Cover.Arrow ((pullback J f).op.obj (op S)).unop\n\u22a2 \u2191(\u2191(Meq.equiv P ((pullback J f).op.obj (op S)).unop)\n (\u2191(NatTrans.app (diagramPullback J P f) (op S)) (\u2191(Meq.equiv P S).symm x)))\n i =\n \u2191(Meq.pullback x f) i\n[PROOFSTEP]\nsimp only [Functor.op_obj, unop_op, pullback_obj, diagram_obj, Functor.comp_obj, diagramPullback_app, Meq.equiv_apply,\n Meq.pullback_apply]\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\ni : Cover.Arrow ((pullback J f).op.obj (op S)).unop\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (Cover.pullback S f) P) i)\n (\u2191(Multiequalizer.lift (Cover.index ((pullback J f).op.obj (op S)).unop P) ((diagram J P X).obj (op S))\n (fun I => Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.base I))\n (_ :\n \u2200 (b : (Cover.index ((pullback J f).op.obj (op S)).unop P).R),\n (fun I => Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.base I))\n (MulticospanIndex.fstTo (Cover.index ((pullback J f).op.obj (op S)).unop P) b) \u226b\n MulticospanIndex.fst (Cover.index ((pullback J f).op.obj (op S)).unop P) b =\n (fun I => Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.base I))\n (MulticospanIndex.sndTo (Cover.index ((pullback J f).op.obj (op S)).unop P) b) \u226b\n MulticospanIndex.snd (Cover.index ((pullback J f).op.obj (op S)).unop P) b))\n (\u2191(Meq.equiv P S).symm x)) =\n \u2191x { Y := i.Y, f := i.f \u226b f, hf := (_ : (Cover.sieve (Cover.pullback S f)).arrows i.f) }\n[PROOFSTEP]\nerw [\u2190 comp_apply, Multiequalizer.lift_\u03b9, Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\ni : Cover.Arrow ((pullback J f).op.obj (op S)).unop\n\u22a2 \u2191x (Cover.Arrow.base i) = \u2191x { Y := i.Y, f := i.f \u226b f, hf := (_ : (Cover.sieve (Cover.pullback S f)).arrows i.f) }\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.a.h.mk\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nY X : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nf : Y \u27f6 X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 Y\nhf\u271d : (Cover.sieve ((pullback J f).op.obj (op S)).unop).arrows f\u271d\n\u22a2 \u2191x (Cover.Arrow.base { Y := Y\u271d, f := f\u271d, hf := hf\u271d }) =\n \u2191x\n { Y := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y, f := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f \u226b f,\n hf := (_ : (Cover.sieve (Cover.pullback S f)).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f) }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(NatTrans.app (toPlus J P) (op X)) x = mk (Meq.mk S x)\n[PROOFSTEP]\ndsimp [mk, toPlus]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P X) (op \u22a4)) x =\n \u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\nlet e : S \u27f6 \u22a4 := homOfLE (OrderTop.le_top _)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191(Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P X) (op \u22a4)) x =\n \u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\nrw [\u2190 colimit.w _ e.op]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191(Cover.toMultiequalizer \u22a4 P \u226b (diagram J P X).map e.op \u226b colimit.\u03b9 (diagram J P X) (op S)) x =\n \u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\ndelta Cover.toMultiequalizer\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I) \u226b\n (diagram J P X).map e.op \u226b colimit.\u03b9 (diagram J P X) (op S))\n x =\n \u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\nerw [comp_apply, comp_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191(colimit.\u03b9 (diagram J P X) (op S))\n (\u2191((diagram J P X).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x)) =\n \u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191((diagram J P X).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x) =\n \u2191(Meq.equiv P S).symm (Meq.mk S x)\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2191(Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index \u22a4 P))\n (fun I => Multiequalizer.\u03b9 (Cover.index \u22a4 P) (Cover.Arrow.map I (homOfLE (_ : S \u2264 \u22a4))))\n (_ :\n \u2200 (I : (Cover.index (op S).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.fstTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.sndTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x) =\n \u2191(Meq.equiv P S).symm (Meq.mk S x)\n[PROOFSTEP]\napply Concrete.multiequalizer_ext\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\n\u22a2 \u2200 (t : (Cover.index S P).L),\n \u2191(Multiequalizer.\u03b9 (Cover.index S P) t)\n (\u2191(Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index \u22a4 P))\n (fun I => Multiequalizer.\u03b9 (Cover.index \u22a4 P) (Cover.Arrow.map I (homOfLE (_ : S \u2264 \u22a4))))\n (_ :\n \u2200 (I : (Cover.index (op S).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.fstTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.sndTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index S P) t) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\ni : (Cover.index S P).L\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index S P) i)\n (\u2191(Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index \u22a4 P))\n (fun I => Multiequalizer.\u03b9 (Cover.index \u22a4 P) (Cover.Arrow.map I (homOfLE (_ : S \u2264 \u22a4))))\n (_ :\n \u2200 (I : (Cover.index (op S).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.fstTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.sndTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I (homOfLE (_ : S \u2264 \u22a4)).op.unop)))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index S P) i) (\u2191(Meq.equiv P S).symm (Meq.mk S x))\n[PROOFSTEP]\nsimp only [\u2190 comp_apply, Category.assoc, Multiequalizer.lift_\u03b9, Category.comp_id, Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : (forget D).obj (P.obj (op X))\ne : S \u27f6 \u22a4 := homOfLE (_ : S \u2264 \u22a4)\ni : (Cover.index S P).L\n\u22a2 \u2191(P.map (Cover.Arrow.map i (homOfLE (_ : S \u2264 \u22a4))).f.op) x = \u2191(Meq.mk S x) i\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(NatTrans.app (toPlus J P) (op I.Y)) (\u2191x I) = \u2191((plusObj J P).map I.f.op) (mk x)\n[PROOFSTEP]\ndsimp only [toPlus, plusObj]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P (op I.Y).unop) (op \u22a4)) (\u2191x I) =\n \u2191(colimMap (diagramPullback J P I.f.op.unop) \u226b colimit.pre (diagram J P (op I.Y).unop) (pullback J I.f.op.unop).op)\n (mk x)\n[PROOFSTEP]\ndelta Cover.toMultiequalizer\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op (op I.Y).unop)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1) \u226b\n colimit.\u03b9 (diagram J P (op I.Y).unop) (op \u22a4))\n (\u2191x I) =\n \u2191(colimMap (diagramPullback J P I.f.op.unop) \u226b colimit.pre (diagram J P (op I.Y).unop) (pullback J I.f.op.unop).op)\n (mk x)\n[PROOFSTEP]\ndsimp [mk]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1) \u226b\n colimit.\u03b9 (diagram J P I.Y) (op \u22a4))\n (\u2191x I) =\n \u2191(colimMap (diagramPullback J P I.f) \u226b colimit.pre (diagram J P I.Y) (pullback J I.f).op)\n (\u2191(colimit.\u03b9 (diagram J P X) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nerw [\u2190 comp_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1) \u226b\n colimit.\u03b9 (diagram J P I.Y) (op \u22a4))\n (\u2191x I) =\n \u2191(colimit.\u03b9 (diagram J P X) (op S) \u226b\n colimMap (diagramPullback J P I.f) \u226b colimit.pre (diagram J P I.Y) (pullback J I.f).op)\n (\u2191(Meq.equiv P S).symm x)\n[PROOFSTEP]\nrw [\u03b9_colimMap_assoc, colimit.\u03b9_pre, comp_apply, comp_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(colimit.\u03b9 (diagram J P I.Y) (op \u22a4))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I)) =\n \u2191(colimit.\u03b9 (diagram J P I.Y) ((pullback J I.f).op.obj (op S)))\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\ndsimp only [Functor.op]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\n\u22a2 \u2191(colimit.\u03b9 (diagram J P I.Y) (op \u22a4))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I)) =\n \u2191(colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nlet e : (J.pullback I.f).obj (unop (op S)) \u27f6 \u22a4 := homOfLE (OrderTop.le_top _)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\n\u22a2 \u2191(colimit.\u03b9 (diagram J P I.Y) (op \u22a4))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I)) =\n \u2191(colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nrw [\u2190 colimit.w _ e.op]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\n\u22a2 \u2191((diagram J P I.Y).map e.op \u226b colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I)) =\n \u2191(colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nerw [comp_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\n\u22a2 \u2191(colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191((diagram J P I.Y).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I))) =\n \u2191(colimit.\u03b9 (diagram J P I.Y) (op ((pullback J I.f).obj (op S).unop)))\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\n\u22a2 \u2191((diagram J P I.Y).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I)) =\n \u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x)\n[PROOFSTEP]\napply Concrete.multiequalizer_ext\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\n\u22a2 \u2200 (t : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L),\n \u2191(Multiequalizer.\u03b9 (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P) t)\n (\u2191((diagram J P I.Y).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I))) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P) t)\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P) i)\n (\u2191((diagram J P I.Y).map e.op)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I))) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P) i)\n (\u2191(NatTrans.app (diagramPullback J P I.f) (op S)) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (Cover.pullback S I.f) P) i)\n (\u2191(Multiequalizer.lift (Cover.index (Cover.pullback S I.f) P) (multiequalizer (Cover.index \u22a4 P))\n (fun I_1 =>\n Multiequalizer.\u03b9 (Cover.index \u22a4 P) (Cover.Arrow.map I_1 (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4))))\n (_ :\n \u2200 (I_1 : (Cover.index (op (Cover.pullback S I.f)).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.fstTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4)).op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op \u22a4).unop P)\n (MulticospanIndex.sndTo (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4)).op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op \u22a4).unop P)\n (Cover.Relation.map I_1 (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4)).op.unop)))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op I.Y)) (fun I_1 => P.map I_1.f.op)\n (_ :\n \u2200 (I_1 : (Cover.index \u22a4 P).R),\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I_1 =\n (fun I_2 => P.map I_2.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I_1) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I_1))\n (\u2191x I))) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (Cover.pullback S I.f) P) i)\n (\u2191(Multiequalizer.lift (Cover.index (Cover.pullback S I.f) P) (multiequalizer (Cover.index S P))\n (fun I_1 => Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.base I_1))\n (_ :\n \u2200 (I_1 : (Cover.index ((pullback J I.f).op.obj (op S)).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op S).unop P)\n (MulticospanIndex.fstTo (Cover.index (op S).unop P) (Cover.Relation.base I_1)) \u226b\n MulticospanIndex.fst (Cover.index (op S).unop P) (Cover.Relation.base I_1) =\n Multiequalizer.\u03b9 (Cover.index (op S).unop P)\n (MulticospanIndex.sndTo (Cover.index (op S).unop P) (Cover.Relation.base I_1)) \u226b\n MulticospanIndex.snd (Cover.index (op S).unop P) (Cover.Relation.base I_1)))\n (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_apply, \u2190 comp_apply, Multiequalizer.lift_\u03b9, Multiequalizer.lift_\u03b9, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\n\u22a2 \u2191(P.map (Cover.Arrow.map i (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4))).f.op) (\u2191x I) =\n \u2191(Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.base i)) (\u2191(Meq.equiv P S).symm x)\n[PROOFSTEP]\nerw [Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\n\u22a2 \u2191(P.map (Cover.Arrow.map i (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4))).f.op) (\u2191x I) = \u2191x (Cover.Arrow.base i)\n[PROOFSTEP]\nlet RR : S.Relation := \u27e8_, _, _, i.f, \ud835\udfd9 _, I.f, i.f \u226b I.f, I.hf, Sieve.downward_closed _ I.hf _, by simp\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\n\u22a2 i.f \u226b I.f = \ud835\udfd9 i.Y \u226b i.f \u226b I.f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\nRR : Cover.Relation S :=\n { Y\u2081 := I.Y, Y\u2082 := i.Y, Z := i.Y, g\u2081 := i.f, g\u2082 := \ud835\udfd9 i.Y, f\u2081 := I.f, f\u2082 := i.f \u226b I.f,\n h\u2081 := (_ : (Cover.sieve S).arrows I.f), h\u2082 := (_ : (Cover.sieve S).arrows (i.f \u226b I.f)),\n w := (_ : i.f \u226b I.f = \ud835\udfd9 i.Y \u226b i.f \u226b I.f) }\n\u22a2 \u2191(P.map (Cover.Arrow.map i (homOfLE (_ : Cover.pullback S I.f \u2264 \u22a4))).f.op) (\u2191x I) = \u2191x (Cover.Arrow.base i)\n[PROOFSTEP]\nerw [x.condition RR]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\nRR : Cover.Relation S :=\n { Y\u2081 := I.Y, Y\u2082 := i.Y, Z := i.Y, g\u2081 := i.f, g\u2082 := \ud835\udfd9 i.Y, f\u2081 := I.f, f\u2082 := i.f \u226b I.f,\n h\u2081 := (_ : (Cover.sieve S).arrows I.f), h\u2082 := (_ : (Cover.sieve S).arrows (i.f \u226b I.f)),\n w := (_ : i.f \u226b I.f = \ud835\udfd9 i.Y \u226b i.f \u226b I.f) }\n\u22a2 \u2191(P.map RR.g\u2082.op) (\u2191x (MulticospanIndex.sndTo (Cover.index S P) RR)) = \u2191x (Cover.Arrow.base i)\n[PROOFSTEP]\nsimp only [unop_op, pullback_obj, op_id, Functor.map_id, id_apply]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx : Meq P S\nI : Cover.Arrow S\ne : (pullback J I.f).obj (op S).unop \u27f6 \u22a4 := homOfLE (_ : (pullback J I.f).obj (op S).unop \u2264 \u22a4)\ni : (Cover.index (op ((pullback J I.f).obj (op S).unop)).unop P).L\nRR : Cover.Relation S :=\n { Y\u2081 := I.Y, Y\u2082 := i.Y, Z := i.Y, g\u2081 := i.f, g\u2082 := \ud835\udfd9 i.Y, f\u2081 := I.f, f\u2082 := i.f \u226b I.f,\n h\u2081 := (_ : (Cover.sieve S).arrows I.f), h\u2082 := (_ : (Cover.sieve S).arrows (i.f \u226b I.f)),\n w := (_ : i.f \u226b I.f = \ud835\udfd9 i.Y \u226b i.f \u226b I.f) }\n\u22a2 \u2191x\n (MulticospanIndex.sndTo (Cover.index S P)\n { Y\u2081 := I.Y, Y\u2082 := i.Y, Z := i.Y, g\u2081 := i.f, g\u2082 := \ud835\udfd9 i.Y, f\u2081 := I.f, f\u2082 := i.f \u226b I.f,\n h\u2081 := (_ : (Cover.sieve S).arrows I.f), h\u2082 := (_ : (Cover.sieve S).arrows (i.f \u226b I.f)),\n w := (_ : i.f \u226b I.f = \ud835\udfd9 i.Y \u226b i.f \u226b I.f) }) =\n \u2191x (Cover.Arrow.base i)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(NatTrans.app (toPlus J P) (op X)) x = mk (Meq.mk \u22a4 x)\n[PROOFSTEP]\ndsimp [mk, toPlus]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P X) (op \u22a4)) x =\n \u2191(colimit.\u03b9 (diagram J P X) (op \u22a4)) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))\n[PROOFSTEP]\ndelta Cover.toMultiequalizer\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I) \u226b\n colimit.\u03b9 (diagram J P X) (op \u22a4))\n x =\n \u2191(colimit.\u03b9 (diagram J P X) (op \u22a4)) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))\n[PROOFSTEP]\nsimp only [comp_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(colimit.\u03b9 (diagram J P X) (op \u22a4))\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x) =\n \u2191(colimit.\u03b9 (diagram J P X) (op \u22a4)) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x =\n \u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x)\n[PROOFSTEP]\napply (Meq.equiv P \u22a4).injective\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\n\u22a2 \u2191(Meq.equiv P \u22a4)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x) =\n \u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))\n[PROOFSTEP]\next i\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\ni : Cover.Arrow \u22a4\n\u22a2 \u2191(\u2191(Meq.equiv P \u22a4)\n (\u2191(Multiequalizer.lift (Cover.index \u22a4 P) (P.obj (op X)) (fun I => P.map I.f.op)\n (_ :\n \u2200 (I : (Cover.index \u22a4 P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.fst (Cover.index \u22a4 P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n MulticospanIndex.snd (Cover.index \u22a4 P) I))\n x))\n i =\n \u2191(\u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))) i\n[PROOFSTEP]\nrw [Meq.equiv_apply, Equiv.apply_symm_apply, \u2190 comp_apply, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category.{max v u, w} D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj (op X))\ni : Cover.Arrow \u22a4\n\u22a2 \u2191(P.map i.f.op) x = \u2191(Meq.mk \u22a4 x) i\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj ((plusObj J P).obj (op X))\n\u22a2 \u2203 S y, x = mk y\n[PROOFSTEP]\nobtain \u27e8S, y, h\u27e9 := Concrete.colimit_exists_rep (J.diagram P X) x\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj ((plusObj J P).obj (op X))\nS : (Cover J X)\u1d52\u1d56\ny : (forget D).obj ((diagram J P X).obj S)\nh : \u2191(colimit.\u03b9 (diagram J P X) S) y = x\n\u22a2 \u2203 S y, x = mk y\n[PROOFSTEP]\nuse S.unop, Meq.equiv _ _ y\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj ((plusObj J P).obj (op X))\nS : (Cover J X)\u1d52\u1d56\ny : (forget D).obj ((diagram J P X).obj S)\nh : \u2191(colimit.\u03b9 (diagram J P X) S) y = x\n\u22a2 x = mk (\u2191(Meq.equiv P S.unop) y)\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj ((plusObj J P).obj (op X))\nS : (Cover J X)\u1d52\u1d56\ny : (forget D).obj ((diagram J P X).obj S)\nh : \u2191(colimit.\u03b9 (diagram J P X) S) y = x\n\u22a2 \u2191(colimit.\u03b9 (diagram J P X) S) y = mk (\u2191(Meq.equiv P S.unop) y)\n[PROOFSTEP]\ndsimp [mk]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj ((plusObj J P).obj (op X))\nS : (Cover J X)\u1d52\u1d56\ny : (forget D).obj ((diagram J P X).obj S)\nh : \u2191(colimit.\u03b9 (diagram J P X) S) y = x\n\u22a2 \u2191(colimit.\u03b9 (diagram J P X) S) y =\n \u2191(colimit.\u03b9 (diagram J P X) S) (\u2191(Meq.equiv P S.unop).symm (\u2191(Meq.equiv P S.unop) y))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\n\u22a2 mk x = mk y \u2194 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\n\u22a2 mk x = mk y \u2192 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\n\u22a2 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nobtain \u27e8W, h1, h2, hh\u27e9 := Concrete.colimit_exists_of_rep_eq _ _ _ h\n[GOAL]\ncase mp.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nhh : \u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x) = \u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y)\n\u22a2 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nuse W.unop, h1.unop, h2.unop\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nhh : \u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x) = \u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y)\n\u22a2 Meq.refine x h1.unop = Meq.refine y h2.unop\n[PROOFSTEP]\next I\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nhh : \u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x) = \u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y)\nI : Cover.Arrow W.unop\n\u22a2 \u2191(Meq.refine x h1.unop) I = \u2191(Meq.refine y h2.unop) I\n[PROOFSTEP]\napply_fun Multiequalizer.\u03b9 (W.unop.index P) I at hh \n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\n\u22a2 \u2191(Meq.refine x h1.unop) I = \u2191(Meq.refine y h2.unop) I\n[PROOFSTEP]\nconvert hh\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191(Meq.refine x h1.unop) I =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191(Meq.refine y h2.unop) I =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\nall_goals\n dsimp [diagram]\n erw [\u2190 comp_apply, Multiequalizer.lift_\u03b9, Meq.equiv_symm_eq_apply]\n cases I; rfl\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191(Meq.refine x h1.unop) I =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191x { Y := I.Y, f := I.f, hf := (_ : ((fun a => \u2191a) S).arrows I.f) } =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I)\n (\u2191(Multiequalizer.lift (Cover.index W.unop P) (multiequalizer (Cover.index S P))\n (fun I => Multiequalizer.\u03b9 (Cover.index S P) (Cover.Arrow.map I h1.unop))\n (_ :\n \u2200 (I : (Cover.index W.unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op S).unop P)\n (MulticospanIndex.fstTo (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop) =\n Multiequalizer.\u03b9 (Cover.index (op S).unop P)\n (MulticospanIndex.sndTo (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op S).unop P) (Cover.Relation.map I h1.unop)))\n (\u2191(Meq.equiv P S).symm x))\n[PROOFSTEP]\nerw [\u2190 comp_apply, Multiequalizer.lift_\u03b9, Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191x { Y := I.Y, f := I.f, hf := (_ : ((fun a => \u2191a) S).arrows I.f) } = \u2191x (Cover.Arrow.map I h1.unop)\n[PROOFSTEP]\ncases I\n[GOAL]\ncase h.e'_2.h.mk\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : (Cover.sieve W.unop).arrows f\u271d\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d })\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d })\n (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191x\n { Y := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y, f := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f,\n hf := (_ : ((fun a => \u2191a) S).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f) } =\n \u2191x (Cover.Arrow.map { Y := Y\u271d, f := f\u271d, hf := hf\u271d } h1.unop)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191(Meq.refine y h2.unop) I =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191y { Y := I.Y, f := I.f, hf := (_ : ((fun a => \u2191a) T).arrows I.f) } =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I)\n (\u2191(Multiequalizer.lift (Cover.index W.unop P) (multiequalizer (Cover.index T P))\n (fun I => Multiequalizer.\u03b9 (Cover.index T P) (Cover.Arrow.map I h2.unop))\n (_ :\n \u2200 (I : (Cover.index W.unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op T).unop P)\n (MulticospanIndex.fstTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop) =\n Multiequalizer.\u03b9 (Cover.index (op T).unop P)\n (MulticospanIndex.sndTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op T).unop P) (Cover.Relation.map I h2.unop)))\n (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\nerw [\u2190 comp_apply, Multiequalizer.lift_\u03b9, Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nI : Cover.Arrow W.unop\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) I) (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op I.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) I))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191y { Y := I.Y, f := I.f, hf := (_ : ((fun a => \u2191a) T).arrows I.f) } = \u2191y (Cover.Arrow.map I h2.unop)\n[PROOFSTEP]\ncases I\n[GOAL]\ncase h.e'_3.h.mk\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\nh : mk x = mk y\nW : (Cover J X)\u1d52\u1d56\nh1 : op S \u27f6 W\nh2 : op T \u27f6 W\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : (Cover.sieve W.unop).arrows f\u271d\nhh :\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d })\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d })\n (\u2191((diagram J P X).map h2) (\u2191(Meq.equiv P T).symm y))\ne_1\u271d :\n (forget D).obj (P.obj (op { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y)) =\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index W.unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }))\n (\u2191((diagram J P X).map h1) (\u2191(Meq.equiv P S).symm x))\n\u22a2 \u2191y\n { Y := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y, f := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f,\n hf := (_ : ((fun a => \u2191a) T).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f) } =\n \u2191y (Cover.Arrow.map { Y := Y\u271d, f := f\u271d, hf := hf\u271d } h2.unop)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS T : Cover J X\nx : Meq P S\ny : Meq P T\n\u22a2 (\u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2) \u2192 mk x = mk y\n[PROOFSTEP]\nrintro \u27e8S, h1, h2, e\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ne : Meq.refine x h1 = Meq.refine y h2\n\u22a2 mk x = mk y\n[PROOFSTEP]\napply Concrete.colimit_rep_eq_of_exists\n[GOAL]\ncase mpr.intro.intro.intro.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ne : Meq.refine x h1 = Meq.refine y h2\n\u22a2 \u2203 k f g, \u2191((diagram J P X).map f) (\u2191(Meq.equiv P S\u271d).symm x) = \u2191((diagram J P X).map g) (\u2191(Meq.equiv P T).symm y)\n[PROOFSTEP]\nuse op S, h1.op, h2.op\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ne : Meq.refine x h1 = Meq.refine y h2\n\u22a2 \u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x) = \u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y)\n[PROOFSTEP]\napply Concrete.multiequalizer_ext\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ne : Meq.refine x h1 = Meq.refine y h2\n\u22a2 \u2200 (t : (Cover.index (op S).unop P).L),\n \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) t) (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) t) (\u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ne : Meq.refine x h1 = Meq.refine y h2\ni : (Cover.index (op S).unop P).L\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\napply_fun fun ee => ee i at e \n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y))\n[PROOFSTEP]\nconvert e\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191(Meq.refine x h1) i\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y)) =\n \u2191(Meq.refine y h2) i\n[PROOFSTEP]\nall_goals\n dsimp [diagram]\n rw [\u2190 comp_apply, Multiequalizer.lift_\u03b9]\n erw [Meq.equiv_symm_eq_apply]\n cases i; rfl\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191(Meq.refine x h1) i\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index S P) i)\n (\u2191(Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index S\u271d P))\n (fun I => Multiequalizer.\u03b9 (Cover.index S\u271d P) (Cover.Arrow.map I h1))\n (_ :\n \u2200 (I : (Cover.index (op S).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op S\u271d).unop P)\n (MulticospanIndex.fstTo (Cover.index (op S\u271d).unop P) (Cover.Relation.map I h1.op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op S\u271d).unop P) (Cover.Relation.map I h1.op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op S\u271d).unop P)\n (MulticospanIndex.sndTo (Cover.index (op S\u271d).unop P) (Cover.Relation.map I h1.op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op S\u271d).unop P) (Cover.Relation.map I h1.op.unop)))\n (\u2191(Meq.equiv P S\u271d).symm x)) =\n \u2191x { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) S\u271d).arrows i.f) }\n[PROOFSTEP]\nrw [\u2190 comp_apply, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index S\u271d P) (Cover.Arrow.map i h1)) (\u2191(Meq.equiv P S\u271d).symm x) =\n \u2191x { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) S\u271d).arrows i.f) }\n[PROOFSTEP]\nerw [Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191x (Cover.Arrow.map i h1) = \u2191x { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) S\u271d).arrows i.f) }\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.e'_2.h.mk\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : (Cover.sieve (op S).unop).arrows f\u271d\ne : \u2191(Meq.refine x h1) { Y := Y\u271d, f := f\u271d, hf := hf\u271d } = \u2191(Meq.refine y h2) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y))\n\u22a2 \u2191x (Cover.Arrow.map { Y := Y\u271d, f := f\u271d, hf := hf\u271d } h1) =\n \u2191x\n { Y := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y, f := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f,\n hf := (_ : ((fun a => \u2191a) S\u271d).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index (op S).unop P) i) (\u2191((diagram J P X).map h2.op) (\u2191(Meq.equiv P T).symm y)) =\n \u2191(Meq.refine y h2) i\n[PROOFSTEP]\ndsimp [diagram]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index S P) i)\n (\u2191(Multiequalizer.lift (Cover.index S P) (multiequalizer (Cover.index T P))\n (fun I => Multiequalizer.\u03b9 (Cover.index T P) (Cover.Arrow.map I h2))\n (_ :\n \u2200 (I : (Cover.index (op S).unop P).R),\n Multiequalizer.\u03b9 (Cover.index (op T).unop P)\n (MulticospanIndex.fstTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)) \u226b\n MulticospanIndex.fst (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop) =\n Multiequalizer.\u03b9 (Cover.index (op T).unop P)\n (MulticospanIndex.sndTo (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)) \u226b\n MulticospanIndex.snd (Cover.index (op T).unop P) (Cover.Relation.map I h2.op.unop)))\n (\u2191(Meq.equiv P T).symm y)) =\n \u2191y { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) T).arrows i.f) }\n[PROOFSTEP]\nrw [\u2190 comp_apply, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index T P) (Cover.Arrow.map i h2)) (\u2191(Meq.equiv P T).symm y) =\n \u2191y { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) T).arrows i.f) }\n[PROOFSTEP]\nerw [Meq.equiv_symm_eq_apply]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\ni : (Cover.index (op S).unop P).L\ne : \u2191(Meq.refine x h1) i = \u2191(Meq.refine y h2) i\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) i))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op i.Y))\n\u22a2 \u2191y (Cover.Arrow.map i h2) = \u2191y { Y := i.Y, f := i.f, hf := (_ : ((fun a => \u2191a) T).arrows i.f) }\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.e'_3.h.mk\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS\u271d T : Cover J X\nx : Meq P S\u271d\ny : Meq P T\nS : Cover J X\nh1 : S \u27f6 S\u271d\nh2 : S \u27f6 T\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : (Cover.sieve (op S).unop).arrows f\u271d\ne : \u2191(Meq.refine x h1) { Y := Y\u271d, f := f\u271d, hf := hf\u271d } = \u2191(Meq.refine y h2) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }\ne_1\u271d :\n (fun x => (forget D).obj (MulticospanIndex.left (Cover.index (op S).unop P) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }))\n (\u2191((diagram J P X).map h1.op) (\u2191(Meq.equiv P S\u271d).symm x)) =\n (forget D).obj (P.obj (op { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y))\n\u22a2 \u2191y (Cover.Arrow.map { Y := Y\u271d, f := f\u271d, hf := hf\u271d } h2) =\n \u2191y\n { Y := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y, f := { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f,\n hf := (_ : ((fun a => \u2191a) T).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f) }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nh : \u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8Sx, x, rfl\u27e9 := exists_rep x\n[GOAL]\ncase intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS : Cover J X\ny : (forget D).obj ((plusObj J P).obj (op X))\nSx : Cover J X\nx : Meq P Sx\nh : \u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) (mk x) = \u2191((plusObj J P).map I.f.op) y\n\u22a2 mk x = y\n[PROOFSTEP]\nobtain \u27e8Sy, y, rfl\u27e9 := exists_rep y\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) (mk x) = \u2191((plusObj J P).map I.f.op) (mk y)\n\u22a2 mk x = mk y\n[PROOFSTEP]\nsimp only [res_mk_eq_mk_pullback] at h \n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\n\u22a2 mk x = mk y\n[PROOFSTEP]\nchoose W h1 h2 hh using fun I : S.Arrow =>\n (eq_mk_iff_exists _ _).mp\n (h I)\n -- To prove equality, it suffices to prove that there exists a cover over which\n -- the representatives become equal.\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\n\u22a2 mk x = mk y\n[PROOFSTEP]\nrw [eq_mk_iff_exists]\n -- Construct the cover over which the representatives become equal by combining the various\n -- covers chosen above.\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\n\u22a2 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nlet B : J.Cover X := S.bind W\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\n\u22a2 \u2203 W h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nuse B\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\n\u22a2 \u2203 h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nlet ex : B \u27f6 Sx :=\n homOfLE\n (by\n rintro Y f \u27e8Z, e1, e2, he2, he1, hee\u27e9\n rw [\u2190 hee]\n apply leOfHom (h1 \u27e8_, _, he2\u27e9)\n exact he1)\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\n\u22a2 B \u2264 Sx\n[PROOFSTEP]\nrintro Y f \u27e8Z, e1, e2, he2, he1, hee\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) Sx).arrows f\n[PROOFSTEP]\nrw [\u2190 hee]\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) Sx).arrows (e1 \u226b e2)\n[PROOFSTEP]\napply leOfHom (h1 \u27e8_, _, he2\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.intro.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) (W { Y := Z, f := e2, hf := he2 })).arrows e1\n[PROOFSTEP]\nexact he1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\n\u22a2 \u2203 h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nlet ey : B \u27f6 Sy :=\n homOfLE\n (by\n rintro Y f \u27e8Z, e1, e2, he2, he1, hee\u27e9\n rw [\u2190 hee]\n apply leOfHom (h2 \u27e8_, _, he2\u27e9)\n exact he1)\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\n\u22a2 B \u2264 Sy\n[PROOFSTEP]\nrintro Y f \u27e8Z, e1, e2, he2, he1, hee\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) Sy).arrows f\n[PROOFSTEP]\nrw [\u2190 hee]\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) Sy).arrows (e1 \u226b e2)\n[PROOFSTEP]\napply leOfHom (h2 \u27e8_, _, he2\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.intro.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\nY : C\nf : Y \u27f6 X\nZ : C\ne1 : Y \u27f6 Z\ne2 : Z \u27f6 X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 \u226b e2 = f\n\u22a2 ((fun a => \u2191a) (W { Y := Z, f := e2, hf := he2 })).arrows e1\n[PROOFSTEP]\nexact he1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\n\u22a2 \u2203 h1 h2, Meq.refine x h1 = Meq.refine y h2\n[PROOFSTEP]\nuse ex, ey\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\n\u22a2 Meq.refine x ex = Meq.refine y ey\n[PROOFSTEP]\next1 I\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine y ey) I\n[PROOFSTEP]\nlet IS : S.Arrow := I.fromMiddle\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nhh : \u2200 (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine y ey) I\n[PROOFSTEP]\nspecialize hh IS\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nhh : Meq.refine (Meq.pullback x IS.f) (h1 IS) = Meq.refine (Meq.pullback y IS.f) (h2 IS)\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine y ey) I\n[PROOFSTEP]\nlet IW : (W IS).Arrow := I.toMiddle\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nhh : Meq.refine (Meq.pullback x IS.f) (h1 IS) = Meq.refine (Meq.pullback y IS.f) (h2 IS)\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine y ey) I\n[PROOFSTEP]\napply_fun fun e => e IW at hh \n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine y ey) I\n[PROOFSTEP]\nconvert hh using 1\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW\n[PROOFSTEP]\nlet Rx : Sx.Relation :=\n \u27e8I.Y, I.Y, I.Y, \ud835\udfd9 _, \ud835\udfd9 _, I.f, I.toMiddleHom \u226b I.fromMiddleHom, leOfHom ex _ I.hf, by\n simpa only [I.middle_spec] using leOfHom ex _ I.hf, by simp [I.middle_spec]\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I)\n[PROOFSTEP]\nsimpa only [I.middle_spec] using leOfHom ex _ I.hf\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 \ud835\udfd9 I.Y \u226b I.f = \ud835\udfd9 I.Y \u226b Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I\n[PROOFSTEP]\nsimp [I.middle_spec]\n[GOAL]\ncase h.e'_2.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\nRx : Cover.Relation Sx :=\n { Y\u2081 := I.Y, Y\u2082 := I.Y, Z := I.Y, g\u2081 := \ud835\udfd9 I.Y, g\u2082 := \ud835\udfd9 I.Y, f\u2081 := I.f,\n f\u2082 := Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I, h\u2081 := (_ : ((fun a => \u2191a) Sx).arrows I.f),\n h\u2082 := (_ : (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I)),\n w := (_ : \ud835\udfd9 I.Y \u226b I.f = \ud835\udfd9 I.Y \u226b Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I) }\n\u22a2 \u2191(Meq.refine x ex) I = \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW\n[PROOFSTEP]\nsimpa [id_apply] using x.condition Rx\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 \u2191(Meq.refine y ey) I = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\n[PROOFSTEP]\nlet Ry : Sy.Relation :=\n \u27e8I.Y, I.Y, I.Y, \ud835\udfd9 _, \ud835\udfd9 _, I.f, I.toMiddleHom \u226b I.fromMiddleHom, leOfHom ey _ I.hf, by\n simpa only [I.middle_spec] using leOfHom ey _ I.hf, by simp [I.middle_spec]\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I)\n[PROOFSTEP]\nsimpa only [I.middle_spec] using leOfHom ey _ I.hf\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\n\u22a2 \ud835\udfd9 I.Y \u226b I.f = \ud835\udfd9 I.Y \u226b Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I\n[PROOFSTEP]\nsimp [I.middle_spec]\n[GOAL]\ncase h.e'_3.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : \u2200 (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) \u2192 Cover J I.Y\nh1 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) \u2192 W I \u27f6 (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B \u27f6 Sx := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sx).arrows f)\ney : B \u27f6 Sy := homOfLE (_ : \u2200 \u2983Y : C\u2984 (f : Y \u27f6 X), ((fun a => \u2191a) B).arrows f \u2192 ((fun a => \u2191a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : \u2191(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1\u271d : (forget D).obj (P.obj (op I.Y)) = (forget D).obj (P.obj (op IW.Y))\nRy : Cover.Relation Sy :=\n { Y\u2081 := I.Y, Y\u2082 := I.Y, Z := I.Y, g\u2081 := \ud835\udfd9 I.Y, g\u2082 := \ud835\udfd9 I.Y, f\u2081 := I.f,\n f\u2082 := Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I, h\u2081 := (_ : ((fun a => \u2191a) Sy).arrows I.f),\n h\u2082 := (_ : (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I)),\n w := (_ : \ud835\udfd9 I.Y \u226b I.f = \ud835\udfd9 I.Y \u226b Cover.Arrow.toMiddleHom I \u226b Cover.Arrow.fromMiddleHom I) }\n\u22a2 \u2191(Meq.refine y ey) I = \u2191(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\n[PROOFSTEP]\nsimpa [id_apply] using y.condition Ry\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\n\u22a2 Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\n[PROOFSTEP]\nintro x y h\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nh : \u2191(NatTrans.app (toPlus J P) (op X)) x = \u2191(NatTrans.app (toPlus J P) (op X)) y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [toPlus_eq_mk] at h \n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nh : mk (Meq.mk \u22a4 x) = mk (Meq.mk \u22a4 y)\n\u22a2 x = y\n[PROOFSTEP]\nrw [eq_mk_iff_exists] at h \n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nh : \u2203 W h1 h2, Meq.refine (Meq.mk \u22a4 x) h1 = Meq.refine (Meq.mk \u22a4 y) h2\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8W, h1, h2, hh\u27e9 := h\n[GOAL]\ncase intro.intro.intro\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nW : Cover J X\nh1 h2 : W \u27f6 \u22a4\nhh : Meq.refine (Meq.mk \u22a4 x) h1 = Meq.refine (Meq.mk \u22a4 y) h2\n\u22a2 x = y\n[PROOFSTEP]\napply hsep X W\n[GOAL]\ncase intro.intro.intro.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nW : Cover J X\nh1 h2 : W \u27f6 \u22a4\nhh : Meq.refine (Meq.mk \u22a4 x) h1 = Meq.refine (Meq.mk \u22a4 y) h2\n\u22a2 \u2200 (I : Cover.Arrow W), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y\n[PROOFSTEP]\nintro I\n[GOAL]\ncase intro.intro.intro.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nW : Cover J X\nh1 h2 : W \u27f6 \u22a4\nhh : Meq.refine (Meq.mk \u22a4 x) h1 = Meq.refine (Meq.mk \u22a4 y) h2\nI : Cover.Arrow W\n\u22a2 \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y\n[PROOFSTEP]\napply_fun fun e => e I at hh \n[GOAL]\ncase intro.intro.intro.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nx y : (forget D).obj (P.obj (op X))\nW : Cover J X\nh1 h2 : W \u27f6 \u22a4\nI : Cover.Arrow W\nhh : \u2191(Meq.refine (Meq.mk \u22a4 x) h1) I = \u2191(Meq.refine (Meq.mk \u22a4 y) h2) I\n\u22a2 \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y\n[PROOFSTEP]\nexact hh\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\n\u22a2 \u2200 (I : Cover.Relation (Cover.bind S T)),\n \u2191(P.map I.g\u2081.op) ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst I)) =\n \u2191(P.map I.g\u2082.op) ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd I))\n[PROOFSTEP]\nintro II\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 \u2191(P.map II.g\u2081.op) ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II)) =\n \u2191(P.map II.g\u2082.op) ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II))\n[PROOFSTEP]\napply inj_of_sep P hsep\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 \u2191(NatTrans.app (toPlus J P) (op II.Z))\n (\u2191(P.map II.g\u2081.op)\n ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II))) =\n \u2191(NatTrans.app (toPlus J P) (op II.Z))\n (\u2191(P.map II.g\u2082.op) ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II)))\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_apply, (J.toPlus P).naturality, (J.toPlus P).naturality, comp_apply, comp_apply]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 \u2191((plusObj J P).map II.g\u2081.op)\n (\u2191(NatTrans.app (toPlus J P) (op II.Y\u2081))\n ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.fst II))) =\n \u2191((plusObj J P).map II.g\u2082.op)\n (\u2191(NatTrans.app (toPlus J P) (op II.Y\u2082))\n ((fun I => \u2191(t (Cover.Arrow.fromMiddle I)) (Cover.Arrow.toMiddle I)) (Cover.Relation.snd II)))\n[PROOFSTEP]\nerw [toPlus_apply (T II.fst.fromMiddle) (t II.fst.fromMiddle) II.fst.toMiddle,\n toPlus_apply (T II.snd.fromMiddle) (t II.snd.fromMiddle) II.snd.toMiddle, \u2190 ht, \u2190 ht, \u2190 comp_apply, \u2190 comp_apply, \u2190\n (J.plusObj P).map_comp, \u2190 (J.plusObj P).map_comp]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 \u2191((plusObj J P).map ((Cover.Arrow.toMiddle (Cover.Relation.fst II)).f.op \u226b II.g\u2081.op))\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.fst II))) =\n \u2191((plusObj J P).map ((Cover.Arrow.toMiddle (Cover.Relation.snd II)).f.op \u226b II.g\u2082.op))\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.snd II)))\n[PROOFSTEP]\nrw [\u2190 op_comp, \u2190 op_comp]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 \u2191((plusObj J P).map (II.g\u2081 \u226b (Cover.Arrow.toMiddle (Cover.Relation.fst II)).f).op)\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.fst II))) =\n \u2191((plusObj J P).map (II.g\u2082 \u226b (Cover.Arrow.toMiddle (Cover.Relation.snd II)).f).op)\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.snd II)))\n[PROOFSTEP]\nlet IR : S.Relation :=\n \u27e8_, _, _, II.g\u2081 \u226b II.fst.toMiddleHom, II.g\u2082 \u226b II.snd.toMiddleHom, II.fst.fromMiddleHom, II.snd.fromMiddleHom,\n II.fst.from_middle_condition, II.snd.from_middle_condition, by\n simpa only [Category.assoc, II.fst.middle_spec, II.snd.middle_spec] using II.w\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\n\u22a2 (II.g\u2081 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.fst II)) \u226b Cover.Arrow.fromMiddleHom (Cover.Relation.fst II) =\n (II.g\u2082 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.snd II)) \u226b Cover.Arrow.fromMiddleHom (Cover.Relation.snd II)\n[PROOFSTEP]\nsimpa only [Category.assoc, II.fst.middle_spec, II.snd.middle_spec] using II.w\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nII : Cover.Relation (Cover.bind S T)\nIR : Cover.Relation S :=\n { Y\u2081 := Cover.Arrow.middle (Cover.Relation.fst II), Y\u2082 := Cover.Arrow.middle (Cover.Relation.snd II), Z := II.Z,\n g\u2081 := II.g\u2081 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.fst II),\n g\u2082 := II.g\u2082 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.snd II),\n f\u2081 := Cover.Arrow.fromMiddleHom (Cover.Relation.fst II), f\u2082 := Cover.Arrow.fromMiddleHom (Cover.Relation.snd II),\n h\u2081 := (_ : (Cover.sieve S).arrows (Cover.Arrow.fromMiddleHom (Cover.Relation.fst II))),\n h\u2082 := (_ : (Cover.sieve S).arrows (Cover.Arrow.fromMiddleHom (Cover.Relation.snd II))),\n w :=\n (_ :\n (II.g\u2081 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.fst II)) \u226b Cover.Arrow.fromMiddleHom (Cover.Relation.fst II) =\n (II.g\u2082 \u226b Cover.Arrow.toMiddleHom (Cover.Relation.snd II)) \u226b\n Cover.Arrow.fromMiddleHom (Cover.Relation.snd II)) }\n\u22a2 \u2191((plusObj J P).map (II.g\u2081 \u226b (Cover.Arrow.toMiddle (Cover.Relation.fst II)).f).op)\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.fst II))) =\n \u2191((plusObj J P).map (II.g\u2082 \u226b (Cover.Arrow.toMiddle (Cover.Relation.snd II)).f).op)\n (\u2191s (Cover.Arrow.fromMiddle (Cover.Relation.snd II)))\n[PROOFSTEP]\nexact s.condition IR\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nhave inj : \u2200 X : C, Function.Injective ((J.toPlus P).app (op X)) := inj_of_sep _ hsep\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nchoose T t ht using fun I =>\n exists_rep\n (s I)\n -- Construct a large cover over which we will define a representative that will\n -- provide the gluing of the given local sections.\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nlet B : J.Cover X := S.bind T\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nchoose Z e1 e2 he2 _ _ using fun I : B.Arrow => I.hf\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nlet w : Meq P B := meqOfSep P hsep X S s T t ht\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\n\u22a2 \u2203 t, Meq.mk S t = s\n[PROOFSTEP]\nuse mk w\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\n\u22a2 Meq.mk S (mk w) = s\n[PROOFSTEP]\next I\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\n\u22a2 \u2191(Meq.mk S (mk w)) I = \u2191s I\n[PROOFSTEP]\ndsimp [Meq.mk]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\n\u22a2 \u2191((plusObj J P).map I.f.op) (mk (meqOfSep P hsep X S s T t ht)) = \u2191s I\n[PROOFSTEP]\nerw [ht, res_mk_eq_mk_pullback]\n -- Use the separatedness of `P\u207a` to prove that this is indeed a gluing of our\n -- original local sections.\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\n\u22a2 mk (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) = mk (t I)\n[PROOFSTEP]\napply sep P (T I)\n[GOAL]\ncase h.h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\n\u22a2 \u2200 (I_1 : Cover.Arrow (T I)),\n \u2191((plusObj J P).map I_1.f.op) (mk (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f)) =\n \u2191((plusObj J P).map I_1.f.op) (mk (t I))\n[PROOFSTEP]\nintro II\n[GOAL]\ncase h.h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\n\u22a2 \u2191((plusObj J P).map II.f.op) (mk (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f)) =\n \u2191((plusObj J P).map II.f.op) (mk (t I))\n[PROOFSTEP]\nsimp only [res_mk_eq_mk_pullback, eq_mk_iff_exists]\n -- It suffices to prove equality for representatives over a\n -- convenient sufficiently large cover...\n[GOAL]\ncase h.h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\n\u22a2 \u2203 W h1 h2,\n Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) h1 =\n Meq.refine (Meq.pullback (t I) II.f) h2\n[PROOFSTEP]\nuse(J.pullback II.f).obj (T I)\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\n\u22a2 \u2203 h1 h2,\n Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) h1 =\n Meq.refine (Meq.pullback (t I) II.f) h2\n[PROOFSTEP]\nlet e0 : (J.pullback II.f).obj (T I) \u27f6 (J.pullback II.f).obj ((J.pullback I.f).obj B) :=\n homOfLE\n (by\n intro Y f hf\n apply Sieve.le_pullback_bind _ _ _ I.hf\n \u00b7 cases I\n exact hf)\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\n\u22a2 (pullback J II.f).obj (T I) \u2264 (pullback J II.f).obj ((pullback J I.f).obj B)\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\nY : C\nf : Y \u27f6 II.Y\nhf : ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f\n\u22a2 ((fun a => \u2191a) ((pullback J II.f).obj ((pullback J I.f).obj B))).arrows f\n[PROOFSTEP]\napply Sieve.le_pullback_bind _ _ _ I.hf\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\nY : C\nf : Y \u27f6 II.Y\nhf : ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f\n\u22a2 ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) I.Y I.f (_ : (Cover.sieve S).arrows I.f)).arrows\n (f \u226b II.f)\n[PROOFSTEP]\ncases I\n[GOAL]\ncase a.mk\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nY Y\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf\u271d : (Cover.sieve S).arrows f\u271d\nII : Cover.Arrow (T { Y := Y\u271d, f := f\u271d, hf := hf\u271d })\nf : Y \u27f6 II.Y\nhf : ((fun a => \u2191a) ((pullback J II.f).obj (T { Y := Y\u271d, f := f\u271d, hf := hf\u271d }))).arrows f\n\u22a2 ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.Y\n { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f (_ : (Cover.sieve S).arrows { Y := Y\u271d, f := f\u271d, hf := hf\u271d }.f)).arrows\n (f \u226b II.f)\n[PROOFSTEP]\nexact hf\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\n\u22a2 \u2203 h1 h2,\n Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) h1 =\n Meq.refine (Meq.pullback (t I) II.f) h2\n[PROOFSTEP]\nuse e0, \ud835\udfd9 _\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\n\u22a2 Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0 =\n Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))\n[PROOFSTEP]\next IV\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\n\u22a2 \u2191(Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0) IV =\n \u2191(Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))) IV\n[PROOFSTEP]\nlet IA : B.Arrow := \u27e8_, (IV.f \u226b II.f) \u226b I.f, \u27e8I.Y, _, _, I.hf, Sieve.downward_closed _ II.hf _, rfl\u27e9\u27e9\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\n\u22a2 \u2191(Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0) IV =\n \u2191(Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))) IV\n[PROOFSTEP]\nlet IB : S.Arrow := IA.fromMiddle\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\n\u22a2 \u2191(Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0) IV =\n \u2191(Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))) IV\n[PROOFSTEP]\nlet IC : (T IB).Arrow := IA.toMiddle\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\n\u22a2 \u2191(Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0) IV =\n \u2191(Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))) IV\n[PROOFSTEP]\nlet ID : (T I).Arrow := \u27e8IV.Y, IV.f \u226b II.f, Sieve.downward_closed (T I).sieve II.hf IV.f\u27e9\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\nID : Cover.Arrow (T I) := { Y := IV.Y, f := IV.f \u226b II.f, hf := (_ : (Cover.sieve (T I)).arrows (IV.f \u226b II.f)) }\n\u22a2 \u2191(Meq.refine (Meq.pullback (Meq.pullback (meqOfSep P hsep X S s T t ht) I.f) II.f) e0) IV =\n \u2191(Meq.refine (Meq.pullback (t I) II.f) (\ud835\udfd9 ((pullback J II.f).obj (T I)))) IV\n[PROOFSTEP]\nchange t IB IC = t I ID\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\nID : Cover.Arrow (T I) := { Y := IV.Y, f := IV.f \u226b II.f, hf := (_ : (Cover.sieve (T I)).arrows (IV.f \u226b II.f)) }\n\u22a2 \u2191(t IB) IC = \u2191(t I) ID\n[PROOFSTEP]\napply inj IV.Y\n[GOAL]\ncase h.h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\nID : Cover.Arrow (T I) := { Y := IV.Y, f := IV.f \u226b II.f, hf := (_ : (Cover.sieve (T I)).arrows (IV.f \u226b II.f)) }\n\u22a2 \u2191(NatTrans.app (toPlus J P) (op IV.Y)) (\u2191(t IB) IC) = \u2191(NatTrans.app (toPlus J P) (op IV.Y)) (\u2191(t I) ID)\n[PROOFSTEP]\nerw [toPlus_apply (T I) (t I) ID, toPlus_apply (T IB) (t IB) IC, \u2190 ht, \u2190 ht]\n -- Conclude by constructing the relation showing equality...\n[GOAL]\ncase h.h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\nID : Cover.Arrow (T I) := { Y := IV.Y, f := IV.f \u226b II.f, hf := (_ : (Cover.sieve (T I)).arrows (IV.f \u226b II.f)) }\n\u22a2 \u2191((plusObj J P).map IC.f.op) (\u2191s IB) = \u2191((plusObj J P).map ID.f.op) (\u2191s I)\n[PROOFSTEP]\nlet IR : S.Relation := \u27e8_, _, IV.Y, IC.f, ID.f, IB.f, I.f, IB.hf, I.hf, IA.middle_spec\u27e9\n[GOAL]\ncase h.h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2075 : Category.{max v u, w} D\ninst\u271d\u2074 : ConcreteCategory D\ninst\u271d\u00b3 : PreservesLimits (forget D)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\ns : Meq (plusObj J P) S\ninj : \u2200 (X : C), Function.Injective \u2191(NatTrans.app (toPlus J P) (op X))\nT : (I : Cover.Arrow S) \u2192 Cover J I.Y\nt : (I : Cover.Arrow S) \u2192 Meq P (T I)\nht : \u2200 (I : Cover.Arrow S), \u2191s I = mk (t I)\nB : Cover J X := Cover.bind S T\nZ : Cover.Arrow B \u2192 C\ne1 : (I : Cover.Arrow B) \u2192 I.Y \u27f6 Z I\ne2 : (I : Cover.Arrow B) \u2192 Z I \u27f6 X\nhe2 : \u2200 (I : Cover.Arrow B), (Cover.sieve S).arrows (e2 I)\nh\u271d :\n \u2200 (I : Cover.Arrow B),\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) (Z I) (e2 I)\n (_ : (Cover.sieve S).arrows (e2 I)) (e1 I)\n\u271d : \u2200 (I : Cover.Arrow B), e1 I \u226b e2 I = I.f\nw : Meq P B := meqOfSep P hsep X S s T t ht\nI : Cover.Arrow S\nII : Cover.Arrow (T I)\ne0 : (pullback J II.f).obj (T I) \u27f6 (pullback J II.f).obj ((pullback J I.f).obj B) :=\n homOfLE\n (_ :\n \u2200 \u2983Y : C\u2984 (f : Y \u27f6 II.Y),\n ((fun a => \u2191a) ((pullback J II.f).obj (T I))).arrows f \u2192\n (Sieve.pullback I.f\n (Sieve.bind (Cover.sieve S).arrows fun Y f H =>\n (fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f H)).arrows\n (f \u226b II.f))\nIV : Cover.Arrow ((pullback J II.f).obj (T I))\nIA : Cover.Arrow B :=\n { Y := IV.Y, f := (IV.f \u226b II.f) \u226b I.f,\n hf :=\n (_ :\n \u2203 Y g f H,\n (fun Y f h => ((fun Y f hf => Cover.sieve (T { Y := Y, f := f, hf := hf })) Y f h).arrows) Y f H g \u2227\n g \u226b f = (IV.f \u226b II.f) \u226b I.f) }\nIB : Cover.Arrow S := Cover.Arrow.fromMiddle IA\nIC : Cover.Arrow (T IB) := Cover.Arrow.toMiddle IA\nID : Cover.Arrow (T I) := { Y := IV.Y, f := IV.f \u226b II.f, hf := (_ : (Cover.sieve (T I)).arrows (IV.f \u226b II.f)) }\nIR : Cover.Relation S :=\n { Y\u2081 := IB.Y, Y\u2082 := I.Y, Z := IV.Y, g\u2081 := IC.f, g\u2082 := ID.f, f\u2081 := IB.f, f\u2082 := I.f,\n h\u2081 := (_ : (Cover.sieve S).arrows IB.f), h\u2082 := (_ : (Cover.sieve S).arrows I.f),\n w := (_ : Cover.Arrow.toMiddleHom IA \u226b Cover.Arrow.fromMiddleHom IA = IA.f) }\n\u22a2 \u2191((plusObj J P).map IC.f.op) (\u2191s IB) = \u2191((plusObj J P).map ID.f.op) (\u2191s I)\n[PROOFSTEP]\nexact s.condition IR\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\n\u22a2 Presheaf.IsSheaf J (plusObj J P)\n[PROOFSTEP]\nrw [Presheaf.isSheaf_iff_multiequalizer]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\n\u22a2 \u2200 (X : C) (S : Cover J X), IsIso (Cover.toMultiequalizer S (plusObj J P))\n[PROOFSTEP]\nintro X S\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\n\u22a2 IsIso (Cover.toMultiequalizer S (plusObj J P))\n[PROOFSTEP]\napply @isIso_of_reflects_iso _ _ _ _ _ _ _ (forget D) ?_\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\n\u22a2 IsIso ((forget D).map (Cover.toMultiequalizer S (plusObj J P)))\n[PROOFSTEP]\nrw [isIso_iff_bijective]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\n\u22a2 Function.Bijective ((forget D).map (Cover.toMultiequalizer S (plusObj J P)))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\n\u22a2 Function.Injective ((forget D).map (Cover.toMultiequalizer S (plusObj J P)))\n[PROOFSTEP]\nintro x y h\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nh :\n (forget D).map (Cover.toMultiequalizer S (plusObj J P)) x = (forget D).map (Cover.toMultiequalizer S (plusObj J P)) y\n\u22a2 x = y\n[PROOFSTEP]\napply sep P S _ _\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nh :\n (forget D).map (Cover.toMultiequalizer S (plusObj J P)) x = (forget D).map (Cover.toMultiequalizer S (plusObj J P)) y\n\u22a2 \u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y\n[PROOFSTEP]\nintro I\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nh :\n (forget D).map (Cover.toMultiequalizer S (plusObj J P)) x = (forget D).map (Cover.toMultiequalizer S (plusObj J P)) y\nI : Cover.Arrow S\n\u22a2 \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y\n[PROOFSTEP]\napply_fun Meq.equiv _ _ at h \n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x) =\n \u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)\n\u22a2 \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y\n[PROOFSTEP]\napply_fun fun e => e I at h \n[GOAL]\ncase left\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n\u22a2 \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n\u22a2 \u2191((plusObj J P).map I.f.op) x =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I\n[PROOFSTEP]\nerw [Meq.equiv_apply, \u2190 comp_apply, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n\u22a2 \u2191((plusObj J P).map I.f.op) y =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n[PROOFSTEP]\nerw [Meq.equiv_apply, \u2190 comp_apply, Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.e'_2\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n\u22a2 \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\nI : Cover.Arrow S\nh :\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) x)) I =\n \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) y)) I\n\u22a2 \u2191((plusObj J P).map I.f.op) y = \u2191((plusObj J P).map I.f.op) y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\n\u22a2 Function.Surjective ((forget D).map (Cover.toMultiequalizer S (plusObj J P)))\n[PROOFSTEP]\nrintro (x : (multiequalizer (S.index _) : D))\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\n\u22a2 \u2203 a, (forget D).map (Cover.toMultiequalizer S (plusObj J P)) a = x\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := exists_of_sep P hsep X S (Meq.equiv _ _ x)\n[GOAL]\ncase right.intro\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\n\u22a2 \u2203 a, (forget D).map (Cover.toMultiequalizer S (plusObj J P)) a = x\n[PROOFSTEP]\nuse t\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\n\u22a2 (forget D).map (Cover.toMultiequalizer S (plusObj J P)) t = x\n[PROOFSTEP]\napply (Meq.equiv _ _).injective\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\n\u22a2 \u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) t) =\n \u2191(Meq.equiv (plusObj J P) S) x\n[PROOFSTEP]\nrw [\u2190 ht]\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\n\u22a2 \u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) t) = Meq.mk S t\n[PROOFSTEP]\next i\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\ni : Cover.Arrow S\n\u22a2 \u2191(\u2191(Meq.equiv (plusObj J P) S) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) t)) i = \u2191(Meq.mk S t) i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\ni : Cover.Arrow S\n\u22a2 \u2191(Multiequalizer.\u03b9 (Cover.index S (plusObj J P)) i) ((forget D).map (Cover.toMultiequalizer S (plusObj J P)) t) =\n \u2191(Meq.mk S t) i\n[PROOFSTEP]\nerw [\u2190 comp_apply]\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\ni : Cover.Arrow S\n\u22a2 \u2191(Cover.toMultiequalizer S (plusObj J P) \u226b Multiequalizer.\u03b9 (Cover.index S (plusObj J P)) i) t = \u2191(Meq.mk S t) i\n[PROOFSTEP]\nrw [Multiequalizer.lift_\u03b9]\n[GOAL]\ncase h.a.h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nhsep :\n \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj (P.obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191(P.map I.f.op) x = \u2191(P.map I.f.op) y) \u2192 x = y\nX : C\nS : Cover J X\nx : (forget D).obj (multiequalizer (Cover.index S (plusObj J P)))\nt : (forget D).obj ((plusObj J P).obj (op X))\nht : Meq.mk S t = \u2191(Meq.equiv (plusObj J P) S) x\ni : Cover.Arrow S\n\u22a2 \u2191((plusObj J P).map i.f.op) t = \u2191(Meq.mk S t) i\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 Presheaf.IsSheaf J (plusObj J (plusObj J P))\n[PROOFSTEP]\napply isSheaf_of_sep\n[GOAL]\ncase hsep\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 \u2200 (X : C) (S : Cover J X) (x y : (forget D).obj ((plusObj J P).obj (op X))),\n (\u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y) \u2192 x = y\n[PROOFSTEP]\nintro X S x y\n[GOAL]\ncase hsep\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : C\u1d52\u1d56 \u2964 D\nX : C\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj (op X))\n\u22a2 (\u2200 (I : Cover.Arrow S), \u2191((plusObj J P).map I.f.op) x = \u2191((plusObj J P).map I.f.op) y) \u2192 x = y\n[PROOFSTEP]\napply sep\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 sheafifyMap J (\ud835\udfd9 P) = \ud835\udfd9 (sheafify J P)\n[PROOFSTEP]\ndsimp [sheafifyMap, sheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 plusMap J (plusMap J (\ud835\udfd9 P)) = \ud835\udfd9 (plusObj J (plusObj J P))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\n\u22a2 sheafifyMap J (\u03b7 \u226b \u03b3) = sheafifyMap J \u03b7 \u226b sheafifyMap J \u03b3\n[PROOFSTEP]\ndsimp [sheafifyMap, sheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\n\u22a2 plusMap J (plusMap J (\u03b7 \u226b \u03b3)) = plusMap J (plusMap J \u03b7) \u226b plusMap J (plusMap J \u03b3)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u22a2 \u03b7 \u226b toSheafify J Q = toSheafify J P \u226b sheafifyMap J \u03b7\n[PROOFSTEP]\ndsimp [sheafifyMap, sheafify, toSheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u22a2 \u03b7 \u226b toPlus J Q \u226b plusMap J (toPlus J Q) = (toPlus J P \u226b plusMap J (toPlus J P)) \u226b plusMap J (plusMap J \u03b7)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\n\u22a2 IsIso (toSheafify J P)\n[PROOFSTEP]\ndsimp [toSheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\n\u22a2 IsIso (toPlus J P \u226b plusMap J (toPlus J P))\n[PROOFSTEP]\nhaveI := isIso_toPlus_of_isSheaf J P hP\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\nthis : IsIso (toPlus J P)\n\u22a2 IsIso (toPlus J P \u226b plusMap J (toPlus J P))\n[PROOFSTEP]\nchange (IsIso (toPlus J P \u226b (J.plusFunctor D).map (toPlus J P)))\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\nthis : IsIso (toPlus J P)\n\u22a2 IsIso (toPlus J P \u226b (plusFunctor J D).map (toPlus J P))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u22a2 toSheafify J P \u226b sheafifyLift J \u03b7 hQ = \u03b7\n[PROOFSTEP]\ndsimp only [sheafifyLift, toSheafify]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u22a2 (toPlus J P \u226b plusMap J (toPlus J P)) \u226b plusLift J (plusLift J \u03b7 hQ) hQ = \u03b7\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u03b3 : sheafify J P \u27f6 Q\n\u22a2 toSheafify J P \u226b \u03b3 = \u03b7 \u2192 \u03b3 = sheafifyLift J \u03b7 hQ\n[PROOFSTEP]\nintro h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u03b3 : sheafify J P \u27f6 Q\nh : toSheafify J P \u226b \u03b3 = \u03b7\n\u22a2 \u03b3 = sheafifyLift J \u03b7 hQ\n[PROOFSTEP]\napply plusLift_unique\n[GOAL]\ncase h\u03b3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u03b3 : sheafify J P \u27f6 Q\nh : toSheafify J P \u226b \u03b3 = \u03b7\n\u22a2 toPlus J (plusObj J P) \u226b \u03b3 = plusLift J \u03b7 hQ\n[PROOFSTEP]\napply plusLift_unique\n[GOAL]\ncase h\u03b3.h\u03b3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u03b3 : sheafify J P \u27f6 Q\nh : toSheafify J P \u226b \u03b3 = \u03b7\n\u22a2 toPlus J P \u226b toPlus J (plusObj J P) \u226b \u03b3 = \u03b7\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 plusMap_toPlus]\n[GOAL]\ncase h\u03b3.h\u03b3\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\n\u03b3 : sheafify J P \u27f6 Q\nh : toSheafify J P \u226b \u03b3 = \u03b7\n\u22a2 (toPlus J P \u226b plusMap J (toPlus J P)) \u226b \u03b3 = \u03b7\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\n\u22a2 (isoSheafify J hP).inv = sheafifyLift J (\ud835\udfd9 P) hP\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nhP : Presheaf.IsSheaf J P\n\u22a2 toSheafify J P \u226b (isoSheafify J hP).inv = \ud835\udfd9 P\n[PROOFSTEP]\nsimp [Iso.comp_inv_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 \u03b3 : sheafify J P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P \u226b \u03b7 = toSheafify J P \u226b \u03b3\n\u22a2 \u03b7 = \u03b3\n[PROOFSTEP]\napply J.plus_hom_ext _ _ hQ\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 \u03b3 : sheafify J P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P \u226b \u03b7 = toSheafify J P \u226b \u03b3\n\u22a2 toPlus J (plusObj J P) \u226b \u03b7 = toPlus J (plusObj J P) \u226b \u03b3\n[PROOFSTEP]\napply J.plus_hom_ext _ _ hQ\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 \u03b3 : sheafify J P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P \u226b \u03b7 = toSheafify J P \u226b \u03b3\n\u22a2 toPlus J P \u226b toPlus J (plusObj J P) \u226b \u03b7 = toPlus J P \u226b toPlus J (plusObj J P) \u226b \u03b3\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 Category.assoc, \u2190 plusMap_toPlus]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q : C\u1d52\u1d56 \u2964 D\n\u03b7 \u03b3 : sheafify J P \u27f6 Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P \u226b \u03b7 = toSheafify J P \u226b \u03b3\n\u22a2 (toPlus J P \u226b plusMap J (toPlus J P)) \u226b \u03b7 = (toPlus J P \u226b plusMap J (toPlus J P)) \u226b \u03b3\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nhR : Presheaf.IsSheaf J R\n\u22a2 sheafifyMap J \u03b7 \u226b sheafifyLift J \u03b3 hR = sheafifyLift J (\u03b7 \u226b \u03b3) hR\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nhR : Presheaf.IsSheaf J R\n\u22a2 toSheafify J P \u226b sheafifyMap J \u03b7 \u226b sheafifyLift J \u03b3 hR = \u03b7 \u226b \u03b3\n[PROOFSTEP]\nrw [\u2190 Category.assoc, \u2190 J.toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]\n[GOAL]\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget D)\ninst\u271d : Preadditive D\nF G : C\u1d52\u1d56 \u2964 D\n\u22a2 (presheafToSheaf J D).map 0 = 0\n[PROOFSTEP]\next : 3\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget D)\ninst\u271d : Preadditive D\nF G : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app ((presheafToSheaf J D).map 0).val x\u271d = NatTrans.app 0.val x\u271d\n[PROOFSTEP]\nrefine' colimit.hom_ext (fun j => _)\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2078 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2077 : Category.{max v u, w} D\ninst\u271d\u2076 : ConcreteCategory D\ninst\u271d\u2075 : PreservesLimits (forget D)\ninst\u271d\u2074 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b3 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b2 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d\u00b9 : ReflectsIsomorphisms (forget D)\ninst\u271d : Preadditive D\nF G : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\nj : (GrothendieckTopology.Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (GrothendieckTopology.diagram J (GrothendieckTopology.plusObj J F) x\u271d.unop) j \u226b\n NatTrans.app ((presheafToSheaf J D).map 0).val x\u271d =\n colimit.\u03b9 (GrothendieckTopology.diagram J (GrothendieckTopology.plusObj J F) x\u271d.unop) j \u226b NatTrans.app 0.val x\u271d\n[PROOFSTEP]\nerw [colimit.\u03b9_map, comp_zero, J.plusMap_zero, J.diagramNatTrans_zero, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\n\u22a2 \u2200 {X' X : C\u1d52\u1d56 \u2964 D} {Y : Sheaf J D} (f : X' \u27f6 X) (g : X \u27f6 (sheafToPresheaf J D).obj Y),\n \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n X' Y).symm\n (f \u226b g) =\n (presheafToSheaf J D).map f \u226b\n \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n X Y).symm\n g\n[PROOFSTEP]\nintro P Q R \u03b7 \u03b3\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP Q : C\u1d52\u1d56 \u2964 D\nR : Sheaf J D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 (sheafToPresheaf J D).obj R\n\u22a2 \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n P R).symm\n (\u03b7 \u226b \u03b3) =\n (presheafToSheaf J D).map \u03b7 \u226b\n \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n Q R).symm\n \u03b3\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP Q : C\u1d52\u1d56 \u2964 D\nR : Sheaf J D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 (sheafToPresheaf J D).obj R\n\u22a2 (\u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n P R).symm\n (\u03b7 \u226b \u03b3)).val =\n ((presheafToSheaf J D).map \u03b7 \u226b\n \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n Q R).symm\n \u03b3).val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP Q : C\u1d52\u1d56 \u2964 D\nR : Sheaf J D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 (sheafToPresheaf J D).obj R\n\u22a2 GrothendieckTopology.sheafifyLift J (\u03b7 \u226b \u03b3) (_ : Presheaf.IsSheaf J R.val) =\n GrothendieckTopology.sheafifyMap J \u03b7 \u226b GrothendieckTopology.sheafifyLift J \u03b3 (_ : Presheaf.IsSheaf J R.val)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP Q : C\u1d52\u1d56 \u2964 D\nR : Sheaf J D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 (sheafToPresheaf J D).obj R\n\u22a2 GrothendieckTopology.sheafifyMap J \u03b7 \u226b GrothendieckTopology.sheafifyLift J \u03b3 (_ : Presheaf.IsSheaf J R.val) =\n GrothendieckTopology.sheafifyLift J (\u03b7 \u226b \u03b3) (_ : Presheaf.IsSheaf J R.val)\n[PROOFSTEP]\napply J.sheafifyMap_sheafifyLift\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nX\u271d : C\u1d52\u1d56 \u2964 D\nY\u271d Y'\u271d : Sheaf J D\n\u03b7 : (presheafToSheaf J D).obj X\u271d \u27f6 Y\u271d\n\u03b3 : Y\u271d \u27f6 Y'\u271d\n\u22a2 \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n X\u271d Y'\u271d)\n (\u03b7 \u226b \u03b3) =\n \u2191((fun P Q =>\n { toFun := fun e => GrothendieckTopology.toSheafify J P \u226b e.val,\n invFun := fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) },\n left_inv :=\n (_ :\n \u2200 (e : (presheafToSheaf J D).obj P \u27f6 Q),\n (fun e => { val := GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) })\n ((fun e => GrothendieckTopology.toSheafify J P \u226b e.val) e) =\n e),\n right_inv :=\n (_ :\n \u2200 (e : P \u27f6 (sheafToPresheaf J D).obj Q),\n GrothendieckTopology.toSheafify J P \u226b\n GrothendieckTopology.sheafifyLift J e (_ : Presheaf.IsSheaf J Q.val) =\n e) })\n X\u271d Y\u271d)\n \u03b7 \u226b\n (sheafToPresheaf J D).map \u03b3\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nX\u271d : C\u1d52\u1d56 \u2964 D\nY\u271d Y'\u271d : Sheaf J D\n\u03b7 : (presheafToSheaf J D).obj X\u271d \u27f6 Y\u271d\n\u03b3 : Y\u271d \u27f6 Y'\u271d\n\u22a2 GrothendieckTopology.toSheafify J X\u271d \u226b \u03b7.val \u226b \u03b3.val = (GrothendieckTopology.toSheafify J X\u271d \u226b \u03b7.val) \u226b \u03b3.val\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF G : Sheaf J D\nf : F \u27f6 G\nm : Mono f\n\u22a2 Mono f.val\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF G : Sheaf J D\nf : F \u27f6 G\nm : Mono f.val\n\u22a2 Mono f\n[PROOFSTEP]\nexact Sheaf.Hom.mono_of_presheaf_mono J D f\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : Sheaf J D\n\u22a2 { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).hom } \u226b\n { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).inv } =\n \ud835\udfd9 P\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : Sheaf J D\n\u22a2 ({ val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).hom } \u226b\n { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).inv }).val =\n (\ud835\udfd9 P).val\n[PROOFSTEP]\napply (J.isoSheafify P.2).hom_inv_id\n[GOAL]\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : Sheaf J D\n\u22a2 { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).inv } \u226b\n { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).hom } =\n \ud835\udfd9 ((presheafToSheaf J D).obj P.val)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2077 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category.{max v u, w} D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nP : Sheaf J D\n\u22a2 ({ val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).inv } \u226b\n { val := (GrothendieckTopology.isoSheafify J (_ : Presheaf.IsSheaf J P.val)).hom }).val =\n (\ud835\udfd9 ((presheafToSheaf J D).obj P.val)).val\n[PROOFSTEP]\napply (J.isoSheafify P.2).inv_hom_id\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Sheafification", "llama_tokens": 123866, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.26050057014291067}} {"text": "[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Nontrivial R\n\u22a2 \u00acSeparable 0\n[PROOFSTEP]\nrintro \u27e8x, y, h\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Nontrivial R\nx y : R[X]\nh : x * 0 + y * \u2191derivative 0 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp only [derivative_zero, mul_zero, add_zero, zero_ne_one] at h \n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Subsingleton R\nf : R[X]\n\u22a2 Separable f\n[PROOFSTEP]\nsimp [Separable, IsCoprime]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\na : R\n\u22a2 Separable (X + \u2191C a)\n[PROOFSTEP]\nrw [separable_def, derivative_add, derivative_X, derivative_C, add_zero]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\na : R\n\u22a2 IsCoprime (X + \u2191C a) 1\n[PROOFSTEP]\nexact isCoprime_one_right\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\n\u22a2 Separable X\n[PROOFSTEP]\nrw [separable_def, derivative_X]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\n\u22a2 IsCoprime X 1\n[PROOFSTEP]\nexact isCoprime_one_right\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nr : R\n\u22a2 Separable (\u2191C r) \u2194 IsUnit r\n[PROOFSTEP]\nrw [separable_def, derivative_C, isCoprime_zero_right, isUnit_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\n\u22a2 Separable f\n[PROOFSTEP]\nhave := h.of_mul_left_left\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (\u2191derivative (f * g))\n\u22a2 Separable f\n[PROOFSTEP]\nrw [derivative_mul] at this \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (\u2191derivative f * g + f * \u2191derivative g)\n\u22a2 Separable f\n[PROOFSTEP]\nexact IsCoprime.of_mul_right_left (IsCoprime.of_add_mul_left_right this)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\n\u22a2 Separable g\n[PROOFSTEP]\nrw [mul_comm] at h \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (g * f)\n\u22a2 Separable g\n[PROOFSTEP]\nexact h.of_mul_left\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nhf : Separable f\nhfg : g \u2223 f\n\u22a2 Separable g\n[PROOFSTEP]\nrcases hfg with \u27e8f', rfl\u27e9\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\ng f' : R[X]\nhf : Separable (g * f')\n\u22a2 Separable g\n[PROOFSTEP]\nexact Separable.of_mul_left hf\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\n\u22a2 IsCoprime f g\n[PROOFSTEP]\nhave := h.of_mul_left_left\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (\u2191derivative (f * g))\n\u22a2 IsCoprime f g\n[PROOFSTEP]\nrw [derivative_mul] at this \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (\u2191derivative f * g + f * \u2191derivative g)\n\u22a2 IsCoprime f g\n[PROOFSTEP]\nexact IsCoprime.of_mul_right_right (IsCoprime.of_add_mul_left_right this)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf : R[X]\nn : \u2115\nh : Separable (f ^ (n + 2))\n\u22a2 IsUnit f \u2228 Separable f \u2227 n + 2 = 1 \u2228 n + 2 = 0\n[PROOFSTEP]\nrw [pow_succ, pow_succ] at h \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\nf : R[X]\nn : \u2115\nh : Separable (f * (f * f ^ n))\n\u22a2 IsUnit f \u2228 Separable f \u2227 n + 2 = 1 \u2228 n + 2 = 0\n[PROOFSTEP]\nexact Or.inl (isCoprime_self.1 h.isCoprime.of_mul_right_left)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np : R[X]\nh : Separable p\nf : R \u2192+* S\na b : R[X]\nH : a * p + b * \u2191derivative p = 1\n\u22a2 Polynomial.map f a * Polynomial.map f p + Polynomial.map f b * \u2191derivative (Polynomial.map f p) = 1\n[PROOFSTEP]\nrw [derivative_map, \u2190 Polynomial.map_mul, \u2190 Polynomial.map_mul, \u2190 Polynomial.map_add, H, Polynomial.map_one]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhp : Separable p\nhq : q * q \u2223 p\n\u22a2 IsUnit q\n[PROOFSTEP]\nobtain \u27e8p, rfl\u27e9 := hq\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : Separable (q * q * p)\n\u22a2 IsUnit q\n[PROOFSTEP]\napply isCoprime_self.mp\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : Separable (q * q * p)\n\u22a2 IsCoprime q q\n[PROOFSTEP]\nhave : IsCoprime (q * (q * p)) (q * (derivative q * p + derivative q * p + q * derivative p)) :=\n by\n simp only [\u2190 mul_assoc, mul_add]\n dsimp only [Separable] at hp \n convert hp using 1\n rw [derivative_mul, derivative_mul]\n ring\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : Separable (q * q * p)\n\u22a2 IsCoprime (q * (q * p)) (q * (\u2191derivative q * p + \u2191derivative q * p + q * \u2191derivative p))\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, mul_add]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : Separable (q * q * p)\n\u22a2 IsCoprime (q * q * p) (q * \u2191derivative q * p + q * \u2191derivative q * p + q * q * \u2191derivative p)\n[PROOFSTEP]\ndsimp only [Separable] at hp \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (\u2191derivative (q * q * p))\n\u22a2 IsCoprime (q * q * p) (q * \u2191derivative q * p + q * \u2191derivative q * p + q * q * \u2191derivative p)\n[PROOFSTEP]\nconvert hp using 1\n[GOAL]\ncase h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (\u2191derivative (q * q * p))\n\u22a2 q * \u2191derivative q * p + q * \u2191derivative q * p + q * q * \u2191derivative p = \u2191derivative (q * q * p)\n[PROOFSTEP]\nrw [derivative_mul, derivative_mul]\n[GOAL]\ncase h.e'_4\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : IsCoprime (q * q * p) (\u2191derivative (q * q * p))\n\u22a2 q * \u2191derivative q * p + q * \u2191derivative q * p + q * q * \u2191derivative p =\n (\u2191derivative q * q + q * \u2191derivative q) * p + q * q * \u2191derivative p\n[PROOFSTEP]\nring\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\nq p : R[X]\nhp : Separable (q * q * p)\nthis : IsCoprime (q * (q * p)) (q * (\u2191derivative q * p + \u2191derivative q * p + q * \u2191derivative p))\n\u22a2 IsCoprime q q\n[PROOFSTEP]\nexact IsCoprime.of_mul_right_left (IsCoprime.of_mul_left_left this)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhq : \u00acIsUnit q\nhsep : Separable p\n\u22a2 multiplicity q p \u2264 1\n[PROOFSTEP]\ncontrapose! hq\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n\u22a2 IsUnit q\n[PROOFSTEP]\napply isUnit_of_self_mul_dvd_separable hsep\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n\u22a2 q * q \u2223 p\n[PROOFSTEP]\nrw [\u2190 sq]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n\u22a2 q ^ 2 \u2223 p\n[PROOFSTEP]\napply multiplicity.pow_dvd_of_le_multiplicity\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n\u22a2 \u21912 \u2264 multiplicity q p\n[PROOFSTEP]\nhave h : \u27e8Part.Dom 1 \u2227 Part.Dom 1, fun _ \u21a6 2\u27e9 \u2264 multiplicity q p := PartENat.add_one_le_of_lt hq\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom \u2227 1.Dom, get := fun x => 2 } \u2264 multiplicity q p\n\u22a2 \u21912 \u2264 multiplicity q p\n[PROOFSTEP]\nrw [and_self] at h \n[GOAL]\ncase a\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q\u271d : \u2115\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom, get := fun x => 2 } \u2264 multiplicity q p\n\u22a2 \u21912 \u2264 multiplicity q p\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q : \u2115\np : R[X]\nhsep : Separable p\n\u22a2 Squarefree p\n[PROOFSTEP]\nrw [multiplicity.squarefree_iff_multiplicity_le_one p]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np\u271d q : \u2115\np : R[X]\nhsep : Separable p\n\u22a2 \u2200 (x : R[X]), multiplicity x p \u2264 1 \u2228 IsUnit x\n[PROOFSTEP]\nexact fun f => or_iff_not_imp_right.mpr fun hunit => multiplicity_le_one_of_separable hunit hsep\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nx : R\n\u22a2 Separable (X - \u2191C x)\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, C_neg] using separable_X_add_C (-x)\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nf g : R[X]\nhf : Separable f\nhg : Separable g\nh : IsCoprime f g\n\u22a2 Separable (f * g)\n[PROOFSTEP]\nrw [separable_def, derivative_mul]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nf g : R[X]\nhf : Separable f\nhg : Separable g\nh : IsCoprime f g\n\u22a2 IsCoprime (f * g) (\u2191derivative f * g + f * \u2191derivative g)\n[PROOFSTEP]\nexact ((hf.mul_right h).add_mul_left_right _).mul_left ((h.symm.mul_right hg).mul_add_right_right _)\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns\u271d : Finset \u03b9\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih :\n (\u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 x \u2260 y \u2192 IsCoprime (f x) (f y)) \u2192\n (\u2200 (x : \u03b9), x \u2208 s \u2192 Separable (f x)) \u2192 Separable (\u220f x in s, f x)\nh1 : \u2200 (x : \u03b9), x \u2208 insert a s \u2192 \u2200 (y : \u03b9), y \u2208 insert a s \u2192 x \u2260 y \u2192 IsCoprime (f x) (f y)\nh2 : \u2200 (x : \u03b9), x \u2208 insert a s \u2192 Separable (f x)\n\u22a2 Separable (\u220f x in insert a s, f x)\n[PROOFSTEP]\nsimp_rw [Finset.forall_mem_insert, forall_and] at h1 h2 \n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns\u271d : Finset \u03b9\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih :\n (\u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 x \u2260 y \u2192 IsCoprime (f x) (f y)) \u2192\n (\u2200 (x : \u03b9), x \u2208 s \u2192 Separable (f x)) \u2192 Separable (\u220f x in s, f x)\nh2 : Separable (f a) \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 Separable (f x)\nh1 :\n ((a \u2260 a \u2192 IsCoprime (f a) (f a)) \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 a \u2260 x \u2192 IsCoprime (f a) (f x)) \u2227\n (\u2200 (x : \u03b9), x \u2208 s \u2192 x \u2260 a \u2192 IsCoprime (f x) (f a)) \u2227\n \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (x_2 : \u03b9), x_2 \u2208 s \u2192 x \u2260 x_2 \u2192 IsCoprime (f x) (f x_2)\n\u22a2 Separable (\u220f x in insert a s, f x)\n[PROOFSTEP]\nrw [prod_insert has]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns\u271d : Finset \u03b9\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih :\n (\u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 x \u2260 y \u2192 IsCoprime (f x) (f y)) \u2192\n (\u2200 (x : \u03b9), x \u2208 s \u2192 Separable (f x)) \u2192 Separable (\u220f x in s, f x)\nh2 : Separable (f a) \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 Separable (f x)\nh1 :\n ((a \u2260 a \u2192 IsCoprime (f a) (f a)) \u2227 \u2200 (x : \u03b9), x \u2208 s \u2192 a \u2260 x \u2192 IsCoprime (f a) (f x)) \u2227\n (\u2200 (x : \u03b9), x \u2208 s \u2192 x \u2260 a \u2192 IsCoprime (f x) (f a)) \u2227\n \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (x_2 : \u03b9), x_2 \u2208 s \u2192 x \u2260 x_2 \u2192 IsCoprime (f x) (f x_2)\n\u22a2 Separable (f a * \u220f x in s, f x)\n[PROOFSTEP]\nexact\n h2.1.mul (ih h1.2.2 h2.2) (IsCoprime.prod_right fun i his => h1.1.2 i his <| Ne.symm <| ne_of_mem_of_not_mem his has)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R\ns : Finset \u03b9\nhfs : Separable (\u220f i in s, (X - \u2191C (f i)))\nx y : \u03b9\nhx : x \u2208 s\nhy : y \u2208 s\nhfxy : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nby_contra hxy\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R\ns : Finset \u03b9\nhfs : Separable (\u220f i in s, (X - \u2191C (f i)))\nx y : \u03b9\nhx : x \u2208 s\nhy : y \u2208 s\nhfxy : f x = f y\nhxy : \u00acx = y\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 insert_erase hx, prod_insert (not_mem_erase _ _), \u2190 insert_erase (mem_erase_of_ne_of_mem (Ne.symm hxy) hy),\n prod_insert (not_mem_erase _ _), \u2190 mul_assoc, hfxy, \u2190 sq] at hfs \n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R\ns : Finset \u03b9\nx y : \u03b9\nhfs : Separable ((X - \u2191C (f y)) ^ 2 * \u220f x in Finset.erase (Finset.erase s x) y, (X - \u2191C (f x)))\nhx : x \u2208 s\nhy : y \u2208 s\nhfxy : f x = f y\nhxy : \u00acx = y\n\u22a2 False\n[PROOFSTEP]\ncases (hfs.of_mul_left.of_pow (not_isUnit_X_sub_C _) two_ne_zero).2\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\ns : Multiset R\nhs : Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 Multiset.Nodup s\n[PROOFSTEP]\nrw [Multiset.nodup_iff_ne_cons_cons]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\ns : Multiset R\nhs : Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 \u2200 (a : R) (t : Multiset R), s \u2260 a ::\u2098 a ::\u2098 t\n[PROOFSTEP]\nrintro a t rfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\na : R\nt : Multiset R\nhs : Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (a ::\u2098 a ::\u2098 t)))\n\u22a2 False\n[PROOFSTEP]\nrefine' not_isUnit_X_sub_C a (isUnit_of_self_mul_dvd_separable hs _)\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\na : R\nt : Multiset R\nhs : Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (a ::\u2098 a ::\u2098 t)))\n\u22a2 (X - \u2191C a) * (X - \u2191C a) \u2223 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (a ::\u2098 a ::\u2098 t))\n[PROOFSTEP]\nsimpa only [Multiset.map_cons, Multiset.prod_cons] using mul_dvd_mul_left _ (dvd_mul_right _ _)\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u22a2 Separable (X ^ n - \u2191C \u2191u)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\n\u22a2 Separable (X ^ n - \u2191C \u2191u)\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hpos)\n[GOAL]\ncase inl\nR : Type u\ninst\u271d : CommRing R\nu : R\u02e3\n\u271d : Nontrivial R\nhn : IsUnit \u21910\n\u22a2 Separable (X ^ 0 - \u2191C \u2191u)\n[PROOFSTEP]\nsimp at hn \n[GOAL]\ncase inr\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\n\u22a2 Separable (X ^ n - \u2191C \u2191u)\n[PROOFSTEP]\napply (separable_def' (X ^ n - C (u : R))).2\n[GOAL]\ncase inr\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\n\u22a2 \u2203 a b, a * (X ^ n - \u2191C \u2191u) + b * \u2191derivative (X ^ n - \u2191C \u2191u) = 1\n[PROOFSTEP]\nobtain \u27e8n', hn'\u27e9 := hn.exists_left_inv\n[GOAL]\ncase inr.intro\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 \u2203 a b, a * (X ^ n - \u2191C \u2191u) + b * \u2191derivative (X ^ n - \u2191C \u2191u) = 1\n[PROOFSTEP]\nrefine' \u27e8-C \u2191u\u207b\u00b9, C (\u2191u\u207b\u00b9 : R) * C n' * X, _\u27e9\n[GOAL]\ncase inr.intro\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 -\u2191C \u2191u\u207b\u00b9 * (X ^ n - \u2191C \u2191u) + \u2191C \u2191u\u207b\u00b9 * \u2191C n' * X * \u2191derivative (X ^ n - \u2191C \u2191u) = 1\n[PROOFSTEP]\nrw [derivative_sub, derivative_C, sub_zero, derivative_pow X n, derivative_X, mul_one]\n[GOAL]\ncase inr.intro\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 -\u2191C \u2191u\u207b\u00b9 * (X ^ n - \u2191C \u2191u) + \u2191C \u2191u\u207b\u00b9 * \u2191C n' * X * (\u2191C \u2191n * X ^ (n - 1)) = 1\n[PROOFSTEP]\ncalc\n -C \u2191u\u207b\u00b9 * (X ^ n - C \u2191u) + C \u2191u\u207b\u00b9 * C n' * X * (\u2191n * X ^ (n - 1)) =\n C (\u2191u\u207b\u00b9 * \u2191u) - C \u2191u\u207b\u00b9 * X ^ n + C \u2191u\u207b\u00b9 * C (n' * \u2191n) * (X * X ^ (n - 1)) :=\n by\n simp only [C.map_mul, C_eq_nat_cast]\n ring\n _ = 1 := by\n simp only [Units.inv_mul, hn', C.map_one, mul_one, \u2190 pow_succ, Nat.sub_add_cancel (show 1 \u2264 n from hpos),\n sub_add_cancel]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 -\u2191C \u2191u\u207b\u00b9 * (X ^ n - \u2191C \u2191u) + \u2191C \u2191u\u207b\u00b9 * \u2191C n' * X * (\u2191n * X ^ (n - 1)) =\n \u2191C (\u2191u\u207b\u00b9 * \u2191u) - \u2191C \u2191u\u207b\u00b9 * X ^ n + \u2191C \u2191u\u207b\u00b9 * \u2191C (n' * \u2191n) * (X * X ^ (n - 1))\n[PROOFSTEP]\nsimp only [C.map_mul, C_eq_nat_cast]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 -\u2191C \u2191u\u207b\u00b9 * (X ^ n - \u2191C \u2191u) + \u2191C \u2191u\u207b\u00b9 * \u2191C n' * X * (\u2191n * X ^ (n - 1)) =\n \u2191C \u2191u\u207b\u00b9 * \u2191C \u2191u - \u2191C \u2191u\u207b\u00b9 * X ^ n + \u2191C \u2191u\u207b\u00b9 * (\u2191C n' * \u2191n) * (X * X ^ (n - 1))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nn : \u2115\nu : R\u02e3\nhn : IsUnit \u2191n\n\u271d : Nontrivial R\nhpos : n > 0\nn' : R\nhn' : n' * \u2191n = 1\n\u22a2 \u2191C (\u2191u\u207b\u00b9 * \u2191u) - \u2191C \u2191u\u207b\u00b9 * X ^ n + \u2191C \u2191u\u207b\u00b9 * \u2191C (n' * \u2191n) * (X * X ^ (n - 1)) = 1\n[PROOFSTEP]\nsimp only [Units.inv_mul, hn', C.map_one, mul_one, \u2190 pow_succ, Nat.sub_add_cancel (show 1 \u2264 n from hpos),\n sub_add_cancel]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\np : R[X]\nhsep : Separable p\nx : R\n\u22a2 rootMultiplicity x p \u2264 1\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\np : R[X]\nhsep : Separable p\nx : R\nhp : p = 0\n\u22a2 rootMultiplicity x p \u2264 1\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\np : R[X]\nhsep : Separable p\nx : R\nhp : \u00acp = 0\n\u22a2 rootMultiplicity x p \u2264 1\n[PROOFSTEP]\nrw [rootMultiplicity_eq_multiplicity, dif_neg hp, \u2190 PartENat.coe_le_coe, PartENat.natCast_get, Nat.cast_one]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\np : R[X]\nhsep : Separable p\nx : R\nhp : \u00acp = 0\n\u22a2 multiplicity (X - \u2191C x) p \u2264 1\n[PROOFSTEP]\nexact multiplicity_le_one_of_separable (not_isUnit_X_sub_C _) hsep\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhsep : Separable p\nx : R\n\u22a2 Multiset.count x (roots p) \u2264 1\n[PROOFSTEP]\nrw [count_roots p]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhsep : Separable p\nx : R\n\u22a2 rootMultiplicity x p \u2264 1\n[PROOFSTEP]\nexact rootMultiplicity_le_one_of_separable hsep x\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nh : \u2191derivative f \u2260 0\ng : F[X]\nhg1 : g \u2208 nonunits F[X]\n_hg2 : g \u2260 0\nx\u271d : g \u2223 f\nhg4 : g \u2223 \u2191derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]\u02e3\nhu : \u2191u = p\n\u22a2 f \u2223 \u2191derivative f\n[PROOFSTEP]\nconv_lhs => rw [hg3, \u2190 hu]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nh : \u2191derivative f \u2260 0\ng : F[X]\nhg1 : g \u2208 nonunits F[X]\n_hg2 : g \u2260 0\nx\u271d : g \u2223 f\nhg4 : g \u2223 \u2191derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]\u02e3\nhu : \u2191u = p\n| f\n[PROOFSTEP]\nrw [hg3, \u2190 hu]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nh : \u2191derivative f \u2260 0\ng : F[X]\nhg1 : g \u2208 nonunits F[X]\n_hg2 : g \u2260 0\nx\u271d : g \u2223 f\nhg4 : g \u2223 \u2191derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]\u02e3\nhu : \u2191u = p\n| f\n[PROOFSTEP]\nrw [hg3, \u2190 hu]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nh : \u2191derivative f \u2260 0\ng : F[X]\nhg1 : g \u2208 nonunits F[X]\n_hg2 : g \u2260 0\nx\u271d : g \u2223 f\nhg4 : g \u2223 \u2191derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]\u02e3\nhu : \u2191u = p\n| f\n[PROOFSTEP]\nrw [hg3, \u2190 hu]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nh : \u2191derivative f \u2260 0\ng : F[X]\nhg1 : g \u2208 nonunits F[X]\n_hg2 : g \u2260 0\nx\u271d : g \u2223 f\nhg4 : g \u2223 \u2191derivative f\np : F[X]\nhg3 : f = g * p\nu : F[X]\u02e3\nhu : \u2191u = p\n\u22a2 g * \u2191u \u2223 \u2191derivative f\n[PROOFSTEP]\nrwa [Units.mul_right_dvd]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F \u2192+* K\np : F[X]\n\u22a2 Separable (map f p) \u2194 Separable p\n[PROOFSTEP]\nsimp_rw [separable_def, derivative_map, isCoprime_map]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\n\u03b9 : Type u_1\nf : \u03b9 \u2192 F\ns : Finset \u03b9\nH : \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 f x = f y \u2192 x = y\n\u22a2 Separable (\u220f i in s, (X - \u2191C (f i)))\n[PROOFSTEP]\nrw [\u2190 prod_attach]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\n\u03b9 : Type u_1\nf : \u03b9 \u2192 F\ns : Finset \u03b9\nH : \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 f x = f y \u2192 x = y\n\u22a2 Separable (\u220f x in attach s, (X - \u2191C (f \u2191x)))\n[PROOFSTEP]\nexact\n separable_prod'\n (fun x _hx y _hy hxy =>\n @pairwise_coprime_X_sub_C _ _ { x // x \u2208 s } (fun x => f x) (fun x y hxy => Subtype.eq <| H x.1 x.2 y.1 y.2 hxy) _\n _ hxy)\n fun _ _ => separable_X_sub_C\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 F\n\u22a2 (\u2200 (x : \u03b9), x \u2208 univ \u2192 \u2200 (y : \u03b9), y \u2208 univ \u2192 f x = f y \u2192 x = y) \u2194 Function.Injective f\n[PROOFSTEP]\nsimp_rw [mem_univ, true_imp_iff, Function.Injective]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F p) g = f\n[PROOFSTEP]\nrcases p.eq_zero_or_pos with (rfl | hp)\n[GOAL]\ncase inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nHF : CharP F 0\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F 0) g = f\n[PROOFSTEP]\nhaveI := CharP.charP_to_charZero F\n[GOAL]\ncase inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nHF : CharP F 0\nthis : CharZero F\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F 0) g = f\n[PROOFSTEP]\nhave := natDegree_eq_zero_of_derivative_eq_zero H\n[GOAL]\ncase inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nHF : CharP F 0\nthis\u271d : CharZero F\nthis : natDegree f = 0\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F 0) g = f\n[PROOFSTEP]\nhave := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'\n[GOAL]\ncase inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nHF : CharP F 0\nthis\u271d\u00b9 : CharZero F\nthis\u271d : natDegree f = 0\nthis : natDegree f \u2260 0\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F 0) g = f\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase inr\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nhp : p > 0\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F p) g = f\n[PROOFSTEP]\nhaveI := isLocalRingHom_expand F hp\n[GOAL]\ncase inr\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nhp : p > 0\nthis : IsLocalRingHom \u2191(expand F p)\n\u22a2 Separable f \u2228 \u00acSeparable f \u2227 \u2203 g, Irreducible g \u2227 \u2191(expand F p) g = f\n[PROOFSTEP]\nexact\n Or.inr\n \u27e8by rw [separable_iff_derivative_ne_zero hf, Classical.not_not, H], contract p f,\n of_irreducible_map (expand F p : F[X] \u2192+* F[X]) (by rwa [\u2190 expand_contract p H hp.ne'] at hf ),\n expand_contract p H hp.ne'\u27e9\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nhp : p > 0\nthis : IsLocalRingHom \u2191(expand F p)\n\u22a2 \u00acSeparable f\n[PROOFSTEP]\nrw [separable_iff_derivative_ne_zero hf, Classical.not_not, H]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nH : \u2191derivative f = 0\nhp : p > 0\nthis : IsLocalRingHom \u2191(expand F p)\n\u22a2 Irreducible (\u2191\u2191(expand F p) (contract p f))\n[PROOFSTEP]\nrwa [\u2190 expand_contract p H hp.ne'] at hf \n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : p \u2260 0\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nreplace hp : p.Prime := (CharP.char_is_prime_or_zero F p).resolve_right hp\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : Nat.Prime p\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\ninduction' hn : f.natDegree using Nat.strong_induction_on with N ih generalizing f\n[GOAL]\ncase h\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nN : \u2115\nih : \u2200 (m : \u2115), m < N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nf : F[X]\nhf : Irreducible f\nhn : natDegree f = N\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrcases separable_or p hf with (h | \u27e8h1, g, hg, hgf\u27e9)\n[GOAL]\ncase h.inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nN : \u2115\nih : \u2200 (m : \u2115), m < N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nf : F[X]\nhf : Irreducible f\nhn : natDegree f = N\nh : Separable f\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrefine' \u27e80, f, h, _\u27e9\n[GOAL]\ncase h.inl\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nN : \u2115\nih : \u2200 (m : \u2115), m < N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nf : F[X]\nhf : Irreducible f\nhn : natDegree f = N\nh : Separable f\n\u22a2 \u2191(expand F (p ^ 0)) f = f\n[PROOFSTEP]\nrw [pow_zero, expand_one]\n[GOAL]\ncase h.inr.intro.intro.intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nN : \u2115\nih : \u2200 (m : \u2115), m < N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nf : F[X]\nhf : Irreducible f\nhn : natDegree f = N\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\ncases' N with N\n[GOAL]\ncase h.inr.intro.intro.intro.zero\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.zero\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrw [natDegree_eq_zero_iff_degree_le_zero, degree_le_zero_iff] at hn \n[GOAL]\ncase h.inr.intro.intro.intro.zero\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : f = \u2191C (coeff f 0)\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrw [hn, separable_C, isUnit_iff_ne_zero, Classical.not_not] at h1 \n[GOAL]\ncase h.inr.intro.intro.intro.zero\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : coeff f 0 = 0\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : f = \u2191C (coeff f 0)\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nhave hf0 : f \u2260 0 := hf.ne_zero\n[GOAL]\ncase h.inr.intro.intro.intro.zero\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : coeff f 0 = 0\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : f = \u2191C (coeff f 0)\nhf0 : f \u2260 0\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrw [h1, C_0] at hn \n[GOAL]\ncase h.inr.intro.intro.intro.zero\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : coeff f 0 = 0\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nih :\n \u2200 (m : \u2115),\n m < Nat.zero \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : f = 0\nhf0 : f \u2260 0\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nexact absurd hn hf0\n[GOAL]\ncase h.inr.intro.intro.intro.succ\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nhave hg1 : g.natDegree * p = N.succ := by rwa [\u2190 natDegree_expand, hgf]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\n\u22a2 natDegree g * p = Nat.succ N\n[PROOFSTEP]\nrwa [\u2190 natDegree_expand, hgf]\n[GOAL]\ncase h.inr.intro.intro.intro.succ\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nhave hg2 : g.natDegree \u2260 0 := by\n intro this\n rw [this, zero_mul] at hg1 \n cases hg1\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\n\u22a2 natDegree g \u2260 0\n[PROOFSTEP]\nintro this\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\nthis : natDegree g = 0\n\u22a2 False\n[PROOFSTEP]\nrw [this, zero_mul] at hg1 \n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : 0 = Nat.succ N\nthis : natDegree g = 0\n\u22a2 False\n[PROOFSTEP]\ncases hg1\n[GOAL]\ncase h.inr.intro.intro.intro.succ\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\nhg2 : natDegree g \u2260 0\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nhave hg3 : g.natDegree < N.succ := by\n rw [\u2190 mul_one g.natDegree, \u2190 hg1]\n exact Nat.mul_lt_mul_of_pos_left hp.one_lt hg2.bot_lt\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\nhg2 : natDegree g \u2260 0\n\u22a2 natDegree g < Nat.succ N\n[PROOFSTEP]\nrw [\u2190 mul_one g.natDegree, \u2190 hg1]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\nhg2 : natDegree g \u2260 0\n\u22a2 natDegree g * 1 < natDegree g * p\n[PROOFSTEP]\nexact Nat.mul_lt_mul_of_pos_left hp.one_lt hg2.bot_lt\n[GOAL]\ncase h.inr.intro.intro.intro.succ\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\ng : F[X]\nhg : Irreducible g\nhgf : \u2191(expand F p) g = f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nhg1 : natDegree g * p = Nat.succ N\nhg2 : natDegree g \u2260 0\nhg3 : natDegree g < Nat.succ N\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrcases ih _ hg3 hg rfl with \u27e8n, g, hg4, rfl\u27e9\n[GOAL]\ncase h.inr.intro.intro.intro.succ.intro.intro.intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nn : \u2115\ng : F[X]\nhg4 : Separable g\nhg : Irreducible (\u2191(expand F (p ^ n)) g)\nhgf : \u2191(expand F p) (\u2191(expand F (p ^ n)) g) = f\nhg1 : natDegree (\u2191(expand F (p ^ n)) g) * p = Nat.succ N\nhg2 : natDegree (\u2191(expand F (p ^ n)) g) \u2260 0\nhg3 : natDegree (\u2191(expand F (p ^ n)) g) < Nat.succ N\n\u22a2 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\n[PROOFSTEP]\nrefine' \u27e8n + 1, g, hg4, _\u27e9\n[GOAL]\ncase h.inr.intro.intro.intro.succ.intro.intro.intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf\u271d : F[X]\nhf\u271d : Irreducible f\u271d\nhp : Nat.Prime p\nx\u271d : \u2115\nhn\u271d : natDegree f\u271d = x\u271d\nf : F[X]\nhf : Irreducible f\nh1 : \u00acSeparable f\nN : \u2115\nih :\n \u2200 (m : \u2115),\n m < Nat.succ N \u2192 \u2200 {f : F[X]}, Irreducible f \u2192 natDegree f = m \u2192 \u2203 n g, Separable g \u2227 \u2191(expand F (p ^ n)) g = f\nhn : natDegree f = Nat.succ N\nn : \u2115\ng : F[X]\nhg4 : Separable g\nhg : Irreducible (\u2191(expand F (p ^ n)) g)\nhgf : \u2191(expand F p) (\u2191(expand F (p ^ n)) g) = f\nhg1 : natDegree (\u2191(expand F (p ^ n)) g) * p = Nat.succ N\nhg2 : natDegree (\u2191(expand F (p ^ n)) g) \u2260 0\nhg3 : natDegree (\u2191(expand F (p ^ n)) g) < Nat.succ N\n\u22a2 \u2191(expand F (p ^ (n + 1))) g = f\n[PROOFSTEP]\nrw [\u2190 hgf, expand_expand, pow_succ]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : Separable (\u2191(expand F (p ^ n)) f)\n\u22a2 IsUnit f \u2228 n = 0\n[PROOFSTEP]\nrw [or_iff_not_imp_right]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : Separable (\u2191(expand F (p ^ n)) f)\n\u22a2 \u00acn = 0 \u2192 IsUnit f\n[PROOFSTEP]\nrintro hn : n \u2260 0\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : Separable (\u2191(expand F (p ^ n)) f)\nhn : n \u2260 0\n\u22a2 IsUnit f\n[PROOFSTEP]\nhave hf2 : derivative (expand F (p ^ n) f) = 0 := by\n rw [derivative_expand, Nat.cast_pow, CharP.cast_eq_zero, zero_pow hn.bot_lt, zero_mul, mul_zero]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : Separable (\u2191(expand F (p ^ n)) f)\nhn : n \u2260 0\n\u22a2 \u2191derivative (\u2191(expand F (p ^ n)) f) = 0\n[PROOFSTEP]\nrw [derivative_expand, Nat.cast_pow, CharP.cast_eq_zero, zero_pow hn.bot_lt, zero_mul, mul_zero]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : Separable (\u2191(expand F (p ^ n)) f)\nhn : n \u2260 0\nhf2 : \u2191derivative (\u2191(expand F (p ^ n)) f) = 0\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [separable_def, hf2, isCoprime_zero_right, isUnit_iff] at hf \n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhf : \u2203 r, IsUnit r \u2227 \u2191C r = \u2191(expand F (p ^ n)) f\nhn : n \u2260 0\nhf2 : \u2191derivative (\u2191(expand F (p ^ n)) f) = 0\n\u22a2 IsUnit f\n[PROOFSTEP]\nrcases hf with \u27e8r, hr, hrf\u27e9\n[GOAL]\ncase intro.intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhn : n \u2260 0\nhf2 : \u2191derivative (\u2191(expand F (p ^ n)) f) = 0\nr : F\nhr : IsUnit r\nhrf : \u2191C r = \u2191(expand F (p ^ n)) f\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [eq_comm, expand_eq_C (pow_pos hp _)] at hrf \n[GOAL]\ncase intro.intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nn : \u2115\nhp : 0 < p\nhn : n \u2260 0\nhf2 : \u2191derivative (\u2191(expand F (p ^ n)) f) = 0\nr : F\nhr : IsUnit r\nhrf : f = \u2191C r\n\u22a2 IsUnit f\n[PROOFSTEP]\nrwa [hrf, isUnit_C]\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 : \u2115\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\nn\u2082 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhgf\u2082 : \u2191(expand F (p ^ n\u2082)) g\u2082 = f\n\u22a2 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nrevert g\u2081 g\u2082\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\n\u22a2 \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192 \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nclear! K\n[GOAL]\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\n\u22a2 \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192 \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nwlog hn : n\u2081 \u2264 n\u2082\n[GOAL]\ncase inr\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nthis :\n \u2200 {F : Type u} [inst : Field F] (p : \u2115) [HF : CharP F p] {f : F[X]},\n Irreducible f \u2192\n 0 < p \u2192\n \u2200 (n\u2081 n\u2082 : \u2115),\n n\u2081 \u2264 n\u2082 \u2192\n \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192\n \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\nhn : \u00acn\u2081 \u2264 n\u2082\n\u22a2 \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192 \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nintro g\u2081 hg\u2081 Hg\u2081 g\u2082 hg\u2082 Hg\u2082\n[GOAL]\ncase inr\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nthis :\n \u2200 {F : Type u} [inst : Field F] (p : \u2115) [HF : CharP F p] {f : F[X]},\n Irreducible f \u2192\n 0 < p \u2192\n \u2200 (n\u2081 n\u2082 : \u2115),\n n\u2081 \u2264 n\u2082 \u2192\n \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192\n \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\nhn : \u00acn\u2081 \u2264 n\u2082\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nHg\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nHg\u2082 : \u2191(expand F (p ^ n\u2082)) g\u2082 = f\n\u22a2 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nsimpa only [eq_comm] using this p hf hp n\u2082 n\u2081 (le_of_not_le hn) g\u2082 hg\u2082 Hg\u2082 g\u2081 hg\u2081 Hg\u2081\n[GOAL]\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nhn : n\u2081 \u2264 n\u2082\n\u22a2 \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192 \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nhave hf0 : f \u2260 0 := hf.ne_zero\n[GOAL]\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nhn : n\u2081 \u2264 n\u2082\nhf0 : f \u2260 0\n\u22a2 \u2200 (g\u2081 : F[X]),\n Separable g\u2081 \u2192\n \u2191(expand F (p ^ n\u2081)) g\u2081 = f \u2192 \u2200 (g\u2082 : F[X]), Separable g\u2082 \u2192 \u2191(expand F (p ^ n\u2082)) g\u2082 = f \u2192 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nintros g\u2081 hg\u2081 hgf\u2081 g\u2082 hg\u2082 hgf\u2082\n[GOAL]\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nhn : n\u2081 \u2264 n\u2082\nhf0 : f \u2260 0\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhgf\u2082 : \u2191(expand F (p ^ n\u2082)) g\u2082 = f\n\u22a2 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nrw [le_iff_exists_add] at hn \n[GOAL]\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 n\u2082 : \u2115\nhn : \u2203 c, n\u2082 = n\u2081 + c\nhf0 : f \u2260 0\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhgf\u2082 : \u2191(expand F (p ^ n\u2082)) g\u2082 = f\n\u22a2 n\u2081 = n\u2082 \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nrcases hn with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 : \u2115\nhf0 : f \u2260 0\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhgf\u2082 : \u2191(expand F (p ^ (n\u2081 + k))) g\u2082 = f\n\u22a2 n\u2081 = n\u2081 + k \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nrw [\u2190 hgf\u2081, pow_add, expand_mul, expand_inj (pow_pos hp n\u2081)] at hgf\u2082 \n[GOAL]\ncase intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 : \u2115\nhf0 : f \u2260 0\ng\u2081 : F[X]\nhg\u2081 : Separable g\u2081\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) g\u2081 = f\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhgf\u2082 : \u2191(expand F (p ^ k)) g\u2082 = g\u2081\n\u22a2 n\u2081 = n\u2081 + k \u2227 g\u2081 = g\u2082\n[PROOFSTEP]\nsubst hgf\u2082\n[GOAL]\ncase intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn\u2081 : \u2115\nhf0 : f \u2260 0\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) g\u2082)\nhgf\u2081 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082) = f\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) g\u2082 = g\u2082\n[PROOFSTEP]\nsubst hgf\u2081\n[GOAL]\ncase intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082) \u2260 0\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) g\u2082 = g\u2082\n[PROOFSTEP]\nrcases isUnit_or_eq_zero_of_separable_expand p k hp hg\u2081 with (h | rfl)\n[GOAL]\ncase intro.inl\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082) \u2260 0\nh : IsUnit g\u2082\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) g\u2082 = g\u2082\n[PROOFSTEP]\nrw [isUnit_iff] at h \n[GOAL]\ncase intro.inl\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nk : \u2115\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) g\u2082) \u2260 0\nh : \u2203 r, IsUnit r \u2227 \u2191C r = g\u2082\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) g\u2082 = g\u2082\n[PROOFSTEP]\nrcases h with \u27e8r, hr, rfl\u27e9\n[GOAL]\ncase intro.inl.intro.intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 k : \u2115\nr : F\nhr : IsUnit r\nhg\u2082 : Separable (\u2191C r)\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) (\u2191C r))\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) (\u2191C r)))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) (\u2191C r)) \u2260 0\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) (\u2191C r) = \u2191C r\n[PROOFSTEP]\nsimp_rw [expand_C] at hf \n[GOAL]\ncase intro.inl.intro.intro\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 k : \u2115\nr : F\nhr : IsUnit r\nhg\u2082 : Separable (\u2191C r)\nhg\u2081 : Separable (\u2191(expand F (p ^ k)) (\u2191C r))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ k)) (\u2191C r)) \u2260 0\nhf : Irreducible (\u2191C r)\n\u22a2 n\u2081 = n\u2081 + k \u2227 \u2191(expand F (p ^ k)) (\u2191C r) = \u2191C r\n[PROOFSTEP]\nexact absurd (isUnit_C.2 hr) hf.1\n[GOAL]\ncase intro.inr\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhg\u2081 : Separable (\u2191(expand F (p ^ 0)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082) \u2260 0\n\u22a2 n\u2081 = n\u2081 + 0 \u2227 \u2191(expand F (p ^ 0)) g\u2082 = g\u2082\n[PROOFSTEP]\nrw [add_zero, pow_zero, expand_one]\n[GOAL]\ncase intro.inr\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhg\u2081 : Separable (\u2191(expand F (p ^ 0)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082) \u2260 0\n\u22a2 n\u2081 = n\u2081 \u2227 g\u2082 = g\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.inr.left\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhg\u2081 : Separable (\u2191(expand F (p ^ 0)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082) \u2260 0\n\u22a2 n\u2081 = n\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.inr.right\nF\u271d : Type u\ninst\u271d\u00b9 : Field F\u271d\np\u271d : \u2115\nF : Type u\ninst\u271d : Field F\np : \u2115\nHF : CharP F p\nhp : 0 < p\nn\u2081 : \u2115\ng\u2082 : F[X]\nhg\u2082 : Separable g\u2082\nhg\u2081 : Separable (\u2191(expand F (p ^ 0)) g\u2082)\nhf : Irreducible (\u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082))\nhf0 : \u2191(expand F (p ^ n\u2081)) (\u2191(expand F (p ^ 0)) g\u2082) \u2260 0\n\u22a2 g\u2082 = g\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nn : \u2115\n\u22a2 Separable (X ^ n - 1) \u2194 \u2191n \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, fun h => separable_X_pow_sub_C_unit 1 (IsUnit.mk0 (\u2191n) h)\u27e9\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nn : \u2115\n\u22a2 Separable (X ^ n - 1) \u2192 \u2191n \u2260 0\n[PROOFSTEP]\nrw [separable_def', derivative_sub, derivative_X_pow, derivative_one, sub_zero]\n -- Suppose `(n : F) = 0`, then the derivative is `0`, so `X ^ n - 1` is a unit, contradiction.\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nn : \u2115\n\u22a2 (\u2203 a b, a * (X ^ n - 1) + b * (\u2191C \u2191n * X ^ (n - 1)) = 1) \u2192 \u2191n \u2260 0\n[PROOFSTEP]\nrintro (h : IsCoprime _ _) hn'\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nn : \u2115\nh : IsCoprime (X ^ n - 1) (\u2191C \u2191n * X ^ (n - 1))\nhn' : \u2191n = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hn', C_0, zero_mul, isCoprime_zero_right] at h \n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\nn : \u2115\nh : IsUnit (X ^ n - 1)\nhn' : \u2191n = 0\n\u22a2 False\n[PROOFSTEP]\nexact not_isUnit_X_pow_sub_one F n h\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n\u22a2 Fintype.card \u2191(rootSet p K) = natDegree p\n[PROOFSTEP]\nsimp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n\u22a2 card (Multiset.toFinset (roots (map (algebraMap F K) p))) = natDegree p\n[PROOFSTEP]\nrw [Multiset.toFinset_card_of_nodup, \u2190 natDegree_eq_card_roots hsplit]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra F K\np : F[X]\nhsep : Separable p\nhsplit : Splits (algebraMap F K) p\n\u22a2 Multiset.Nodup (roots (map (algebraMap F K) p))\n[PROOFSTEP]\nexact nodup_roots hsep.map\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\n\u22a2 h = \u2191C (leadingCoeff h) * (X - \u2191C x)\n[PROOFSTEP]\nhave h_ne_zero : h \u2260 0 := by\n rintro rfl\n exact not_separable_zero h_sep\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\n\u22a2 h \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh_sep : Separable 0\nh_root : eval x 0 = 0\nh_splits : Splits i 0\nh_roots : \u2200 (y : K), y \u2208 roots (map i 0) \u2192 y = \u2191i x\n\u22a2 False\n[PROOFSTEP]\nexact not_separable_zero h_sep\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 h = \u2191C (leadingCoeff h) * (X - \u2191C x)\n[PROOFSTEP]\napply Polynomial.eq_X_sub_C_of_splits_of_single_root i h_splits\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 roots (map i h) = {\u2191i x}\n[PROOFSTEP]\napply Finset.mk.inj\n[GOAL]\ncase x\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 { val := roots (map i h), nodup := ?nodup } = { val := {\u2191i x}, nodup := ?nodup }\n[PROOFSTEP]\nchange _ = {i x}\n[GOAL]\ncase x\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 { val := roots (map i h), nodup := ?nodup } = {\u2191i x}\n[PROOFSTEP]\nrw [Finset.eq_singleton_iff_unique_mem]\n[GOAL]\ncase x\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 \u2191i x \u2208 { val := roots (map i h), nodup := ?nodup } \u2227\n \u2200 (x_1 : K), x_1 \u2208 { val := roots (map i h), nodup := ?nodup } \u2192 x_1 = \u2191i x\ncase nodup\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 Multiset.Nodup (roots (map i h))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase x.left\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 \u2191i x \u2208 { val := roots (map i h), nodup := ?nodup }\n[PROOFSTEP]\napply Finset.mem_mk.mpr\n[GOAL]\ncase x.left\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 \u2191i x \u2208 roots (map i h)\n[PROOFSTEP]\nrw [mem_roots (show h.map i \u2260 0 from map_ne_zero h_ne_zero)]\n[GOAL]\ncase x.left\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 IsRoot (map i h) (\u2191i x)\n[PROOFSTEP]\nrw [IsRoot.def, \u2190 eval\u2082_eq_eval_map, eval\u2082_hom, h_root]\n[GOAL]\ncase x.left\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 \u2191i 0 = 0\n[PROOFSTEP]\nexact RingHom.map_zero i\n[GOAL]\ncase nodup\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 Multiset.Nodup (roots (map i h))\n[PROOFSTEP]\nexact nodup_roots (Separable.map h_sep)\n[GOAL]\ncase x.right\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni : F \u2192+* K\nx : F\nh : F[X]\nh_sep : Separable h\nh_root : eval x h = 0\nh_splits : Splits i h\nh_roots : \u2200 (y : K), y \u2208 roots (map i h) \u2192 y = \u2191i x\nh_ne_zero : h \u2260 0\n\u22a2 \u2200 (x_1 : K), x_1 \u2208 { val := roots (map i h), nodup := (_ : Multiset.Nodup (roots (map i h))) } \u2192 x_1 = \u2191i x\n[PROOFSTEP]\nexact h_roots\n[GOAL]\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\n\u22a2 \u2203 s, map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in s, (X - \u2191C a)\n[PROOFSTEP]\nobtain \u27e8s, h\u27e9 := (splits_iff_exists_multiset _).1 sp\n[GOAL]\ncase intro\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 \u2203 s, map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in s, (X - \u2191C a)\n[PROOFSTEP]\nuse s.toFinset\n[GOAL]\ncase h\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in Multiset.toFinset s, (X - \u2191C a)\n[PROOFSTEP]\nrw [h, Finset.prod_eq_multiset_prod, \u2190 Multiset.toFinset_eq]\n[GOAL]\ncase h\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 Multiset.Nodup s\n[PROOFSTEP]\napply nodup_of_separable_prod\n[GOAL]\ncase h.hs\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) s))\n[PROOFSTEP]\napply Separable.of_mul_right\n[GOAL]\ncase h.hs.h\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 Separable (?h.hs.f * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s))\ncase h.hs.f\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 K[X]\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase h.hs.h\nF : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\ni\u271d i : F \u2192+* K\nf : F[X]\nsep : Separable f\nsp : Splits i f\ns : Multiset K\nh : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\n\u22a2 Separable (map i f)\n[PROOFSTEP]\nexact sep.map\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\n\u22a2 Separable f\n[PROOFSTEP]\nrw [separable_iff_derivative_ne_zero hf, Ne, \u2190 degree_eq_bot, degree_derivative_eq]\n[GOAL]\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\n\u22a2 \u00ac\u2191(natDegree f - 1) = \u22a5\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\n\u22a2 0 < natDegree f\n[PROOFSTEP]\nrw [pos_iff_ne_zero, Ne, natDegree_eq_zero_iff_degree_le_zero, degree_le_zero_iff]\n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\n\u22a2 \u00acf = \u2191C (coeff f 0)\n[PROOFSTEP]\nrefine' fun hf1 => hf.not_unit _\n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\nhf1 : f = \u2191C (coeff f 0)\n\u22a2 IsUnit f\n[PROOFSTEP]\nrw [hf1, isUnit_C, isUnit_iff_ne_zero]\n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\nhf1 : f = \u2191C (coeff f 0)\n\u22a2 coeff f 0 \u2260 0\n[PROOFSTEP]\nintro hf2\n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\nhf1 : f = \u2191C (coeff f 0)\nhf2 : coeff f 0 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hf2, C_0] at hf1 \n[GOAL]\ncase hp\nF : Type u\ninst\u271d\u00b2 : Field F\nK : Type v\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero F\nf : F[X]\nhf : Irreducible f\nhf1 : f = 0\nhf2 : coeff f 0 = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd hf1 hf.ne_zero\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx : F\n\u22a2 Separable (minpoly F x)\n[PROOFSTEP]\nrw [minpoly.eq_X_sub_C']\n[GOAL]\nF : Type u_1\ninst\u271d : Field F\nx : F\n\u22a2 Separable (X - \u2191C x)\n[PROOFSTEP]\nexact separable_X_sub_C\n[GOAL]\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : IsScalarTower F K E\nh : IsSeparable F E\nx : K\n\u22a2 IsIntegral F x \u2227 Separable (minpoly F x)\n[PROOFSTEP]\nrefine' (isSeparable_iff.1 h (algebraMap K E x)).imp isIntegral_tower_bot_of_isIntegral_field fun hs => _\n[GOAL]\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (\u2191(algebraMap K E) x))\n\u22a2 Separable (minpoly F x)\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := minpoly.dvd F x ((aeval_algebraMap_eq_zero_iff _ _ _).mp (minpoly.aeval F ((algebraMap K E) x)))\n[GOAL]\ncase intro\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (\u2191(algebraMap K E) x))\nq : F[X]\nhq : minpoly F (\u2191(algebraMap K E) x) = minpoly F x * q\n\u22a2 Separable (minpoly F x)\n[PROOFSTEP]\nrw [hq] at hs \n[GOAL]\ncase intro\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nq : F[X]\nhs : Separable (minpoly F x * q)\nhq : minpoly F (\u2191(algebraMap K E) x) = minpoly F x * q\n\u22a2 Separable (minpoly F x)\n[PROOFSTEP]\nexact hs.of_mul_left\n[GOAL]\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F K\ninst\u271d\u2075 : Algebra F E\ninst\u271d\u2074 : Algebra K E\ninst\u271d\u00b3 : IsScalarTower F K E\nE' : Type u_4\ninst\u271d\u00b2 : Field E'\ninst\u271d\u00b9 : Algebra F E'\nf : E \u2192\u2090[F] E'\ninst\u271d : IsSeparable F E'\n\u22a2 IsSeparable F E\n[PROOFSTEP]\nletI : Algebra E E' := RingHom.toAlgebra f.toRingHom\n[GOAL]\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F K\ninst\u271d\u2075 : Algebra F E\ninst\u271d\u2074 : Algebra K E\ninst\u271d\u00b3 : IsScalarTower F K E\nE' : Type u_4\ninst\u271d\u00b2 : Field E'\ninst\u271d\u00b9 : Algebra F E'\nf : E \u2192\u2090[F] E'\ninst\u271d : IsSeparable F E'\nthis : Algebra E E' := RingHom.toAlgebra \u2191f\n\u22a2 IsSeparable F E\n[PROOFSTEP]\nhaveI : IsScalarTower F E E' := IsScalarTower.of_algebraMap_eq fun x => (f.commutes x).symm\n[GOAL]\nF : Type u_1\nK : Type u_2\nE : Type u_3\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F K\ninst\u271d\u2075 : Algebra F E\ninst\u271d\u2074 : Algebra K E\ninst\u271d\u00b3 : IsScalarTower F K E\nE' : Type u_4\ninst\u271d\u00b2 : Field E'\ninst\u271d\u00b9 : Algebra F E'\nf : E \u2192\u2090[F] E'\ninst\u271d : IsSeparable F E'\nthis\u271d : Algebra E E' := RingHom.toAlgebra \u2191f\nthis : IsScalarTower F E E'\n\u22a2 IsSeparable F E\n[PROOFSTEP]\nexact isSeparable_tower_bot_of_isSeparable F E E'\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2075 : CommRing S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K L\npb : PowerBasis K S\nh_sep : Separable (minpoly K pb.gen)\nh_splits : Splits (algebraMap K L) (minpoly K pb.gen)\n\u22a2 Fintype.card (S \u2192\u2090[K] L) = pb.dim\n[PROOFSTEP]\nlet s := ((minpoly K pb.gen).map (algebraMap K L)).roots.toFinset\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2075 : CommRing S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K L\npb : PowerBasis K S\nh_sep : Separable (minpoly K pb.gen)\nh_splits : Splits (algebraMap K L) (minpoly K pb.gen)\ns : Finset L := Multiset.toFinset (roots (Polynomial.map (algebraMap K L) (minpoly K pb.gen)))\n\u22a2 Fintype.card (S \u2192\u2090[K] L) = pb.dim\n[PROOFSTEP]\nlet _ := (PowerBasis.AlgHom.fintype pb : Fintype (S \u2192\u2090[K] L))\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2075 : CommRing S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K L\npb : PowerBasis K S\nh_sep : Separable (minpoly K pb.gen)\nh_splits : Splits (algebraMap K L) (minpoly K pb.gen)\ns : Finset L := Multiset.toFinset (roots (Polynomial.map (algebraMap K L) (minpoly K pb.gen)))\nx\u271d : Fintype (S \u2192\u2090[K] L) := PowerBasis.AlgHom.fintype pb\n\u22a2 Fintype.card (S \u2192\u2090[K] L) = pb.dim\n[PROOFSTEP]\nrw [Fintype.card_congr pb.liftEquiv', Fintype.card_of_subtype s (fun x => Multiset.mem_toFinset), \u2190\n pb.natDegree_minpoly, natDegree_eq_card_roots h_splits, Multiset.toFinset_card_of_nodup]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2075 : CommRing S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K L\npb : PowerBasis K S\nh_sep : Separable (minpoly K pb.gen)\nh_splits : Splits (algebraMap K L) (minpoly K pb.gen)\ns : Finset L := Multiset.toFinset (roots (Polynomial.map (algebraMap K L) (minpoly K pb.gen)))\nx\u271d : Fintype (S \u2192\u2090[K] L) := PowerBasis.AlgHom.fintype pb\n\u22a2 Multiset.Nodup (roots (Polynomial.map (algebraMap K L) (minpoly K pb.gen)))\n[PROOFSTEP]\nexact nodup_roots ((separable_map (algebraMap K L)).mpr h_sep)\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Separable", "llama_tokens": 33055, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.2605005701429106}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 (f.op \u226b (inv f).op).unop = (\ud835\udfd9 (op Y)).unop\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 ((inv f).op \u226b f.op).unop = (\ud835\udfd9 (op X)).unop\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f.op\n\u22a2 (f \u226b (inv f.op).unop).op = (\ud835\udfd9 X).op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f.op\n\u22a2 ((inv f.op).unop \u226b f).op = (\ud835\udfd9 Y).op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 IsIso f.unop \u2194 IsIso f\n[PROOFSTEP]\nrw [\u2190 isIso_op_iff f.unop, Quiver.Hom.op_unop]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 (inv f).op = inv f.op\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase hom_inv_id\nC : Type u\u2081\ninst : Category.{v\u2081, u\u2081} C\nX Y : C\nf : X \u27f6 Y\ninst_1 : IsIso f\n\u22a2 f.op \u226b (inv f).op = \ud835\udfd9 (op Y)\n[PROOFSTEP]\nrw [\u2190 op_comp, IsIso.inv_hom_id, op_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\ninst\u271d : IsIso f\n\u22a2 (inv f).unop = inv f.unop\n[PROOFSTEP]\naesop_cat_nonterminal\n[GOAL]\ncase hom_inv_id\nC : Type u\u2081\ninst : Category.{v\u2081, u\u2081} C\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\ninst_1 : IsIso f\n\u22a2 f.unop \u226b (inv f).unop = \ud835\udfd9 Y.unop\n[PROOFSTEP]\nrw [\u2190 unop_comp, IsIso.inv_hom_id, unop_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\nF : C \u2964 D\ninst\u271d : Faithful F\nX\u271d Y\u271d : C\u1d52\u1d56\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : F.op.map a\u2081\u271d = F.op.map a\u2082\u271d\n\u22a2 a\u2081\u271d.unop = a\u2082\u271d.unop\n[PROOFSTEP]\nsimpa using map_injective F (Quiver.Hom.op_inj 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\u1d52\u1d56 \u2964 D\n\u22a2 F.rightOp.leftOp = F\n[PROOFSTEP]\ncases F\n[GOAL]\ncase mk\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\ntoPrefunctor\u271d : C\u1d52\u1d56 \u2964q D\nmap_id\u271d : \u2200 (X : C\u1d52\u1d56), toPrefunctor\u271d.map (\ud835\udfd9 X) = \ud835\udfd9 (toPrefunctor\u271d.obj X)\nmap_comp\u271d :\n \u2200 {X Y Z : C\u1d52\u1d56} (f : X \u27f6 Y) (g : Y \u27f6 Z), toPrefunctor\u271d.map (f \u226b g) = toPrefunctor\u271d.map f \u226b toPrefunctor\u271d.map g\n\u22a2 (mk toPrefunctor\u271d).rightOp.leftOp = mk toPrefunctor\u271d\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\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (G.op.map f \u226b (fun X => (app \u03b1 X.unop).op) Y).unop = ((fun X => (app \u03b1 X.unop).op) X \u226b F.op.map f).unop\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 G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u27f6 G\nX Y : C\nf : X \u27f6 Y\n\u22a2 ((Functor.unop G).map f \u226b (fun X => (app \u03b1 (op X)).unop) Y).op =\n ((fun X => (app \u03b1 (op X)).unop) X \u226b (Functor.unop F).map f).op\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\n\u03b1 : F.op \u27f6 G.op\nX Y : C\nf : X \u27f6 Y\n\u22a2 (G.map f \u226b (fun X => (app \u03b1 (op X)).unop) Y).op = ((fun X => (app \u03b1 (op X)).unop) X \u226b F.map f).op\n[PROOFSTEP]\nsimpa only [Functor.op_map] using (\u03b1.naturality f.op).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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : Functor.unop F \u27f6 Functor.unop G\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (G.map f \u226b (fun X => (app \u03b1 X.unop).op) Y).unop = ((fun X => (app \u03b1 X.unop).op) X \u226b F.map f).unop\n[PROOFSTEP]\nsimpa only [Functor.unop_map] using (\u03b1.naturality f.unop).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 G H : C \u2964 D\u1d52\u1d56\n\u03b1 : F \u27f6 G\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (G.leftOp.map f \u226b (fun X => (app \u03b1 X.unop).unop) Y).op = ((fun X => (app \u03b1 X.unop).unop) X \u226b F.leftOp.map f).op\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 H : C \u2964 D\u1d52\u1d56\n\u03b1 : F.leftOp \u27f6 G.leftOp\nX Y : C\nf : X \u27f6 Y\n\u22a2 (G.map f \u226b (fun X => (app \u03b1 (op X)).op) Y).unop = ((fun X => (app \u03b1 (op X)).op) X \u226b F.map f).unop\n[PROOFSTEP]\nsimpa only [Functor.leftOp_map] using (\u03b1.naturality f.op).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 G H : C\u1d52\u1d56 \u2964 D\n\u03b1 : F \u27f6 G\nX Y : C\nf : X \u27f6 Y\n\u22a2 (G.rightOp.map f \u226b (fun X => (app \u03b1 (op X)).op) Y).unop = ((fun X => (app \u03b1 (op X)).op) X \u226b F.rightOp.map f).unop\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 H : C\u1d52\u1d56 \u2964 D\n\u03b1 : F.rightOp \u27f6 G.rightOp\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 (G.map f \u226b (fun X => (app \u03b1 X.unop).unop) Y).op = ((fun X => (app \u03b1 X.unop).unop) X \u226b F.map f).op\n[PROOFSTEP]\nsimpa only [Functor.rightOp_map] using (\u03b1.naturality f.unop).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : C\nX Y : C\u1d52\u1d56\nf : X \u2245 Y\n\u22a2 f.hom.unop \u226b f.inv.unop = \ud835\udfd9 Y.unop\n[PROOFSTEP]\nsimp only [\u2190 unop_comp, f.inv_hom_id, unop_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : C\nX Y : C\u1d52\u1d56\nf : X \u2245 Y\n\u22a2 f.inv.unop \u226b f.hom.unop = \ud835\udfd9 X.unop\n[PROOFSTEP]\nsimp only [\u2190 unop_comp, f.hom_inv_id, unop_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : C\nX Y : C\u1d52\u1d56\nf : X \u2245 Y\n\u22a2 Iso.op (unop f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d : C\nX Y : C\u1d52\u1d56\nf : X \u2245 Y\n\u22a2 (Iso.op (unop f)).hom = f.hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d X Y : C\nf : X \u2245 Y\n\u22a2 unop (Iso.op f) = f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d X Y : C\nf : X \u2245 Y\n\u22a2 (unop (Iso.op f)).hom = f.hom\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 \u2245 G\n\u22a2 NatTrans.op \u03b1.hom \u226b NatTrans.op \u03b1.inv = \ud835\udfd9 G.op\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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.op \u03b1.hom \u226b NatTrans.op \u03b1.inv) x\u271d = NatTrans.app (\ud835\udfd9 G.op) 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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (NatTrans.app \u03b1.hom x\u271d.unop).op \u226b (NatTrans.app \u03b1.inv x\u271d.unop).op = \ud835\udfd9 (op (G.obj x\u271d.unop))\n[PROOFSTEP]\nrw [\u2190 op_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\nF G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (NatTrans.app \u03b1.inv x\u271d.unop \u226b NatTrans.app \u03b1.hom x\u271d.unop).op = \ud835\udfd9 (op (G.obj x\u271d.unop))\n[PROOFSTEP]\nrw [\u03b1.inv_hom_id_app]\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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (\ud835\udfd9 (G.obj x\u271d.unop)).op = \ud835\udfd9 (op (G.obj x\u271d.unop))\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 \u2245 G\n\u22a2 NatTrans.op \u03b1.inv \u226b NatTrans.op \u03b1.hom = \ud835\udfd9 F.op\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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.op \u03b1.inv \u226b NatTrans.op \u03b1.hom) x\u271d = NatTrans.app (\ud835\udfd9 F.op) 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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (NatTrans.app \u03b1.inv x\u271d.unop).op \u226b (NatTrans.app \u03b1.hom x\u271d.unop).op = \ud835\udfd9 (op (F.obj x\u271d.unop))\n[PROOFSTEP]\nrw [\u2190 op_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\nF G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (NatTrans.app \u03b1.hom x\u271d.unop \u226b NatTrans.app \u03b1.inv x\u271d.unop).op = \ud835\udfd9 (op (F.obj x\u271d.unop))\n[PROOFSTEP]\nrw [\u03b1.hom_inv_id_app]\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 G : C \u2964 D\n\u03b1 : F \u2245 G\nx\u271d : C\u1d52\u1d56\n\u22a2 (\ud835\udfd9 (F.obj x\u271d.unop)).op = \ud835\udfd9 (op (F.obj x\u271d.unop))\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.op \u2245 G.op\n\u22a2 NatTrans.removeOp \u03b1.hom \u226b NatTrans.removeOp \u03b1.inv = \ud835\udfd9 G\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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 NatTrans.app (NatTrans.removeOp \u03b1.hom \u226b NatTrans.removeOp \u03b1.inv) x\u271d = NatTrans.app (\ud835\udfd9 G) 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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.hom (op x\u271d)).unop \u226b (NatTrans.app \u03b1.inv (op x\u271d)).unop = \ud835\udfd9 (G.obj x\u271d)\n[PROOFSTEP]\nrw [\u2190 unop_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\nF G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.inv (op x\u271d) \u226b NatTrans.app \u03b1.hom (op x\u271d)).unop = \ud835\udfd9 (G.obj x\u271d)\n[PROOFSTEP]\nrw [\u03b1.inv_hom_id_app]\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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (\ud835\udfd9 (G.op.obj (op x\u271d))).unop = \ud835\udfd9 (G.obj x\u271d)\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.op \u2245 G.op\n\u22a2 NatTrans.removeOp \u03b1.inv \u226b NatTrans.removeOp \u03b1.hom = \ud835\udfd9 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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 NatTrans.app (NatTrans.removeOp \u03b1.inv \u226b NatTrans.removeOp \u03b1.hom) x\u271d = NatTrans.app (\ud835\udfd9 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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.inv (op x\u271d)).unop \u226b (NatTrans.app \u03b1.hom (op x\u271d)).unop = \ud835\udfd9 (F.obj x\u271d)\n[PROOFSTEP]\nrw [\u2190 unop_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\nF G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.hom (op x\u271d) \u226b NatTrans.app \u03b1.inv (op x\u271d)).unop = \ud835\udfd9 (F.obj x\u271d)\n[PROOFSTEP]\nrw [\u03b1.hom_inv_id_app]\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 G : C \u2964 D\n\u03b1 : F.op \u2245 G.op\nx\u271d : C\n\u22a2 (\ud835\udfd9 (F.op.obj (op x\u271d))).unop = \ud835\udfd9 (F.obj x\u271d)\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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\n\u22a2 NatTrans.unop \u03b1.hom \u226b NatTrans.unop \u03b1.inv = \ud835\udfd9 (Functor.unop G)\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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 NatTrans.app (NatTrans.unop \u03b1.hom \u226b NatTrans.unop \u03b1.inv) x\u271d = NatTrans.app (\ud835\udfd9 (Functor.unop G)) 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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.hom (op x\u271d)).unop \u226b (NatTrans.app \u03b1.inv (op x\u271d)).unop = \ud835\udfd9 (G.obj (op x\u271d)).unop\n[PROOFSTEP]\nrw [\u2190 unop_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\nF\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.inv (op x\u271d) \u226b NatTrans.app \u03b1.hom (op x\u271d)).unop = \ud835\udfd9 (G.obj (op x\u271d)).unop\n[PROOFSTEP]\nrw [\u03b1.inv_hom_id_app]\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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (\ud835\udfd9 (G.obj (op x\u271d))).unop = \ud835\udfd9 (G.obj (op x\u271d)).unop\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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\n\u22a2 NatTrans.unop \u03b1.inv \u226b NatTrans.unop \u03b1.hom = \ud835\udfd9 (Functor.unop 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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 NatTrans.app (NatTrans.unop \u03b1.inv \u226b NatTrans.unop \u03b1.hom) x\u271d = NatTrans.app (\ud835\udfd9 (Functor.unop 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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.inv (op x\u271d)).unop \u226b (NatTrans.app \u03b1.hom (op x\u271d)).unop = \ud835\udfd9 (F.obj (op x\u271d)).unop\n[PROOFSTEP]\nrw [\u2190 unop_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\nF\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (NatTrans.app \u03b1.hom (op x\u271d) \u226b NatTrans.app \u03b1.inv (op x\u271d)).unop = \ud835\udfd9 (F.obj (op x\u271d)).unop\n[PROOFSTEP]\nrw [\u03b1.hom_inv_id_app]\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\u271d G\u271d : C \u2964 D\nF G : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\n\u03b1 : F \u2245 G\nx\u271d : C\n\u22a2 (\ud835\udfd9 (F.obj (op x\u271d))).unop = \ud835\udfd9 (F.obj (op x\u271d)).unop\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\ne : C \u224c D\nX : C\u1d52\u1d56\n\u22a2 e.functor.op.map (NatTrans.app (NatIso.op e.unitIso).symm.hom X) \u226b\n NatTrans.app (NatIso.op e.counitIso).symm.hom (e.functor.op.obj X) =\n \ud835\udfd9 (e.functor.op.obj X)\n[PROOFSTEP]\napply Quiver.Hom.unop_inj\n[GOAL]\ncase 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\ne : C \u224c D\nX : C\u1d52\u1d56\n\u22a2 (e.functor.op.map (NatTrans.app (NatIso.op e.unitIso).symm.hom X) \u226b\n NatTrans.app (NatIso.op e.counitIso).symm.hom (e.functor.op.obj X)).unop =\n (\ud835\udfd9 (e.functor.op.obj X)).unop\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase 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\ne : C \u224c D\nX : C\u1d52\u1d56\n\u22a2 NatTrans.app e.counitIso.inv (e.functor.obj X.unop) \u226b e.functor.map (NatTrans.app e.unitIso.inv X.unop) =\n \ud835\udfd9 (e.functor.obj X.unop)\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\ne : C\u1d52\u1d56 \u224c D\u1d52\u1d56\nX : C\n\u22a2 (Functor.unop e.functor).map (NatTrans.app (NatIso.unop e.unitIso).symm.hom X) \u226b\n NatTrans.app (NatIso.unop e.counitIso).symm.hom ((Functor.unop e.functor).obj X) =\n \ud835\udfd9 ((Functor.unop e.functor).obj X)\n[PROOFSTEP]\napply Quiver.Hom.op_inj\n[GOAL]\ncase 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\ne : C\u1d52\u1d56 \u224c D\u1d52\u1d56\nX : C\n\u22a2 ((Functor.unop e.functor).map (NatTrans.app (NatIso.unop e.unitIso).symm.hom X) \u226b\n NatTrans.app (NatIso.unop e.counitIso).symm.hom ((Functor.unop e.functor).obj X)).op =\n (\ud835\udfd9 ((Functor.unop e.functor).obj X)).op\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase 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\ne : C\u1d52\u1d56 \u224c D\u1d52\u1d56\nX : C\n\u22a2 NatTrans.app e.counitIso.inv (e.functor.obj (Opposite.op X)) \u226b\n e.functor.map (NatTrans.app e.unitIso.inv (Opposite.op X)) =\n \ud835\udfd9 (e.functor.obj (Opposite.op X))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nA B : C\u1d52\u1d56\nx\u271d : A \u2245 B\n\u22a2 (fun g => Iso.op g) ((fun f => Iso.unop f) x\u271d) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nA B : C\u1d52\u1d56\nx\u271d : A \u2245 B\n\u22a2 ((fun g => Iso.op g) ((fun f => Iso.unop f) x\u271d)).hom = x\u271d.hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nA B : C\u1d52\u1d56\nx\u271d : B.unop \u2245 A.unop\n\u22a2 (fun f => Iso.unop f) ((fun g => Iso.op g) x\u271d) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nA B : C\u1d52\u1d56\nx\u271d : B.unop \u2245 A.unop\n\u22a2 ((fun f => Iso.unop f) ((fun g => Iso.op g) x\u271d)).hom = x\u271d.hom\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\n\u22a2 \u2200 {X Y : (C \u2964 D)\u1d52\u1d56} (f : X \u27f6 Y),\n (\ud835\udfed (C \u2964 D)\u1d52\u1d56).map f \u226b ((fun F => Iso.op (opUnopIso F.unop)) Y).hom =\n ((fun F => Iso.op (opUnopIso F.unop)) X).hom \u226b (opHom C D \u22d9 opInv C D).map f\n[PROOFSTEP]\nintro F G 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)\u1d52\u1d56\nf : F \u27f6 G\n\u22a2 (\ud835\udfed (C \u2964 D)\u1d52\u1d56).map f \u226b ((fun F => Iso.op (opUnopIso F.unop)) G).hom =\n ((fun F => Iso.op (opUnopIso F.unop)) F).hom \u226b (opHom C D \u22d9 opInv C D).map f\n[PROOFSTEP]\ndsimp [opUnopIso]\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)\u1d52\u1d56\nf : F \u27f6 G\n\u22a2 f \u226b (NatIso.ofComponents fun X => Iso.refl (G.unop.obj X)).hom.op =\n (NatIso.ofComponents fun X => Iso.refl (F.unop.obj X)).hom.op \u226b (NatTrans.mk fun X => NatTrans.app f.unop X).op\n[PROOFSTEP]\nrw [show f = f.unop.op by simp, \u2190 op_comp, \u2190 op_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\nF G : (C \u2964 D)\u1d52\u1d56\nf : F \u27f6 G\n\u22a2 f = f.unop.op\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)\u1d52\u1d56\nf : F \u27f6 G\n\u22a2 ((NatIso.ofComponents fun X => Iso.refl (G.unop.obj X)).hom \u226b f.unop).op =\n ((NatTrans.mk fun X => NatTrans.app f.unop.op.unop X) \u226b\n (NatIso.ofComponents fun X => Iso.refl (F.unop.obj X)).hom).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\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)\u1d52\u1d56\nf : F \u27f6 G\n\u22a2 (NatIso.ofComponents fun X => Iso.refl (G.unop.obj X)).hom \u226b f.unop =\n (NatTrans.mk fun X => NatTrans.app f.unop.op.unop X) \u226b (NatIso.ofComponents fun X => Iso.refl (F.unop.obj X)).hom\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\n\u22a2 \u2200 {X Y : (C\u1d52\u1d56 \u2964 D)\u1d52\u1d56} (f : X \u27f6 Y),\n (\ud835\udfed (C\u1d52\u1d56 \u2964 D)\u1d52\u1d56).map f \u226b ((fun F => Iso.op (rightOpLeftOpIso F.unop)) Y).hom =\n ((fun F => Iso.op (rightOpLeftOpIso F.unop)) X).hom \u226b\n (mk { obj := fun F => F.unop.rightOp, map := fun {X Y} \u03b7 => NatTrans.rightOp \u03b7.unop } \u22d9\n mk { obj := fun F => op F.leftOp, map := fun {X Y} \u03b7 => (NatTrans.leftOp \u03b7).op }).map\n f\n[PROOFSTEP]\nintro F G \u03b7\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\u1d52\u1d56 \u2964 D)\u1d52\u1d56\n\u03b7 : F \u27f6 G\n\u22a2 (\ud835\udfed (C\u1d52\u1d56 \u2964 D)\u1d52\u1d56).map \u03b7 \u226b ((fun F => Iso.op (rightOpLeftOpIso F.unop)) G).hom =\n ((fun F => Iso.op (rightOpLeftOpIso F.unop)) F).hom \u226b\n (mk { obj := fun F => F.unop.rightOp, map := fun {X Y} \u03b7 => NatTrans.rightOp \u03b7.unop } \u22d9\n mk { obj := fun F => op F.leftOp, map := fun {X Y} \u03b7 => (NatTrans.leftOp \u03b7).op }).map\n \u03b7\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 G : (C\u1d52\u1d56 \u2964 D)\u1d52\u1d56\n\u03b7 : F \u27f6 G\n\u22a2 \u03b7 \u226b (rightOpLeftOpIso G.unop).hom.op =\n (rightOpLeftOpIso F.unop).hom.op \u226b (NatTrans.leftOp (NatTrans.rightOp \u03b7.unop)).op\n[PROOFSTEP]\nrw [show \u03b7 = \u03b7.unop.op by simp, \u2190 op_comp, \u2190 op_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\nF G : (C\u1d52\u1d56 \u2964 D)\u1d52\u1d56\n\u03b7 : F \u27f6 G\n\u22a2 \u03b7 = \u03b7.unop.op\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\u1d52\u1d56 \u2964 D)\u1d52\u1d56\n\u03b7 : F \u27f6 G\n\u22a2 ((rightOpLeftOpIso G.unop).hom \u226b \u03b7.unop).op =\n (NatTrans.leftOp (NatTrans.rightOp \u03b7.unop.op.unop) \u226b (rightOpLeftOpIso F.unop).hom).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\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\u1d52\u1d56 \u2964 D)\u1d52\u1d56\n\u03b7 : F \u27f6 G\n\u22a2 (rightOpLeftOpIso G.unop).hom \u226b \u03b7.unop =\n NatTrans.leftOp (NatTrans.rightOp \u03b7.unop.op.unop) \u226b (rightOpLeftOpIso F.unop).hom\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Opposites", "llama_tokens": 11231, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.25993056361716305}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u22a2 Function.LeftInverse\n (fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) }\n[PROOFSTEP]\nintro \u03b7\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b7 : (composeAndSheafify J G).obj X \u27f6 Y\n\u22a2 (fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n ((fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) })\n \u03b7) =\n \u03b7\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b7 : (composeAndSheafify J G).obj X \u27f6 Y\n\u22a2 ((fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n ((fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) })\n \u03b7)).val =\n \u03b7.val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b7 : (composeAndSheafify J G).obj X \u27f6 Y\n\u22a2 sheafifyLift J\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val).symm\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val) (toSheafify J (X.val \u22d9 G) \u226b \u03b7.val)))\n (_ : Presheaf.IsSheaf J Y.val) =\n \u03b7.val\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b7 : (composeAndSheafify J G).obj X \u27f6 Y\n\u22a2 \u03b7.val =\n sheafifyLift J\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val).symm\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val) (toSheafify J (X.val \u22d9 G) \u226b \u03b7.val)))\n (_ : Presheaf.IsSheaf J Y.val)\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase h.a\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b7 : (composeAndSheafify J G).obj X \u27f6 Y\n\u22a2 toSheafify J (X.val \u22d9 G) \u226b \u03b7.val =\n \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val).symm\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val) (toSheafify J (X.val \u22d9 G) \u226b \u03b7.val))\n[PROOFSTEP]\nrw [Equiv.symm_apply_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u22a2 Function.RightInverse\n (fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) }\n[PROOFSTEP]\nintro \u03b3\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b3 : X \u27f6 (sheafCompose J F).obj Y\n\u22a2 (fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) })\n ((fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n \u03b3) =\n \u03b3\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b3 : X \u27f6 (sheafCompose J F).obj Y\n\u22a2 ((fun \u03b7 =>\n {\n val :=\n \u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val)\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 E D).obj G).obj ((sheafToPresheaf J E).obj X)) \u226b \u03b7.val) })\n ((fun \u03b3 =>\n {\n val :=\n sheafifyLift J\n (\u2191(Adjunction.homEquiv A ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map \u03b3))\n (_ : Presheaf.IsSheaf J Y.val) })\n \u03b3)).val =\n \u03b3.val\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX : Sheaf J E\nY : Sheaf J D\nA : (whiskeringRight C\u1d52\u1d56 E D).obj G \u22a3 (whiskeringRight C\u1d52\u1d56 D E).obj F := Adjunction.whiskerRight C\u1d52\u1d56 adj\n\u03b3 : X \u27f6 (sheafCompose J F).obj Y\n\u22a2 \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val)\n (toSheafify J (X.val \u22d9 G) \u226b\n sheafifyLift J (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X.val Y.val).symm \u03b3.val)\n (_ : Presheaf.IsSheaf J Y.val)) =\n \u03b3.val\n[PROOFSTEP]\nrw [J.toSheafify_sheafifyLift, Equiv.apply_symm_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 \u2191(composeEquiv J adj X'\u271d Y\u271d).symm (f \u226b g) = (composeAndSheafify J G).map f \u226b \u2191(composeEquiv J adj X\u271d Y\u271d).symm g\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 (\u2191(composeEquiv J adj X'\u271d Y\u271d).symm (f \u226b g)).val =\n ((composeAndSheafify J G).map f \u226b \u2191(composeEquiv J adj X\u271d Y\u271d).symm g).val\n[PROOFSTEP]\ndsimp [composeEquiv]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 sheafifyLift J (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X'\u271d.val Y\u271d.val).symm (f.val \u226b g.val))\n (_ : Presheaf.IsSheaf J Y\u271d.val) =\n sheafifyMap J (whiskerRight f.val G) \u226b\n sheafifyLift J (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val)\n[PROOFSTEP]\nrw [sheafifyMap_sheafifyLift]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 sheafifyLift J (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X'\u271d.val Y\u271d.val).symm (f.val \u226b g.val))\n (_ : Presheaf.IsSheaf J Y\u271d.val) =\n sheafifyLift J\n (whiskerRight f.val G \u226b \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val)\n[PROOFSTEP]\nerw [Adjunction.homEquiv_naturality_left_symm]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 sheafifyLift J\n (((whiskeringRight C\u1d52\u1d56 E D).obj G).map f.val \u226b\n \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val) =\n sheafifyLift J\n (whiskerRight f.val G \u226b \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val)\n[PROOFSTEP]\nrw [whiskeringRight_obj_map]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX'\u271d X\u271d : Sheaf J E\nY\u271d : Sheaf J D\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (sheafCompose J F).obj Y\u271d\n\u22a2 sheafifyLift J\n (whiskerRight f.val G \u226b \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val) =\n sheafifyLift J\n (whiskerRight f.val G \u226b \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val).symm g.val)\n (_ : Presheaf.IsSheaf J Y\u271d.val)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX\u271d : Sheaf J E\nY\u271d Y'\u271d : Sheaf J D\nf : (composeAndSheafify J G).obj X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Y'\u271d\n\u22a2 \u2191(composeEquiv J adj X\u271d Y'\u271d) (f \u226b g) = \u2191(composeEquiv J adj X\u271d Y\u271d) f \u226b (sheafCompose J F).map g\n[PROOFSTEP]\next\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX\u271d : Sheaf J E\nY\u271d Y'\u271d : Sheaf J D\nf : (composeAndSheafify J G).obj X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Y'\u271d\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app (\u2191(composeEquiv J adj X\u271d Y'\u271d) (f \u226b g)).val x\u271d =\n NatTrans.app (\u2191(composeEquiv J adj X\u271d Y\u271d) f \u226b (sheafCompose J F).map g).val x\u271d\n[PROOFSTEP]\ndsimp [composeEquiv]\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX\u271d : Sheaf J E\nY\u271d Y'\u271d : Sheaf J D\nf : (composeAndSheafify J G).obj X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Y'\u271d\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y'\u271d.val)\n (toSheafify J (X\u271d.val \u22d9 G) \u226b f.val \u226b g.val))\n x\u271d =\n NatTrans.app\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) X\u271d.val Y\u271d.val) (toSheafify J (X\u271d.val \u22d9 G) \u226b f.val))\n x\u271d \u226b\n F.map (NatTrans.app g.val x\u271d)\n[PROOFSTEP]\nerw [Adjunction.homEquiv_unit, Adjunction.homEquiv_unit]\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX\u271d : Sheaf J E\nY\u271d Y'\u271d : Sheaf J D\nf : (composeAndSheafify J G).obj X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Y'\u271d\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app\n (NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit X\u271d.val \u226b\n ((whiskeringRight C\u1d52\u1d56 D E).obj F).map (toSheafify J (X\u271d.val \u22d9 G) \u226b f.val \u226b g.val))\n x\u271d =\n NatTrans.app\n (NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit X\u271d.val \u226b\n ((whiskeringRight C\u1d52\u1d56 D E).obj F).map (toSheafify J (X\u271d.val \u22d9 G) \u226b f.val))\n x\u271d \u226b\n F.map (NatTrans.app g.val x\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w.h\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nadj : G \u22a3 F\nX\u271d : Sheaf J E\nY\u271d Y'\u271d : Sheaf J D\nf : (composeAndSheafify J G).obj X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Y'\u271d\nx\u271d : C\u1d52\u1d56\n\u22a2 NatTrans.app (NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit X\u271d.val) x\u271d \u226b\n F.map (NatTrans.app (toSheafify J (X\u271d.val \u22d9 G)) x\u271d \u226b NatTrans.app f.val x\u271d \u226b NatTrans.app g.val x\u271d) =\n (NatTrans.app (NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit X\u271d.val) x\u271d \u226b\n F.map (NatTrans.app (toSheafify J (X\u271d.val \u22d9 G)) x\u271d \u226b NatTrans.app f.val x\u271d)) \u226b\n F.map (NatTrans.app g.val x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nY : SheafOfTypes J\n\u22a2 (NatTrans.app (adjunctionToTypes J adj).unit Y).val =\n NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit ((sheafOfTypesToPresheaf J).obj Y) \u226b\n whiskerRight\n (toSheafify J (((whiskeringRight C\u1d52\u1d56 (Type (max v u)) D).obj G).obj ((sheafOfTypesToPresheaf J).obj Y)))\n (forget D)\n[PROOFSTEP]\ndsimp [adjunctionToTypes, Adjunction.comp]\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nY : SheafOfTypes J\n\u22a2 (NatTrans.app (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit Y).val \u226b\n (NatTrans.app (adjunction J adj).unit ((sheafEquivSheafOfTypes J).inverse.obj Y)).val \u226b\n \ud835\udfd9 (sheafify J (Y.val \u22d9 G) \u22d9 forget D) =\n NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit Y.val \u226b whiskerRight (toSheafify J (Y.val \u22d9 G)) (forget D)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nY : SheafOfTypes J\n\u22a2 (NatTrans.app (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit Y).val \u226b\n \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) Y.val (sheafify J (Y.val \u22d9 G)))\n (toSheafify J (Y.val \u22d9 G)) =\n NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).unit Y.val \u226b whiskerRight (toSheafify J (Y.val \u22d9 G)) (forget D)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\n\u22a2 (NatTrans.app (adjunctionToTypes J adj).counit X).val =\n sheafifyLift J\n ((Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1)\n (_ : Presheaf.IsSheaf J X.val)\n[PROOFSTEP]\napply J.sheafifyLift_unique\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\n\u22a2 toSheafify J\n (((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) \u226b\n (NatTrans.app (adjunctionToTypes J adj).counit X).val =\n (Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1\n[PROOFSTEP]\ndsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app, instCategorySheaf_comp_val, instCategorySheaf_id_val]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\n\u22a2 toSheafify J\n (((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) \u226b\n (NatTrans.app\n (Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor \u22d9 composeAndSheafify J G)).hom\n X).val \u226b\n (NatTrans.app\n (whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G)))\n X).val \u226b\n (NatTrans.app (adjunction J adj).counit X).val =\n (Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1\n[PROOFSTEP]\nrw [adjunction_counit_app_val]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\n\u22a2 toSheafify J\n (((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) \u226b\n (NatTrans.app\n (Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor \u22d9 composeAndSheafify J G)).hom\n X).val \u226b\n (NatTrans.app\n (whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G)))\n X).val \u226b\n sheafifyLift J\n (\u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) (X.val \u22d9 forget D) X.val).symm\n (\ud835\udfd9 (X.val \u22d9 forget D)))\n (_ : Presheaf.IsSheaf J X.val) =\n (Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1\n[PROOFSTEP]\nerw [Category.id_comp, J.sheafifyMap_sheafifyLift, J.toSheafify_sheafifyLift]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\n\u22a2 ((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n (NatTrans.app (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n ((sheafCompose J (forget D)).obj X))) \u226b\n \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) (X.val \u22d9 forget D) X.val).symm (\ud835\udfd9 (X.val \u22d9 forget D)) =\n (Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1\n[PROOFSTEP]\next\n[GOAL]\ncase a.w.h.w\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n (forget D).obj\n ((((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse \u22d9\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n (((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n (NatTrans.app (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n ((sheafCompose J (forget D)).obj X))) \u226b\n \u2191(Adjunction.homEquiv (Adjunction.whiskerRight C\u1d52\u1d56 adj) (X.val \u22d9 forget D) X.val).symm\n (\ud835\udfd9 (X.val \u22d9 forget D)))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n ((Functor.associator X.1 (forget D) G).hom \u226b NatTrans.app (Adjunction.whiskerRight C\u1d52\u1d56 adj).counit X.1) x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\ndsimp [sheafEquivSheafOfTypes, Equivalence.symm, Equivalence.toAdjunction, NatIso.ofComponents, Adjunction.whiskerRight,\n Adjunction.mkOfUnitCounit]\n[GOAL]\ncase a.w.h.w\nC : Type u\ninst\u271d\u2079 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\u2081\ninst\u271d\u2078 : Category.{max v u, w\u2081} D\nE : Type w\u2082\ninst\u271d\u2077 : Category.{max v u, w\u2082} E\nF : D \u2964 E\nG\u271d : E \u2964 D\ninst\u271d\u2076 : (X : C) \u2192 (S : Cover J X) \u2192 (P : C\u1d52\u1d56 \u2964 D) \u2192 PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nG : Type (max v u) \u2964 D\nadj : G \u22a3 forget D\nX : Sheaf J D\nx\u271d\u00b9 : C\u1d52\u1d56\nx\u271d :\n (forget D).obj\n ((((whiskeringRight C\u1d52\u1d56 (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse \u22d9\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x\u271d\u00b9)\n\u22a2 \u2191(G.map (\ud835\udfd9 ((forget D).obj (X.val.obj x\u271d\u00b9))) \u226b\n G.map (\ud835\udfd9 ((forget D).obj (X.val.obj x\u271d\u00b9))) \u226b\n \ud835\udfd9 (G.obj ((forget D).obj (X.val.obj x\u271d\u00b9))) \u226b NatTrans.app adj.counit (X.val.obj x\u271d\u00b9) \u226b \ud835\udfd9 (X.val.obj x\u271d\u00b9))\n x\u271d =\n \u2191(\ud835\udfd9 (G.obj ((forget D).obj (X.1.obj x\u271d\u00b9))) \u226b\n \ud835\udfd9 (G.obj ((forget D).obj (X.1.obj x\u271d\u00b9))) \u226b NatTrans.app adj.counit (X.1.obj x\u271d\u00b9) \u226b \ud835\udfd9 (X.val.obj x\u271d\u00b9))\n x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Adjunction", "llama_tokens": 18319, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.39233683016710835, "lm_q1q2_score": 0.25969672292608315}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 EventuallyConst f l \u2194 \u2203 x, Tendsto f l (pure x)\n[PROOFSTEP]\nsimp_rw [EventuallyConst, EventuallyEq, tendsto_pure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.805\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\n\u22a2 EventuallyConst p l \u2194 (p =\u1da0[l] fun x => False) \u2228 p =\u1da0[l] fun x => True\n[PROOFSTEP]\nsimp only [EventuallyConst, Prop.exists_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.958\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\np : \u03b1 \u2192 Prop\n\u22a2 EventuallyConst p l \u2194 (\u2200\u1da0 (x : \u03b1) in l, p x) \u2228 \u2200\u1da0 (x : \u03b1) in l, \u00acp x\n[PROOFSTEP]\nsimp [eventuallyConst_pred', or_comm, EventuallyEq]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Nonempty \u03b2\n\u22a2 EventuallyConst f \u22a5\n[PROOFSTEP]\nsimp [EventuallyConst, EventuallyEq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Nonempty \u03b2\n\u22a2 EventuallyConst f l\n[PROOFSTEP]\nrcases isEmpty_or_nonempty \u03b1 with h | h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Nonempty \u03b2\nh : IsEmpty \u03b1\n\u22a2 EventuallyConst f l\n[PROOFSTEP]\nsimp only [l.filter_eq_bot_of_isEmpty, EventuallyConst.bot]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Nonempty \u03b2\nh : Nonempty \u03b1\n\u22a2 EventuallyConst f l\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Subsingleton \u03b1\ninst\u271d : Nonempty \u03b2\nh : Nonempty \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 EventuallyConst f l\n[PROOFSTEP]\nrefine \u27e8f default, eventually_of_forall fun x \u21a6 congr_arg f <| Subsingleton.elim _ _\u27e9\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nh : EventuallyConst (mulIndicator s fun x => c) l\n\u22a2 EventuallyConst s l\n[PROOFSTEP]\nrw [eventuallyConst_set]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nh : EventuallyConst (mulIndicator s fun x => c) l\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, x \u2208 s) \u2228 \u2200\u1da0 (x : \u03b1) in l, \u00acx \u2208 s\n[PROOFSTEP]\nrcases h with \u27e8d, hd\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nd : \u03b2\nhd : (mulIndicator s fun x => c) =\u1da0[l] fun x => d\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, x \u2208 s) \u2228 \u2200\u1da0 (x : \u03b1) in l, \u00acx \u2208 s\n[PROOFSTEP]\nrcases eq_or_ne d 1 with rfl | hd\u2081\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nhd : (mulIndicator s fun x => c) =\u1da0[l] fun x => 1\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, x \u2208 s) \u2228 \u2200\u1da0 (x : \u03b1) in l, \u00acx \u2208 s\n[PROOFSTEP]\nrefine .inr <| hd.mono fun x hx \u21a6 ?_\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nhd : (mulIndicator s fun x => c) =\u1da0[l] fun x => 1\nx : \u03b1\nhx : mulIndicator s (fun x => c) x = (fun x => 1) x\n\u22a2 \u00acx \u2208 s\n[PROOFSTEP]\nsimpa only [mulIndicator_apply_eq_one, hc] using hx\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nd : \u03b2\nhd : (mulIndicator s fun x => c) =\u1da0[l] fun x => d\nhd\u2081 : d \u2260 1\n\u22a2 (\u2200\u1da0 (x : \u03b1) in l, x \u2208 s) \u2228 \u2200\u1da0 (x : \u03b1) in l, \u00acx \u2208 s\n[PROOFSTEP]\nrefine .inl <| hd.mono fun x hx \u21a6 ?_\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : One \u03b2\ns : Set \u03b1\nc : \u03b2\nhc : c \u2260 1\nd : \u03b2\nhd : (mulIndicator s fun x => c) =\u1da0[l] fun x => d\nhd\u2081 : d \u2260 1\nx : \u03b1\nhx : mulIndicator s (fun x => c) x = (fun x => d) x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nsimpa [hc] using ne_of_eq_of_ne hx hd\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 EventuallyConst f atTop \u2194 \u2203 i, \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 EventuallyConst f atTop \u2192 \u2203 i, \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nc : \u03b2\nhc : f =\u1da0[atTop] fun x => c\n\u22a2 \u2203 i, \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i\n[PROOFSTEP]\nrcases eventually_atTop.1 hc with \u27e8i, hi\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\nc : \u03b2\nhc : f =\u1da0[atTop] fun x => c\ni : \u03b1\nhi : \u2200 (b : \u03b1), b \u2265 i \u2192 f b = (fun x => c) b\n\u22a2 \u2203 i, \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i\n[PROOFSTEP]\nexact \u27e8i, fun j hj \u21a6 (hi j hj).trans (hi i le_rfl).symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 (\u2203 i, \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i) \u2192 EventuallyConst f atTop\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : Nonempty \u03b1\ni : \u03b1\nhi : \u2200 (j : \u03b1), i \u2264 j \u2192 f j = f i\n\u22a2 EventuallyConst f atTop\n[PROOFSTEP]\nexact \u27e8f i, eventually_atTop.2 \u27e8i, hi\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\n\u22a2 EventuallyConst f atTop \u2194 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\n[PROOFSTEP]\nrw [eventuallyConst_atTop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\n\u22a2 (\u2203 i, \u2200 (j : \u2115), i \u2264 j \u2192 f j = f i) \u2194 \u2203 n, \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\n[PROOFSTEP]\nrefine exists_congr fun n \u21a6 \u27e8fun h m hm \u21a6 ?_, fun h m hm \u21a6 ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (j : \u2115), n \u2264 j \u2192 f j = f n\nm : \u2115\nhm : n \u2264 m\n\u22a2 f (m + 1) = f m\n[PROOFSTEP]\nexact (h (m + 1) (hm.trans m.le_succ)).trans (h m hm).symm\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm : \u2115\nhm : n \u2264 m\n\u22a2 f m = f n\n[PROOFSTEP]\ninduction m, hm using Nat.le_induction with\n| base => rfl\n| succ m hm ihm => exact (h m hm).trans ihm\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm : \u2115\nhm : n \u2264 m\n\u22a2 f m = f n\n[PROOFSTEP]\ninduction m, hm using Nat.le_induction with\n| base => rfl\n| succ m hm ihm => exact (h m hm).trans ihm\n[GOAL]\ncase refine_2.base\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm : \u2115\n\u22a2 f n = f n\n[PROOFSTEP]\n\n| base => rfl\n[GOAL]\ncase refine_2.base\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm : \u2115\n\u22a2 f n = f n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_2.succ\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm\u271d m : \u2115\nhm : n \u2264 m\nihm : f m = f n\n\u22a2 f (m + 1) = f n\n[PROOFSTEP]\n\n| succ m hm ihm => exact (h m hm).trans ihm\n[GOAL]\ncase refine_2.succ\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.10621\nl : Filter \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nf : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 (m : \u2115), n \u2264 m \u2192 f (m + 1) = f m\nm\u271d m : \u2115\nhm : n \u2264 m\nihm : f m = f n\n\u22a2 f (m + 1) = f n\n[PROOFSTEP]\nexact (h m hm).trans ihm\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.EventuallyConst", "llama_tokens": 3678, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.531209388216861, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.2593807236915521}} {"text": "[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 Injective boxes\n[PROOFSTEP]\nrintro \u27e8s\u2081, h\u2081, h\u2081'\u27e9 \u27e8s\u2082, h\u2082, h\u2082'\u27e9 (rfl : s\u2081 = s\u2082)\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\ns\u2081 : Finset (Box \u03b9)\nh\u2081 : \u2200 (J : Box \u03b9), J \u2208 s\u2081 \u2192 J \u2264 I\nh\u2081' : Set.Pairwise (\u2191s\u2081) (Disjoint on Box.toSet)\nh\u2082 : \u2200 (J : Box \u03b9), J \u2208 s\u2081 \u2192 J \u2264 I\nh\u2082' : Set.Pairwise (\u2191s\u2081) (Disjoint on Box.toSet)\n\u22a2 { boxes := s\u2081, le_of_mem' := h\u2081, pairwiseDisjoint := h\u2081' } =\n { boxes := s\u2081, le_of_mem' := h\u2082, pairwiseDisjoint := h\u2082' }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nI\u271d J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\u271d\nx : \u03b9 \u2192 \u211d\nI J : Box \u03b9\nh : J \u2264 I\n\u22a2 \u2200 (J_1 : Box \u03b9), J_1 \u2208 {J} \u2192 J_1 \u2264 I\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\nI\u271d J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\u271d\nx : \u03b9 \u2192 \u211d\nI J : Box \u03b9\nh : J \u2264 I\n\u22a2 Set.Pairwise (\u2191{J}) (Disjoint on Box.toSet)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 \u2200 (a b : Prepartition I), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nsuffices : \u2200 {\u03c0\u2081 \u03c0\u2082 : Prepartition I}, \u03c0\u2081 \u2264 \u03c0\u2082 \u2192 \u03c0\u2082 \u2264 \u03c0\u2081 \u2192 \u03c0\u2081.boxes \u2286 \u03c0\u2082.boxes\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nthis : \u2200 {\u03c0\u2081 \u03c0\u2082 : Prepartition I}, \u03c0\u2081 \u2264 \u03c0\u2082 \u2192 \u03c0\u2082 \u2264 \u03c0\u2081 \u2192 \u03c0\u2081.boxes \u2286 \u03c0\u2082.boxes\n\u22a2 \u2200 (a b : Prepartition I), a \u2264 b \u2192 b \u2264 a \u2192 a = b\ncase this\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 \u2200 {\u03c0\u2081 \u03c0\u2082 : Prepartition I}, \u03c0\u2081 \u2264 \u03c0\u2082 \u2192 \u03c0\u2082 \u2264 \u03c0\u2081 \u2192 \u03c0\u2081.boxes \u2286 \u03c0\u2082.boxes\n[PROOFSTEP]\nexact fun \u03c0\u2081 \u03c0\u2082 h\u2081 h\u2082 => injective_boxes (Subset.antisymm (this h\u2081 h\u2082) (this h\u2082 h\u2081))\n[GOAL]\ncase this\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 \u2200 {\u03c0\u2081 \u03c0\u2082 : Prepartition I}, \u03c0\u2081 \u2264 \u03c0\u2082 \u2192 \u03c0\u2082 \u2264 \u03c0\u2081 \u2192 \u03c0\u2081.boxes \u2286 \u03c0\u2082.boxes\n[PROOFSTEP]\nintro \u03c0\u2081 \u03c0\u2082 h\u2081 h\u2082 J hJ\n[GOAL]\ncase this\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\n\u22a2 J \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nrcases h\u2081 hJ with \u27e8J', hJ', hle\u27e9\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\n\u22a2 J \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nrcases h\u2082 hJ' with \u27e8J'', hJ'', hle'\u27e9\n[GOAL]\ncase this.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\nJ'' : Box \u03b9\nhJ'' : J'' \u2208 \u03c0\u2081\nhle' : J' \u2264 J''\n\u22a2 J \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nobtain rfl : J = J''\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\nJ'' : Box \u03b9\nhJ'' : J'' \u2208 \u03c0\u2081\nhle' : J' \u2264 J''\n\u22a2 J = J''\ncase this.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\nhJ'' : J \u2208 \u03c0\u2081\nhle' : J' \u2264 J\n\u22a2 J \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nexact \u03c0\u2081.eq_of_le hJ hJ'' (hle.trans hle')\n[GOAL]\ncase this.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\nhJ'' : J \u2208 \u03c0\u2081\nhle' : J' \u2264 J\n\u22a2 J \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nobtain rfl : J' = J\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081.boxes\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhle : J \u2264 J'\nhJ'' : J \u2208 \u03c0\u2081\nhle' : J' \u2264 J\n\u22a2 J' = J\ncase this.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhJ : J' \u2208 \u03c0\u2081.boxes\nhle : J' \u2264 J'\nhJ'' : J' \u2208 \u03c0\u2081\nhle' : J' \u2264 J'\n\u22a2 J' \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nexact le_antisymm \u2039_\u203a \u2039_\u203a\n[GOAL]\ncase this.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh\u2081 : \u03c0\u2081 \u2264 \u03c0\u2082\nh\u2082 : \u03c0\u2082 \u2264 \u03c0\u2081\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nhJ : J' \u2208 \u03c0\u2081.boxes\nhle : J' \u2264 J'\nhJ'' : J' \u2208 \u03c0\u2081\nhle' : J' \u2264 J'\n\u22a2 J' \u2208 \u03c0\u2082.boxes\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0 : Prepartition I\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 I \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\n\u22a2 InjOn (fun J => {i | Box.lower J i = x i}) {J | J \u2208 \u03c0 \u2227 x \u2208 \u2191Box.Icc J}\n[PROOFSTEP]\nrintro J\u2081 \u27e8h\u2081, hx\u2081\u27e9 J\u2082 \u27e8h\u2082, hx\u2082\u27e9 (H : {i | J\u2081.lower i = x i} = {i | J\u2082.lower i = x i})\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\nH : {i | Box.lower J\u2081 i = x i} = {i | Box.lower J\u2082 i = x i}\n\u22a2 J\u2081 = J\u2082\n[PROOFSTEP]\nsuffices \u2200 i, (Ioc (J\u2081.lower i) (J\u2081.upper i) \u2229 Ioc (J\u2082.lower i) (J\u2082.upper i)).Nonempty\n by\n choose y hy\u2081 hy\u2082 using this\n exact \u03c0.eq_of_mem_of_mem h\u2081 h\u2082 hy\u2081 hy\u2082\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\nH : {i | Box.lower J\u2081 i = x i} = {i | Box.lower J\u2082 i = x i}\nthis : \u2200 (i : \u03b9), Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n\u22a2 J\u2081 = J\u2082\n[PROOFSTEP]\nchoose y hy\u2081 hy\u2082 using this\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\nH : {i | Box.lower J\u2081 i = x i} = {i | Box.lower J\u2082 i = x i}\ny : \u03b9 \u2192 \u211d\nhy\u2081 : \u2200 (i : \u03b9), y i \u2208 Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i)\nhy\u2082 : \u2200 (i : \u03b9), y i \u2208 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i)\n\u22a2 J\u2081 = J\u2082\n[PROOFSTEP]\nexact \u03c0.eq_of_mem_of_mem h\u2081 h\u2082 hy\u2081 hy\u2082\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\nH : {i | Box.lower J\u2081 i = x i} = {i | Box.lower J\u2082 i = x i}\n\u22a2 \u2200 (i : \u03b9), Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\nH : {i | Box.lower J\u2081 i = x i} = {i | Box.lower J\u2082 i = x i}\ni : \u03b9\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nsimp only [Set.ext_iff, mem_setOf] at H \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\ncases' (hx\u2081.1 i).eq_or_lt with hi\u2081 hi\u2081\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nhave hi\u2082 : J\u2082.lower i = x i := (H _).1 hi\u2081\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nhave H\u2081 : x i < J\u2081.upper i := by simpa only [hi\u2081] using J\u2081.lower_lt_upper i\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\n\u22a2 x i < Box.upper J\u2081 i\n[PROOFSTEP]\nsimpa only [hi\u2081] using J\u2081.lower_lt_upper i\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\nH\u2081 : x i < Box.upper J\u2081 i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nhave H\u2082 : x i < J\u2082.upper i := by simpa only [hi\u2082] using J\u2082.lower_lt_upper i\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\nH\u2081 : x i < Box.upper J\u2081 i\n\u22a2 x i < Box.upper J\u2082 i\n[PROOFSTEP]\nsimpa only [hi\u2082] using J\u2082.lower_lt_upper i\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\nH\u2081 : x i < Box.upper J\u2081 i\nH\u2082 : x i < Box.upper J\u2082 i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nrw [Ioc_inter_Ioc, hi\u2081, hi\u2082, sup_idem, Set.nonempty_Ioc]\n[GOAL]\ncase intro.intro.inl\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i = x i\nhi\u2082 : Box.lower J\u2082 i = x i\nH\u2081 : x i < Box.upper J\u2081 i\nH\u2082 : x i < Box.upper J\u2082 i\n\u22a2 x i < Box.upper J\u2081 i \u2293 Box.upper J\u2082 i\n[PROOFSTEP]\nexact lt_min H\u2081 H\u2082\n[GOAL]\ncase intro.intro.inr\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i < x i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nhave hi\u2082 : J\u2082.lower i < x i := (hx\u2082.1 i).lt_of_ne (mt (H _).2 hi\u2081.ne)\n[GOAL]\ncase intro.intro.inr\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d x : \u03b9 \u2192 \u211d\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx\u2081 : x \u2208 \u2191Box.Icc J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0\nhx\u2082 : x \u2208 \u2191Box.Icc J\u2082\ni : \u03b9\nH : \u2200 (x_1 : \u03b9), Box.lower J\u2081 x_1 = x x_1 \u2194 Box.lower J\u2082 x_1 = x x_1\nhi\u2081 : Box.lower J\u2081 i < x i\nhi\u2082 : Box.lower J\u2082 i < x i\n\u22a2 Set.Nonempty (Set.Ioc (Box.lower J\u2081 i) (Box.upper J\u2081 i) \u2229 Set.Ioc (Box.lower J\u2082 i) (Box.upper J\u2082 i))\n[PROOFSTEP]\nexact \u27e8x i, \u27e8hi\u2081, hx\u2081.2 i\u27e9, \u27e8hi\u2082, hx\u2082.2 i\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nx : \u03b9 \u2192 \u211d\n\u22a2 card (filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes) \u2264 2 ^ Fintype.card \u03b9\n[PROOFSTEP]\nrw [\u2190 Fintype.card_set]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nx : \u03b9 \u2192 \u211d\n\u22a2 card (filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes) \u2264 Fintype.card (Set \u03b9)\n[PROOFSTEP]\nrefine' Finset.card_le_card_of_inj_on (fun J : Box \u03b9 => {i | J.lower i = x i}) (fun _ _ => Finset.mem_univ _) _\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nx : \u03b9 \u2192 \u211d\n\u22a2 \u2200 (a\u2081 : Box \u03b9),\n a\u2081 \u2208 filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes \u2192\n \u2200 (a\u2082 : Box \u03b9),\n a\u2082 \u2208 filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes \u2192\n (fun J => {i | Box.lower J i = x i}) a\u2081 = (fun J => {i | Box.lower J i = x i}) a\u2082 \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsimpa only [Finset.mem_filter] using \u03c0.injOn_setOf_mem_Icc_setOf_lower_eq x\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 x \u2208 Prepartition.iUnion \u03c0 \u2194 \u2203 J, J \u2208 \u03c0 \u2227 x \u2208 J\n[PROOFSTEP]\nconvert Set.mem_iUnion\u2082\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nx\u271d : Box \u03b9\n\u22a2 x\u271d \u2208 \u03c0 \u2227 x \u2208 x\u271d \u2194 \u2203 j, x \u2208 \u2191x\u271d\n[PROOFSTEP]\nrw [Box.mem_coe, exists_prop]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nh : J \u2264 I\n\u22a2 Prepartition.iUnion (single I J h) = \u2191J\n[PROOFSTEP]\nsimp [iUnion_def]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 Prepartition.iUnion \u22a4 = \u2191I\n[PROOFSTEP]\nsimp [Prepartition.iUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 Prepartition.iUnion \u03c0\u2081 = \u2205 \u2194 \u03c0\u2081 = \u22a5\n[PROOFSTEP]\nsimp [\u2190 injective_boxes.eq_iff, Finset.ext_iff, Prepartition.iUnion, imp_false]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 \u03c0\u2081 \u2264 \u03c0\u2082 \u2194\n (\u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J') \u2227\n Prepartition.iUnion \u03c0\u2081 \u2286 Prepartition.iUnion \u03c0\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 \u03c0\u2081 \u2264 \u03c0\u2082 \u2192\n (\u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J') \u2227\n Prepartition.iUnion \u03c0\u2081 \u2286 Prepartition.iUnion \u03c0\u2082\n[PROOFSTEP]\nrefine' fun H => \u27e8fun J hJ J' hJ' Hne => _, iUnion_mono H\u27e9\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nH : \u03c0\u2081 \u2264 \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nHne : Set.Nonempty (\u2191J \u2229 \u2191J')\n\u22a2 J \u2264 J'\n[PROOFSTEP]\nrcases H hJ with \u27e8J'', hJ'', Hle\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nH : \u03c0\u2081 \u2264 \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nHne : Set.Nonempty (\u2191J \u2229 \u2191J')\nJ'' : Box \u03b9\nhJ'' : J'' \u2208 \u03c0\u2082\nHle : J \u2264 J''\n\u22a2 J \u2264 J'\n[PROOFSTEP]\nrcases Hne with \u27e8x, hx, hx'\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\nH : \u03c0\u2081 \u2264 \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0\u2082\nJ'' : Box \u03b9\nhJ'' : J'' \u2208 \u03c0\u2082\nHle : J \u2264 J''\nx : \u03b9 \u2192 \u211d\nhx : x \u2208 \u2191J\nhx' : x \u2208 \u2191J'\n\u22a2 J \u2264 J'\n[PROOFSTEP]\nrwa [\u03c0\u2082.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)]\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u22a2 (\u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J') \u2227\n Prepartition.iUnion \u03c0\u2081 \u2286 Prepartition.iUnion \u03c0\u2082 \u2192\n \u03c0\u2081 \u2264 \u03c0\u2082\n[PROOFSTEP]\nrintro \u27e8H, HU\u27e9 J hJ\n[GOAL]\ncase mpr.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J'\nHU : Prepartition.iUnion \u03c0\u2081 \u2286 Prepartition.iUnion \u03c0\u2082\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\n\u22a2 \u2203 I', I' \u2208 \u03c0\u2082 \u2227 J \u2264 I'\n[PROOFSTEP]\nsimp only [Set.subset_def, mem_iUnion] at HU \n[GOAL]\ncase mpr.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\nHU : \u2200 (x : \u03b9 \u2192 \u211d), (\u2203 J, J \u2208 \u03c0\u2081 \u2227 x \u2208 J) \u2192 \u2203 J, J \u2208 \u03c0\u2082 \u2227 x \u2208 J\n\u22a2 \u2203 I', I' \u2208 \u03c0\u2082 \u2227 J \u2264 I'\n[PROOFSTEP]\nrcases HU J.upper \u27e8J, hJ, J.upper_mem\u27e9 with \u27e8J\u2082, hJ\u2082, hx\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u2200 (J' : Box \u03b9), J' \u2208 \u03c0\u2082 \u2192 Set.Nonempty (\u2191J \u2229 \u2191J') \u2192 J \u2264 J'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\u2081\nHU : \u2200 (x : \u03b9 \u2192 \u211d), (\u2203 J, J \u2208 \u03c0\u2081 \u2227 x \u2208 J) \u2192 \u2203 J, J \u2208 \u03c0\u2082 \u2227 x \u2208 J\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0\u2082\nhx : J.upper \u2208 J\u2082\n\u22a2 \u2203 I', I' \u2208 \u03c0\u2082 \u2227 J \u2264 I'\n[PROOFSTEP]\nexact \u27e8J\u2082, hJ\u2082, H _ hJ _ hJ\u2082 \u27e8_, J.upper_mem, hx\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\nhJ : J \u2208 Finset.biUnion \u03c0.boxes fun J => (\u03c0i J).boxes\n\u22a2 J \u2264 I\n[PROOFSTEP]\nsimp only [Finset.mem_biUnion, exists_prop, mem_boxes] at hJ \n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\nhJ : \u2203 a, a \u2208 \u03c0 \u2227 J \u2208 \u03c0i a\n\u22a2 J \u2264 I\n[PROOFSTEP]\nrcases hJ with \u27e8J', hJ', hJ\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ J' : Box \u03b9\nhJ' : J' \u2208 \u03c0\nhJ : J \u2208 \u03c0i J'\n\u22a2 J \u2264 I\n[PROOFSTEP]\nexact ((\u03c0i J').le_of_mem hJ).trans (\u03c0.le_of_mem hJ')\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 Set.Pairwise (\u2191(Finset.biUnion \u03c0.boxes fun J => (\u03c0i J).boxes)) (Disjoint on Box.toSet)\n[PROOFSTEP]\nsimp only [Set.Pairwise, Finset.mem_coe, Finset.mem_biUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 \u2200 \u2983x : Box \u03b9\u2984,\n (\u2203 a, a \u2208 \u03c0.boxes \u2227 x \u2208 (\u03c0i a).boxes) \u2192\n \u2200 \u2983y : Box \u03b9\u2984, (\u2203 a, a \u2208 \u03c0.boxes \u2227 y \u2208 (\u03c0i a).boxes) \u2192 x \u2260 y \u2192 (Disjoint on Box.toSet) x y\n[PROOFSTEP]\nrintro J\u2081' \u27e8J\u2081, hJ\u2081, hJ\u2081'\u27e9 J\u2082' \u27e8J\u2082, hJ\u2082, hJ\u2082'\u27e9 Hne\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' J\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2082).boxes\nHne : J\u2081' \u2260 J\u2082'\n\u22a2 (Disjoint on Box.toSet) J\u2081' J\u2082'\n[PROOFSTEP]\nrw [Function.onFun, Set.disjoint_left]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' J\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2082).boxes\nHne : J\u2081' \u2260 J\u2082'\n\u22a2 \u2200 \u2983a : \u03b9 \u2192 \u211d\u2984, a \u2208 \u2191J\u2081' \u2192 \u00aca \u2208 \u2191J\u2082'\n[PROOFSTEP]\nrintro x hx\u2081 hx\u2082\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' J\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2082).boxes\nHne : J\u2081' \u2260 J\u2082'\nx : \u03b9 \u2192 \u211d\nhx\u2081 : x \u2208 \u2191J\u2081'\nhx\u2082 : x \u2208 \u2191J\u2082'\n\u22a2 False\n[PROOFSTEP]\napply Hne\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' J\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2082).boxes\nHne : J\u2081' \u2260 J\u2082'\nx : \u03b9 \u2192 \u211d\nhx\u2081 : x \u2208 \u2191J\u2081'\nhx\u2082 : x \u2208 \u2191J\u2082'\n\u22a2 J\u2081' = J\u2082'\n[PROOFSTEP]\nobtain rfl : J\u2081 = J\u2082\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' J\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2082).boxes\nHne : J\u2081' \u2260 J\u2082'\nx : \u03b9 \u2192 \u211d\nhx\u2081 : x \u2208 \u2191J\u2081'\nhx\u2082 : x \u2208 \u2191J\u2082'\n\u22a2 J\u2081 = J\u2082\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' : Box \u03b9\nHne : J\u2081' \u2260 J\u2082'\nx : \u03b9 \u2192 \u211d\nhx\u2081 : x \u2208 \u2191J\u2081'\nhx\u2082 : x \u2208 \u2191J\u2082'\nhJ\u2082 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2081).boxes\n\u22a2 J\u2081' = J\u2082'\n[PROOFSTEP]\nexact \u03c0.eq_of_mem_of_mem hJ\u2081 hJ\u2082 ((\u03c0i J\u2081).le_of_mem hJ\u2081' hx\u2081) ((\u03c0i J\u2082).le_of_mem hJ\u2082' hx\u2082)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ\u2081' J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2081' : J\u2081' \u2208 (\u03c0i J\u2081).boxes\nJ\u2082' : Box \u03b9\nHne : J\u2081' \u2260 J\u2082'\nx : \u03b9 \u2192 \u211d\nhx\u2081 : x \u2208 \u2191J\u2081'\nhx\u2082 : x \u2208 \u2191J\u2082'\nhJ\u2082 : J\u2081 \u2208 \u03c0.boxes\nhJ\u2082' : J\u2082' \u2208 (\u03c0i J\u2081).boxes\n\u22a2 J\u2081' = J\u2082'\n[PROOFSTEP]\nexact (\u03c0i J\u2081).eq_of_mem_of_mem hJ\u2081' hJ\u2082' hx\u2081 hx\u2082\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 J \u2208 biUnion \u03c0 \u03c0i \u2194 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J'\n[PROOFSTEP]\nsimp [biUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 (biUnion \u03c0 fun x => \u22a4) = \u03c0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nJ\u271d : Box \u03b9\n\u22a2 (J\u271d \u2208 biUnion \u03c0 fun x => \u22a4) \u2194 J\u271d \u2208 \u03c0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : \u03c0\u2081 = \u03c0\u2082\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\n\u22a2 biUnion \u03c0\u2081 \u03c0i\u2081 = biUnion \u03c0\u2082 \u03c0i\u2082\n[PROOFSTEP]\nsubst \u03c0\u2082\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\n\u22a2 biUnion \u03c0\u2081 \u03c0i\u2081 = biUnion \u03c0\u2081 \u03c0i\u2082\n[PROOFSTEP]\next J\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\nJ : Box \u03b9\n\u22a2 J \u2208 biUnion \u03c0\u2081 \u03c0i\u2081 \u2194 J \u2208 biUnion \u03c0\u2081 \u03c0i\u2082\n[PROOFSTEP]\nsimp only [mem_biUnion]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\nJ : Box \u03b9\n\u22a2 (\u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2081 J') \u2194 \u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2082 J'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\nJ : Box \u03b9\n\u22a2 (\u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2081 J') \u2192 \u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2082 J'\n[PROOFSTEP]\nexact fun \u27e8J', h\u2081, h\u2082\u27e9 => \u27e8J', h\u2081, hi J' h\u2081 \u25b8 h\u2082\u27e9\n[GOAL]\ncase h.mpr\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhi : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081 \u2192 \u03c0i\u2081 J = \u03c0i\u2082 J\nJ : Box \u03b9\n\u22a2 (\u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2082 J') \u2192 \u2203 J', J' \u2208 \u03c0\u2081 \u2227 J \u2208 \u03c0i\u2081 J'\n[PROOFSTEP]\nexact fun \u27e8J', h\u2081, h\u2082\u27e9 => \u27e8J', h\u2081, hi J' h\u2081 \u25b8 h\u2082\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 Prepartition.iUnion (biUnion \u03c0 \u03c0i) = \u22c3 (J : Box \u03b9) (_ : J \u2208 \u03c0), Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nsimp [Prepartition.iUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nM : Type u_2\ninst\u271d : AddCommMonoid M\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nf : Box \u03b9 \u2192 M\n\u22a2 \u2211 J in Finset.biUnion \u03c0.boxes fun J => (\u03c0i J).boxes, f J = \u2211 J in \u03c0.boxes, \u2211 J' in (\u03c0i J).boxes, f J'\n[PROOFSTEP]\nrefine' Finset.sum_biUnion fun J\u2081 h\u2081 J\u2082 h\u2082 hne => Finset.disjoint_left.2 fun J' h\u2081' h\u2082' => _\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nM : Type u_2\ninst\u271d : AddCommMonoid M\n\u03c0 : Prepartition I\n\u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nf : Box \u03b9 \u2192 M\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u2191\u03c0.boxes\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u2191\u03c0.boxes\nhne : J\u2081 \u2260 J\u2082\nJ' : Box \u03b9\nh\u2081' : J' \u2208 (fun J => (\u03c0i J).boxes) J\u2081\nh\u2082' : J' \u2208 (fun J => (\u03c0i J).boxes) J\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hne (\u03c0.eq_of_le_of_le h\u2081 h\u2082 ((\u03c0i J\u2081).le_of_mem h\u2081') ((\u03c0i J\u2082).le_of_mem h\u2082'))\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 biUnionIndex \u03c0 \u03c0i J \u2208 \u03c0\n[PROOFSTEP]\nrw [biUnionIndex, dif_pos hJ]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J') \u2208 \u03c0\n[PROOFSTEP]\nexact (\u03c0.mem_biUnion.1 hJ).choose_spec.1\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\n\u22a2 biUnionIndex \u03c0 \u03c0i J \u2264 I\n[PROOFSTEP]\nby_cases hJ : J \u2208 \u03c0.biUnion \u03c0i\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 biUnionIndex \u03c0 \u03c0i J \u2264 I\n[PROOFSTEP]\nexact \u03c0.le_of_mem (\u03c0.biUnionIndex_mem hJ)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\nhJ : \u00acJ \u2208 biUnion \u03c0 \u03c0i\n\u22a2 biUnionIndex \u03c0 \u03c0i J \u2264 I\n[PROOFSTEP]\nrw [biUnionIndex, dif_neg hJ]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 J \u2208 \u03c0i (biUnionIndex \u03c0 \u03c0i J)\n[PROOFSTEP]\nconvert (\u03c0.mem_biUnion.1 hJ).choose_spec.2\n[GOAL]\ncase h.e'_2.h.e'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 biUnionIndex \u03c0 \u03c0i J = Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J')\n[PROOFSTEP]\nexact dif_pos hJ\n[GOAL]\ncase h.e'_3.e'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\ne_2\u271d : Prepartition (biUnionIndex \u03c0 \u03c0i J) = Prepartition (Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J'))\n\u22a2 biUnionIndex \u03c0 \u03c0i J = Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J')\n[PROOFSTEP]\nexact dif_pos hJ\n[GOAL]\ncase h.e'_5.e'_1\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 biUnion \u03c0 \u03c0i\ne_2\u271d : Prepartition (biUnionIndex \u03c0 \u03c0i J) = Prepartition (Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J'))\n\u22a2 biUnionIndex \u03c0 \u03c0i J = Exists.choose (_ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J')\n[PROOFSTEP]\nexact dif_pos hJ\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 (biUnion \u03c0 fun J => biUnion (\u03c0i J) (\u03c0i' J)) = biUnion (biUnion \u03c0 \u03c0i) fun J => \u03c0i' (biUnionIndex \u03c0 \u03c0i J) J\n[PROOFSTEP]\next J\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\n\u22a2 (J \u2208 biUnion \u03c0 fun J => biUnion (\u03c0i J) (\u03c0i' J)) \u2194 J \u2208 biUnion (biUnion \u03c0 \u03c0i) fun J => \u03c0i' (biUnionIndex \u03c0 \u03c0i J) J\n[PROOFSTEP]\nsimp only [mem_biUnion, exists_prop]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\n\u22a2 (\u2203 J', J' \u2208 \u03c0 \u2227 \u2203 J'_1, J'_1 \u2208 \u03c0i J' \u2227 J \u2208 \u03c0i' J' J'_1) \u2194\n \u2203 J', (\u2203 J'_1, J'_1 \u2208 \u03c0 \u2227 J' \u2208 \u03c0i J'_1) \u2227 J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J') J'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\n\u22a2 (\u2203 J', J' \u2208 \u03c0 \u2227 \u2203 J'_1, J'_1 \u2208 \u03c0i J' \u2227 J \u2208 \u03c0i' J' J'_1) \u2192\n \u2203 J', (\u2203 J'_1, J'_1 \u2208 \u03c0 \u2227 J' \u2208 \u03c0i J'_1) \u2227 J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J') J'\n[PROOFSTEP]\nrintro \u27e8J\u2081, hJ\u2081, J\u2082, hJ\u2082, hJ\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0i J\u2081\nhJ : J \u2208 \u03c0i' J\u2081 J\u2082\n\u22a2 \u2203 J', (\u2203 J'_1, J'_1 \u2208 \u03c0 \u2227 J' \u2208 \u03c0i J'_1) \u2227 J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J') J'\n[PROOFSTEP]\nrefine' \u27e8J\u2082, \u27e8J\u2081, hJ\u2081, hJ\u2082\u27e9, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ J\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0i J\u2081\nhJ : J \u2208 \u03c0i' J\u2081 J\u2082\n\u22a2 J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J\u2082) J\u2082\n[PROOFSTEP]\nrwa [\u03c0.biUnionIndex_of_mem hJ\u2081 hJ\u2082]\n[GOAL]\ncase h.mpr\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ : Box \u03b9\n\u22a2 (\u2203 J', (\u2203 J'_1, J'_1 \u2208 \u03c0 \u2227 J' \u2208 \u03c0i J'_1) \u2227 J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J') J') \u2192\n \u2203 J', J' \u2208 \u03c0 \u2227 \u2203 J'_1, J'_1 \u2208 \u03c0i J' \u2227 J \u2208 \u03c0i' J' J'_1\n[PROOFSTEP]\nrintro \u27e8J\u2081, \u27e8J\u2082, hJ\u2082, hJ\u2081\u27e9, hJ\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ J\u2081 : Box \u03b9\nhJ : J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J\u2081) J\u2081\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0\nhJ\u2081 : J\u2081 \u2208 \u03c0i J\u2082\n\u22a2 \u2203 J', J' \u2208 \u03c0 \u2227 \u2203 J'_1, J'_1 \u2208 \u03c0i J' \u2227 J \u2208 \u03c0i' J' J'_1\n[PROOFSTEP]\nrefine' \u27e8J\u2082, hJ\u2082, J\u2081, hJ\u2081, _\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0i' : Box \u03b9 \u2192 (J : Box \u03b9) \u2192 Prepartition J\nJ J\u2081 : Box \u03b9\nhJ : J \u2208 \u03c0i' (biUnionIndex \u03c0 \u03c0i J\u2081) J\u2081\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0\nhJ\u2081 : J\u2081 \u2208 \u03c0i J\u2082\n\u22a2 J \u2208 \u03c0i' J\u2082 J\u2081\n[PROOFSTEP]\nrwa [\u03c0.biUnionIndex_of_mem hJ\u2082 hJ\u2081] at hJ \n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nJ : Box \u03b9\nhJ : J \u2208 \u2191eraseNone boxes\n\u22a2 J \u2264 I\n[PROOFSTEP]\nrw [mem_eraseNone] at hJ \n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nJ : Box \u03b9\nhJ : some J \u2208 boxes\n\u22a2 J \u2264 I\n[PROOFSTEP]\nsimpa only [WithBot.some_eq_coe, WithBot.coe_le_coe] using le_of_mem _ hJ\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u2191(\u2191eraseNone boxes)\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u2191(\u2191eraseNone boxes)\nhne : J\u2081 \u2260 J\u2082\n\u22a2 (Disjoint on Box.toSet) J\u2081 J\u2082\n[PROOFSTEP]\nsimp only [mem_coe, mem_eraseNone] at h\u2081 h\u2082 \n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nJ\u2081 J\u2082 : Box \u03b9\nhne : J\u2081 \u2260 J\u2082\nh\u2081 : some J\u2081 \u2208 boxes\nh\u2082 : some J\u2082 \u2208 boxes\n\u22a2 (Disjoint on Box.toSet) J\u2081 J\u2082\n[PROOFSTEP]\nexact Box.disjoint_coe.1 (pairwise_disjoint h\u2081 h\u2082 (mt Option.some_inj.1 hne))\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\n\u22a2 Prepartition.iUnion (ofWithBot boxes le_of_mem pairwise_disjoint) = \u22c3 (J : WithBot (Box \u03b9)) (_ : J \u2208 boxes), \u2191J\n[PROOFSTEP]\nsuffices \u22c3 (J : Box \u03b9) (_ : \u2191J \u2208 boxes), \u2191J = \u22c3 J \u2208 boxes, (J : Set (\u03b9 \u2192 \u211d)) by simpa [ofWithBot, Prepartition.iUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nthis : \u22c3 (J : Box \u03b9) (_ : \u2191J \u2208 boxes), \u2191J = \u22c3 (J : WithBot (Box \u03b9)) (_ : J \u2208 boxes), \u2191J\n\u22a2 Prepartition.iUnion (ofWithBot boxes le_of_mem pairwise_disjoint) = \u22c3 (J : WithBot (Box \u03b9)) (_ : J \u2208 boxes), \u2191J\n[PROOFSTEP]\nsimpa [ofWithBot, Prepartition.iUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\n\u22a2 \u22c3 (J : Box \u03b9) (_ : \u2191J \u2208 boxes), \u2191J = \u22c3 (J : WithBot (Box \u03b9)) (_ : J \u2208 boxes), \u2191J\n[PROOFSTEP]\nsimp only [\u2190 Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box \u03b9), @iUnion_comm _ _ (@Eq _ _ _), iUnion_iUnion_eq_right]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2260 \u22a5 \u2192 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2264 \u2191J'\n\u22a2 ofWithBot boxes le_of_mem pairwise_disjoint \u2264 \u03c0\n[PROOFSTEP]\nhave : \u2200 J : Box \u03b9, \u2191J \u2208 boxes \u2192 \u2203 J' \u2208 \u03c0, J \u2264 J' := fun J hJ => by\n simpa only [WithBot.coe_le_coe] using H J hJ WithBot.coe_ne_bot\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2260 \u22a5 \u2192 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2264 \u2191J'\nJ : Box \u03b9\nhJ : \u2191J \u2208 boxes\n\u22a2 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2264 J'\n[PROOFSTEP]\nsimpa only [WithBot.coe_le_coe] using H J hJ WithBot.coe_ne_bot\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2260 \u22a5 \u2192 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2264 \u2191J'\nthis : \u2200 (J : Box \u03b9), \u2191J \u2208 boxes \u2192 \u2203 J', J' \u2208 \u03c0 \u2227 J \u2264 J'\n\u22a2 ofWithBot boxes le_of_mem pairwise_disjoint \u2264 \u03c0\n[PROOFSTEP]\nsimpa [ofWithBot, le_def]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 J', J' \u2208 boxes \u2227 \u2191J \u2264 J'\n\u22a2 \u03c0 \u2264 ofWithBot boxes le_of_mem pairwise_disjoint\n[PROOFSTEP]\nintro J hJ\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 J', J' \u2208 boxes \u2227 \u2191J \u2264 J'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 \u2203 I', I' \u2208 ofWithBot boxes le_of_mem pairwise_disjoint \u2227 J \u2264 I'\n[PROOFSTEP]\nrcases H J hJ with \u27e8J', J'mem, hle\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 J', J' \u2208 boxes \u2227 \u2191J \u2264 J'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\nJ' : WithBot (Box \u03b9)\nJ'mem : J' \u2208 boxes\nhle : \u2191J \u2264 J'\n\u22a2 \u2203 I', I' \u2208 ofWithBot boxes le_of_mem pairwise_disjoint \u2227 J \u2264 I'\n[PROOFSTEP]\nlift J' to Box \u03b9 using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hle\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nboxes : Finset (WithBot (Box \u03b9))\nle_of_mem : \u2200 (J : WithBot (Box \u03b9)), J \u2208 boxes \u2192 J \u2264 \u2191I\npairwise_disjoint : Set.Pairwise (\u2191boxes) Disjoint\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 J', J' \u2208 boxes \u2227 \u2191J \u2264 J'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\nJ' : Box \u03b9\nJ'mem : \u2191J' \u2208 boxes\nhle : \u2191J \u2264 \u2191J'\n\u22a2 \u2203 I', I' \u2208 ofWithBot boxes le_of_mem pairwise_disjoint \u2227 J \u2264 I'\n[PROOFSTEP]\nexact \u27e8J', mem_ofWithBot.2 J'mem, WithBot.coe_le_coe.1 hle\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ : Box \u03b9\nJ' : WithBot (Box \u03b9)\nhJ' : J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0.boxes\n\u22a2 J' \u2264 \u2191J\n[PROOFSTEP]\nrcases Finset.mem_image.1 hJ' with \u27e8J', -, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ J' : Box \u03b9\nhJ' : \u2191J \u2293 \u2191J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0.boxes\n\u22a2 \u2191J \u2293 \u2191J' \u2264 \u2191J\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ : Box \u03b9\n\u22a2 Set.Pairwise (\u2191(Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0.boxes)) Disjoint\n[PROOFSTEP]\nsimp only [Set.Pairwise, onFun, Finset.mem_coe, Finset.mem_image]\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ : Box \u03b9\n\u22a2 \u2200 \u2983x : WithBot (Box \u03b9)\u2984,\n (\u2203 a, a \u2208 \u03c0.boxes \u2227 \u2191J \u2293 \u2191a = x) \u2192 \u2200 \u2983y : WithBot (Box \u03b9)\u2984, (\u2203 a, a \u2208 \u03c0.boxes \u2227 \u2191J \u2293 \u2191a = y) \u2192 x \u2260 y \u2192 Disjoint x y\n[PROOFSTEP]\nrintro _ \u27e8J\u2081, h\u2081, rfl\u27e9 _ \u27e8J\u2082, h\u2082, rfl\u27e9 Hne\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0.boxes\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0.boxes\nHne : \u2191J \u2293 \u2191J\u2081 \u2260 \u2191J \u2293 \u2191J\u2082\n\u22a2 Disjoint (\u2191J \u2293 \u2191J\u2081) (\u2191J \u2293 \u2191J\u2082)\n[PROOFSTEP]\nhave : J\u2081 \u2260 J\u2082 := by\n rintro rfl\n exact Hne rfl\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0.boxes\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0.boxes\nHne : \u2191J \u2293 \u2191J\u2081 \u2260 \u2191J \u2293 \u2191J\u2082\n\u22a2 J\u2081 \u2260 J\u2082\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ J\u2081 : Box \u03b9\nh\u2081 h\u2082 : J\u2081 \u2208 \u03c0.boxes\nHne : \u2191J \u2293 \u2191J\u2081 \u2260 \u2191J \u2293 \u2191J\u2081\n\u22a2 False\n[PROOFSTEP]\nexact Hne rfl\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0.boxes\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 \u03c0.boxes\nHne : \u2191J \u2293 \u2191J\u2081 \u2260 \u2191J \u2293 \u2191J\u2082\nthis : J\u2081 \u2260 J\u2082\n\u22a2 Disjoint (\u2191J \u2293 \u2191J\u2081) (\u2191J \u2293 \u2191J\u2082)\n[PROOFSTEP]\nexact ((Box.disjoint_coe.2 <| \u03c0.disjoint_coe_of_mem h\u2081 h\u2082 this).inf_left' _).inf_right' _\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 J\u2081 \u2208 restrict \u03c0 J \u2194 \u2203 J', J' \u2208 \u03c0 \u2227 \u2191J\u2081 = \u2191J \u2293 \u2191J'\n[PROOFSTEP]\nsimp [restrict, eq_comm]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 J\u2081 \u2208 restrict \u03c0 J \u2194 \u2203 J', J' \u2208 \u03c0 \u2227 \u2191J\u2081 = \u2191J \u2229 \u2191J'\n[PROOFSTEP]\nsimp only [mem_restrict, \u2190 Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nHle : \u03c0\u2081 \u2264 \u03c0\u2082\n\u22a2 restrict \u03c0\u2081 J \u2264 restrict \u03c0\u2082 J\n[PROOFSTEP]\nrefine' ofWithBot_mono fun J\u2081 hJ\u2081 hne => _\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nHle : \u03c0\u2081 \u2264 \u03c0\u2082\nJ\u2081 : WithBot (Box \u03b9)\nhJ\u2081 : J\u2081 \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0\u2081.boxes\nhne : J\u2081 \u2260 \u22a5\n\u22a2 \u2203 J', J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0\u2082.boxes \u2227 J\u2081 \u2264 J'\n[PROOFSTEP]\nrw [Finset.mem_image] at hJ\u2081 \n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nHle : \u03c0\u2081 \u2264 \u03c0\u2082\nJ\u2081 : WithBot (Box \u03b9)\nhJ\u2081 : \u2203 a, a \u2208 \u03c0\u2081.boxes \u2227 \u2191J \u2293 \u2191a = J\u2081\nhne : J\u2081 \u2260 \u22a5\n\u22a2 \u2203 J', J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0\u2082.boxes \u2227 J\u2081 \u2264 J'\n[PROOFSTEP]\nrcases hJ\u2081 with \u27e8J\u2081, hJ\u2081, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nHle : \u03c0\u2081 \u2264 \u03c0\u2082\nJ\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0\u2081.boxes\nhne : \u2191J \u2293 \u2191J\u2081 \u2260 \u22a5\n\u22a2 \u2203 J', J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0\u2082.boxes \u2227 \u2191J \u2293 \u2191J\u2081 \u2264 J'\n[PROOFSTEP]\nrcases Hle hJ\u2081 with \u27e8J\u2082, hJ\u2082, hle\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nHle : \u03c0\u2081 \u2264 \u03c0\u2082\nJ\u2081 : Box \u03b9\nhJ\u2081 : J\u2081 \u2208 \u03c0\u2081.boxes\nhne : \u2191J \u2293 \u2191J\u2081 \u2260 \u22a5\nJ\u2082 : Box \u03b9\nhJ\u2082 : J\u2082 \u2208 \u03c0\u2082\nhle : J\u2081 \u2264 J\u2082\n\u22a2 \u2203 J', J' \u2208 Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0\u2082.boxes \u2227 \u2191J \u2293 \u2191J\u2081 \u2264 J'\n[PROOFSTEP]\nexact \u27e8_, Finset.mem_image_of_mem _ hJ\u2082, inf_le_inf_left _ <| WithBot.coe_le_coe.2 hle\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\n\u22a2 (restrict \u03c0 J).boxes = \u03c0.boxes\n[PROOFSTEP]\nsimp only [restrict, ofWithBot, eraseNone_eq_biUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\n\u22a2 Finset.biUnion (Finset.image (fun J' => \u2191J \u2293 \u2191J') \u03c0.boxes) Option.toFinset = \u03c0.boxes\n[PROOFSTEP]\nrefine' Finset.image_biUnion.trans _\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\n\u22a2 (Finset.biUnion \u03c0.boxes fun a => Option.toFinset (\u2191J \u2293 \u2191a)) = \u03c0.boxes\n[PROOFSTEP]\nrefine' (Finset.biUnion_congr rfl _).trans Finset.biUnion_singleton_eq_self\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\n\u22a2 \u2200 (a : Box \u03b9), a \u2208 \u03c0.boxes \u2192 Option.toFinset (\u2191J \u2293 \u2191a) = {a}\n[PROOFSTEP]\nintro J' hJ'\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0.boxes\n\u22a2 Option.toFinset (\u2191J \u2293 \u2191J') = {J'}\n[PROOFSTEP]\nrw [inf_of_le_right, \u2190 WithBot.some_eq_coe, Option.toFinset_some]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\nh : I \u2264 J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0.boxes\n\u22a2 \u2191J' \u2264 \u2191J\n[PROOFSTEP]\nexact WithBot.coe_le_coe.2 ((\u03c0.le_of_mem hJ').trans h)\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u22a2 Prepartition.iUnion (restrict \u03c0 J) = \u2191J \u2229 Prepartition.iUnion \u03c0\n[PROOFSTEP]\nsimp [restrict, \u2190 inter_iUnion, \u2190 iUnion_def]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\n\u22a2 restrict (biUnion \u03c0 \u03c0i) J = \u03c0i J\n[PROOFSTEP]\nrefine' (eq_of_boxes_subset_iUnion_superset (fun J\u2081 h\u2081 => _) _).symm\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 (\u03c0i J).boxes\n\u22a2 J\u2081 \u2208 (restrict (biUnion \u03c0 \u03c0i) J).boxes\n[PROOFSTEP]\nrefine' (mem_restrict _).2 \u27e8J\u2081, \u03c0.mem_biUnion.2 \u27e8J, hJ, h\u2081\u27e9, (inf_of_le_right _).symm\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 (\u03c0i J).boxes\n\u22a2 \u2191J\u2081 \u2264 \u2191J\n[PROOFSTEP]\nexact WithBot.coe_le_coe.2 (le_of_mem _ h\u2081)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\n\u22a2 Prepartition.iUnion (restrict (biUnion \u03c0 \u03c0i) J) \u2286 Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nsimp only [iUnion_restrict, iUnion_biUnion, Set.subset_def, Set.mem_inter_iff, Set.mem_iUnion]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\n\u22a2 \u2200 (x : \u03b9 \u2192 \u211d), (x \u2208 \u2191J \u2227 \u2203 i i_1, x \u2208 Prepartition.iUnion (\u03c0i i)) \u2192 x \u2208 Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nrintro x \u27e8hxJ, J\u2081, h\u2081, hx\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nx : \u03b9 \u2192 \u211d\nhxJ : x \u2208 \u2191J\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx : x \u2208 Prepartition.iUnion (\u03c0i J\u2081)\n\u22a2 x \u2208 Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nobtain rfl : J = J\u2081\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nx : \u03b9 \u2192 \u211d\nhxJ : x \u2208 \u2191J\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhx : x \u2208 Prepartition.iUnion (\u03c0i J\u2081)\n\u22a2 J = J\u2081\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nx : \u03b9 \u2192 \u211d\nhxJ : x \u2208 \u2191J\nh\u2081 : J \u2208 \u03c0\nhx : x \u2208 Prepartition.iUnion (\u03c0i J)\n\u22a2 x \u2208 Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nexact \u03c0.eq_of_mem_of_mem hJ h\u2081 hxJ (iUnion_subset _ hx)\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx\u271d : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\nhJ : J \u2208 \u03c0\nx : \u03b9 \u2192 \u211d\nhxJ : x \u2208 \u2191J\nh\u2081 : J \u2208 \u03c0\nhx : x \u2208 Prepartition.iUnion (\u03c0i J)\n\u22a2 x \u2208 Prepartition.iUnion (\u03c0i J)\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\n\u22a2 biUnion \u03c0 \u03c0i \u2264 \u03c0' \u2194 \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\n\u22a2 biUnion \u03c0 \u03c0i \u2264 \u03c0' \u2192 \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\n[PROOFSTEP]\nintro H J hJ\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\n\u22a2 (\u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J) \u2192 biUnion \u03c0 \u03c0i \u2264 \u03c0'\n[PROOFSTEP]\nintro H J hJ\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : biUnion \u03c0 \u03c0i \u2264 \u03c0'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 \u03c0i J \u2264 restrict \u03c0' J\n[PROOFSTEP]\nrw [\u2190 \u03c0.restrict_biUnion \u03c0i hJ]\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : biUnion \u03c0 \u03c0i \u2264 \u03c0'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 restrict (biUnion \u03c0 \u03c0i) J \u2264 restrict \u03c0' J\n[PROOFSTEP]\nexact restrict_mono H\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\nJ : Box \u03b9\nhJ : J \u2208 biUnion \u03c0 \u03c0i\n\u22a2 \u2203 I', I' \u2208 \u03c0' \u2227 J \u2264 I'\n[PROOFSTEP]\nrw [mem_biUnion] at hJ \n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\nJ : Box \u03b9\nhJ : \u2203 J', J' \u2208 \u03c0 \u2227 J \u2208 \u03c0i J'\n\u22a2 \u2203 I', I' \u2208 \u03c0' \u2227 J \u2264 I'\n[PROOFSTEP]\nrcases hJ with \u27e8J\u2081, h\u2081, hJ\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhJ : J \u2208 \u03c0i J\u2081\n\u22a2 \u2203 I', I' \u2208 \u03c0' \u2227 J \u2264 I'\n[PROOFSTEP]\nrcases H J\u2081 h\u2081 hJ with \u27e8J\u2082, h\u2082, Hle\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhJ : J \u2208 \u03c0i J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 restrict \u03c0' J\u2081\nHle : J \u2264 J\u2082\n\u22a2 \u2203 I', I' \u2208 \u03c0' \u2227 J \u2264 I'\n[PROOFSTEP]\nrcases \u03c0'.mem_restrict.mp h\u2082 with \u27e8J\u2083, h\u2083, H\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081\u271d J\u2082\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH\u271d : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u03c0i J \u2264 restrict \u03c0' J\nJ J\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 \u03c0\nhJ : J \u2208 \u03c0i J\u2081\nJ\u2082 : Box \u03b9\nh\u2082 : J\u2082 \u2208 restrict \u03c0' J\u2081\nHle : J \u2264 J\u2082\nJ\u2083 : Box \u03b9\nh\u2083 : J\u2083 \u2208 \u03c0'\nH : \u2191J\u2082 = \u2191J\u2081 \u2293 \u2191J\u2083\n\u22a2 \u2203 I', I' \u2208 \u03c0' \u2227 J \u2264 I'\n[PROOFSTEP]\nexact \u27e8J\u2083, h\u2083, Hle.trans <| WithBot.coe_le_coe.1 <| H.trans_le inf_le_right\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\n\u22a2 \u03c0' \u2264 biUnion \u03c0 \u03c0i \u2194 \u03c0' \u2264 \u03c0 \u2227 \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J\n[PROOFSTEP]\nrefine' \u27e8fun H => \u27e8H.trans (\u03c0.biUnion_le \u03c0i), fun J hJ => _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 biUnion \u03c0 \u03c0i\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 restrict \u03c0' J \u2264 \u03c0i J\n[PROOFSTEP]\nrw [\u2190 \u03c0.restrict_biUnion \u03c0i hJ]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 biUnion \u03c0 \u03c0i\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\n\u22a2 restrict \u03c0' J \u2264 restrict (biUnion \u03c0 \u03c0i) J\n[PROOFSTEP]\nexact restrict_mono H\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\n\u22a2 (\u03c0' \u2264 \u03c0 \u2227 \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J) \u2192 \u03c0' \u2264 biUnion \u03c0 \u03c0i\n[PROOFSTEP]\nrintro \u27e8H, Hi\u27e9 J' hJ'\n[GOAL]\ncase refine'_2.intro\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 \u03c0\nHi : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0'\n\u22a2 \u2203 I', I' \u2208 biUnion \u03c0 \u03c0i \u2227 J' \u2264 I'\n[PROOFSTEP]\nrcases H hJ' with \u27e8J, hJ, hle\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 \u03c0\nHi : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\nhle : J' \u2264 J\n\u22a2 \u2203 I', I' \u2208 biUnion \u03c0 \u03c0i \u2227 J' \u2264 I'\n[PROOFSTEP]\nhave : J' \u2208 \u03c0'.restrict J := \u03c0'.mem_restrict.2 \u27e8J', hJ', (inf_of_le_right <| WithBot.coe_le_coe.2 hle).symm\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 \u03c0\nHi : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\nhle : J' \u2264 J\nthis : J' \u2208 restrict \u03c0' J\n\u22a2 \u2203 I', I' \u2208 biUnion \u03c0 \u03c0i \u2227 J' \u2264 I'\n[PROOFSTEP]\nrcases Hi J hJ this with \u27e8Ji, hJi, hlei\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i\u271d \u03c0i\u2081 \u03c0i\u2082 \u03c0i : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0' : Prepartition I\nH : \u03c0' \u2264 \u03c0\nHi : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 restrict \u03c0' J \u2264 \u03c0i J\nJ' : Box \u03b9\nhJ' : J' \u2208 \u03c0'\nJ : Box \u03b9\nhJ : J \u2208 \u03c0\nhle : J' \u2264 J\nthis : J' \u2208 restrict \u03c0' J\nJi : Box \u03b9\nhJi : Ji \u2208 \u03c0i J\nhlei : J' \u2264 Ji\n\u22a2 \u2203 I', I' \u2208 biUnion \u03c0 \u03c0i \u2227 J' \u2264 I'\n[PROOFSTEP]\nexact \u27e8Ji, \u03c0.mem_biUnion.2 \u27e8J, hJ, hJi\u27e9, hlei\u27e9\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\n\u22a2 J \u2208 \u03c0\u2081 \u2293 \u03c0\u2082 \u2194 \u2203 J\u2081, J\u2081 \u2208 \u03c0\u2081 \u2227 \u2203 J\u2082, J\u2082 \u2208 \u03c0\u2082 \u2227 \u2191J = \u2191J\u2081 \u2293 \u2191J\u2082\n[PROOFSTEP]\nsimp only [inf_def, mem_biUnion, mem_restrict]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\n\u22a2 Prepartition.iUnion (\u03c0\u2081 \u2293 \u03c0\u2082) = Prepartition.iUnion \u03c0\u2081 \u2229 Prepartition.iUnion \u03c0\u2082\n[PROOFSTEP]\nsimp only [inf_def, iUnion_biUnion, iUnion_restrict, \u2190 iUnion_inter, \u2190 iUnion_def]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\np : Box \u03b9 \u2192 Prop\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 p J\n\u22a2 filter \u03c0 p = \u03c0\n[PROOFSTEP]\next J\n[GOAL]\ncase h\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\np : Box \u03b9 \u2192 Prop\nhp : \u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 p J\nJ : Box \u03b9\n\u22a2 J \u2208 filter \u03c0 p \u2194 J \u2208 \u03c0\n[PROOFSTEP]\nsimpa using hp J\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 Prepartition.iUnion (filter \u03c0 fun J => \u00acp J) = Prepartition.iUnion \u03c0 \\ Prepartition.iUnion (filter \u03c0 p)\n[PROOFSTEP]\nsimp only [Prepartition.iUnion]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 \u22c3 (J : Box \u03b9) (_ : J \u2208 filter \u03c0 fun J => \u00acp J), \u2191J =\n (\u22c3 (J : Box \u03b9) (_ : J \u2208 \u03c0), \u2191J) \\ \u22c3 (J : Box \u03b9) (_ : J \u2208 filter \u03c0 p), \u2191J\n[PROOFSTEP]\nconvert (@Set.biUnion_diff_biUnion_eq _ (Box \u03b9) \u03c0.boxes (\u03c0.filter p).boxes (\u2191) _).symm\n[GOAL]\ncase h.e'_2.h.e'_3.h.pq.a.a\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\nx\u271d : Box \u03b9\n\u22a2 (x\u271d \u2208 filter \u03c0 fun J => \u00acp J) \u2194 x\u271d \u2208 \u2191\u03c0.boxes \\ \u2191(filter \u03c0 p).boxes\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 PairwiseDisjoint (\u2191\u03c0.boxes \u222a \u2191(filter \u03c0 p).boxes) Box.toSet\n[PROOFSTEP]\nrw [Set.PairwiseDisjoint]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 Set.Pairwise (\u2191\u03c0.boxes \u222a \u2191(filter \u03c0 p).boxes) (Disjoint on Box.toSet)\n[PROOFSTEP]\nconvert \u03c0.pairwiseDisjoint\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 \u2191\u03c0.boxes \u222a \u2191(filter \u03c0 p).boxes = \u2191\u03c0.boxes\n[PROOFSTEP]\nrw [Set.union_eq_left_iff_subset, filter_boxes, coe_filter]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\np : Box \u03b9 \u2192 Prop\n\u22a2 {x | x \u2208 \u03c0.boxes \u2227 p x} \u2286 \u2191\u03c0.boxes\n[PROOFSTEP]\nexact fun _ \u27e8h, _\u27e9 => h\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03b1 : Type u_2\nM : Type u_3\ninst\u271d : AddCommMonoid M\n\u03c0 : Prepartition I\nf : Box \u03b9 \u2192 \u03b1\ng : Box \u03b9 \u2192 M\n\u22a2 \u2211 y in Finset.image f \u03c0.boxes, \u2211 J in (filter \u03c0 fun J => f J = y).boxes, g J = \u2211 J in \u03c0.boxes, g J\n[PROOFSTEP]\nconvert sum_fiberwise_of_maps_to (fun _ => Finset.mem_image_of_mem f) g\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Disjoint (Prepartition.iUnion \u03c0\u2081) (Prepartition.iUnion \u03c0\u2082)\nthis : \u2200 (J\u2081 : Box \u03b9), J\u2081 \u2208 \u03c0\u2081 \u2192 \u2200 (J\u2082 : Box \u03b9), J\u2082 \u2208 \u03c0\u2082 \u2192 J\u2081 \u2260 J\u2082 \u2192 Disjoint \u2191J\u2081 \u2191J\u2082\n\u22a2 Set.Pairwise (\u2191(\u03c0\u2081.boxes \u222a \u03c0\u2082.boxes)) (Disjoint on Box.toSet)\n[PROOFSTEP]\nsimpa [pairwise_union_of_symmetric (symmetric_disjoint.comap _), pairwiseDisjoint]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : Disjoint (Prepartition.iUnion \u03c0\u2081) (Prepartition.iUnion \u03c0\u2082)\n\u22a2 Prepartition.iUnion (disjUnion \u03c0\u2081 \u03c0\u2082 h) = Prepartition.iUnion \u03c0\u2081 \u222a Prepartition.iUnion \u03c0\u2082\n[PROOFSTEP]\nsimp [disjUnion, Prepartition.iUnion, iUnion_or, iUnion_union_distrib]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0\u271d \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\n\u03c0 : Prepartition I\n\u22a2 IsPartition \u03c0 \u2194 Prepartition.iUnion \u03c0 = \u2191I\n[PROOFSTEP]\nsimp_rw [IsPartition, Set.Subset.antisymm_iff, \u03c0.iUnion_subset, true_and_iff, Set.subset_def, mem_iUnion, Box.mem_coe]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : J \u2264 I\n\u22a2 IsPartition (single I J h) \u2194 J = I\n[PROOFSTEP]\nsimp [isPartition_iff_iUnion_eq]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : IsPartition \u03c0\nhx : x \u2208 I\n\u22a2 \u2203! J x_1, x \u2208 J\n[PROOFSTEP]\nrcases h x hx with \u27e8J, h, hx\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nI J\u271d J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh\u271d : IsPartition \u03c0\nhx\u271d : x \u2208 I\nJ : Box \u03b9\nh : J \u2208 \u03c0\nhx : x \u2208 J\n\u22a2 \u2203! J x_1, x \u2208 J\n[PROOFSTEP]\nexact ExistsUnique.intro\u2082 J h hx fun J' h' hx' => \u03c0.eq_of_mem_of_mem h' h hx' hx\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : IsPartition \u03c0\nhJ : J \u2264 I\n\u22a2 Prepartition.iUnion (restrict \u03c0 J) = \u2191J\n[PROOFSTEP]\nsimp [h.iUnion_eq, hJ]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh\u2081 : IsPartition \u03c0\u2081\nh\u2082 : IsPartition \u03c0\u2082\n\u22a2 Prepartition.iUnion (\u03c0\u2081 \u2293 \u03c0\u2082) = \u2191I\n[PROOFSTEP]\nsimp [h\u2081.iUnion_eq, h\u2082.iUnion_eq]\n[GOAL]\n\u03b9 : Type u_1\nI J J\u2081 J\u2082 : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : Prepartition I\nx : \u03b9 \u2192 \u211d\n\u03c0i \u03c0i\u2081 \u03c0i\u2082 : (J : Box \u03b9) \u2192 Prepartition J\nh : Prepartition.iUnion \u03c0\u2082 = \u2191I \\ Prepartition.iUnion \u03c0\u2081\n\u22a2 Prepartition.iUnion \u03c0\u2081 \u222a Prepartition.iUnion \u03c0\u2082 = \u2191I\n[PROOFSTEP]\nsimp [h, \u03c0\u2081.iUnion_subset]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.BoxIntegral.Partition.Basic", "llama_tokens": 32436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.2593807164439105}} {"text": "[GOAL]\nn : \u2115\nc : { c // Composition.length c < n + 2 }\n\u22a2 (invImage (fun a => sizeOf a) instWellFoundedRelation).1 (Composition.length \u2191c) (Nat.succ (Nat.succ n))\n[PROOFSTEP]\nexact c.2\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 leftInv p i 0 = 0\n[PROOFSTEP]\nrw [leftInv]\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 leftInv p i 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c F E)) \u2191(ContinuousLinearEquiv.symm i)\n[PROOFSTEP]\nrw [leftInv]\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 leftInv (removeZero p) i = leftInv p i\n[PROOFSTEP]\next1 n\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\n\u22a2 leftInv (removeZero p) i n = leftInv p i n\n[PROOFSTEP]\ninduction' n using Nat.strongRec' with n IH\n[GOAL]\ncase h.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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 leftInv (removeZero p) i m = leftInv p i m\n\u22a2 leftInv (removeZero p) i n = leftInv p i n\n[PROOFSTEP]\nmatch n with\n| 0 =>\n simp\n -- if one replaces `simp` with `refl`, the proof times out in the kernel.\n| 1 =>\n simp\n -- TODO: why?\n| n + 2 =>\n simp only [leftInv, neg_inj]\n refine' Finset.sum_congr rfl fun c cuniv => _\n rcases c with \u27e8c, hc\u27e9\n ext v\n dsimp\n simp [IH _ hc]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < 0 \u2192 leftInv (removeZero p) i m = leftInv p i m\n\u22a2 leftInv (removeZero p) i 0 = leftInv p i 0\n[PROOFSTEP]\nsimp\n -- if one replaces `simp` with `refl`, the proof times out in the kernel.\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < 1 \u2192 leftInv (removeZero p) i m = leftInv p i m\n\u22a2 leftInv (removeZero p) i 1 = leftInv p i 1\n[PROOFSTEP]\nsimp\n -- TODO: why?\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\n\u22a2 leftInv (removeZero p) i (n + 2) = leftInv p i (n + 2)\n[PROOFSTEP]\nsimp only [leftInv, neg_inj]\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\n\u22a2 \u2211 c : { c // Composition.length c < n + 2 },\n ContinuousMultilinearMap.compAlongComposition\n (compContinuousLinearMap (removeZero p) \u2191(ContinuousLinearEquiv.symm i)) (\u2191c)\n (leftInv (removeZero p) i (Composition.length \u2191c)) =\n \u2211 c : { c // Composition.length c < n + 2 },\n ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) (\u2191c)\n (leftInv p i (Composition.length \u2191c))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun c cuniv => _\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\nc : { c // Composition.length c < n + 2 }\ncuniv : c \u2208 univ\n\u22a2 ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap (removeZero p) \u2191(ContinuousLinearEquiv.symm i))\n (\u2191c) (leftInv (removeZero p) i (Composition.length \u2191c)) =\n ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) (\u2191c)\n (leftInv p i (Composition.length \u2191c))\n[PROOFSTEP]\nrcases c with \u27e8c, hc\u27e9\n[GOAL]\ncase mk\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\nc : Composition (n + 2)\nhc : Composition.length c < n + 2\ncuniv : { val := c, property := hc } \u2208 univ\n\u22a2 ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap (removeZero p) \u2191(ContinuousLinearEquiv.symm i))\n (\u2191{ val := c, property := hc }) (leftInv (removeZero p) i (Composition.length \u2191{ val := c, property := hc })) =\n ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i))\n (\u2191{ val := c, property := hc }) (leftInv p i (Composition.length \u2191{ val := c, property := hc }))\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\nc : Composition (n + 2)\nhc : Composition.length c < n + 2\ncuniv : { val := c, property := hc } \u2208 univ\nv : Fin (n + 2) \u2192 F\n\u22a2 \u2191(ContinuousMultilinearMap.compAlongComposition\n (compContinuousLinearMap (removeZero p) \u2191(ContinuousLinearEquiv.symm i)) (\u2191{ val := c, property := hc })\n (leftInv (removeZero p) i (Composition.length \u2191{ val := c, property := hc })))\n v =\n \u2191(ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i))\n (\u2191{ val := c, property := hc }) (leftInv p i (Composition.length \u2191{ val := c, property := hc })))\n v\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 leftInv (removeZero p) i m = leftInv p i m\nc : Composition (n + 2)\nhc : Composition.length c < n + 2\ncuniv : { val := c, property := hc } \u2208 univ\nv : Fin (n + 2) \u2192 F\n\u22a2 \u2191(leftInv (removeZero p) i (Composition.length c))\n (applyComposition (compContinuousLinearMap (removeZero p) \u2191(ContinuousLinearEquiv.symm i)) c v) =\n \u2191(leftInv p i (Composition.length c))\n (applyComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) c v)\n[PROOFSTEP]\nsimp [IH _ hc]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 FormalMultilinearSeries.comp (leftInv p i) p = id \ud835\udd5c E\n[PROOFSTEP]\next (n v)\n[GOAL]\ncase h.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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn : \u2115\nv : Fin n \u2192 E\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p n) v = \u2191(id \ud835\udd5c E n) v\n[PROOFSTEP]\nmatch n with\n| 0 =>\n simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def, not_false_iff,\n zero_ne_one, comp_coeff_zero']\n| 1 =>\n simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,\n ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]\n|\nn +\n 2 =>\n have A :\n (Finset.univ : Finset (Composition (n + 2))) =\n {c | Composition.length c < n + 2}.toFinset \u222a {Composition.ones (n + 2)} :=\n by\n refine' Subset.antisymm (fun c _ => _) (subset_univ _)\n by_cases h : c.length < n + 2\n \u00b7 simp [h, Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\n \u00b7 simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)]\n have B :\n Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset {Composition.ones (n + 2)} := by\n simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\n have C :\n ((p.leftInv i (Composition.ones (n + 2)).length) fun j : Fin (Composition.ones n.succ.succ).length =>\n p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) =\n p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j :=\n by\n apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_\n exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr\n have D :\n (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =\n -\u2211 c : Composition (n + 2) in {c : Composition (n + 2) | c.length < n + 2}.toFinset,\n (p.leftInv i c.length) (p.applyComposition c v) :=\n by\n simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj, ContinuousMultilinearMap.sum_apply]\n convert\n (sum_toFinset_eq_subtype (fun c : Composition (n + 2) => c.length < n + 2)\n (fun c : Composition (n + 2) =>\n (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap (i.symm : F \u2192L[\ud835\udd5c] E)) c\n (p.leftInv i c.length))\n fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans\n _\n simp only [compContinuousLinearMap_applyComposition, ContinuousMultilinearMap.compAlongComposition_apply]\n congr\n ext c\n congr\n ext k\n simp [h, Function.comp]\n simp [FormalMultilinearSeries.comp, show n + 2 \u2260 1 by norm_num, A, Finset.sum_union B, applyComposition_ones, C, D,\n -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn : \u2115\nv : Fin 0 \u2192 E\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p 0) v = \u2191(id \ud835\udd5c E 0) v\n[PROOFSTEP]\nsimp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def, not_false_iff, zero_ne_one,\n comp_coeff_zero']\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn : \u2115\nv : Fin 1 \u2192 E\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p 1) v = \u2191(id \ud835\udd5c E 1) v\n[PROOFSTEP]\nsimp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,\n ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v\n[PROOFSTEP]\nhave A :\n (Finset.univ : Finset (Composition (n + 2))) =\n {c | Composition.length c < n + 2}.toFinset \u222a {Composition.ones (n + 2)} :=\n by\n refine' Subset.antisymm (fun c _ => _) (subset_univ _)\n by_cases h : c.length < n + 2\n \u00b7 simp [h, Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\n \u00b7 simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\n\u22a2 univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n[PROOFSTEP]\nrefine' Subset.antisymm (fun c _ => _) (subset_univ _)\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nc : Composition (n + 2)\nx\u271d : c \u2208 univ\n\u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n[PROOFSTEP]\nby_cases h : c.length < n + 2\n[GOAL]\ncase pos\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\ni : E \u2243L[\ud835\udd5c] F\nh\u271d : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nc : Composition (n + 2)\nx\u271d : c \u2208 univ\nh : Composition.length c < n + 2\n\u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n[PROOFSTEP]\nsimp [h, Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\n[GOAL]\ncase neg\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\ni : E \u2243L[\ud835\udd5c] F\nh\u271d : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nc : Composition (n + 2)\nx\u271d : c \u2208 univ\nh : \u00acComposition.length c < n + 2\n\u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n[PROOFSTEP]\nsimp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v\n[PROOFSTEP]\nhave B :\n Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset {Composition.ones (n + 2)} := by\n simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\n\u22a2 Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\n[PROOFSTEP]\nsimp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v\n[PROOFSTEP]\nhave C :\n ((p.leftInv i (Composition.ones (n + 2)).length) fun j : Fin (Composition.ones n.succ.succ).length =>\n p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) =\n p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j :=\n by\n apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_\n exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\n\u22a2 (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n[PROOFSTEP]\napply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nj : \u2115\nhj1 : j < Composition.length (Composition.ones (n + 2))\nhj2 : j < n + 2\n\u22a2 (\u2191(p 1) fun x =>\n v\n (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2)\n { val := j, isLt := hj1 })) =\n \u2191(p 1) fun x => v { val := j, isLt := hj2 }\n[PROOFSTEP]\nexact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nj : \u2115\nhj1 : j < Composition.length (Composition.ones (n + 2))\nhj2 : j < n + 2\nk : \u2115\nx\u271d\u00b9 x\u271d : k < 1\n\u22a2 v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) { val := j, isLt := hj1 }) =\n v { val := j, isLt := hj2 }\n[PROOFSTEP]\ncongr\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v\n[PROOFSTEP]\nhave D :\n (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =\n -\u2211 c : Composition (n + 2) in {c : Composition (n + 2) | c.length < n + 2}.toFinset,\n (p.leftInv i c.length) (p.applyComposition c v) :=\n by\n simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj, ContinuousMultilinearMap.sum_apply]\n convert\n (sum_toFinset_eq_subtype (fun c : Composition (n + 2) => c.length < n + 2)\n (fun c : Composition (n + 2) =>\n (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap (i.symm : F \u2192L[\ud835\udd5c] E)) c\n (p.leftInv i c.length))\n fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans\n _\n simp only [compContinuousLinearMap_applyComposition, ContinuousMultilinearMap.compAlongComposition_apply]\n congr\n ext c\n congr\n ext k\n simp [h, Function.comp]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) =\n -\u2211 c in Set.toFinset {c | Composition.length c < n + 2},\n \u2191(leftInv p i (Composition.length c)) (applyComposition p c v)\n[PROOFSTEP]\nsimp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj, ContinuousMultilinearMap.sum_apply]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 (\u2211 x : { c // Composition.length c < n + 2 },\n \u2191(ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) (\u2191x)\n (leftInv p i (Composition.length \u2191x)))\n fun j => \u2191(p 1) fun x => v j) =\n \u2211 x in Set.toFinset {c | Composition.length c < n + 2},\n \u2191(leftInv p i (Composition.length x)) (applyComposition p x v)\n[PROOFSTEP]\nconvert\n (sum_toFinset_eq_subtype (fun c : Composition (n + 2) => c.length < n + 2)\n (fun c : Composition (n + 2) =>\n (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap (i.symm : F \u2192L[\ud835\udd5c] E)) c\n (p.leftInv i c.length))\n fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans\n _\n[GOAL]\ncase convert_2\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 (\u2211 a in Set.toFinset {x | Composition.length x < n + 2},\n \u2191(ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) a\n (leftInv p i (Composition.length a)))\n fun j => \u2191(p 1) fun x => v j) =\n \u2211 x in Set.toFinset {c | Composition.length c < n + 2},\n \u2191(leftInv p i (Composition.length x)) (applyComposition p x v)\n[PROOFSTEP]\nsimp only [compContinuousLinearMap_applyComposition, ContinuousMultilinearMap.compAlongComposition_apply]\n[GOAL]\ncase convert_2\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 \u2211 x in Set.toFinset {x | Composition.length x < n + 2},\n \u2191(leftInv p i (Composition.length x))\n (applyComposition p x (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j)) =\n \u2211 x in Set.toFinset {x | Composition.length x < n + 2},\n \u2191(leftInv p i (Composition.length x)) (applyComposition p x v)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase convert_2.e_f\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\n\u22a2 (fun x =>\n \u2191(leftInv p i (Composition.length x))\n (applyComposition p x (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j))) =\n fun x => \u2191(leftInv p i (Composition.length x)) (applyComposition p x v)\n[PROOFSTEP]\next c\n[GOAL]\ncase convert_2.e_f.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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\nc : Composition (n + 2)\n\u22a2 \u2191(leftInv p i (Composition.length c))\n (applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j)) =\n \u2191(leftInv p i (Composition.length c)) (applyComposition p c v)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase convert_2.e_f.h.h.e_6.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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\nc : Composition (n + 2)\n\u22a2 applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j) = applyComposition p c v\n[PROOFSTEP]\next k\n[GOAL]\ncase convert_2.e_f.h.h.e_6.h.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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\nc : Composition (n + 2)\nk : Fin (Composition.length c)\n\u22a2 applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j) k = applyComposition p c v k\n[PROOFSTEP]\nsimp [h, Function.comp]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\nD :\n (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) =\n -\u2211 c in Set.toFinset {c | Composition.length c < n + 2},\n \u2191(leftInv p i (Composition.length c)) (applyComposition p c v)\n\u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v\n[PROOFSTEP]\nsimp [FormalMultilinearSeries.comp, show n + 2 \u2260 1 by norm_num, A, Finset.sum_union B, applyComposition_ones, C, D,\n -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 E\nA : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)}\nB : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)}\nC :\n (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j =>\n \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) =\n \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j\nD :\n (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) =\n -\u2211 c in Set.toFinset {c | Composition.length c < n + 2},\n \u2191(leftInv p i (Composition.length c)) (applyComposition p c v)\n\u22a2 n + 2 \u2260 1\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 rightInv p i 0 = 0\n[PROOFSTEP]\nrw [rightInv]\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 rightInv p i 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c F E)) \u2191(ContinuousLinearEquiv.symm i)\n[PROOFSTEP]\nrw [rightInv]\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\ni : E \u2243L[\ud835\udd5c] F\n\u22a2 rightInv (removeZero p) i = rightInv p i\n[PROOFSTEP]\next1 n\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\n\u22a2 rightInv (removeZero p) i n = rightInv p i n\n[PROOFSTEP]\ninduction' n using Nat.strongRec' with n IH\n[GOAL]\ncase h.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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 rightInv (removeZero p) i n = rightInv p i n\n[PROOFSTEP]\nmatch n with\n| 0 => simp only [rightInv_coeff_zero]\n| 1 => simp only [rightInv_coeff_one]\n| n + 2 =>\n simp only [rightInv, neg_inj]\n rw [removeZero_comp_of_pos _ _ (add_pos_of_nonneg_of_pos n.zero_le zero_lt_two)]\n congr (config := { closePost := false }) 2 with k\n by_cases hk : k < n + 2 <;> simp [hk, IH]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < 0 \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 rightInv (removeZero p) i 0 = rightInv p i 0\n[PROOFSTEP]\nsimp only [rightInv_coeff_zero]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nIH : \u2200 (m : \u2115), m < 1 \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 rightInv (removeZero p) i 1 = rightInv p i 1\n[PROOFSTEP]\nsimp only [rightInv_coeff_one]\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 rightInv (removeZero p) i (n + 2) = rightInv p i (n + 2)\n[PROOFSTEP]\nsimp only [rightInv, neg_inj]\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (FormalMultilinearSeries.comp (removeZero p) (fun k => if k < n + 2 then rightInv (removeZero p) i k else 0)\n (n + 2)) =\n ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2))\n[PROOFSTEP]\nrw [removeZero_comp_of_pos _ _ (add_pos_of_nonneg_of_pos n.zero_le zero_lt_two)]\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\n\u22a2 ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv (removeZero p) i k else 0) (n + 2)) =\n ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2))\n[PROOFSTEP]\ncongr (config := { closePost := false }) 2 with k\n[GOAL]\ncase e_f.e_p.h.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\nk : \u2115\nx\u271d : Fin k \u2192 F\n\u22a2 \u2191(if k < n + 2 then rightInv (removeZero p) i k else 0) x\u271d = \u2191(if k < n + 2 then rightInv p i k else 0) x\u271d\n[PROOFSTEP]\nby_cases hk : k < n + 2\n[GOAL]\ncase pos\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\nk : \u2115\nx\u271d : Fin k \u2192 F\nhk : k < n + 2\n\u22a2 \u2191(if k < n + 2 then rightInv (removeZero p) i k else 0) x\u271d = \u2191(if k < n + 2 then rightInv p i k else 0) x\u271d\n[PROOFSTEP]\nsimp [hk, IH]\n[GOAL]\ncase neg\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nIH : \u2200 (m : \u2115), m < n + 2 \u2192 rightInv (removeZero p) i m = rightInv p i m\nk : \u2115\nx\u271d : Fin k \u2192 F\nhk : \u00ack < n + 2\n\u22a2 \u2191(if k < n + 2 then rightInv (removeZero p) i k else 0) x\u271d = \u2191(if k < n + 2 then rightInv p i k else 0) x\u271d\n[PROOFSTEP]\nsimp [hk, IH]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p q n) v =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c}, \u2191(p (Composition.length c)) (applyComposition q c v) +\n \u2191(p 1) fun x => \u2191(q n) v\n[PROOFSTEP]\nhave A : (Finset.univ : Finset (Composition n)) = {c | 1 < Composition.length c}.toFinset \u222a {Composition.single n hn} :=\n by\n refine' Subset.antisymm (fun c _ => _) (subset_univ _)\n by_cases h : 1 < c.length\n \u00b7 simp [h, Set.mem_toFinset (s := {c | 1 < Composition.length c})]\n \u00b7 have : c.length = 1 := by refine' (eq_iff_le_not_lt.2 \u27e8_, h\u27e9).symm; exact c.length_pos_of_pos hn\n rw [\u2190 Composition.eq_single_iff_length hn] at this \n simp [this]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\n\u22a2 univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nrefine' Subset.antisymm (fun c _ => _) (subset_univ _)\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\n\u22a2 c \u2208 Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nby_cases h : 1 < c.length\n[GOAL]\ncase pos\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : 1 < Composition.length c\n\u22a2 c \u2208 Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nsimp [h, Set.mem_toFinset (s := {c | 1 < Composition.length c})]\n[GOAL]\ncase neg\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : \u00ac1 < Composition.length c\n\u22a2 c \u2208 Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nhave : c.length = 1 := by refine' (eq_iff_le_not_lt.2 \u27e8_, h\u27e9).symm; exact c.length_pos_of_pos 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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : \u00ac1 < Composition.length c\n\u22a2 Composition.length c = 1\n[PROOFSTEP]\nrefine' (eq_iff_le_not_lt.2 \u27e8_, h\u27e9).symm\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : \u00ac1 < Composition.length c\n\u22a2 1 \u2264 Composition.length c\n[PROOFSTEP]\nexact c.length_pos_of_pos hn\n[GOAL]\ncase neg\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : \u00ac1 < Composition.length c\nthis : Composition.length c = 1\n\u22a2 c \u2208 Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nrw [\u2190 Composition.eq_single_iff_length hn] at this \n[GOAL]\ncase neg\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nc : Composition n\nx\u271d : c \u2208 univ\nh : \u00ac1 < Composition.length c\nthis : c = Composition.single n hn\n\u22a2 c \u2208 Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n[PROOFSTEP]\nsimp [this]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n\u22a2 \u2191(FormalMultilinearSeries.comp p q n) v =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c}, \u2191(p (Composition.length c)) (applyComposition q c v) +\n \u2191(p 1) fun x => \u2191(q n) v\n[PROOFSTEP]\nhave B : Disjoint ({c | 1 < Composition.length c} : Set (Composition n)).toFinset {Composition.single n hn} := by\n simp [Set.mem_toFinset (s := {c | 1 < Composition.length c})]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\n\u22a2 Disjoint (Set.toFinset {c | 1 < Composition.length c}) {Composition.single n hn}\n[PROOFSTEP]\nsimp [Set.mem_toFinset (s := {c | 1 < Composition.length c})]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\nB : Disjoint (Set.toFinset {c | 1 < Composition.length c}) {Composition.single n hn}\n\u22a2 \u2191(FormalMultilinearSeries.comp p q n) v =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c}, \u2191(p (Composition.length c)) (applyComposition q c v) +\n \u2191(p 1) fun x => \u2191(q n) v\n[PROOFSTEP]\nhave C :\n p (Composition.single n hn).length (q.applyComposition (Composition.single n hn) v) = p 1 fun _ : Fin 1 => q n v :=\n by\n apply p.congr (Composition.single_length hn) fun j hj1 _ => ?_\n simp [applyComposition_single]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\nB : Disjoint (Set.toFinset {c | 1 < Composition.length c}) {Composition.single n hn}\n\u22a2 \u2191(p (Composition.length (Composition.single n hn))) (applyComposition q (Composition.single n hn) v) =\n \u2191(p 1) fun x => \u2191(q n) v\n[PROOFSTEP]\napply p.congr (Composition.single_length hn) fun j hj1 _ => ?_\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\nB : Disjoint (Set.toFinset {c | 1 < Composition.length c}) {Composition.single n hn}\nj : \u2115\nhj1 : j < Composition.length (Composition.single n hn)\nx\u271d : j < 1\n\u22a2 applyComposition q (Composition.single n hn) v { val := j, isLt := hj1 } = \u2191(q n) v\n[PROOFSTEP]\nsimp [applyComposition_single]\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\nn : \u2115\nhn : 0 < n\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c F E\nv : Fin n \u2192 F\nA : univ = Set.toFinset {c | 1 < Composition.length c} \u222a {Composition.single n hn}\nB : Disjoint (Set.toFinset {c | 1 < Composition.length c}) {Composition.single n hn}\nC :\n \u2191(p (Composition.length (Composition.single n hn))) (applyComposition q (Composition.single n hn) v) =\n \u2191(p 1) fun x => \u2191(q n) v\n\u22a2 \u2191(FormalMultilinearSeries.comp p q n) v =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c}, \u2191(p (Composition.length c)) (applyComposition q c v) +\n \u2191(p 1) fun x => \u2191(q n) v\n[PROOFSTEP]\nsimp [FormalMultilinearSeries.comp, A, Finset.sum_union B, C, -Set.toFinset_setOf, -add_right_inj,\n -Composition.single_length]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\n\u22a2 \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u2191(p (Composition.length c)) (applyComposition (fun k => if k < n + 2 then rightInv p i k else 0) c v) =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u2191(p (Composition.length c)) (applyComposition (rightInv p i) c v)\n[PROOFSTEP]\nhave N : 0 < n + 2 := by norm_num\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\n\u22a2 0 < n + 2\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\n\u22a2 \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u2191(p (Composition.length c)) (applyComposition (fun k => if k < n + 2 then rightInv p i k else 0) c v) =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u2191(p (Composition.length c)) (applyComposition (rightInv p i) c v)\n[PROOFSTEP]\nrefine' sum_congr rfl fun c hc => p.congr rfl fun j hj1 hj2 => _\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\nc : Composition (n + 2)\nhc : c \u2208 Set.toFinset {c | 1 < Composition.length c}\nj : \u2115\nhj1 hj2 : j < Composition.length c\n\u22a2 applyComposition (fun k => if k < n + 2 then rightInv p i k else 0) c v { val := j, isLt := hj1 } =\n applyComposition (rightInv p i) c v { val := j, isLt := hj2 }\n[PROOFSTEP]\nhave : \u2200 k, c.blocksFun k < n + 2 :=\n by\n simp only [Set.mem_toFinset (s := {c : Composition (n + 2) | 1 < c.length}), Set.mem_setOf_eq] at hc \n simp [\u2190 Composition.ne_single_iff N, Composition.eq_single_iff_length, ne_of_gt hc]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\nc : Composition (n + 2)\nhc : c \u2208 Set.toFinset {c | 1 < Composition.length c}\nj : \u2115\nhj1 hj2 : j < Composition.length c\n\u22a2 \u2200 (k : Fin (Composition.length c)), Composition.blocksFun c k < n + 2\n[PROOFSTEP]\nsimp only [Set.mem_toFinset (s := {c : Composition (n + 2) | 1 < c.length}), Set.mem_setOf_eq] at hc \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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\nc : Composition (n + 2)\nj : \u2115\nhj1 hj2 : j < Composition.length c\nhc : 1 < Composition.length c\n\u22a2 \u2200 (k : Fin (Composition.length c)), Composition.blocksFun c k < n + 2\n[PROOFSTEP]\nsimp [\u2190 Composition.ne_single_iff N, Composition.eq_single_iff_length, ne_of_gt hc]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\nc : Composition (n + 2)\nhc : c \u2208 Set.toFinset {c | 1 < Composition.length c}\nj : \u2115\nhj1 hj2 : j < Composition.length c\nthis : \u2200 (k : Fin (Composition.length c)), Composition.blocksFun c k < n + 2\n\u22a2 applyComposition (fun k => if k < n + 2 then rightInv p i k else 0) c v { val := j, isLt := hj1 } =\n applyComposition (rightInv p i) c v { val := j, isLt := hj2 }\n[PROOFSTEP]\nsimp [applyComposition, this]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 FormalMultilinearSeries.comp p (rightInv p i) = id \ud835\udd5c F\n[PROOFSTEP]\next (n v)\n[GOAL]\ncase h.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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn : \u2115\nv : Fin n \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p (rightInv p i) n) v = \u2191(id \ud835\udd5c F n) v\n[PROOFSTEP]\nmatch n with\n| 0 =>\n simp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def, not_false_iff, zero_ne_one,\n comp_coeff_zero']\n| 1 =>\n simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply, id_apply_one,\n ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply]\n| n + 2 =>\n have N : 0 < n + 2 := by norm_num\n simp [comp_rightInv_aux1 N, h, rightInv, lt_irrefl n, show n + 2 \u2260 1 by norm_num, \u2190 sub_eq_add_neg, sub_eq_zero,\n comp_rightInv_aux2, -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn : \u2115\nv : Fin 0 \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p (rightInv p i) 0) v = \u2191(id \ud835\udd5c F 0) v\n[PROOFSTEP]\nsimp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def, not_false_iff, zero_ne_one,\n comp_coeff_zero']\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn : \u2115\nv : Fin 1 \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p (rightInv p i) 1) v = \u2191(id \ud835\udd5c F 1) v\n[PROOFSTEP]\nsimp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply, id_apply_one,\n ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p (rightInv p i) (n + 2)) v = \u2191(id \ud835\udd5c F (n + 2)) v\n[PROOFSTEP]\nhave N : 0 < n + 2 := by norm_num\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 F\n\u22a2 0 < n + 2\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\n\u22a2 \u2191(FormalMultilinearSeries.comp p (rightInv p i) (n + 2)) v = \u2191(id \ud835\udd5c F (n + 2)) v\n[PROOFSTEP]\nsimp [comp_rightInv_aux1 N, h, rightInv, lt_irrefl n, show n + 2 \u2260 1 by norm_num, \u2190 sub_eq_add_neg, sub_eq_zero,\n comp_rightInv_aux2, -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\nn\u271d n : \u2115\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\n\u22a2 n + 2 \u2260 1\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nhn : 2 \u2264 n\n\u22a2 rightInv p i n =\n -ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\n[PROOFSTEP]\nmatch n with\n| 0 => exact False.elim (zero_lt_two.not_le hn)\n| 1 => exact False.elim (one_lt_two.not_le hn)\n| n + 2 =>\n simp only [rightInv, neg_inj]\n congr (config := { closePost := false }) 1\n ext v\n have N : 0 < n + 2 := by norm_num\n have : ((p 1) fun i : Fin 1 => 0) = 0 := ContinuousMultilinearMap.map_zero _\n simp [comp_rightInv_aux1 N, lt_irrefl n, this, comp_rightInv_aux2, -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nhn : 2 \u2264 0\n\u22a2 rightInv p i 0 =\n -ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\n[PROOFSTEP]\nexact False.elim (zero_lt_two.not_le 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\ni : E \u2243L[\ud835\udd5c] F\nn : \u2115\nhn : 2 \u2264 1\n\u22a2 rightInv p i 1 =\n -ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\n[PROOFSTEP]\nexact False.elim (one_lt_two.not_le 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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\n\u22a2 rightInv p i (n + 2) =\n -ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\n[PROOFSTEP]\nsimp only [rightInv, neg_inj]\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\n\u22a2 ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2)) =\n ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\n[PROOFSTEP]\ncongr (config := { closePost := false }) 1\n[GOAL]\ncase e_f\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\n\u22a2 FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2) =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c\n[PROOFSTEP]\next v\n[GOAL]\ncase e_f.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\nv : Fin (n + 2) \u2192 F\n\u22a2 \u2191(FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2)) v =\n \u2191(\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c) v\n[PROOFSTEP]\nhave N : 0 < n + 2 := by norm_num\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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\nv : Fin (n + 2) \u2192 F\n\u22a2 0 < n + 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_f.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\n\u22a2 \u2191(FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2)) v =\n \u2191(\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c) v\n[PROOFSTEP]\nhave : ((p 1) fun i : Fin 1 => 0) = 0 := ContinuousMultilinearMap.map_zero _\n[GOAL]\ncase e_f.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\ni : E \u2243L[\ud835\udd5c] F\nn\u271d n : \u2115\nhn : 2 \u2264 n + 2\nv : Fin (n + 2) \u2192 F\nN : 0 < n + 2\nthis : (\u2191(p 1) fun i => 0) = 0\n\u22a2 \u2191(FormalMultilinearSeries.comp p (fun k => if k < n + 2 then rightInv p i k else 0) (n + 2)) v =\n \u2191(\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c) v\n[PROOFSTEP]\nsimp [comp_rightInv_aux1 N, lt_irrefl n, this, comp_rightInv_aux2, -Set.toFinset_setOf]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 leftInv p i = FormalMultilinearSeries.comp (leftInv p i) (id \ud835\udd5c F)\n[PROOFSTEP]\nsimp\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 FormalMultilinearSeries.comp (leftInv p i) (id \ud835\udd5c F) =\n FormalMultilinearSeries.comp (leftInv p i) (FormalMultilinearSeries.comp p (rightInv p i))\n[PROOFSTEP]\nrw [comp_rightInv p i h h0]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 FormalMultilinearSeries.comp (leftInv p i) (FormalMultilinearSeries.comp p (rightInv p i)) =\n FormalMultilinearSeries.comp (FormalMultilinearSeries.comp (leftInv p i) p) (rightInv p i)\n[PROOFSTEP]\nrw [comp_assoc]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 FormalMultilinearSeries.comp (FormalMultilinearSeries.comp (leftInv p i) p) (rightInv p i) =\n FormalMultilinearSeries.comp (id \ud835\udd5c E) (rightInv p i)\n[PROOFSTEP]\nrw [leftInv_comp p i h]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\nh0 : p 0 = 0\n\u22a2 FormalMultilinearSeries.comp (id \ud835\udd5c E) (rightInv p i) = rightInv p i\n[PROOFSTEP]\nsimp\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 leftInv p i = leftInv (removeZero p) i\n[PROOFSTEP]\nrw [leftInv_removeZero]\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 leftInv (removeZero p) i = rightInv (removeZero p) i\n[PROOFSTEP]\napply leftInv_eq_rightInv_aux\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 removeZero p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h0\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 removeZero p 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 p (0 + 1) = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n[PROOFSTEP]\nexact h\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\ni : E \u2243L[\ud835\udd5c] F\nh : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i\n\u22a2 rightInv (removeZero p) i = rightInv p i\n[PROOFSTEP]\nrw [rightInv_removeZero]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length c * \u220f j : Fin (Composition.length c), p (Composition.blocksFun c j) =\n \u2211 k in Ico 2 (n + 1),\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u220f j : Fin (Composition.length c), r * (a ^ Composition.blocksFun c j * p (Composition.blocksFun c j))\n[PROOFSTEP]\nsimp_rw [mul_sum]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x in Ico 2 (n + 1),\n \u2211 x_1 in Set.toFinset {c | 1 < Composition.length c},\n a ^ x * (r ^ Composition.length x_1 * \u220f j : Fin (Composition.length x_1), p (Composition.blocksFun x_1 j)) =\n \u2211 k in Ico 2 (n + 1),\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u220f j : Fin (Composition.length c), r * (a ^ Composition.blocksFun c j * p (Composition.blocksFun c j))\n[PROOFSTEP]\napply sum_congr rfl fun k _ => ?_\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nx\u271d : k \u2208 Ico 2 (n + 1)\n\u22a2 \u2211 x in Set.toFinset {c | 1 < Composition.length c},\n a ^ k * (r ^ Composition.length x * \u220f j : Fin (Composition.length x), p (Composition.blocksFun x j)) =\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u220f j : Fin (Composition.length c), r * (a ^ Composition.blocksFun c j * p (Composition.blocksFun c j))\n[PROOFSTEP]\napply sum_congr rfl fun c _ => ?_\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nx\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\nx\u271d : c \u2208 Set.toFinset {c | 1 < Composition.length c}\n\u22a2 a ^ k * (r ^ Composition.length c * \u220f j : Fin (Composition.length c), p (Composition.blocksFun c j)) =\n \u220f j : Fin (Composition.length c), r * (a ^ Composition.blocksFun c j * p (Composition.blocksFun c j))\n[PROOFSTEP]\nrw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, Composition.sum_blocksFun, prod_const, card_fin]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nx\u271d\u00b9 : k \u2208 Ico 2 (n + 1)\nc : Composition k\nx\u271d : c \u2208 Set.toFinset {c | 1 < Composition.length c}\n\u22a2 a ^ k * (r ^ Composition.length c * \u220f j : Fin (Composition.length c), p (Composition.blocksFun c j)) =\n r ^ Composition.length c * (a ^ k * \u220f x : Fin (Composition.length c), p (Composition.blocksFun c x))\n[PROOFSTEP]\nring\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 k in Ico 2 (n + 1),\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n \u220f j : Fin (Composition.length c), r * (a ^ Composition.blocksFun c j * p (Composition.blocksFun c j)) \u2264\n \u2211 d in compPartialSumTarget 2 (n + 1) n,\n \u220f j : Fin (Composition.length d.snd), r * (a ^ Composition.blocksFun d.snd j * p (Composition.blocksFun d.snd j))\n[PROOFSTEP]\nrw [sum_sigma']\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 x in Finset.sigma (Ico 2 (n + 1)) fun k => Set.toFinset {c | 1 < Composition.length c},\n \u220f j : Fin (Composition.length x.snd),\n r * (a ^ Composition.blocksFun x.snd j * p (Composition.blocksFun x.snd j)) \u2264\n \u2211 d in compPartialSumTarget 2 (n + 1) n,\n \u220f j : Fin (Composition.length d.snd), r * (a ^ Composition.blocksFun d.snd j * p (Composition.blocksFun d.snd j))\n[PROOFSTEP]\nrefine'\n sum_le_sum_of_subset_of_nonneg _ fun x _ _ =>\n prod_nonneg fun j _ => mul_nonneg hr (mul_nonneg (pow_nonneg ha _) (hp _))\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 (Finset.sigma (Ico 2 (n + 1)) fun k => Set.toFinset {c | 1 < Composition.length c}) \u2286 compPartialSumTarget 2 (n + 1) n\n[PROOFSTEP]\nrintro \u27e8k, c\u27e9 hd\n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : { fst := k, snd := c } \u2208 Finset.sigma (Ico 2 (n + 1)) fun k => Set.toFinset {c | 1 < Composition.length c}\n\u22a2 { fst := k, snd := c } \u2208 compPartialSumTarget 2 (n + 1) n\n[PROOFSTEP]\nsimp only [Set.mem_toFinset (s := {c | 1 < Composition.length c}), mem_Ico, mem_sigma, Set.mem_setOf_eq] at hd \n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\n\u22a2 { fst := k, snd := c } \u2208 compPartialSumTarget 2 (n + 1) n\n[PROOFSTEP]\nsimp only [mem_compPartialSumTarget_iff]\n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\n\u22a2 2 \u2264 Composition.length c \u2227\n Composition.length c < n + 1 \u2227\n \u2200 (j : Fin (Composition.length { fst := k, snd := c }.snd)), Composition.blocksFun c j < n\n[PROOFSTEP]\nrefine' \u27e8hd.2, c.length_le.trans_lt hd.1.2, fun j => _\u27e9\n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\nj : Fin (Composition.length { fst := k, snd := c }.snd)\n\u22a2 Composition.blocksFun c j < n\n[PROOFSTEP]\nhave : c \u2260 Composition.single k (zero_lt_two.trans_le hd.1.1) := by\n simp [Composition.eq_single_iff_length, ne_of_gt hd.2]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\nj : Fin (Composition.length { fst := k, snd := c }.snd)\n\u22a2 c \u2260 Composition.single k (_ : 0 < k)\n[PROOFSTEP]\nsimp [Composition.eq_single_iff_length, ne_of_gt hd.2]\n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\nj : Fin (Composition.length { fst := k, snd := c }.snd)\nthis : c \u2260 Composition.single k (_ : 0 < k)\n\u22a2 Composition.blocksFun c j < n\n[PROOFSTEP]\nrw [Composition.ne_single_iff] at this \n[GOAL]\ncase mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nc : Composition k\nhd : (2 \u2264 k \u2227 k < n + 1) \u2227 1 < Composition.length c\nj : Fin (Composition.length { fst := k, snd := c }.snd)\nthis : \u2200 (i : Fin (Composition.length c)), Composition.blocksFun c i < k\n\u22a2 Composition.blocksFun c j < n\n[PROOFSTEP]\nexact (this j).trans_le (Nat.lt_succ_iff.mp hd.1.2)\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 d in compPartialSumTarget 2 (n + 1) n,\n \u220f j : Fin (Composition.length d.snd),\n r * (a ^ Composition.blocksFun d.snd j * p (Composition.blocksFun d.snd j)) =\n \u2211 e in compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ Sigma.snd e j * p (Sigma.snd e j))\n[PROOFSTEP]\nsymm\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 e in compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ Sigma.snd e j * p (Sigma.snd e j)) =\n \u2211 d in compPartialSumTarget 2 (n + 1) n,\n \u220f j : Fin (Composition.length d.snd), r * (a ^ Composition.blocksFun d.snd j * p (Composition.blocksFun d.snd j))\n[PROOFSTEP]\napply compChangeOfVariables_sum\n[GOAL]\ncase 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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2200 (e : (n : \u2115) \u00d7 (Fin n \u2192 \u2115)) (he : e \u2208 compPartialSumSource 2 (n + 1) n),\n \u220f j : Fin e.fst, r * (a ^ Sigma.snd e j * p (Sigma.snd e j)) =\n \u220f j : Fin (Composition.length (compChangeOfVariables 2 (n + 1) n e he).snd),\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n e he).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n e he).snd j))\n[PROOFSTEP]\nrintro \u27e8k, blocks_fun\u27e9 H\n[GOAL]\ncase h.mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\n\u22a2 \u220f j : Fin { fst := k, snd := blocks_fun }.fst,\n r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)) =\n \u220f j : Fin (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd),\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j))\n[PROOFSTEP]\nhave K : (compChangeOfVariables 2 (n + 1) n \u27e8k, blocks_fun\u27e9 H).snd.length = k := by simp\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\n\u22a2 Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mk\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 \u220f j : Fin { fst := k, snd := blocks_fun }.fst,\n r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)) =\n \u220f j : Fin (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd),\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.mk.h.e_2.e_n\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 { fst := k, snd := blocks_fun }.fst =\n Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n[PROOFSTEP]\ntry rw [K]\n[GOAL]\ncase h.mk.h.e_2.e_n\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 { fst := k, snd := blocks_fun }.fst =\n Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n[PROOFSTEP]\nrw [K]\n[GOAL]\ncase h.mk.h.e_4.e_1\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 Fin { fst := k, snd := blocks_fun }.fst =\n Fin (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd)\n[PROOFSTEP]\ntry rw [K]\n[GOAL]\ncase h.mk.h.e_4.e_1\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 Fin { fst := k, snd := blocks_fun }.fst =\n Fin (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd)\n[PROOFSTEP]\nrw [K]\n[GOAL]\ncase h.mk.h.e_4.e_2\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 HEq (Fin.fintype { fst := k, snd := blocks_fun }.fst)\n (Fin.fintype (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd))\n[PROOFSTEP]\ntry rw [K]\n[GOAL]\ncase h.mk.h.e_4.e_2\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 HEq (Fin.fintype { fst := k, snd := blocks_fun }.fst)\n (Fin.fintype (Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd))\n[PROOFSTEP]\nrw [K]\n[GOAL]\ncase h.mk.h.e_5\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 HEq (fun j => r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)))\n fun j =>\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j))\n[PROOFSTEP]\ntry rw [K]\n[GOAL]\ncase h.mk.h.e_5\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 HEq (fun j => r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)))\n fun j =>\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j))\n[PROOFSTEP]\nrw [K]\n[GOAL]\ncase h.mk.h.e_5\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 HEq (fun j => r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)))\n fun j =>\n r *\n (a ^ Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j *\n p (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd j))\n[PROOFSTEP]\nrw [Fin.heq_fun_iff K.symm]\n[GOAL]\ncase h.mk.h.e_5\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\n\u22a2 \u2200 (i : Fin k),\n r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } i * p (Sigma.snd { fst := k, snd := blocks_fun } i)) =\n r *\n (a ^\n Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n { val := \u2191i,\n isLt :=\n (_ :\n \u2191i <\n Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd) } *\n p\n (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n { val := \u2191i,\n isLt :=\n (_ :\n \u2191i <\n Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd) }))\n[PROOFSTEP]\nintro j\n[GOAL]\ncase h.mk.h.e_5\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nk : \u2115\nblocks_fun : Fin k \u2192 \u2115\nH : { fst := k, snd := blocks_fun } \u2208 compPartialSumSource 2 (n + 1) n\nK : Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd = k\nj : Fin k\n\u22a2 r * (a ^ Sigma.snd { fst := k, snd := blocks_fun } j * p (Sigma.snd { fst := k, snd := blocks_fun } j)) =\n r *\n (a ^\n Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n { val := \u2191j,\n isLt :=\n (_ :\n \u2191j < Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd) } *\n p\n (Composition.blocksFun (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd\n { val := \u2191j,\n isLt :=\n (_ :\n \u2191j < Composition.length (compChangeOfVariables 2 (n + 1) n { fst := k, snd := blocks_fun } H).snd) }))\n[PROOFSTEP]\nrw [compChangeOfVariables_blocksFun]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 e in compPartialSumSource 2 (n + 1) n, \u220f j : Fin e.fst, r * (a ^ Sigma.snd e j * p (Sigma.snd e j)) =\n \u2211 j in Ico 2 (n + 1), r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\nrw [compPartialSumSource, \u2190\n sum_sigma' (Ico 2 (n + 1)) (fun k : \u2115 => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k \u2192 \u2115)))\n (fun n e => \u220f j : Fin n, r * (a ^ e j * p (e j)))]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\n\u22a2 \u2211 a_1 in Ico 2 (n + 1), \u2211 s in Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin a_1, r * (a ^ s j * p (s j)) =\n \u2211 j in Ico 2 (n + 1), r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\napply sum_congr rfl fun j _ => ?_\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\nx\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2211 s in Fintype.piFinset fun x => Ico 1 n, \u220f j : Fin j, r * (a ^ s j * p (s j)) =\n r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\nsimp only [\u2190 @MultilinearMap.mkPiAlgebra_apply \u211d (Fin j) _ \u211d]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\nx\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 (\u2211 x in Fintype.piFinset fun x => Ico 1 n,\n \u2191(MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun j => r * (a ^ x j * p (x j))) =\n r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\nsimp only [\u2190 MultilinearMap.map_sum_finset (MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun _ (m : \u2115) => r * (a ^ m * p m)]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\nx\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 (\u2191(MultilinearMap.mkPiAlgebra \u211d (Fin j) \u211d) fun i => \u2211 j in Ico 1 n, r * (a ^ j * p j)) =\n r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\nsimp only [MultilinearMap.mkPiAlgebra_apply]\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\nn : \u2115\np : \u2115 \u2192 \u211d\nhp : \u2200 (k : \u2115), 0 \u2264 p k\nr a : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nj : \u2115\nx\u271d : j \u2208 Ico 2 (n + 1)\n\u22a2 \u220f i : Fin j, \u2211 j in Ico 1 n, r * (a ^ j * p j) = r ^ j * (\u2211 k in Ico 1 n, a ^ k * p k) ^ j\n[PROOFSTEP]\nsimp [prod_const, \u2190 mul_sum, mul_pow]\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 \u2211 k in Ico 1 (n + 1), a ^ k * \u2016rightInv p i k\u2016 = a * I + \u2211 k in Ico 2 (n + 1), a ^ k * \u2016rightInv p i k\u2016\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.norm_map, pow_one, rightInv_coeff_one,\n show Ico (1 : \u2115) 2 = {1} from Nat.Ico_succ_singleton 1, sum_singleton, \u2190 sum_Ico_consecutive _ one_le_two 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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 a * I + \u2211 k in Ico 2 (n + 1), a ^ k * \u2016rightInv p i k\u2016 =\n a * I +\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2016ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 \u2211 k in Ico 2 (n + 1), a ^ k * \u2016rightInv p i k\u2016 =\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2016ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\u2016\n[PROOFSTEP]\napply sum_congr rfl fun j hj => ?_\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\n\u22a2 a ^ j * \u2016rightInv p i j\u2016 =\n a ^ j *\n \u2016ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\u2016\n[PROOFSTEP]\nrw [rightInv_coeff _ _ _ (mem_Ico.1 hj).1, norm_neg]\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 a * I +\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2016ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\u2016 \u2264\n a * \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 +\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n (I *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c *\n \u220f j : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j)\u2016)\n[PROOFSTEP]\napply_rules [add_le_add, le_refl, sum_le_sum fun j hj => ?_, mul_le_mul_of_nonneg_left, pow_nonneg, ha]\n[GOAL]\ncase 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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2016ContinuousLinearMap.compContinuousMultilinearMap (\u2191(ContinuousLinearEquiv.symm i))\n (\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c)\u2016 \u2264\n I *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply (ContinuousLinearMap.norm_compContinuousMultilinearMap_le _ _).trans\n[GOAL]\ncase 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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 *\n \u2016\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c\u2016 \u2264\n I *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2016\u2211 c in Set.toFinset {c | 1 < Composition.length c}, compAlongComposition p (rightInv p i) c\u2016 \u2264\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply (norm_sum_le _ _).trans\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\n\u22a2 \u2211 i_1 in Set.toFinset {c | 1 < Composition.length c}, \u2016compAlongComposition p (rightInv p i) i_1\u2016 \u2264\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply sum_le_sum fun c _ => ?_\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\nc : Composition j\nx\u271d : c \u2208 Set.toFinset {c | 1 < Composition.length c}\n\u22a2 \u2016compAlongComposition p (rightInv p i) c\u2016 \u2264\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply (compAlongComposition_norm _ _ _).trans\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\nc : Composition j\nx\u271d : c \u2208 Set.toFinset {c | 1 < Composition.length c}\n\u22a2 \u2016p (Composition.length c)\u2016 * \u220f i_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c i_1)\u2016 \u2264\n C * r ^ Composition.length c * \u220f j_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j_1)\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right (hp _)\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\nj : \u2115\nhj : j \u2208 Ico 2 (n + 1)\nc : Composition j\nx\u271d : c \u2208 Set.toFinset {c | 1 < Composition.length c}\n\u22a2 0 \u2264 \u220f i_1 : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c i_1)\u2016\n[PROOFSTEP]\nexact prod_nonneg fun j _ => norm_nonneg _\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 a * \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 +\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n (I *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n C * r ^ Composition.length c *\n \u220f j : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j)\u2016) =\n I * a +\n I * C *\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length c * \u220f j : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j)\u2016\n[PROOFSTEP]\nsimp_rw [mul_assoc C, \u2190 mul_sum, \u2190 mul_assoc, mul_comm _ \u2016(i.symm : F \u2192L[\ud835\udd5c] E)\u2016, mul_assoc, \u2190 mul_sum, \u2190 mul_assoc,\n mul_comm _ C, mul_assoc, \u2190 mul_sum]\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 * a +\n \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 *\n (C *\n \u2211 x in Ico 2 (n + 1),\n a ^ x *\n \u2211 x_1 in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length x_1 *\n \u220f x_2 : Fin (Composition.length x_1), \u2016rightInv p i (Composition.blocksFun x_1 x_2)\u2016) =\n \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 * a +\n C *\n (\u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 *\n \u2211 x in Ico 2 (n + 1),\n a ^ x *\n \u2211 x_1 in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length x_1 *\n \u220f x_2 : Fin (Composition.length x_1), \u2016rightInv p i (Composition.blocksFun x_1 x_2)\u2016)\n[PROOFSTEP]\nring\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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 I * a +\n I * C *\n \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length c * \u220f j : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j)\u2016 \u2264\n I * a + I * C * \u2211 k in Ico 2 (n + 1), (r * \u2211 j in Ico 1 n, a ^ j * \u2016rightInv p i j\u2016) ^ k\n[PROOFSTEP]\napply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, norm_nonneg, hC, mul_nonneg]\n[GOAL]\ncase h\u2082.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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 \u2211 k in Ico 2 (n + 1),\n a ^ k *\n \u2211 c in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length c * \u220f j : Fin (Composition.length c), \u2016rightInv p i (Composition.blocksFun c j)\u2016 \u2264\n \u2211 k in Ico 2 (n + 1), (r * \u2211 j in Ico 1 n, a ^ j * \u2016rightInv p i j\u2016) ^ k\n[PROOFSTEP]\nsimp_rw [mul_pow]\n[GOAL]\ncase h\u2082.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\nn : \u2115\nhn : 2 \u2264 n + 1\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nr a C : \u211d\nhr : 0 \u2264 r\nha : 0 \u2264 a\nhC : 0 \u2264 C\nhp : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 \u2211 x in Ico 2 (n + 1),\n a ^ x *\n \u2211 x_1 in Set.toFinset {c | 1 < Composition.length c},\n r ^ Composition.length x_1 *\n \u220f x_2 : Fin (Composition.length x_1), \u2016rightInv p i (Composition.blocksFun x_1 x_2)\u2016 \u2264\n \u2211 x in Ico 2 (n + 1), r ^ x * (\u2211 x in Ico 1 n, a ^ x * \u2016rightInv p i x\u2016) ^ x\n[PROOFSTEP]\napply radius_right_inv_pos_of_radius_pos_aux1 n (fun k => \u2016p.rightInv i k\u2016) (fun k => norm_nonneg _) hr ha\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nobtain \u27e8C, r, Cpos, rpos, ple\u27e9 : \u2203 (C r : _) (_ : 0 < C) (_ : 0 < r), \u2200 n : \u2115, \u2016p n\u2016 \u2264 C * r ^ n :=\n le_mul_pow_of_radius_pos p hp\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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nlet I :=\n \u2016(i.symm : F \u2192L[\ud835\udd5c] E)\u2016\n -- choose `a` small enough to make sure that `\u2211_{k \u2264 n} a\u1d4f Q\u2096` will be controllable by\n -- induction\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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nobtain \u27e8a, apos, ha1, ha2\u27e9 :\n \u2203 (a : _) (apos : 0 < a), 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2 :=\n by\n have : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n tendsto_const_nhds.mul tendsto_id\n have A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by apply (tendsto_order.1 this).2; simp [zero_lt_one]\n have : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) := tendsto_const_nhds.mul tendsto_id\n have B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by apply (tendsto_order.1 this).2; simp [zero_lt_one]\n have C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by filter_upwards [self_mem_nhdsWithin] with _ ha using ha\n rcases(C.and ((A.and B).filter_mono inf_le_left)).exists with \u27e8a, ha\u27e9\n exact\n \u27e8a, ha.1, ha.2.1.le, ha.2.2.le\u27e9\n -- check by induction that the partial sums are suitably bounded, using the choice of `a` and the\n -- inductive control from Lemma `radius_rightInv_pos_of_radius_pos_aux2`.\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\n\u22a2 \u2203 a apos, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nhave : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=\n tendsto_const_nhds.mul tendsto_id\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 \u2203 a apos, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nhave A : \u2200\u1da0 a in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1 := by apply (tendsto_order.1 this).2; simp [zero_lt_one]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\n[PROOFSTEP]\napply (tendsto_order.1 this).2\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\n\u22a2 1 > 2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0\n[PROOFSTEP]\nsimp [zero_lt_one]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\n\u22a2 \u2203 a apos, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nhave : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0)) := tendsto_const_nhds.mul tendsto_id\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 \u2203 a apos, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nhave B : \u2200\u1da0 a in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2 := by apply (tendsto_order.1 this).2; simp [zero_lt_one]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n[PROOFSTEP]\napply (tendsto_order.1 this).2\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\n\u22a2 1 / 2 > r * (I + 1) * 0\n[PROOFSTEP]\nsimp [zero_lt_one]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a apos, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nhave C : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d), (0 : \u211d) < a := by filter_upwards [self_mem_nhdsWithin] with _ ha using ha\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd[Set.Ioi 0] 0, 0 < a\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with _ ha using ha\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[Set.Ioi 0] 0, 0 < a\n\u22a2 \u2203 a apos, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nrcases(C.and ((A.and B).filter_mono inf_le_left)).exists with \u27e8a, ha\u27e9\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC\u271d r : \u211d\nCpos : 0 < C\u271d\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C\u271d * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\nthis\u271d : Tendsto (fun a => 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a) (\ud835\udcdd 0) (\ud835\udcdd (2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * 0))\nA : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1\nthis : Tendsto (fun a => r * (I + 1) * a) (\ud835\udcdd 0) (\ud835\udcdd (r * (I + 1) * 0))\nB : \u2200\u1da0 (a : \u211d) in \ud835\udcdd 0, r * (I + 1) * a < 1 / 2\nC : \u2200\u1da0 (a : \u211d) in \ud835\udcdd[Set.Ioi 0] 0, 0 < a\na : \u211d\nha : 0 < a \u2227 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a < 1 \u2227 r * (I + 1) * a < 1 / 2\n\u22a2 \u2203 a apos, 2 * I * C\u271d * r ^ 2 * (I + 1) ^ 2 * a \u2264 1 \u2227 r * (I + 1) * a \u2264 1 / 2\n[PROOFSTEP]\nexact\n \u27e8a, ha.1, ha.2.1.le, ha.2.2.le\u27e9\n -- check by induction that the partial sums are suitably bounded, using the choice of `a` and the\n -- inductive control from Lemma `radius_rightInv_pos_of_radius_pos_aux2`.\n[GOAL]\ncase intro.intro.intro.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nlet S n := \u2211 k in Ico 1 n, a ^ k * \u2016p.rightInv i k\u2016\n[GOAL]\ncase intro.intro.intro.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nhave IRec : \u2200 n, 1 \u2264 n \u2192 S n \u2264 (I + 1) * a := by\n apply Nat.le_induction\n \u00b7 simp only\n rw [Ico_eq_empty_of_le (le_refl 1), sum_empty]\n exact mul_nonneg (add_nonneg (norm_nonneg _) zero_le_one) apos.le\n \u00b7 intro n one_le_n hn\n have In : 2 \u2264 n + 1 := by linarith only [one_le_n]\n have Snonneg : 0 \u2264 S n := sum_nonneg fun x _ => mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _)\n have rSn : r * S n \u2264 1 / 2 :=\n calc\n r * S n \u2264 r * ((I + 1) * a) := mul_le_mul_of_nonneg_left hn rpos.le\n _ \u2264 1 / 2 := by rwa [\u2190 mul_assoc]\n calc\n S (n + 1) \u2264 I * a + I * C * \u2211 k in Ico 2 (n + 1), (r * S n) ^ k :=\n radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple\n _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by rw [geom_sum_Ico' _ In];\n exact ne_of_lt (rSn.trans_lt (by norm_num))\n _ \u2264 I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) :=\n by\n apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le]\n refine' div_le_div (sq_nonneg _) _ (by norm_num) (by linarith only [rSn])\n simp only [sub_le_self_iff]\n apply pow_nonneg (mul_nonneg rpos.le Snonneg)\n _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring\n _ \u2264 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by\n apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le, zero_le_two,\n pow_le_pow_of_le_left, rpos.le]\n _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring\n _ \u2264 (I + 1) * a := by\n apply_rules [mul_le_mul_of_nonneg_right, apos.le, add_le_add, le_refl]\n -- conclude that all coefficients satisfy `a\u207f Q\u2099 \u2264 (I + 1) a`.\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\n[PROOFSTEP]\napply Nat.le_induction\n[GOAL]\ncase base\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 S 1 \u2264 (I + 1) * a\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase base\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 \u2211 x in Ico 1 1, a ^ x * \u2016rightInv p i x\u2016 \u2264 (\u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 + 1) * a\n[PROOFSTEP]\nrw [Ico_eq_empty_of_le (le_refl 1), sum_empty]\n[GOAL]\ncase base\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 0 \u2264 (\u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 + 1) * a\n[PROOFSTEP]\nexact mul_nonneg (add_nonneg (norm_nonneg _) zero_le_one) apos.le\n[GOAL]\ncase succ\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\n\u22a2 \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a \u2192 S (n + 1) \u2264 (I + 1) * a\n[PROOFSTEP]\nintro n one_le_n hn\n[GOAL]\ncase succ\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\n\u22a2 S (n + 1) \u2264 (I + 1) * a\n[PROOFSTEP]\nhave In : 2 \u2264 n + 1 := by linarith only [one_le_n]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\n\u22a2 2 \u2264 n + 1\n[PROOFSTEP]\nlinarith only [one_le_n]\n[GOAL]\ncase succ\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\n\u22a2 S (n + 1) \u2264 (I + 1) * a\n[PROOFSTEP]\nhave Snonneg : 0 \u2264 S n := sum_nonneg fun x _ => mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _)\n[GOAL]\ncase succ\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\n\u22a2 S (n + 1) \u2264 (I + 1) * a\n[PROOFSTEP]\nhave rSn : r * S n \u2264 1 / 2 :=\n calc\n r * S n \u2264 r * ((I + 1) * a) := mul_le_mul_of_nonneg_left hn rpos.le\n _ \u2264 1 / 2 := by rwa [\u2190 mul_assoc]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\n\u22a2 r * ((I + 1) * a) \u2264 1 / 2\n[PROOFSTEP]\nrwa [\u2190 mul_assoc]\n[GOAL]\ncase succ\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 S (n + 1) \u2264 (I + 1) * a\n[PROOFSTEP]\ncalc\n S (n + 1) \u2264 I * a + I * C * \u2211 k in Ico 2 (n + 1), (r * S n) ^ k :=\n radius_rightInv_pos_of_radius_pos_aux2 In p i rpos.le apos.le Cpos.le ple\n _ = I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) := by rw [geom_sum_Ico' _ In];\n exact ne_of_lt (rSn.trans_lt (by norm_num))\n _ \u2264 I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) :=\n by\n apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le]\n refine' div_le_div (sq_nonneg _) _ (by norm_num) (by linarith only [rSn])\n simp only [sub_le_self_iff]\n apply pow_nonneg (mul_nonneg rpos.le Snonneg)\n _ = I * a + 2 * I * C * (r * S n) ^ 2 := by ring\n _ \u2264 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 := by\n apply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le, zero_le_two,\n pow_le_pow_of_le_left, rpos.le]\n _ = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a := by ring\n _ \u2264 (I + 1) * a := by\n apply_rules [mul_le_mul_of_nonneg_right, apos.le, add_le_add, le_refl]\n -- conclude that all coefficients satisfy `a\u207f Q\u2099 \u2264 (I + 1) a`.\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * \u2211 k in Ico 2 (n + 1), (r * S n) ^ k =\n I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))\n[PROOFSTEP]\nrw [geom_sum_Ico' _ In]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 r * S n \u2260 1\n[PROOFSTEP]\nexact ne_of_lt (rSn.trans_lt (by norm_num))\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n)) \u2264 I * a + I * C * ((r * S n) ^ 2 / (1 / 2))\n[PROOFSTEP]\napply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le]\n[GOAL]\ncase h\u2082.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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 ((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n) \u2264 (r * S n) ^ 2 / (1 / 2)\n[PROOFSTEP]\nrefine' div_le_div (sq_nonneg _) _ (by norm_num) (by linarith only [rSn])\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 1 / 2 \u2264 1 - r * S n\n[PROOFSTEP]\nlinarith only [rSn]\n[GOAL]\ncase h\u2082.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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 (r * S n) ^ 2 - (r * S n) ^ (n + 1) \u2264 (r * S n) ^ 2\n[PROOFSTEP]\nsimp only [sub_le_self_iff]\n[GOAL]\ncase h\u2082.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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\ni_symm : F \u2243L[\ud835\udd5c] E\n\u22a2 0 \u2264 (r * \u2211 x in Ico 1 n, a ^ x * \u2016rightInv p i x\u2016) ^ (n + 1)\n[PROOFSTEP]\napply pow_nonneg (mul_nonneg rpos.le Snonneg)\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + I * C * ((r * S n) ^ 2 / (1 / 2)) = I * a + 2 * I * C * (r * S n) ^ 2\n[PROOFSTEP]\nring\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + 2 * I * C * (r * S n) ^ 2 \u2264 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2\n[PROOFSTEP]\napply_rules [add_le_add, le_refl, mul_le_mul_of_nonneg_left, mul_nonneg, norm_nonneg, Cpos.le, zero_le_two,\n pow_le_pow_of_le_left, rpos.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 I * a + 2 * I * C * (r * ((I + 1) * a)) ^ 2 = (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a\n[PROOFSTEP]\nring\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nn : \u2115\none_le_n : 1 \u2264 n\nhn : S n \u2264 (I + 1) * a\nIn : 2 \u2264 n + 1\nSnonneg : 0 \u2264 S n\nrSn : r * S n \u2264 1 / 2\n\u22a2 (I + 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) * a \u2264 (I + 1) * a\n[PROOFSTEP]\napply_rules [mul_le_mul_of_nonneg_right, apos.le, add_le_add, le_refl]\n -- conclude that all coefficients satisfy `a\u207f Q\u2099 \u2264 (I + 1) a`.\n[GOAL]\ncase intro.intro.intro.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nlet a' : NNReal := \u27e8a, apos.le\u27e9\n[GOAL]\ncase intro.intro.intro.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\nsuffices H : (a' : ENNReal) \u2264 (p.rightInv i).radius\n[GOAL]\ncase intro.intro.intro.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\np : FormalMultilinearSeries \ud835\udd5c E F\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nH : \u2191a' \u2264 radius (rightInv p i)\n\u22a2 0 < radius (rightInv p i)\n[PROOFSTEP]\napply lt_of_lt_of_le _ H\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nH : \u2191a' \u2264 radius (rightInv p i)\n\u22a2 0 < \u2191a'\n[PROOFSTEP]\nexact_mod_cast apos\n[GOAL]\ncase 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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\n\u22a2 \u2191a' \u2264 radius (rightInv p i)\n[PROOFSTEP]\napply le_radius_of_bound _ ((I + 1) * a) fun n => ?_\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\n\u22a2 \u2016rightInv p i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 \u2016rightInv p i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n[PROOFSTEP]\nhave : \u2016p.rightInv i n\u2016 = \u2016p.rightInv i 0\u2016 := by congr <;> try rw [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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 \u2016rightInv p i n\u2016 = \u2016rightInv p i 0\u2016\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_1.h.e_3\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => F) fun i => F\n[PROOFSTEP]\ntry rw [hn]\n[GOAL]\ncase h.e_1.h.e_3\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => F) fun i => F\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase h.e_1.h.e_6\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => AddCommGroup.toAddCommMonoid) fun i => AddCommGroup.toAddCommMonoid\n[PROOFSTEP]\ntry rw [hn]\n[GOAL]\ncase h.e_1.h.e_6\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => AddCommGroup.toAddCommMonoid) fun i => AddCommGroup.toAddCommMonoid\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase h.e_1.h.e_8\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => NormedSpace.toModule) fun i => NormedSpace.toModule\n[PROOFSTEP]\ntry rw [hn]\n[GOAL]\ncase h.e_1.h.e_8\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => NormedSpace.toModule) fun i => NormedSpace.toModule\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase h.e_1.h.e_10\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => UniformSpace.toTopologicalSpace) fun i => UniformSpace.toTopologicalSpace\n[PROOFSTEP]\ntry rw [hn]\n[GOAL]\ncase h.e_1.h.e_10\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\n\u22a2 HEq (fun i => UniformSpace.toTopologicalSpace) fun i => UniformSpace.toTopologicalSpace\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase pos\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\nthis : \u2016rightInv p i n\u2016 = \u2016rightInv p i 0\u2016\n\u22a2 \u2016rightInv p i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n[PROOFSTEP]\nsimp only [this, norm_zero, zero_mul, rightInv_coeff_zero]\n[GOAL]\ncase pos\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : n = 0\nthis : \u2016rightInv p i n\u2016 = \u2016rightInv p i 0\u2016\n\u22a2 0 \u2264 (\u2016\u2191(ContinuousLinearEquiv.symm i)\u2016 + 1) * a\n[PROOFSTEP]\napply_rules [mul_nonneg, add_nonneg, norm_nonneg, zero_le_one, apos.le]\n[GOAL]\ncase neg\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u2016rightInv p i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n[PROOFSTEP]\nhave one_le_n : 1 \u2264 n := bot_lt_iff_ne_bot.2 hn\n[GOAL]\ncase neg\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\n\u22a2 \u2016rightInv p i n\u2016 * \u2191a' ^ n \u2264 (I + 1) * a\n[PROOFSTEP]\ncalc\n \u2016p.rightInv i n\u2016 * (a' : \u211d) ^ n = a ^ n * \u2016p.rightInv i n\u2016 := mul_comm _ _\n _ \u2264 \u2211 k in Ico 1 (n + 1), a ^ k * \u2016p.rightInv i k\u2016 :=\n (haveI : \u2200 k \u2208 Ico 1 (n + 1), 0 \u2264 a ^ k * \u2016p.rightInv i k\u2016 := fun k _ =>\n mul_nonneg (pow_nonneg apos.le _) (norm_nonneg _)\n single_le_sum this (by simp [one_le_n]))\n _ \u2264 (I + 1) * a := IRec (n + 1) (by norm_num)\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\nthis : \u2200 (k : \u2115), k \u2208 Ico 1 (n + 1) \u2192 0 \u2264 a ^ k * \u2016rightInv p i k\u2016\n\u22a2 n \u2208 Ico 1 (n + 1)\n[PROOFSTEP]\nsimp [one_le_n]\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\ni : E \u2243L[\ud835\udd5c] F\nhp : 0 < radius p\nC r : \u211d\nCpos : 0 < C\nrpos : 0 < r\nple : \u2200 (n : \u2115), \u2016p n\u2016 \u2264 C * r ^ n\nI : \u211d := \u2016\u2191(ContinuousLinearEquiv.symm i)\u2016\na : \u211d\napos : 0 < a\nha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a \u2264 1\nha2 : r * (I + 1) * a \u2264 1 / 2\nS : \u2115 \u2192 \u211d := fun n => \u2211 k in Ico 1 n, a ^ k * \u2016rightInv p i k\u2016\nIRec : \u2200 (n : \u2115), 1 \u2264 n \u2192 S n \u2264 (I + 1) * a\na' : NNReal := { val := a, property := (_ : 0 \u2264 a) }\nn : \u2115\nhn : \u00acn = 0\none_le_n : 1 \u2264 n\n\u22a2 1 \u2264 n + 1\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Analytic.Inverse", "llama_tokens": 78202, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.42632159254749036, "lm_q1q2_score": 0.25905993029561447}} {"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\n\u22a2 \u2191\u22a4 \u2286 \u2191(span K (range (restrict (LinearIndependent.extend hs (_ : s \u2286 univ)) id)))\n[PROOFSTEP]\nsimpa using hs.subset_span_extend (subset_univ s)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\n\u22a2 range \u2191(extend hs) = LinearIndependent.extend hs (_ : s \u2286 univ)\n[PROOFSTEP]\nrw [coe_extend, Subtype.range_coe_subtype, setOf_mem_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : \u2191(ofVectorSpaceIndex K V)\n\u22a2 \u2191(ofVectorSpace K V) x = \u2191x\n[PROOFSTEP]\nunfold ofVectorSpace\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : \u2191(ofVectorSpaceIndex K V)\n\u22a2 \u2191(extend (_ : LinearIndependent K fun x => \u2191x)) x = \u2191x\n[PROOFSTEP]\nexact Basis.mk_apply _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 LinearIndependent K Subtype.val\n[PROOFSTEP]\nconvert (ofVectorSpace K V).linearIndependent\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 Subtype.val = \u2191(ofVectorSpace K V)\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_4.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : { x // x \u2208 ofVectorSpaceIndex K V }\n\u22a2 \u2191x = \u2191(ofVectorSpace K V) x\n[PROOFSTEP]\nrw [ofVectorSpace_apply_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module K V\ninst\u271d\u00b2 : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Fintype V\n\u22a2 \u2203 n, card V = card K ^ n\n[PROOFSTEP]\nclassical exact \u27e8card (Basis.ofVectorSpaceIndex K V), Module.card_fintype (Basis.ofVectorSpace K V)\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module K V\ninst\u271d\u00b2 : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Fintype V\n\u22a2 \u2203 n, card V = card K ^ n\n[PROOFSTEP]\nexact \u27e8card (Basis.ofVectorSpaceIndex K V), Module.card_fintype (Basis.ofVectorSpace K V)\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\n\u22a2 IsAtom (span K {v})\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\n\u22a2 span K {v} \u2260 \u22a5\n[PROOFSTEP]\nrw [Submodule.ne_bot_iff]\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\n\u22a2 \u2203 x, x \u2208 span K {v} \u2227 x \u2260 0\n[PROOFSTEP]\nexact \u27e8v, \u27e8mem_span_singleton_self v, hv\u27e9\u27e9\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\n\u22a2 \u2200 (b : Submodule K V), b < span K {v} \u2192 b = \u22a5\n[PROOFSTEP]\nintro T hT\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\n\u22a2 T = \u22a5\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\nh : \u00acT = \u22a5\n\u22a2 False\n[PROOFSTEP]\napply hT.2\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\nh : \u00acT = \u22a5\n\u22a2 \u2191(span K {v}) \u2286 \u2191T\n[PROOFSTEP]\nchange span K { v } \u2264 T\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\nh : \u00acT = \u22a5\n\u22a2 span K {v} \u2264 T\n[PROOFSTEP]\nsimp_rw [span_singleton_le_iff_mem, \u2190 Ne.def, Submodule.ne_bot_iff] at *\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\nh : \u2203 x, x \u2208 T \u2227 x \u2260 0\n\u22a2 v \u2208 T\n[PROOFSTEP]\nrcases h with \u27e8s, \u27e8hs, hz\u27e9\u27e9\n[GOAL]\ncase right.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns\u271d t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\ns : V\nhs : s \u2208 T\nhz : s \u2260 0\n\u22a2 v \u2208 T\n[PROOFSTEP]\nrcases mem_span_singleton.1 (hT.1 hs) with \u27e8a, rfl\u27e9\n[GOAL]\ncase right.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\na : K\nhs : a \u2022 v \u2208 T\nhz : a \u2022 v \u2260 0\n\u22a2 v \u2208 T\n[PROOFSTEP]\nrcases eq_or_ne a 0 with rfl | h\n[GOAL]\ncase right.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\nhs : 0 \u2022 v \u2208 T\nhz : 0 \u2022 v \u2260 0\n\u22a2 v \u2208 T\n[PROOFSTEP]\nsimp only [zero_smul, ne_eq, not_true] at hz \n[GOAL]\ncase right.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\nT : Submodule K V\nhT : T < span K {v}\na : K\nhs : a \u2022 v \u2208 T\nhz : a \u2022 v \u2260 0\nh : a \u2260 0\n\u22a2 v \u2208 T\n[PROOFSTEP]\nrwa [T.smul_mem_iff h] at hs \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\n\u22a2 IsAtom W \u2194 \u2203 v x, W = span K {v}\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : IsAtom W\n\u22a2 \u2203 v x, W = span K {v}\n[PROOFSTEP]\ncases' h with hbot h\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nh : \u2200 (b : Submodule K V), b < W \u2192 b = \u22a5\n\u22a2 \u2203 v x, W = span K {v}\n[PROOFSTEP]\nrcases(Submodule.ne_bot_iff W).1 hbot with \u27e8v, \u27e8hW, hv\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nh : \u2200 (b : Submodule K V), b < W \u2192 b = \u22a5\nv : V\nhW : v \u2208 W\nhv : v \u2260 0\n\u22a2 \u2203 v x, W = span K {v}\n[PROOFSTEP]\nrefine' \u27e8v, \u27e8hv, _\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nh : \u2200 (b : Submodule K V), b < W \u2192 b = \u22a5\nv : V\nhW : v \u2208 W\nhv : v \u2260 0\n\u22a2 W = span K {v}\n[PROOFSTEP]\nby_contra heq\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nh : \u2200 (b : Submodule K V), b < W \u2192 b = \u22a5\nv : V\nhW : v \u2208 W\nhv : v \u2260 0\nheq : \u00acW = span K {v}\n\u22a2 False\n[PROOFSTEP]\nspecialize h (span K { v })\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nv : V\nhW : v \u2208 W\nhv : v \u2260 0\nheq : \u00acW = span K {v}\nh : span K {v} < W \u2192 span K {v} = \u22a5\n\u22a2 False\n[PROOFSTEP]\nrw [span_singleton_eq_bot, lt_iff_le_and_ne] at h \n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W \u2260 \u22a5\nv : V\nhW : v \u2208 W\nhv : v \u2260 0\nheq : \u00acW = span K {v}\nh : span K {v} \u2264 W \u2227 span K {v} \u2260 W \u2192 v = 0\n\u22a2 False\n[PROOFSTEP]\nexact hv (h \u27e8(span_singleton_le_iff_mem v W).2 hW, Ne.symm heq\u27e9)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : \u2203 v x, W = span K {v}\n\u22a2 IsAtom W\n[PROOFSTEP]\nrcases h with \u27e8v, \u27e8hv, rfl\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z v : V\nhv : v \u2260 0\n\u22a2 IsAtom (span K {v})\n[PROOFSTEP]\nexact nonzero_span_atom v hv\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\n\u22a2 \u2203 s, W = sSup s \u2227 \u2200 (a : Submodule K V), a \u2208 s \u2192 IsAtom a\n[PROOFSTEP]\nrefine \u27e8_, submodule_eq_sSup_le_nonzero_spans W, ?_\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\n\u22a2 \u2200 (a : Submodule K V), a \u2208 {T | \u2203 m x x, T = span K {m}} \u2192 IsAtom a\n[PROOFSTEP]\nrintro _ \u27e8w, \u27e8_, \u27e8hw, rfl\u27e9\u27e9\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\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nW : Submodule K V\nw : V\nw\u271d : w \u2208 W\nhw : w \u2260 0\n\u22a2 IsAtom (span K {w})\n[PROOFSTEP]\nexact nonzero_span_atom w hw\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nlet B := Basis.ofVectorSpaceIndex K V\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nlet hB := Basis.ofVectorSpace K V\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nhave hB\u2080 : _ := hB.linearIndependent.to_subtype_range\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nhave : LinearIndependent K (fun x => x : f '' B \u2192 V') :=\n by\n have h\u2081 : LinearIndependent K ((\u2191) : \u21a5(f '' Set.range (Basis.ofVectorSpace K V)) \u2192 V') :=\n @LinearIndependent.image_subtype _ _ _ _ _ _ _ _ _ f hB\u2080 (show Disjoint _ _ by simp [hf_inj])\n rwa [Basis.range_ofVectorSpace K V] at h\u2081 \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\n\u22a2 LinearIndependent K fun x => \u2191x\n[PROOFSTEP]\nhave h\u2081 : LinearIndependent K ((\u2191) : \u21a5(f '' Set.range (Basis.ofVectorSpace K V)) \u2192 V') :=\n @LinearIndependent.image_subtype _ _ _ _ _ _ _ _ _ f hB\u2080 (show Disjoint _ _ by simp [hf_inj])\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\n\u22a2 Disjoint (span K (Set.range \u2191(Basis.ofVectorSpace K V))) (ker f)\n[PROOFSTEP]\nsimp [hf_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nh\u2081 : LinearIndependent K Subtype.val\n\u22a2 LinearIndependent K fun x => \u2191x\n[PROOFSTEP]\nrwa [Basis.range_ofVectorSpace K V] at h\u2081 \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nlet C := this.extend (subset_univ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nhave BC := this.subset_extend (subset_univ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : \u2191f '' B \u2286 LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nlet hC := Basis.extend this\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : \u2191f '' B \u2286 LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nhaveI Vinh : Inhabited V := \u27e80\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : \u2191f '' B \u2286 LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\n\u22a2 \u2203 g, comp g f = id\n[PROOFSTEP]\nrefine' \u27e8(hC.constr \u2115 : _ \u2192 _) (C.restrict (invFun f)), hB.ext fun b => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : \u2191f '' B \u2286 LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\n\u22a2 \u2191(comp (\u2191(Basis.constr hC \u2115) (Set.restrict C (invFun \u2191f))) f) (\u2191hB b) = \u2191id (\u2191hB b)\n[PROOFSTEP]\nrw [image_subset_iff] at BC \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\n\u22a2 \u2191(comp (\u2191(Basis.constr hC \u2115) (Set.restrict C (invFun \u2191f))) f) (\u2191hB b) = \u2191id (\u2191hB b)\n[PROOFSTEP]\nhave fb_eq : f b = hC \u27e8f b, BC b.2\u27e9 := by\n change f b = Basis.extend this _\n simp_rw [Basis.extend_apply_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\n\u22a2 \u2191f \u2191b = \u2191hC { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) }\n[PROOFSTEP]\nchange f b = Basis.extend this _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\n\u22a2 \u2191f \u2191b =\n \u2191(Basis.extend this)\n { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) }\n[PROOFSTEP]\nsimp_rw [Basis.extend_apply_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\nfb_eq : \u2191f \u2191b = \u2191hC { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) }\n\u22a2 \u2191(comp (\u2191(Basis.constr hC \u2115) (Set.restrict C (invFun \u2191f))) f) (\u2191hB b) = \u2191id (\u2191hB b)\n[PROOFSTEP]\ndsimp []\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\nfb_eq : \u2191f \u2191b = \u2191hC { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) }\n\u22a2 \u2191(\u2191(Basis.constr (Basis.extend this) \u2115)\n (Set.restrict (LinearIndependent.extend this (_ : \u2191f '' Basis.ofVectorSpaceIndex K V \u2286 univ)) (invFun \u2191f)))\n (\u2191f (\u2191(Basis.ofVectorSpace K V) b)) =\n \u2191(Basis.ofVectorSpace K V) b\n[PROOFSTEP]\nrw [Basis.ofVectorSpace_apply_self, fb_eq, hC.constr_basis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_inj : ker f = \u22a5\nB : Set V := Basis.ofVectorSpaceIndex K V\nhB : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhB\u2080 : LinearIndependent K Subtype.val\nthis : LinearIndependent K fun x => \u2191x\nC : Set V' := LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nBC : B \u2286 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)\nhC : Basis (\u2191(LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ))) K V' := Basis.extend this\nVinh : Inhabited V\nb : \u2191(Basis.ofVectorSpaceIndex K V)\nfb_eq : \u2191f \u2191b = \u2191hC { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) }\n\u22a2 Set.restrict (LinearIndependent.extend this (_ : \u2191f '' Basis.ofVectorSpaceIndex K V \u2286 univ)) (invFun \u2191f)\n { val := \u2191f \u2191b, property := (_ : \u2191b \u2208 \u2191f \u207b\u00b9' LinearIndependent.extend this (_ : \u2191f '' B \u2286 univ)) } =\n \u2191b\n[PROOFSTEP]\nexact leftInverse_invFun (LinearMap.ker_eq_bot.1 hf_inj) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\n\u22a2 \u2203 g, comp f g = id\n[PROOFSTEP]\nlet C := Basis.ofVectorSpaceIndex K V'\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\nC : Set V' := Basis.ofVectorSpaceIndex K V'\n\u22a2 \u2203 g, comp f g = id\n[PROOFSTEP]\nlet hC := Basis.ofVectorSpace K V'\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\nC : Set V' := Basis.ofVectorSpaceIndex K V'\nhC : Basis (\u2191(Basis.ofVectorSpaceIndex K V')) K V' := Basis.ofVectorSpace K V'\n\u22a2 \u2203 g, comp f g = id\n[PROOFSTEP]\nhaveI : Inhabited V := \u27e80\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\nC : Set V' := Basis.ofVectorSpaceIndex K V'\nhC : Basis (\u2191(Basis.ofVectorSpaceIndex K V')) K V' := Basis.ofVectorSpace K V'\nthis : Inhabited V\n\u22a2 \u2203 g, comp f g = id\n[PROOFSTEP]\nrefine' \u27e8(hC.constr \u2115 : _ \u2192 _) (C.restrict (invFun f)), hC.ext fun c => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\nC : Set V' := Basis.ofVectorSpaceIndex K V'\nhC : Basis (\u2191(Basis.ofVectorSpaceIndex K V')) K V' := Basis.ofVectorSpace K V'\nthis : Inhabited V\nc : \u2191(Basis.ofVectorSpaceIndex K V')\n\u22a2 \u2191(comp f (\u2191(Basis.constr hC \u2115) (Set.restrict C (invFun \u2191f)))) (\u2191hC c) = \u2191id (\u2191hC c)\n[PROOFSTEP]\nrw [LinearMap.comp_apply, hC.constr_basis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : V \u2192\u2097[K] V'\nhf_surj : range f = \u22a4\nC : Set V' := Basis.ofVectorSpaceIndex K V'\nhC : Basis (\u2191(Basis.ofVectorSpaceIndex K V')) K V' := Basis.ofVectorSpace K V'\nthis : Inhabited V\nc : \u2191(Basis.ofVectorSpaceIndex K V')\n\u22a2 \u2191f (Set.restrict C (invFun \u2191f) c) = \u2191id (\u2191hC c)\n[PROOFSTEP]\nsimp [rightInverse_invFun (LinearMap.range_eq_top.1 hf_surj) c]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nf : { x // x \u2208 p } \u2192\u2097[K] V'\ng : V \u2192\u2097[K] { x // x \u2208 p }\nhg : comp g (Submodule.subtype p) = id\n\u22a2 comp (comp f g) (Submodule.subtype p) = f\n[PROOFSTEP]\nrw [LinearMap.comp_assoc, hg, f.comp_id]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nhp : p < \u22a4\n\u22a2 \u2203 f, f \u2260 0 \u2227 p \u2264 ker f\n[PROOFSTEP]\nrcases SetLike.exists_of_lt hp with \u27e8v, -, hpv\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nhp : p < \u22a4\nv : V\nhpv : \u00acv \u2208 p\n\u22a2 \u2203 f, f \u2260 0 \u2227 p \u2264 ker f\n[PROOFSTEP]\nclear hp\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\n\u22a2 \u2203 f, f \u2260 0 \u2227 p \u2264 ker f\n[PROOFSTEP]\nrcases(LinearPMap.supSpanSingleton \u27e8p, 0\u27e9 v (1 : K) hpv).toFun.exists_extend with \u27e8f, hf\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\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nf : V \u2192\u2097[K] K\nhf :\n LinearMap.comp f (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\n\u22a2 \u2203 f, f \u2260 0 \u2227 p \u2264 ker f\n[PROOFSTEP]\nrefine' \u27e8f, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nf : V \u2192\u2097[K] K\nhf :\n LinearMap.comp f (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\n\u22a2 f \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nhf :\n LinearMap.comp 0 (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\n\u22a2 False\n[PROOFSTEP]\nrw [LinearMap.zero_comp] at hf \n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nhf : 0 = (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\n\u22a2 False\n[PROOFSTEP]\nhave := LinearPMap.supSpanSingleton_apply_mk \u27e8p, 0\u27e9 v (1 : K) hpv 0 p.zero_mem 1\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nhf : 0 = (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\nthis :\n \u2191(LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv)\n { val := 0 + 1 \u2022 v,\n property :=\n (_ :\n 0 + 1 \u2022 v \u2208\n { domain := p, toFun := 0 }.domain \u2294 (LinearPMap.mkSpanSingleton v 1 (_ : v = 0 \u2192 False)).domain) } =\n \u2191{ domain := p, toFun := 0 } { val := 0, property := (_ : 0 \u2208 p) } + 1 \u2022 1\n\u22a2 False\n[PROOFSTEP]\nsimpa using (LinearMap.congr_fun hf _).trans this\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nf : V \u2192\u2097[K] K\nhf :\n LinearMap.comp f (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\n\u22a2 p \u2264 ker f\n[PROOFSTEP]\nrefine' fun x hx => mem_ker.2 _\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nf : V \u2192\u2097[K] K\nhf :\n LinearMap.comp f (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\nx : V\nhx : x \u2208 p\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nhave := LinearPMap.supSpanSingleton_apply_mk \u27e8p, 0\u27e9 v (1 : K) hpv x hx 0\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nK : Type u_3\nV : Type u_4\nV' : Type u_5\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\np : Submodule K V\nv : V\nhpv : \u00acv \u2208 p\nf : V \u2192\u2097[K] K\nhf :\n LinearMap.comp f (Submodule.subtype (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).domain) =\n (LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv).toFun\nx : V\nhx : x \u2208 p\nthis :\n \u2191(LinearPMap.supSpanSingleton { domain := p, toFun := 0 } v 1 hpv)\n { val := x + 0 \u2022 v,\n property :=\n (_ :\n x + 0 \u2022 v \u2208\n { domain := p, toFun := 0 }.domain \u2294 (LinearPMap.mkSpanSingleton v 1 (_ : v = 0 \u2192 False)).domain) } =\n \u2191{ domain := p, toFun := 0 } { val := x, property := hx } + 0 \u2022 1\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nsimpa using (LinearMap.congr_fun hf _).trans this\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Basis.VectorSpace", "llama_tokens": 17519, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5428632979641571, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.25871760111903436}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v, u\u2082} D\ne : C \u224c D\ninst\u271d : WellPowered C\nX : D\n\u22a2 EssentiallySmall (MonoOver ((Equivalence.symm e).functor.obj X))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Subobject.WellPowered", "llama_tokens": 109, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.2581694313878645}} {"text": "[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\n\u22a2 s \u2208 powerset t \u2194 s \u2286 t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ns t\u271d t : Finset \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\n\u22a2 { val := val\u271d, nodup := nodup\u271d } \u2208 powerset t \u2194 { val := val\u271d, nodup := nodup\u271d } \u2286 t\n[PROOFSTEP]\nsimp [powerset, mem_mk, mem_pmap, mk.injEq, mem_powerset, exists_prop, exists_eq_right, \u2190 val_le_iff]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Finset \u03b1\n\u22a2 \u2191(powerset s) = toSet \u207b\u00b9' \ud835\udcab\u2191s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ns\u271d t s x\u271d : Finset \u03b1\n\u22a2 x\u271d \u2208 \u2191(powerset s) \u2194 x\u271d \u2208 toSet \u207b\u00b9' \ud835\udcab\u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns t : Finset \u03b1\n\u22a2 powerset s = {\u2205} \u2194 s = \u2205\n[PROOFSTEP]\nrw [\u2190 powerset_empty, powerset_inj]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\na : \u03b1\nht : t \u2208 powerset s\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 t\n[PROOFSTEP]\napply mt _ h\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\na : \u03b1\nht : t \u2208 powerset s\nh : \u00aca \u2208 s\n\u22a2 a \u2208 t \u2192 a \u2208 s\n[PROOFSTEP]\napply mem_powerset.1 ht\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 powerset (insert a s) = powerset s \u222a image (insert a) (powerset s)\n[PROOFSTEP]\next t\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 t \u2208 powerset (insert a s) \u2194 t \u2208 powerset s \u222a image (insert a) (powerset s)\n[PROOFSTEP]\nsimp only [exists_prop, mem_powerset, mem_image, mem_union, subset_insert_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n[PROOFSTEP]\nby_cases h : a \u2208 t\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pos.mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 erase t a \u2286 s \u2192 t \u2286 s \u2228 \u2203 a_2, a_2 \u2286 s \u2227 insert a a_2 = t\n[PROOFSTEP]\nexact fun H => Or.inr \u27e8_, H, insert_erase h\u27e9\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\n\u22a2 (t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t) \u2192 erase t a \u2286 s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase pos.mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s\n[PROOFSTEP]\ncases' H with H H\n[GOAL]\ncase pos.mpr.inl\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : t \u2286 s\n\u22a2 erase t a \u2286 s\n[PROOFSTEP]\nexact Subset.trans (erase_subset a t) H\n[GOAL]\ncase pos.mpr.inr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nH : \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n\u22a2 erase t a \u2286 s\n[PROOFSTEP]\nrcases H with \u27e8u, hu\u27e9\n[GOAL]\ncase pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s\n[PROOFSTEP]\nrw [\u2190 hu.2]\n[GOAL]\ncase pos.mpr.inr.intro\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : a \u2208 t\nu : Finset \u03b1\nhu : u \u2286 s \u2227 insert a u = t\n\u22a2 erase (insert a u) a \u2286 s\n[PROOFSTEP]\nexact Subset.trans (erase_insert_subset a u) hu.1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n[PROOFSTEP]\nhave : \u00ac\u2203 u : Finset \u03b1, u \u2286 s \u2227 insert a u = t := by simp [Ne.symm (ne_insert_of_not_mem _ _ h)]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\n\u22a2 \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t\n[PROOFSTEP]\nsimp [Ne.symm (ne_insert_of_not_mem _ _ h)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\nt : Finset \u03b1\nh : \u00aca \u2208 t\nthis : \u00ac\u2203 u, u \u2286 s \u2227 insert a u = t\n\u22a2 erase t a \u2286 s \u2194 t \u2286 s \u2228 \u2203 a_1, a_1 \u2286 s \u2227 insert a a_1 = t\n[PROOFSTEP]\nsimp [Finset.erase_eq_of_not_mem h, this]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 t \u2208 ssubsets s \u2194 t \u2282 s\n[PROOFSTEP]\nrw [ssubsets, mem_erase, mem_powerset, ssubset_iff_subset_ne, and_comm]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 \u2205 \u2208 ssubsets s\n[PROOFSTEP]\nrw [mem_ssubsets, ssubset_iff_subset_ne]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 \u2205 \u2286 s \u2227 \u2205 \u2260 s\n[PROOFSTEP]\nexact \u27e8empty_subset s, h.ne_empty.symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\nn : \u2115\ns t : Finset \u03b1\n\u22a2 s \u2208 powersetLen n t \u2194 s \u2286 t \u2227 card s = n\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\nn : \u2115\nt : Finset \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\n\u22a2 { val := val\u271d, nodup := nodup\u271d } \u2208 powersetLen n t \u2194\n { val := val\u271d, nodup := nodup\u271d } \u2286 t \u2227 card { val := val\u271d, nodup := nodup\u271d } = n\n[PROOFSTEP]\nsimp [powersetLen, val_le_iff.symm]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Finset \u03b1\n\u22a2 powersetLen 0 s = {\u2205}\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2208 powersetLen 0 s \u2194 a\u271d \u2208 {\u2205}\n[PROOFSTEP]\nrw [mem_powersetLen, mem_singleton, card_eq_zero]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2286 s \u2227 a\u271d = \u2205 \u2194 a\u271d = \u2205\n[PROOFSTEP]\nrefine'\n \u27e8fun h => h.2, fun h => by\n rw [h]\n exact \u27e8empty_subset s, rfl\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\nh : a\u271d = \u2205\n\u22a2 a\u271d \u2286 s \u2227 a\u271d = \u2205\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\nh : a\u271d = \u2205\n\u22a2 \u2205 \u2286 s \u2227 \u2205 = \u2205\n[PROOFSTEP]\nexact \u27e8empty_subset s, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nh : card s < n\n\u22a2 card (powersetLen n s) = 0\n[PROOFSTEP]\nrw [card_powersetLen, Nat.choose_eq_zero_of_lt h]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\n\u22a2 powersetLen n s = filter (fun x => card x = n) (powerset s)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2208 powersetLen n s \u2194 a\u271d \u2208 filter (fun x => card x = n) (powerset s)\n[PROOFSTEP]\nsimp [mem_powersetLen]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\n\u22a2 powersetLen (Nat.succ n) (insert x s) = powersetLen (Nat.succ n) s \u222a image (insert x) (powersetLen n s)\n[PROOFSTEP]\nrw [powersetLen_eq_filter, powerset_insert, filter_union, \u2190 powersetLen_eq_filter]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\n\u22a2 powersetLen (Nat.succ n) s \u222a filter (fun x => card x = Nat.succ n) (image (insert x) (powerset s)) =\n powersetLen (Nat.succ n) s \u222a image (insert x) (powersetLen n s)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\n\u22a2 filter (fun x => card x = Nat.succ n) (image (insert x) (powerset s)) = image (insert x) (powersetLen n s)\n[PROOFSTEP]\nrw [powersetLen_eq_filter, image_filter]\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\n\u22a2 image (insert x) (filter ((fun x => card x = Nat.succ n) \u2218 insert x) (powerset s)) =\n image (insert x) (filter (fun x => card x = n) (powerset s))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a.e_s\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\n\u22a2 filter ((fun x => card x = Nat.succ n) \u2218 insert x) (powerset s) = filter (fun x => card x = n) (powerset s)\n[PROOFSTEP]\next t\n[GOAL]\ncase e_a.e_s.a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\nt : Finset \u03b1\n\u22a2 t \u2208 filter ((fun x => card x = Nat.succ n) \u2218 insert x) (powerset s) \u2194 t \u2208 filter (fun x => card x = n) (powerset s)\n[PROOFSTEP]\nsimp only [mem_powerset, mem_filter, Function.comp_apply, and_congr_right_iff]\n[GOAL]\ncase e_a.e_s.a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\nt : Finset \u03b1\n\u22a2 t \u2286 s \u2192 (card (insert x t) = Nat.succ n \u2194 card t = n)\n[PROOFSTEP]\nintro ht\n[GOAL]\ncase e_a.e_s.a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\nt : Finset \u03b1\nht : t \u2286 s\n\u22a2 card (insert x t) = Nat.succ n \u2194 card t = n\n[PROOFSTEP]\nhave : x \u2209 t := fun H => h (ht H)\n[GOAL]\ncase e_a.e_s.a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\ns : Finset \u03b1\nh : \u00acx \u2208 s\nn : \u2115\nt : Finset \u03b1\nht : t \u2286 s\nthis : \u00acx \u2208 t\n\u22a2 card (insert x t) = Nat.succ n \u2194 card t = n\n[PROOFSTEP]\nsimp [card_insert_of_not_mem this, Nat.succ_inj']\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nh : n \u2264 card s\n\u22a2 Finset.Nonempty (powersetLen n s)\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction_on with x s hx IH generalizing n\n\u00b7 rw [card_empty, le_zero_iff] at h \n rw [h, powersetLen_zero]\n exact Finset.singleton_nonempty _\n\u00b7 cases n\n \u00b7 simp\n \u00b7 rw [card_insert_of_not_mem hx, Nat.succ_le_succ_iff] at h \n rw [powersetLen_succ_insert hx]\n refine' Nonempty.mono _ ((IH h).image (insert x))\n exact subset_union_right _ _\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn : \u2115\ns : Finset \u03b1\nh : n \u2264 card s\n\u22a2 Finset.Nonempty (powersetLen n s)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with x s hx IH generalizing n\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n \u2264 card \u2205\n\u22a2 Finset.Nonempty (powersetLen n \u2205)\n[PROOFSTEP]\nrw [card_empty, le_zero_iff] at h \n[GOAL]\ncase empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty (powersetLen n \u2205)\n[PROOFSTEP]\nrw [h, powersetLen_zero]\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\nn\u271d : \u2115\ns : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\nn : \u2115\nh : n = 0\n\u22a2 Finset.Nonempty {\u2205}\n[PROOFSTEP]\nexact Finset.singleton_nonempty _\n[GOAL]\ncase insert\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn\u271d : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n\u271d \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nn : \u2115\nh : n \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetLen n (insert x s))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase insert.zero\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nh : Nat.zero \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetLen Nat.zero (insert x s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nn\u271d : \u2115\nh : Nat.succ n\u271d \u2264 card (insert x s)\n\u22a2 Finset.Nonempty (powersetLen (Nat.succ n\u271d) (insert x s))\n[PROOFSTEP]\nrw [card_insert_of_not_mem hx, Nat.succ_le_succ_iff] at h \n[GOAL]\ncase insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetLen (Nat.succ n\u271d) (insert x s))\n[PROOFSTEP]\nrw [powersetLen_succ_insert hx]\n[GOAL]\ncase insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 Finset.Nonempty (powersetLen (Nat.succ n\u271d) s \u222a image (insert x) (powersetLen n\u271d s))\n[PROOFSTEP]\nrefine' Nonempty.mono _ ((IH h).image (insert x))\n[GOAL]\ncase insert.succ\n\u03b1 : Type u_1\ns\u271d\u00b9 t : Finset \u03b1\nn : \u2115\ns\u271d : Finset \u03b1\nh\u271d : n \u2264 card s\u271d\nx : \u03b1\ns : Finset \u03b1\nhx : \u00acx \u2208 s\nIH : \u2200 {n : \u2115}, n \u2264 card s \u2192 Finset.Nonempty (powersetLen n s)\nn\u271d : \u2115\nh : n\u271d \u2264 card s\n\u22a2 image (insert x) (powersetLen n\u271d s) \u2286 powersetLen (Nat.succ n\u271d) s \u222a image (insert x) (powersetLen n\u271d s)\n[PROOFSTEP]\nexact subset_union_right _ _\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Finset \u03b1\n\u22a2 powersetLen (card s) s = {s}\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2208 powersetLen (card s) s \u2194 a\u271d \u2208 {s}\n[PROOFSTEP]\nrw [mem_powersetLen, mem_singleton]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2286 s \u2227 card a\u271d = card s \u2194 a\u271d = s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2286 s \u2227 card a\u271d = card s \u2192 a\u271d = s\n[PROOFSTEP]\nexact fun \u27e8hs, hc\u27e9 => eq_of_subset_of_card_le hs hc.ge\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ns\u271d t s a\u271d : Finset \u03b1\n\u22a2 a\u271d = s \u2192 a\u271d \u2286 s \u2227 card a\u271d = card s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ns t a\u271d : Finset \u03b1\n\u22a2 a\u271d \u2286 a\u271d \u2227 card a\u271d = card a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Finset \u03b1\n\u22a2 powerset s =\n disjiUnion (range (card s + 1)) (fun i => powersetLen i s)\n (_ : Set.Pairwise \u2191(range (card s + 1)) fun i j => _root_.Disjoint (powersetLen i s) (powersetLen j s))\n[PROOFSTEP]\nrefine' ext fun a => \u27e8fun ha => _, fun ha => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ns\u271d t s a : Finset \u03b1\nha : a \u2208 powerset s\n\u22a2 a \u2208\n disjiUnion (range (card s + 1)) (fun i => powersetLen i s)\n (_ : Set.Pairwise \u2191(range (card s + 1)) fun i j => _root_.Disjoint (powersetLen i s) (powersetLen j s))\n[PROOFSTEP]\nrw [mem_disjiUnion]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ns\u271d t s a : Finset \u03b1\nha : a \u2208 powerset s\n\u22a2 \u2203 a_1, a_1 \u2208 range (card s + 1) \u2227 a \u2208 powersetLen a_1 s\n[PROOFSTEP]\nexact\n \u27e8a.card, mem_range.mpr (Nat.lt_succ_of_le (card_le_of_subset (mem_powerset.mp ha))),\n mem_powersetLen.mpr \u27e8mem_powerset.mp ha, rfl\u27e9\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ns\u271d t s a : Finset \u03b1\nha :\n a \u2208\n disjiUnion (range (card s + 1)) (fun i => powersetLen i s)\n (_ : Set.Pairwise \u2191(range (card s + 1)) fun i j => _root_.Disjoint (powersetLen i s) (powersetLen j s))\n\u22a2 a \u2208 powerset s\n[PROOFSTEP]\nrcases mem_disjiUnion.mp ha with \u27e8i, _hi, ha\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\ns\u271d t s a : Finset \u03b1\nha\u271d :\n a \u2208\n disjiUnion (range (card s + 1)) (fun i => powersetLen i s)\n (_ : Set.Pairwise \u2191(range (card s + 1)) fun i j => _root_.Disjoint (powersetLen i s) (powersetLen j s))\ni : \u2115\n_hi : i \u2208 range (card s + 1)\nha : a \u2208 powersetLen i s\n\u22a2 a \u2208 powerset s\n[PROOFSTEP]\nexact mem_powerset.mpr (mem_powersetLen.mp ha).1\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq (Finset \u03b1)\ns : Finset \u03b1\n\u22a2 powerset s = Finset.biUnion (range (card s + 1)) fun i => powersetLen i s\n[PROOFSTEP]\nsimpa only [disjiUnion_eq_biUnion] using powerset_card_disjiUnion s\n[GOAL]\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetLen (Nat.succ n) u) id = u\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 sup (powersetLen (Nat.succ n) u) id \u2264 u\n[PROOFSTEP]\nsimp_rw [Finset.sup_le_iff, mem_powersetLen]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 (b : Finset \u03b1), b \u2286 u \u2227 card b = Nat.succ n \u2192 id b \u2264 u\n[PROOFSTEP]\nrintro x \u27e8h, -\u27e9\n[GOAL]\ncase a.intro\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nx : Finset \u03b1\nh : x \u2286 u\n\u22a2 id x \u2264 u\n[PROOFSTEP]\nexact h\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 u \u2264 sup (powersetLen (Nat.succ n) u) id\n[PROOFSTEP]\nrw [sup_eq_biUnion, le_iff_subset, subset_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetLen (Nat.succ n) u) id\n[PROOFSTEP]\ncases' (Nat.succ_le_of_lt hn).eq_or_lt with h' h'\n[GOAL]\ncase a.inl\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n = card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetLen (Nat.succ n) u) id\n[PROOFSTEP]\nsimp [h']\n[GOAL]\ncase a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 u \u2192 x \u2208 Finset.biUnion (powersetLen (Nat.succ n) u) id\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 x \u2208 Finset.biUnion (powersetLen (Nat.succ n) u) id\n[PROOFSTEP]\nsimp only [mem_biUnion, exists_prop, id.def]\n[GOAL]\ncase a.inr\n\u03b1 : Type u_1\ns t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\n\u22a2 \u2203 a, a \u2208 powersetLen (Nat.succ n) u \u2227 x \u2208 a\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 : \u2203 t, t \u2208 powersetLen n (u.erase x) :=\n powersetLen_nonempty (le_trans (Nat.le_pred_of_lt hn) pred_card_le_card_erase)\n[GOAL]\ncase a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetLen n (erase u x)\n\u22a2 \u2203 a, a \u2208 powersetLen (Nat.succ n) u \u2227 x \u2208 a\n[PROOFSTEP]\nrefine' \u27e8insert x t, _, mem_insert_self _ _\u27e9\n[GOAL]\ncase a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetLen n (erase u x)\n\u22a2 insert x t \u2208 powersetLen (Nat.succ n) u\n[PROOFSTEP]\nrw [\u2190 insert_erase hx, powersetLen_succ_insert (not_mem_erase _ _)]\n[GOAL]\ncase a.inr.intro\n\u03b1 : Type u_1\ns t\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\nu : Finset \u03b1\nn : \u2115\nhn : n < card u\nh' : Nat.succ n < card u\nx : \u03b1\nhx : x \u2208 u\nt : Finset \u03b1\nht : t \u2208 powersetLen n (erase u x)\n\u22a2 insert x t \u2208 powersetLen (Nat.succ n) (erase u x) \u222a image (insert x) (powersetLen n (erase u x))\n[PROOFSTEP]\nexact mem_union_right _ (mem_image_of_mem _ ht)\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t s : Finset \u03b1\ni : \u2115\n\u22a2 Multiset.map val (powersetLen i s).val = Multiset.powersetLen i s.val\n[PROOFSTEP]\nsimp [Finset.powersetLen, map_pmap, pmap_eq_map, map_id']\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 t \u2208 powersetLen n (map f s) \u2194 t \u2208 map (mapEmbedding f).toEmbedding (powersetLen n s)\n[PROOFSTEP]\nsimp only [card_map, mem_powersetLen, le_eq_subset, gt_iff_lt, mem_map, mapEmbedding_apply]\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 t \u2286 map f s \u2227 card t = n \u2194 \u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 t \u2286 map f s \u2227 card t = n \u2192 \u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t\n[PROOFSTEP]\nclassical\nintro h\nhave : map f (filter (fun x => (f x \u2208 t)) s) = t := by\n ext x\n simp only [mem_map, mem_filter, decide_eq_true_eq]\n exact\n \u27e8fun \u27e8_y, \u27e8_hy\u2081, hy\u2082\u27e9, hy\u2083\u27e9 => hy\u2083 \u25b8 hy\u2082, fun hx =>\n let \u27e8y, hy\u27e9 := mem_map.1 (h.1 hx);\n \u27e8y, \u27e8hy.1, hy.2 \u25b8 hx\u27e9, hy.2\u27e9\u27e9\nrefine' \u27e8_, _, this\u27e9\nrw [\u2190 card_map f, this, h.2]; simp\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 t \u2286 map f s \u2227 card t = n \u2192 \u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\n\u22a2 \u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t\n[PROOFSTEP]\nhave : map f (filter (fun x => (f x \u2208 t)) s) = t := by\n ext x\n simp only [mem_map, mem_filter, decide_eq_true_eq]\n exact\n \u27e8fun \u27e8_y, \u27e8_hy\u2081, hy\u2082\u27e9, hy\u2083\u27e9 => hy\u2083 \u25b8 hy\u2082, fun hx =>\n let \u27e8y, hy\u27e9 := mem_map.1 (h.1 hx);\n \u27e8y, \u27e8hy.1, hy.2 \u25b8 hx\u27e9, hy.2\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\n\u22a2 map f (filter (fun x => \u2191f x \u2208 t) s) = t\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\nx : \u03b2\n\u22a2 x \u2208 map f (filter (fun x => \u2191f x \u2208 t) s) \u2194 x \u2208 t\n[PROOFSTEP]\nsimp only [mem_map, mem_filter, decide_eq_true_eq]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\nx : \u03b2\n\u22a2 (\u2203 a, (a \u2208 s \u2227 \u2191f a \u2208 t) \u2227 \u2191f a = x) \u2194 x \u2208 t\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8_y, \u27e8_hy\u2081, hy\u2082\u27e9, hy\u2083\u27e9 => hy\u2083 \u25b8 hy\u2082, fun hx =>\n let \u27e8y, hy\u27e9 := mem_map.1 (h.1 hx);\n \u27e8y, \u27e8hy.1, hy.2 \u25b8 hx\u27e9, hy.2\u27e9\u27e9\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\nthis : map f (filter (fun x => \u2191f x \u2208 t) s) = t\n\u22a2 \u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t\n[PROOFSTEP]\nrefine' \u27e8_, _, this\u27e9\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\nthis : map f (filter (fun x => \u2191f x \u2208 t) s) = t\n\u22a2 filter (fun x => \u2191f x \u2208 t) s \u2286 s \u2227 card (filter (fun x => \u2191f x \u2208 t) s) = n\n[PROOFSTEP]\nrw [\u2190 card_map f, this, h.2]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\nh : t \u2286 map f s \u2227 card t = n\nthis : map f (filter (fun x => \u2191f x \u2208 t) s) = t\n\u22a2 filter (fun x => \u2191f x \u2208 t) s \u2286 s \u2227 n = n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ns\u271d t\u271d : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\nn : \u2115\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 (\u2203 a, (a \u2286 s \u2227 card a = n) \u2227 \u2191(mapEmbedding f).toEmbedding a = t) \u2192 t \u2286 map f s \u2227 card t = n\n[PROOFSTEP]\nrintro \u27e8a, \u27e8has, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\ns a : Finset \u03b1\nhas : a \u2286 s\n\u22a2 \u2191(mapEmbedding f).toEmbedding a \u2286 map f s \u2227 card (\u2191(mapEmbedding f).toEmbedding a) = card a\n[PROOFSTEP]\ndsimp [RelEmbedding.coe_toEmbedding]\n --Porting note: Why is `rw` required here and not `simp`?\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\ns a : Finset \u03b1\nhas : a \u2286 s\n\u22a2 \u2191(mapEmbedding f) a \u2286 map f s \u2227 card (\u2191(mapEmbedding f) a) = card a\n[PROOFSTEP]\nrw [mapEmbedding_apply]\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\n\u03b2 : Type u_2\nf : \u03b1 \u21aa \u03b2\ns a : Finset \u03b1\nhas : a \u2286 s\n\u22a2 map f a \u2286 map f s \u2227 card (map f a) = card a\n[PROOFSTEP]\nsimp [has]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Powerset", "llama_tokens": 11425, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.25787288950272114}} {"text": "[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\n\u22a2 Linear k (FdRep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 AddCommGroup (CoeSort.coe V)\n[PROOFSTEP]\nchange AddCommGroup ((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V).obj\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 AddCommGroup \u2191((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V).obj\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 Module k (CoeSort.coe V)\n[PROOFSTEP]\nchange Module k ((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V).obj\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 Module k \u2191((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V).obj\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 FiniteDimensional k (CoeSort.coe V)\n[PROOFSTEP]\nchange FiniteDimensional k ((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V)\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 FiniteDimensional k \u2191((forget\u2082 (FdRep k G) (FGModuleCat k)).obj V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 \u2191(\u03c1 W) g = \u2191(LinearEquiv.conj (isoToLinearEquiv i)) (\u2191(\u03c1 V) g)\n[PROOFSTEP]\nerw [FdRep.isoToLinearEquiv, \u2190 FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 \u2191(\u03c1 W) g =\n ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv \u226b\n \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom\n[PROOFSTEP]\nrw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom \u226b \u2191(\u03c1 W) g =\n \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom\n[PROOFSTEP]\nexact (i.hom.comm g).symm\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 Rep.\u03c1 ((forget\u2082 (FdRep k G) (Rep k G)).obj V) = \u03c1 V\n[PROOFSTEP]\next g v\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV : FdRep k G\ng : G\nv : CoeSort.coe ((forget\u2082 (FdRep k G) (Rep k G)).obj V)\n\u22a2 \u2191(\u2191(Rep.\u03c1 ((forget\u2082 (FdRep k G) (Rep k G)).obj V)) g) v = \u2191(\u2191(\u03c1 V) g) v\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\n\u22a2 MonoidalCategory (FdRep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\n\u22a2 MonoidalPreadditive (FdRep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\n\u22a2 MonoidalLinear k (FdRep k G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\n\u22a2 HasKernels (FdRep k G)\n[PROOFSTEP]\ninfer_instance\n -- Verify that Schur's lemma applies out of the box.\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nX Y : FdRep k G\nx\u271d : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y\n\u22a2 (fun f => Action.Hom.mk ((forget\u2082 (FGModuleCat k) (ModuleCat k)).map f.hom))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : k) (x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_2 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_2 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nX Y : FdRep k G\nx\u271d\u00b9 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y\nx\u271d : \u2191((forget\u2082 (FdRep k G) (Rep k G)).obj X).V\n\u22a2 \u2191((fun f => Action.Hom.mk ((forget\u2082 (FGModuleCat k) (ModuleCat k)).map f.hom))\n (AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : k) (x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200\n (x x_2 :\n (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) =\n (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200\n (x x_2 :\n (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) =\n (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n x\u271d\u00b9)).hom\n x\u271d =\n \u2191x\u271d\u00b9.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nX Y : FdRep k G\nx\u271d : X \u27f6 Y\n\u22a2 AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : k) (x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_2 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_2 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n ((fun f => Action.Hom.mk ((forget\u2082 (FGModuleCat k) (ModuleCat k)).map f.hom)) x\u271d) =\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.w\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nX Y : FdRep k G\nx\u271d\u00b9 : X \u27f6 Y\nx\u271d : (forget (FGModuleCat k)).obj X.V\n\u22a2 \u2191(AddHom.toFun\n {\n toAddHom :=\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200 (x x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_1) = (fun f => Action.Hom.mk f.hom) (x + x_1)) },\n map_smul' :=\n (_ :\n \u2200 (x : k) (x_1 : (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200\n (x x_2 :\n (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1) =\n AddHom.toFun\n { toFun := fun f => Action.Hom.mk f.hom,\n map_add' :=\n (_ :\n \u2200\n (x x_2 :\n (forget\u2082 (FdRep k G) (Rep k G)).obj X \u27f6 (forget\u2082 (FdRep k G) (Rep k G)).obj Y),\n (fun f => Action.Hom.mk f.hom) (x + x_2) = (fun f => Action.Hom.mk f.hom) (x + x_2)) }\n (x \u2022 x_1)) }.toAddHom\n ((fun f => Action.Hom.mk ((forget\u2082 (FGModuleCat k) (ModuleCat k)).map f.hom)) x\u271d\u00b9)).hom\n x\u271d =\n \u2191x\u271d\u00b9.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Group G\n\u22a2 RightRigidCategory (FdRep k G)\n[PROOFSTEP]\nchange RightRigidCategory (Action (FGModuleCat k) (GroupCat.of G))\n[GOAL]\nk G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Group G\n\u22a2 RightRigidCategory (Action (FGModuleCat k) ((forget\u2082 GroupCat MonCat).obj (GroupCat.of G)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk G V : Type u\ninst\u271d\u2074 : Field k\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : FiniteDimensional k V\n\u03c1V : Representation k G V\nW : FdRep k G\n\u22a2 of (dual \u03c1V) \u2297 W \u2245 of (linHom \u03c1V (\u03c1 W))\n[PROOFSTEP]\nrefine Action.mkIso (dualTensorIsoLinHomAux \u03c1V W) ?_\n[GOAL]\nk G V : Type u\ninst\u271d\u2074 : Field k\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : FiniteDimensional k V\n\u03c1V : Representation k G V\nW : FdRep k G\n\u22a2 \u2200 (g : \u2191(MonCat.of G)),\n \u2191(of (dual \u03c1V) \u2297 W).\u03c1 g \u226b (dualTensorIsoLinHomAux \u03c1V W).hom =\n (dualTensorIsoLinHomAux \u03c1V W).hom \u226b \u2191(of (linHom \u03c1V (\u03c1 W))).\u03c1 g\n[PROOFSTEP]\nconvert dualTensorHom_comm \u03c1V W.\u03c1\n", "meta": {"mathlib_filename": "Mathlib.RepresentationTheory.FdRep", "llama_tokens": 4528, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442251064863695, "lm_q2_score": 0.399811640739795, "lm_q1q2_score": 0.2575686968300846}} {"text": "[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 id <$> x = x\n[PROOFSTEP]\nrw [\u2190 abs_repr x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 id <$> abs (repr x) = abs (repr x)\n[PROOFSTEP]\ncases' repr x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 id <$> abs { fst := a, snd := f } = abs { fst := a, snd := f }\n[PROOFSTEP]\nrw [\u2190 abs_map]\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 abs (id <$> { fst := a, snd := f }) = abs { fst := a, snd := f }\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : F \u03b1\n\u22a2 (g \u2218 f) <$> x = g <$> f <$> x\n[PROOFSTEP]\nrw [\u2190 abs_repr x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : F \u03b1\n\u22a2 (g \u2218 f) <$> abs (repr x) = g <$> f <$> abs (repr x)\n[PROOFSTEP]\ncases' repr x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 \u03b2 \u03b3 : Type u\nf\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 (g \u2218 f\u271d) <$> abs { fst := a, snd := f } = g <$> f\u271d <$> abs { fst := a, snd := f }\n[PROOFSTEP]\nrw [\u2190 abs_map, \u2190 abs_map, \u2190 abs_map]\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 \u03b2 \u03b3 : Type u\nf\u271d : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 abs ((g \u2218 f\u271d) <$> { fst := a, snd := f }) = abs (g <$> f\u271d <$> { fst := a, snd := f })\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2194 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2192 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mp.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\ncases' h : repr y with a f\n[GOAL]\ncase mp.intro.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nuse a, fun i => (f i).val\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i => \u2191(f i) } \u2227 \u2200 (i : PFunctor.B (P F) a), p \u2191(f i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i => \u2191(f i) }\n[PROOFSTEP]\nrw [\u2190 hy, \u2190 abs_repr y, h, \u2190 abs_map]\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs (Subtype.val <$> { fst := a, snd := f }) = abs { fst := a, snd := fun i => \u2191(f i) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2200 (i : PFunctor.B (P F) a), p \u2191(f i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\ni : PFunctor.B (P F) a\n\u22a2 p \u2191(f i)\n[PROOFSTEP]\napply (f i).property\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)) \u2192 Liftp p x\n[PROOFSTEP]\nrintro \u27e8a, f, h\u2080, h\u2081\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 Liftp p x\n[PROOFSTEP]\nuse abs \u27e8a, fun i => \u27e8f i, h\u2081 i\u27e9\u27e9\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 Subtype.val <$> abs { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = x\n[PROOFSTEP]\nrw [\u2190 abs_map, h\u2080]\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : x = abs { fst := a, snd := f }\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 abs (Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } }) =\n abs { fst := a, snd := f }\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2194 \u2203 u, abs u = x \u2227 \u2200 (i : PFunctor.B (P F) u.fst), p (Sigma.snd u i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 Liftp p x \u2192 \u2203 u, abs u = x \u2227 \u2200 (i : PFunctor.B (P F) u.fst), p (Sigma.snd u i)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mp.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\n\u22a2 \u2203 u, abs u = x \u2227 \u2200 (i : PFunctor.B (P F) u.fst), p (Sigma.snd u i)\n[PROOFSTEP]\ncases' h : repr y with a f\n[GOAL]\ncase mp.intro.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2203 u, abs u = x \u2227 \u2200 (i : PFunctor.B (P F) u.fst), p (Sigma.snd u i)\n[PROOFSTEP]\nuse\u27e8a, fun i => (f i).val\u27e9\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs { fst := a, snd := fun i => \u2191(f i) } = x \u2227\n \u2200 (i : PFunctor.B (P F) { fst := a, snd := fun i => \u2191(f i) }.fst),\n p (Sigma.snd { fst := a, snd := fun i => \u2191(f i) } i)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs { fst := a, snd := fun i => \u2191(f i) } = x \u2227 \u2200 (i : PFunctor.B (P F) a), p \u2191(f i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs { fst := a, snd := fun i => \u2191(f i) } = x\n[PROOFSTEP]\nrw [\u2190 hy, \u2190 abs_repr y, h, \u2190 abs_map]\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 abs { fst := a, snd := fun i => \u2191(f i) } = abs (Subtype.val <$> { fst := a, snd := f })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\n\u22a2 \u2200 (i : PFunctor.B (P F) a), p \u2191(f i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\ny : F (Subtype p)\nhy : Subtype.val <$> y = x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Subtype p\nh : repr y = { fst := a, snd := f }\ni : PFunctor.B (P F) a\n\u22a2 p \u2191(f i)\n[PROOFSTEP]\napply (f i).property\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\n\u22a2 (\u2203 u, abs u = x \u2227 \u2200 (i : PFunctor.B (P F) u.fst), p (Sigma.snd u i)) \u2192 Liftp p x\n[PROOFSTEP]\nrintro \u27e8\u27e8a, f\u27e9, h\u2080, h\u2081\u27e9\n[GOAL]\ncase mpr.intro.mk.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : abs { fst := a, snd := f } = x\nh\u2081 : \u2200 (i : PFunctor.B (P F) { fst := a, snd := f }.fst), p (Sigma.snd { fst := a, snd := f } i)\n\u22a2 Liftp p x\n[PROOFSTEP]\ndsimp at *\n[GOAL]\ncase mpr.intro.mk.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : abs { fst := a, snd := f } = x\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 Liftp p x\n[PROOFSTEP]\nuse abs \u27e8a, fun i => \u27e8f i, h\u2081 i\u27e9\u27e9\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : abs { fst := a, snd := f } = x\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 Subtype.val <$> abs { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = x\n[PROOFSTEP]\nrw [\u2190 abs_map, \u2190 h\u2080]\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh\u2080 : abs { fst := a, snd := f } = x\nh\u2081 : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 abs (Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } }) =\n abs { fst := a, snd := f }\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\n\u22a2 Liftr r x y \u2194\n \u2203 a f\u2080 f\u2081,\n x = abs { fst := a, snd := f\u2080 } \u2227 y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\n\u22a2 Liftr r x y \u2192\n \u2203 a f\u2080 f\u2081,\n x = abs { fst := a, snd := f\u2080 } \u2227 y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\nrintro \u27e8u, xeq, yeq\u27e9\n[GOAL]\ncase mp.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\n\u22a2 \u2203 a f\u2080 f\u2081,\n x = abs { fst := a, snd := f\u2080 } \u2227 y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\ncases' h : repr u with a f\n[GOAL]\ncase mp.intro.intro.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 \u2203 a f\u2080 f\u2081,\n x = abs { fst := a, snd := f\u2080 } \u2227 y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\nuse a, fun i => (f i).val.fst, fun i => (f i).val.snd\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i => (\u2191(f i)).fst } \u2227\n y = abs { fst := a, snd := fun i => (\u2191(f i)).snd } \u2227 \u2200 (i : PFunctor.B (P F) a), r (\u2191(f i)).fst (\u2191(f i)).snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 x = abs { fst := a, snd := fun i => (\u2191(f i)).fst }\n[PROOFSTEP]\nrw [\u2190 xeq, \u2190 abs_repr u, h, \u2190 abs_map]\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 abs ((fun t => (\u2191t).fst) <$> { fst := a, snd := f }) = abs { fst := a, snd := fun i => (\u2191(f i)).fst }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 y = abs { fst := a, snd := fun i => (\u2191(f i)).snd } \u2227 \u2200 (i : PFunctor.B (P F) a), r (\u2191(f i)).fst (\u2191(f i)).snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.right.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 y = abs { fst := a, snd := fun i => (\u2191(f i)).snd }\n[PROOFSTEP]\nrw [\u2190 yeq, \u2190 abs_repr u, h, \u2190 abs_map]\n[GOAL]\ncase h.right.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 abs ((fun t => (\u2191t).snd) <$> { fst := a, snd := f }) = abs { fst := a, snd := fun i => (\u2191(f i)).snd }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\n\u22a2 \u2200 (i : PFunctor.B (P F) a), r (\u2191(f i)).fst (\u2191(f i)).snd\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.right.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\nu : F { p // r p.fst p.snd }\nxeq : (fun t => (\u2191t).fst) <$> u = x\nyeq : (fun t => (\u2191t).snd) <$> u = y\na : (P F).A\nf : PFunctor.B (P F) a \u2192 { p // r p.fst p.snd }\nh : repr u = { fst := a, snd := f }\ni : PFunctor.B (P F) a\n\u22a2 r (\u2191(f i)).fst (\u2191(f i)).snd\n[PROOFSTEP]\nexact (f i).property\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\n\u22a2 (\u2203 a f\u2080 f\u2081,\n x = abs { fst := a, snd := f\u2080 } \u2227 y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)) \u2192\n Liftr r x y\n[PROOFSTEP]\nrintro \u27e8a, f\u2080, f\u2081, xeq, yeq, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 Liftr r x y\n[PROOFSTEP]\nuse abs \u27e8a, fun i => \u27e8(f\u2080 i, f\u2081 i), h i\u27e9\u27e9\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (fun t => (\u2191t).fst) <$> abs { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } } =\n x \u2227\n (fun t => (\u2191t).snd) <$>\n abs { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } } =\n y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (fun t => (\u2191t).fst) <$> abs { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } } =\n x\n[PROOFSTEP]\nrw [xeq, \u2190 abs_map]\n[GOAL]\ncase h.left\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 abs\n ((fun t => (\u2191t).fst) <$>\n { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } }) =\n abs { fst := a, snd := f\u2080 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (fun t => (\u2191t).snd) <$> abs { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } } =\n y\n[PROOFSTEP]\nrw [yeq, \u2190 abs_map]\n[GOAL]\ncase h.right\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx y : F \u03b1\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f\u2080 }\nyeq : y = abs { fst := a, snd := f\u2081 }\nh : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 abs\n ((fun t => (\u2191t).snd) <$>\n { fst := a, snd := fun i => { val := (f\u2080 i, f\u2081 i), property := (_ : r (f\u2080 i) (f\u2081 i)) } }) =\n abs { fst := a, snd := f\u2081 }\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : PFunctor.W (P F)\n\u22a2 recF g x = g (abs (recF g <$> PFunctor.W.dest x))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\na\u271d : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d \u2192 WType (P F).B\n\u22a2 recF g (WType.mk a\u271d f\u271d) = g (abs (recF g <$> PFunctor.W.dest (WType.mk a\u271d f\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\n\u22a2 Wequiv x y \u2192 recF u x = recF u y\n[PROOFSTEP]\nintro h\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\nh : Wequiv x y\n\u22a2 recF u x = recF u y\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase ind\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d f'\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a\u271d\u00b9), Wequiv (f\u271d x) (f'\u271d x)\na_ih\u271d : \u2200 (x : PFunctor.B (P F) a\u271d\u00b9), recF u (f\u271d x) = recF u (f'\u271d x)\n\u22a2 recF u (WType.mk a\u271d\u00b9 f\u271d) = recF u (WType.mk a\u271d\u00b9 f'\u271d)\ncase abs\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na'\u271d : (P F).A\nf'\u271d : PFunctor.B (P F) a'\u271d \u2192 PFunctor.W (P F)\na\u271d : abs { fst := a\u271d\u00b9, snd := f\u271d } = abs { fst := a'\u271d, snd := f'\u271d }\n\u22a2 recF u (WType.mk a\u271d\u00b9 f\u271d) = recF u (WType.mk a'\u271d f'\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : recF u u\u271d = recF u v\u271d\na_ih\u271d : recF u v\u271d = recF u w\u271d\n\u22a2 recF u u\u271d = recF u w\u271d\n[PROOFSTEP]\ncase ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a), Wequiv (f x) (f' x)\nih : \u2200 (x : PFunctor.B (P F) a), recF u (f x) = recF u (f' x)\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a f')\n[PROOFSTEP]\ncase ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a), Wequiv (f x) (f' x)\nih : \u2200 (x : PFunctor.B (P F) a), recF u (f x) = recF u (f' x)\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a f')\n[PROOFSTEP]\nsimp only [recF_eq', PFunctor.map_eq, Function.comp, ih]\n[GOAL]\ncase abs\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na'\u271d : (P F).A\nf'\u271d : PFunctor.B (P F) a'\u271d \u2192 PFunctor.W (P F)\na\u271d : abs { fst := a\u271d\u00b9, snd := f\u271d } = abs { fst := a'\u271d, snd := f'\u271d }\n\u22a2 recF u (WType.mk a\u271d\u00b9 f\u271d) = recF u (WType.mk a'\u271d f'\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : recF u u\u271d = recF u v\u271d\na_ih\u271d : recF u v\u271d = recF u w\u271d\n\u22a2 recF u u\u271d = recF u w\u271d\n[PROOFSTEP]\ncase abs a f a' f' h => simp only [recF_eq', abs_map, h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 PFunctor.W (P F)\nh : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a' f')\n[PROOFSTEP]\ncase abs a f a' f' h => simp only [recF_eq', abs_map, h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y : PFunctor.W (P F)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 PFunctor.W (P F)\nh : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\n\u22a2 recF u (WType.mk a f) = recF u (WType.mk a' f')\n[PROOFSTEP]\nsimp only [recF_eq', abs_map, h]\n[GOAL]\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : recF u u\u271d = recF u v\u271d\na_ih\u271d : recF u v\u271d = recF u w\u271d\n\u22a2 recF u u\u271d = recF u w\u271d\n[PROOFSTEP]\ncase trans x y z _ _ ih\u2081 ih\u2082 => exact Eq.trans ih\u2081 ih\u2082\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx\u271d y\u271d x y z : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv x y\na\u271d : Wequiv y z\nih\u2081 : recF u x = recF u y\nih\u2082 : recF u y = recF u z\n\u22a2 recF u x = recF u z\n[PROOFSTEP]\ncase trans x y z _ _ ih\u2081 ih\u2082 => exact Eq.trans ih\u2081 ih\u2082\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nu : F \u03b1 \u2192 \u03b1\nx\u271d y\u271d x y z : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv x y\na\u271d : Wequiv y z\nih\u2081 : recF u x = recF u y\nih\u2082 : recF u y = recF u z\n\u22a2 recF u x = recF u z\n[PROOFSTEP]\nexact Eq.trans ih\u2081 ih\u2082\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\nh : Qpf.abs (PFunctor.W.dest x) = Qpf.abs (PFunctor.W.dest y)\n\u22a2 Wequiv x y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\ny : PFunctor.W (P F)\na\u271d : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d \u2192 WType (P F).B\nh : Qpf.abs (PFunctor.W.dest (WType.mk a\u271d f\u271d)) = Qpf.abs (PFunctor.W.dest y)\n\u22a2 Wequiv (WType.mk a\u271d f\u271d) y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na\u271d\u00b9 : (P F).A\nf\u271d\u00b9 : PFunctor.B (P F) a\u271d\u00b9 \u2192 WType (P F).B\na\u271d : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d \u2192 WType (P F).B\nh : Qpf.abs (PFunctor.W.dest (WType.mk a\u271d\u00b9 f\u271d\u00b9)) = Qpf.abs (PFunctor.W.dest (WType.mk a\u271d f\u271d))\n\u22a2 Wequiv (WType.mk a\u271d\u00b9 f\u271d\u00b9) (WType.mk a\u271d f\u271d)\n[PROOFSTEP]\napply Wequiv.abs\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na\u271d\u00b9 : (P F).A\nf\u271d\u00b9 : PFunctor.B (P F) a\u271d\u00b9 \u2192 WType (P F).B\na\u271d : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d \u2192 WType (P F).B\nh : Qpf.abs (PFunctor.W.dest (WType.mk a\u271d\u00b9 f\u271d\u00b9)) = Qpf.abs (PFunctor.W.dest (WType.mk a\u271d f\u271d))\n\u22a2 Qpf.abs { fst := a\u271d\u00b9, snd := f\u271d\u00b9 } = Qpf.abs { fst := a\u271d, snd := f\u271d }\n[PROOFSTEP]\napply h\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : PFunctor.W (P F)\n\u22a2 Wequiv x x\n[PROOFSTEP]\ncases' x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\n\u22a2 Wequiv (WType.mk a f) (WType.mk a f)\n[PROOFSTEP]\nexact Wequiv.abs a f a f rfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\n\u22a2 Wequiv x y \u2192 Wequiv y x\n[PROOFSTEP]\nintro h\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\nh : Wequiv x y\n\u22a2 Wequiv y x\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase ind\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d f'\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a\u271d\u00b9), Wequiv (f\u271d x) (f'\u271d x)\na_ih\u271d : \u2200 (x : PFunctor.B (P F) a\u271d\u00b9), Wequiv (f'\u271d x) (f\u271d x)\n\u22a2 Wequiv (WType.mk a\u271d\u00b9 f'\u271d) (WType.mk a\u271d\u00b9 f\u271d)\ncase abs\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na'\u271d : (P F).A\nf'\u271d : PFunctor.B (P F) a'\u271d \u2192 PFunctor.W (P F)\na\u271d : Qpf.abs { fst := a\u271d\u00b9, snd := f\u271d } = Qpf.abs { fst := a'\u271d, snd := f'\u271d }\n\u22a2 Wequiv (WType.mk a'\u271d f'\u271d) (WType.mk a\u271d\u00b9 f\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : Wequiv v\u271d u\u271d\na_ih\u271d : Wequiv w\u271d v\u271d\n\u22a2 Wequiv w\u271d u\u271d\n[PROOFSTEP]\ncase ind a f f' _ ih => exact Wequiv.ind _ _ _ ih\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a), Wequiv (f x) (f' x)\nih : \u2200 (x : PFunctor.B (P F) a), Wequiv (f' x) (f x)\n\u22a2 Wequiv (WType.mk a f') (WType.mk a f)\n[PROOFSTEP]\ncase ind a f f' _ ih => exact Wequiv.ind _ _ _ ih\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na\u271d : \u2200 (x : PFunctor.B (P F) a), Wequiv (f x) (f' x)\nih : \u2200 (x : PFunctor.B (P F) a), Wequiv (f' x) (f x)\n\u22a2 Wequiv (WType.mk a f') (WType.mk a f)\n[PROOFSTEP]\nexact Wequiv.ind _ _ _ ih\n[GOAL]\ncase abs\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na\u271d\u00b9 : (P F).A\nf\u271d : PFunctor.B (P F) a\u271d\u00b9 \u2192 PFunctor.W (P F)\na'\u271d : (P F).A\nf'\u271d : PFunctor.B (P F) a'\u271d \u2192 PFunctor.W (P F)\na\u271d : Qpf.abs { fst := a\u271d\u00b9, snd := f\u271d } = Qpf.abs { fst := a'\u271d, snd := f'\u271d }\n\u22a2 Wequiv (WType.mk a'\u271d f'\u271d) (WType.mk a\u271d\u00b9 f\u271d)\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : Wequiv v\u271d u\u271d\na_ih\u271d : Wequiv w\u271d v\u271d\n\u22a2 Wequiv w\u271d u\u271d\n[PROOFSTEP]\ncase abs a f a' f' h => exact Wequiv.abs _ _ _ _ h.symm\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 PFunctor.W (P F)\nh : Qpf.abs { fst := a, snd := f } = Qpf.abs { fst := a', snd := f' }\n\u22a2 Wequiv (WType.mk a' f') (WType.mk a f)\n[PROOFSTEP]\ncase abs a f a' f' h => exact Wequiv.abs _ _ _ _ h.symm\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 PFunctor.W (P F)\nh : Qpf.abs { fst := a, snd := f } = Qpf.abs { fst := a', snd := f' }\n\u22a2 Wequiv (WType.mk a' f') (WType.mk a f)\n[PROOFSTEP]\nexact Wequiv.abs _ _ _ _ h.symm\n[GOAL]\ncase trans\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y u\u271d v\u271d w\u271d : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv u\u271d v\u271d\na\u271d : Wequiv v\u271d w\u271d\na_ih\u271d\u00b9 : Wequiv v\u271d u\u271d\na_ih\u271d : Wequiv w\u271d v\u271d\n\u22a2 Wequiv w\u271d u\u271d\n[PROOFSTEP]\ncase trans x y z _ _ ih\u2081 ih\u2082 => exact Qpf.Wequiv.trans _ _ _ ih\u2082 ih\u2081\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d y\u271d x y z : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv x y\na\u271d : Wequiv y z\nih\u2081 : Wequiv y x\nih\u2082 : Wequiv z y\n\u22a2 Wequiv z x\n[PROOFSTEP]\ncase trans x y z _ _ ih\u2081 ih\u2082 => exact Qpf.Wequiv.trans _ _ _ ih\u2082 ih\u2081\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d y\u271d x y z : PFunctor.W (P F)\na\u271d\u00b9 : Wequiv x y\na\u271d : Wequiv y z\nih\u2081 : Wequiv y x\nih\u2082 : Wequiv z y\n\u22a2 Wequiv z x\n[PROOFSTEP]\nexact Qpf.Wequiv.trans _ _ _ ih\u2082 ih\u2081\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : PFunctor.W (P F)\n\u22a2 Wequiv (Wrepr x) x\n[PROOFSTEP]\ninduction' x with a f ih\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 Wequiv (Wrepr (WType.mk a f)) (WType.mk a f)\n[PROOFSTEP]\napply Wequiv.trans\n[GOAL]\ncase mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 Wequiv (Wrepr (WType.mk a f)) ?mk.v\n[PROOFSTEP]\nchange Wequiv (Wrepr \u27e8a, f\u27e9) (PFunctor.W.mk (Wrepr <$> \u27e8a, f\u27e9))\n[GOAL]\ncase mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 Wequiv (Wrepr (WType.mk a f)) (PFunctor.W.mk (Wrepr <$> { fst := a, snd := f }))\n[PROOFSTEP]\napply Wequiv.abs'\n[GOAL]\ncase mk.a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 abs (PFunctor.W.dest (Wrepr (WType.mk a f))) =\n abs (PFunctor.W.dest (PFunctor.W.mk (Wrepr <$> { fst := a, snd := f })))\n[PROOFSTEP]\nhave : Wrepr \u27e8a, f\u27e9 = PFunctor.W.mk (repr (abs (Wrepr <$> \u27e8a, f\u27e9))) := rfl\n[GOAL]\ncase mk.a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\nthis : Wrepr (WType.mk a f) = PFunctor.W.mk (repr (abs (Wrepr <$> { fst := a, snd := f })))\n\u22a2 abs (PFunctor.W.dest (Wrepr (WType.mk a f))) =\n abs (PFunctor.W.dest (PFunctor.W.mk (Wrepr <$> { fst := a, snd := f })))\n[PROOFSTEP]\nrw [this, PFunctor.W.dest_mk, abs_repr]\n[GOAL]\ncase mk.a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\nthis : Wrepr (WType.mk a f) = PFunctor.W.mk (repr (abs (Wrepr <$> { fst := a, snd := f })))\n\u22a2 abs (Wrepr <$> { fst := a, snd := f }) = abs (PFunctor.W.dest (PFunctor.W.mk (Wrepr <$> { fst := a, snd := f })))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 Wequiv (PFunctor.W.mk (Wrepr <$> { fst := a, snd := f })) (WType.mk a f)\n[PROOFSTEP]\napply Wequiv.ind\n[GOAL]\ncase mk.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), Wequiv (Wrepr (f a)) (f a)\n\u22a2 \u2200 (x : PFunctor.B (P F) a), Wequiv ((Wrepr \u2218 f) x) (f x)\n[PROOFSTEP]\nexact ih\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 rec g (mk x) = g (rec g <$> x)\n[PROOFSTEP]\nhave : recF g \u2218 fixToW = Fix.rec g := by\n apply funext\n apply Quotient.ind\n intro x\n apply recF_eq_of_Wequiv\n rw [fixToW]\n apply Wrepr_equiv\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 recF g \u2218 fixToW = rec g\n[PROOFSTEP]\napply funext\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (x : Fix F), (recF g \u2218 fixToW) x = rec g x\n[PROOFSTEP]\napply Quotient.ind\n[GOAL]\ncase h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\n\u22a2 \u2200 (a : PFunctor.W (P F)), (recF g \u2218 fixToW) (Quotient.mk Wsetoid a) = rec g (Quotient.mk Wsetoid a)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 (recF g \u2218 fixToW) (Quotient.mk Wsetoid x) = rec g (Quotient.mk Wsetoid x)\n[PROOFSTEP]\napply recF_eq_of_Wequiv\n[GOAL]\ncase h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv (fixToW (Quotient.mk Wsetoid x)) x\n[PROOFSTEP]\nrw [fixToW]\n[GOAL]\ncase h.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx\u271d : F (Fix F)\nx : PFunctor.W (P F)\n\u22a2 Wequiv\n (Quotient.lift Wrepr\n (_ :\n \u2200 (x y : PFunctor.W (P F)),\n Wequiv x y \u2192 recF (fun x => PFunctor.W.mk (repr x)) x = recF (fun x => PFunctor.W.mk (repr x)) y)\n (Quotient.mk Wsetoid x))\n x\n[PROOFSTEP]\napply Wrepr_equiv\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 rec g (mk x) = g (rec g <$> x)\n[PROOFSTEP]\nconv =>\n lhs\n rw [Fix.rec, Fix.mk]\n dsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n| rec g (mk x) = g (rec g <$> x)\n[PROOFSTEP]\n lhs\n rw [Fix.rec, Fix.mk]\n dsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n| rec g (mk x) = g (rec g <$> x)\n[PROOFSTEP]\n lhs\n rw [Fix.rec, Fix.mk]\n dsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n| rec g (mk x) = g (rec g <$> x)\n[PROOFSTEP]\nlhs\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n| rec g (mk x)\n[PROOFSTEP]\nrw [Fix.rec, Fix.mk]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n| Quot.lift (recF g) (_ : \u2200 (x y : PFunctor.W (P F)), Wequiv x y \u2192 recF g x = recF g y)\n (Quot.mk Setoid.r (PFunctor.W.mk (fixToW <$> repr x)))\n[PROOFSTEP]\ndsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> repr x)) = g (rec g <$> x)\n[PROOFSTEP]\ncases' h : repr x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nx : F (Fix F)\nthis : recF g \u2218 fixToW = rec g\na : (P F).A\nf : PFunctor.B (P F) a \u2192 Fix F\nh : repr x = { fst := a, snd := f }\n\u22a2 recF g (PFunctor.W.mk (fixToW <$> { fst := a, snd := f })) = g (rec g <$> x)\n[PROOFSTEP]\nrw [PFunctor.map_eq, recF_eq, \u2190 PFunctor.map_eq, PFunctor.W.dest_mk, \u2190 PFunctor.comp_map, abs_map, \u2190 h, abs_repr, this]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (WType.mk a f)\n[PROOFSTEP]\nhave : Fix.mk (abs \u27e8a, fun x => \u27e6f x\u27e7\u27e9) = \u27e6Wrepr \u27e8a, f\u27e9\u27e7 :=\n by\n apply Quot.sound; apply Wequiv.abs'\n rw [PFunctor.W.dest_mk, abs_map, abs_repr, \u2190 abs_map, PFunctor.map_eq]\n conv =>\n rhs\n simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 Setoid.r (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))\n (Wrepr (WType.mk a f))\n[PROOFSTEP]\napply Wequiv.abs'\n[GOAL]\ncase a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs\n (PFunctor.W.dest\n (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))) =\n abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\nrw [PFunctor.W.dest_mk, abs_map, abs_repr, \u2190 abs_map, PFunctor.map_eq]\n[GOAL]\ncase a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\nconv =>\n rhs\n simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n| abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\n rhs\n simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n| abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\n rhs\n simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n| abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\nrhs\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n| abs (PFunctor.W.dest (Wrepr (WType.mk a f)))\n[PROOFSTEP]\nsimp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (WType.mk a f)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Quotient.mk Wsetoid (Wrepr (WType.mk a f)) = Quotient.mk Wsetoid (WType.mk a f)\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Setoid.r (Wrepr (WType.mk a f)) (WType.mk a f)\n[PROOFSTEP]\napply Wrepr_equiv\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\n\u22a2 \u2200 (x : Fix F), g\u2081 x = g\u2082 x\n[PROOFSTEP]\napply Quot.ind\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\n\u22a2 \u2200 (a : PFunctor.W (P F)), g\u2081 (Quot.mk Setoid.r a) = g\u2082 (Quot.mk Setoid.r a)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\nx : PFunctor.W (P F)\n\u22a2 g\u2081 (Quot.mk Setoid.r x) = g\u2082 (Quot.mk Setoid.r x)\n[PROOFSTEP]\ninduction' x with a f ih\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\n\u22a2 g\u2081 (Quot.mk Setoid.r (WType.mk a f)) = g\u2082 (Quot.mk Setoid.r (WType.mk a f))\n[PROOFSTEP]\nchange g\u2081 \u27e6\u27e8a, f\u27e9\u27e7 = g\u2082 \u27e6\u27e8a, f\u27e9\u27e7\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\n\u22a2 g\u2081 (Quotient.mk Wsetoid (WType.mk a f)) = g\u2082 (Quotient.mk Wsetoid (WType.mk a f))\n[PROOFSTEP]\nrw [\u2190 Fix.ind_aux a f]\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\n\u22a2 g\u2081 (mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })) =\n g\u2082 (mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }))\n[PROOFSTEP]\napply h\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\n\u22a2 g\u2081 <$> abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) } =\n g\u2082 <$> abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }\n[PROOFSTEP]\nrw [\u2190 abs_map, \u2190 abs_map, PFunctor.map_eq, PFunctor.map_eq]\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\n\u22a2 abs { fst := a, snd := g\u2081 \u2218 fun x => Quotient.mk Wsetoid (f x) } =\n abs { fst := a, snd := g\u2082 \u2218 fun x => Quotient.mk Wsetoid (f x) }\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase mk.mk.a.e_a.e_snd.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng\u2081 g\u2082 : Fix F \u2192 \u03b1\nh : \u2200 (x : F (Fix F)), g\u2081 <$> x = g\u2082 <$> x \u2192 g\u2081 (mk x) = g\u2082 (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), g\u2081 (Quot.mk Setoid.r (f a)) = g\u2082 (Quot.mk Setoid.r (f a))\nx : PFunctor.B (P F) a\n\u22a2 (g\u2081 \u2218 fun x => Quotient.mk Wsetoid (f x)) x = (g\u2082 \u2218 fun x => Quotient.mk Wsetoid (f x)) x\n[PROOFSTEP]\napply ih\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nh : Fix F \u2192 \u03b1\nhyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x)\n\u22a2 rec g = h\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nh : Fix F \u2192 \u03b1\nhyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n\u22a2 rec g x = h x\n[PROOFSTEP]\napply Fix.ind_rec\n[GOAL]\ncase h.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nh : Fix F \u2192 \u03b1\nhyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n\u22a2 \u2200 (x : F (Fix F)), rec g <$> x = (fun x => h x) <$> x \u2192 rec g (mk x) = h (mk x)\n[PROOFSTEP]\nintro x hyp'\n[GOAL]\ncase h.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : F \u03b1 \u2192 \u03b1\nh : Fix F \u2192 \u03b1\nhyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x)\nx\u271d : Fix F\nx : F (Fix F)\nhyp' : rec g <$> x = (fun x => h x) <$> x\n\u22a2 rec g (mk x) = h (mk x)\n[PROOFSTEP]\nrw [hyp, \u2190 hyp', Fix.rec_eq]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : Fix F\n\u22a2 mk (dest x) = x\n[PROOFSTEP]\nchange (Fix.mk \u2218 Fix.dest) x = id x\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : Fix F\n\u22a2 (mk \u2218 dest) x = id x\n[PROOFSTEP]\napply Fix.ind_rec (mk \u2218 dest) id\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : Fix F\n\u22a2 \u2200 (x : F (Fix F)), (mk \u2218 dest) <$> x = id <$> x \u2192 (mk \u2218 dest) (mk x) = id (mk x)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d : Fix F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = id <$> x \u2192 (mk \u2218 dest) (mk x) = id (mk x)\n[PROOFSTEP]\nrw [Function.comp_apply, id_eq, Fix.dest, Fix.rec_eq, id_map, comp_map]\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d : Fix F\nx : F (Fix F)\n\u22a2 mk <$> rec (Functor.map mk) <$> x = x \u2192 mk (mk <$> rec (Functor.map mk) <$> x) = mk x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d : Fix F\nx : F (Fix F)\nh : mk <$> rec (Functor.map mk) <$> x = x\n\u22a2 mk (mk <$> rec (Functor.map mk) <$> x) = mk x\n[PROOFSTEP]\nrw [h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n\u22a2 dest (mk x) = x\n[PROOFSTEP]\nunfold Fix.dest\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n\u22a2 rec (Functor.map mk) (mk x) = x\n[PROOFSTEP]\nrw [Fix.rec_eq, \u2190 Fix.dest, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = x\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 id_map x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n| (mk \u2218 dest) <$> x = x\n[PROOFSTEP]\n rhs\n rw [\u2190 id_map x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n| (mk \u2218 dest) <$> x = x\n[PROOFSTEP]\n rhs\n rw [\u2190 id_map x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n| (mk \u2218 dest) <$> x = x\n[PROOFSTEP]\nrhs\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n| x\n[PROOFSTEP]\nrw [\u2190 id_map x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx : F (Fix F)\n\u22a2 (mk \u2218 dest) <$> x = id <$> x\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d : F (Fix F)\nx : Fix F\n\u22a2 (mk \u2218 dest) x = id x\n[PROOFSTEP]\napply Fix.mk_dest\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\n\u22a2 \u2200 (x : Fix F), p x\n[PROOFSTEP]\napply Quot.ind\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\n\u22a2 \u2200 (a : PFunctor.W (P F)), p (Quot.mk Setoid.r a)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\nx : PFunctor.W (P F)\n\u22a2 p (Quot.mk Setoid.r x)\n[PROOFSTEP]\ninduction' x with a f ih\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 p (Quot.mk Setoid.r (WType.mk a f))\n[PROOFSTEP]\nchange p \u27e6\u27e8a, f\u27e9\u27e7\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 p (Quotient.mk Wsetoid (WType.mk a f))\n[PROOFSTEP]\nrw [\u2190 Fix.ind_aux a f]\n[GOAL]\ncase mk.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 p (mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }))\n[PROOFSTEP]\napply h\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 Liftp p (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })\n[PROOFSTEP]\nrw [liftp_iff]\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 \u2203 a_1 f_1,\n abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) } = abs { fst := a_1, snd := f_1 } \u2227\n \u2200 (i : PFunctor.B (P F) a_1), p (f_1 i)\n[PROOFSTEP]\nrefine' \u27e8_, _, rfl, _\u27e9\n[GOAL]\ncase mk.mk.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\np : Fix F \u2192 Prop\nh : \u2200 (x : F (Fix F)), Liftp p x \u2192 p (mk x)\na : (P F).A\nf : PFunctor.B (P F) a \u2192 WType (P F).B\nih : \u2200 (a : PFunctor.B (P F) a), p (Quot.mk Setoid.r (f a))\n\u22a2 \u2200 (i : PFunctor.B (P F) a), p (Quotient.mk Wsetoid (f i))\n[PROOFSTEP]\nconvert ih\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 PFunctor.M.dest (corecF g x) = corecF g <$> repr (g x)\n[PROOFSTEP]\nrw [corecF, PFunctor.M.dest_corec]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 \u2200 (a b : PFunctor.M (P F)),\n Mcongr a b \u2192\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) a = (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) b\n[PROOFSTEP]\nrintro x y \u27e8r, pr, rxy\u27e9\n[GOAL]\ncase intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x = (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest x) = Quot.mk Mcongr <$> abs (PFunctor.M.dest 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]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\n\u22a2 \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n[PROOFSTEP]\nintro x y h\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx\u271d y\u271d : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x\u271d y\u271d\nx y : PFunctor.M (P F)\nh : r x y\n\u22a2 Mcongr x y\n[PROOFSTEP]\nexact \u27e8r, pr, h\u27e9\n[GOAL]\ncase intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n\u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest x) = Quot.mk Mcongr <$> abs (PFunctor.M.dest y)\n[PROOFSTEP]\nrw [\u2190 Quot.factor_mk_eq _ _ this]\n[GOAL]\ncase intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest x) =\n (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest y)\n[PROOFSTEP]\nconv =>\n lhs\n rw [comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n| (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest x) =\n (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest y)\n[PROOFSTEP]\n lhs\n rw [comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n| (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest x) =\n (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest y)\n[PROOFSTEP]\n lhs\n rw [comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n| (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest x) =\n (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest y)\n[PROOFSTEP]\nlhs\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nx y : PFunctor.M (P F)\nr : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop\npr : IsPrecongr r\nrxy : r x y\nthis : \u2200 (x y : PFunctor.M (P F)), r x y \u2192 Mcongr x y\n| (Quot.factor (fun x y => r x y) (fun x y => Mcongr x y) this \u2218 Quot.mk fun x y => r x y) <$> abs (PFunctor.M.dest x)\n[PROOFSTEP]\nrw [comp_map, \u2190 abs_map, pr rxy, abs_map, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 dest (corec g x) = corec g <$> g x\n[PROOFSTEP]\nconv =>\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n| dest (corec g x) = corec g <$> g x\n[PROOFSTEP]\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n| dest (corec g x) = corec g <$> g x\n[PROOFSTEP]\n lhs\n rw [Cofix.dest, Cofix.corec];\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n| dest (corec g x) = corec g <$> g x\n[PROOFSTEP]\nlhs\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n| dest (corec g x)\n[PROOFSTEP]\nrw [Cofix.dest, Cofix.corec]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.lift (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x))\n (_ :\n \u2200 (x y : PFunctor.M (P F)),\n Mcongr x y \u2192\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) x =\n (fun x => Quot.mk Mcongr <$> abs (PFunctor.M.dest x)) y)\n (Quot.mk Mcongr (corecF g x)) =\n corec g <$> g x\n[PROOFSTEP]\ndsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 Quot.mk Mcongr <$> abs (PFunctor.M.dest (corecF g x)) = corec g <$> g x\n[PROOFSTEP]\nrw [corecF_eq, abs_map, abs_repr, \u2190 comp_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\ng : \u03b1 \u2192 F \u03b1\nx : \u03b1\n\u22a2 (Quot.mk Mcongr \u2218 corecF g) <$> g x = corec g <$> g x\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y\n[PROOFSTEP]\nintro x\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (y : Cofix F), r x y \u2192 x = y\n[PROOFSTEP]\napply Quot.inductionOn (motive := _) x\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y\n[PROOFSTEP]\nclear x\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y \u2192 Quot.mk Mcongr a = y\n[PROOFSTEP]\nintro x y\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 r (Quot.mk Mcongr x) y \u2192 Quot.mk Mcongr x = y\n[PROOFSTEP]\napply Quot.inductionOn (motive := _) y\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\ny : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\n[PROOFSTEP]\nclear y\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) \u2192 Quot.mk Mcongr x = Quot.mk Mcongr a\n[PROOFSTEP]\nintro y rxy\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Quot.mk Mcongr x = Quot.mk Mcongr y\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y\n[PROOFSTEP]\nlet r' x y := r (Quot.mk _ x) (Quot.mk _ y)\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 Mcongr x y\n[PROOFSTEP]\nhave : IsPrecongr r' := by\n intro a b r'ab\n have h\u2080 :\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n h _ _ r'ab\n have h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v :=\n by\n intro u v cuv\n apply Quot.sound\n simp only\n rw [Quot.sound cuv]\n apply h'\n let f : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (by\n intro c; apply Quot.inductionOn (motive := _) c; clear c\n intro c d; apply Quot.inductionOn (motive := _) d; clear d\n intro d rcd; apply Quot.sound; apply rcd)\n have : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n rw [\u2190 this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map, h\u2080]\n rw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\n\u22a2 IsPrecongr r'\n[PROOFSTEP]\nintro a b r'ab\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)\n[PROOFSTEP]\nhave h\u2080 :\n Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=\n h _ _ r'ab\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)\n[PROOFSTEP]\nhave h\u2081 : \u2200 u v : q.P.M, Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v :=\n by\n intro u v cuv\n apply Quot.sound\n simp only\n rw [Quot.sound cuv]\n apply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\n\u22a2 \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n[PROOFSTEP]\nintro u v cuv\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 Quot.mk r' u = Quot.mk r' v\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r' u v\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr u) (Quot.mk Mcongr v)\n[PROOFSTEP]\nrw [Quot.sound cuv]\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nu v : PFunctor.M (P F)\ncuv : Mcongr u v\n\u22a2 r (Quot.mk Mcongr v) (Quot.mk Mcongr v)\n[PROOFSTEP]\napply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)\n[PROOFSTEP]\nlet f : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (by\n intro c; apply Quot.inductionOn (motive := _) c; clear c\n intro c d; apply Quot.inductionOn (motive := _) d; clear d\n intro d rcd; apply Quot.sound; apply rcd)\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a b : Cofix F), r a b \u2192 Quot.lift (Quot.mk r') h\u2081 a = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nintro c\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\napply Quot.inductionOn (motive := _) c\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nclear c\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\n\u22a2 \u2200 (a : PFunctor.M (P F)) (b : Cofix F),\n r (Quot.mk Mcongr a) b \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h\u2081 b\n[PROOFSTEP]\nintro c d\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n\u22a2 r (Quot.mk Mcongr c) d \u2192 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 d\n[PROOFSTEP]\napply Quot.inductionOn (motive := _) d\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\nd : Cofix F\n\u22a2 \u2200 (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)\n[PROOFSTEP]\nclear d\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc : PFunctor.M (P F)\n\u22a2 \u2200 (a : PFunctor.M (P F)),\n r (Quot.mk Mcongr c) (Quot.mk Mcongr a) \u2192\n Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr a)\n[PROOFSTEP]\nintro d rcd\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h\u2081 (Quot.mk Mcongr d)\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nc d : PFunctor.M (P F)\nrcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)\n\u22a2 r' c d\n[PROOFSTEP]\napply rcd\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (_ : \u2200 (c b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)\n[PROOFSTEP]\nhave : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r' := rfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (_ : \u2200 (c b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\nthis : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r'\n\u22a2 abs (Quot.mk r' <$> PFunctor.M.dest a) = abs (Quot.mk r' <$> PFunctor.M.dest b)\n[PROOFSTEP]\nrw [\u2190 this, PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map, h\u2080]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\na b : PFunctor.M (P F)\nr'ab : r' a b\nh\u2080 : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)\nh\u2081 : \u2200 (u v : PFunctor.M (P F)), Mcongr u v \u2192 Quot.mk r' u = Quot.mk r' v\nf : Quot r \u2192 Quot r' :=\n Quot.lift (Quot.lift (Quot.mk r') h\u2081)\n (_ : \u2200 (c b : Cofix F), r c b \u2192 Quot.lift (Quot.mk r') h\u2081 c = Quot.lift (Quot.mk r') h\u2081 b)\nthis : f \u2218 Quot.mk r \u2218 Quot.mk Mcongr = Quot.mk r'\n\u22a2 f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =\n abs ((f \u2218 Quot.mk r \u2218 Quot.mk Mcongr) <$> PFunctor.M.dest b)\n[PROOFSTEP]\nrw [PFunctor.comp_map _ _ f, PFunctor.comp_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh' : \u2200 (x : Cofix F), r x x\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nx y : PFunctor.M (P F)\nrxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nr' : PFunctor.M (P F) \u2192 PFunctor.M (P F) \u2192 Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)\nthis : IsPrecongr r'\n\u22a2 Mcongr x y\n[PROOFSTEP]\nrefine' \u27e8r', this, rxy\u27e9\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y\n[PROOFSTEP]\nlet r' (x y) := x = y \u2228 r x y\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y\n[PROOFSTEP]\nintro x y rxy\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 x = y\n[PROOFSTEP]\napply Cofix.bisim_aux r'\n[GOAL]\ncase h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x : Cofix F), r' x x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h'\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 r' x x\n[PROOFSTEP]\nleft\n[GOAL]\ncase h'.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y : Cofix F\nrxy : r x\u271d y\nx : Cofix F\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 \u2200 (x y : Cofix F), r' x y \u2192 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n[PROOFSTEP]\nintro x y r'xy\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n[PROOFSTEP]\ncases' r'xy with r'xy r'xy\n[GOAL]\ncase h.inl\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : x = y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n[PROOFSTEP]\nrw [r'xy]\n[GOAL]\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n[PROOFSTEP]\nhave : \u2200 x y, r x y \u2192 r' x y := fun x y h => Or.inr h\n[GOAL]\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.mk r' <$> dest x = Quot.mk r' <$> dest y\n[PROOFSTEP]\nrw [\u2190 Quot.factor_mk_eq _ _ this]\n[GOAL]\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n (Quot.factor (fun x y => r x y) (fun x y => r' x y) this \u2218 Quot.mk fun x y => r x y) <$> dest y\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 (Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this \u2218 Quot.mk fun x y => r x y) <$> dest x =\n (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) <$> dest y\n[PROOFSTEP]\nrw [@comp_map _ _ q _ _ _ (Quot.mk r), @comp_map _ _ q _ _ _ (Quot.mk r)]\n[GOAL]\ncase h.inr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx\u271d y\u271d : Cofix F\nrxy : r x\u271d y\u271d\nx y : Cofix F\nr'xy : r x y\nthis : \u2200 (x y : Cofix F), r x y \u2192 r' x y\n\u22a2 Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest x =\n Quot.factor (fun x y => r x y) (fun x y => x = y \u2228 r x y) this <$> Quot.mk r <$> dest y\n[PROOFSTEP]\nrw [h _ _ r'xy]\n[GOAL]\ncase a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r' x y\n[PROOFSTEP]\nright\n[GOAL]\ncase a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Quot.mk r <$> dest x = Quot.mk r <$> dest y\nr' : Cofix F \u2192 Cofix F \u2192 Prop := fun x y => x = y \u2228 r x y\nx y : Cofix F\nrxy : r x y\n\u22a2 r x y\n[PROOFSTEP]\nexact rxy\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 x = y\n[PROOFSTEP]\napply Cofix.bisim_rel\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\n\u22a2 \u2200 (x y : Cofix F), r x y \u2192 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y\n[PROOFSTEP]\nintro x y rxy\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y\n[PROOFSTEP]\nrcases(liftr_iff r _ _).mp (h x y rxy) with \u27e8a, f\u2080, f\u2081, dxeq, dyeq, h'\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 (Quot.mk fun x y => r x y) <$> dest x = (Quot.mk fun x y => r x y) <$> dest y\n[PROOFSTEP]\nrw [dxeq, dyeq, \u2190 abs_map, \u2190 abs_map, PFunctor.map_eq, PFunctor.map_eq]\n[GOAL]\ncase h.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\n\u22a2 abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2080 } = abs { fst := a, snd := (Quot.mk fun x y => r x y) \u2218 f\u2081 }\n[PROOFSTEP]\ncongr 2 with i\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 ((Quot.mk fun x y => r x y) \u2218 f\u2080) i = ((Quot.mk fun x y => r x y) \u2218 f\u2081) i\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase h.intro.intro.intro.intro.intro.e_a.e_snd.h.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nr : Cofix F \u2192 Cofix F \u2192 Prop\nh : \u2200 (x y : Cofix F), r x y \u2192 Liftr r (dest x) (dest y)\nx y : Cofix F\nrxy : r x y\na : (P F).A\nf\u2080 f\u2081 : PFunctor.B (P F) a \u2192 Cofix F\ndxeq : dest x = abs { fst := a, snd := f\u2080 }\ndyeq : dest y = abs { fst := a, snd := f\u2081 }\nh' : \u2200 (i : PFunctor.B (P F) a), r (f\u2080 i) (f\u2081 i)\ni : PFunctor.B (P F) a\n\u22a2 r (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\napply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n \u2200 (x : \u03b1),\n Q x \u2192\n \u2203 a f f',\n dest (u x) = abs { fst := a, snd := f } \u2227\n dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\n\u22a2 Liftr R (dest x) (dest y)\n[PROOFSTEP]\nrcases h x' Qx' with \u27e8a, f, f', ux'eq, vx'eq, h'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n \u2200 (x : \u03b1),\n Q x \u2192\n \u2203 a f f',\n dest (u x) = abs { fst := a, snd := f } \u2227\n dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 Liftr R (dest x) (dest y)\n[PROOFSTEP]\nrw [liftr_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u_1\nQ : \u03b1 \u2192 Prop\nu v : \u03b1 \u2192 Cofix F\nh :\n \u2200 (x : \u03b1),\n Q x \u2192\n \u2203 a f f',\n dest (u x) = abs { fst := a, snd := f } \u2227\n dest (v x) = abs { fst := a, snd := f' } \u2227 \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\nx\u271d\u00b9 : \u03b1\nQx : Q x\u271d\u00b9\nR : Cofix F \u2192 Cofix F \u2192 Prop := fun w z => \u2203 x', Q x' \u2227 w = u x' \u2227 z = v x'\nx y : Cofix F\nx\u271d : R x y\nx' : \u03b1\nQx' : Q x'\nxeq : x = u x'\nyeq : y = v x'\na : (P F).A\nf f' : PFunctor.B (P F) a \u2192 Cofix F\nux'eq : dest (u x') = abs { fst := a, snd := f }\nvx'eq : dest (v x') = abs { fst := a, snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a), \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x'\n\u22a2 \u2203 a f\u2080 f\u2081,\n dest x = abs { fst := a, snd := f\u2080 } \u2227\n dest y = abs { fst := a, snd := f\u2081 } \u2227 \u2200 (i : PFunctor.B (P F) a), R (f\u2080 i) (f\u2081 i)\n[PROOFSTEP]\nrefine' \u27e8a, f, f', xeq.symm \u25b8 ux'eq, yeq.symm \u25b8 vx'eq, h'\u27e9\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1 \u2192 Functor.Comp F\u2082 F\u2081 \u03b1\n[PROOFSTEP]\ndsimp [Functor.Comp]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1 \u2192 F\u2082 (F\u2081 \u03b1)\n[PROOFSTEP]\nintro p\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\np : PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1\n\u22a2 F\u2082 (F\u2081 \u03b1)\n[PROOFSTEP]\nexact abs \u27e8p.1.1, fun x => abs \u27e8p.1.2 x, fun y => p.2 \u27e8x, y\u27e9\u27e9\u27e9\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 Functor.Comp F\u2082 F\u2081 \u03b1 \u2192 PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1\n[PROOFSTEP]\ndsimp [Functor.Comp]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 F\u2082 (F\u2081 \u03b1) \u2192 PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1\n[PROOFSTEP]\nintro y\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\ny : F\u2082 (F\u2081 \u03b1)\n\u22a2 PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1\n[PROOFSTEP]\nrefine' \u27e8\u27e8(repr y).1, fun u => (repr ((repr y).2 u)).1\u27e9, _\u27e9\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\ny : F\u2082 (F\u2081 \u03b1)\n\u22a2 PFunctor.B (PFunctor.comp (P F\u2082) (P F\u2081)) { fst := (repr y).fst, snd := fun u => (repr (Sigma.snd (repr y) u)).fst } \u2192\n \u03b1\n[PROOFSTEP]\ndsimp [PFunctor.comp]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\ny : F\u2082 (F\u2081 \u03b1)\n\u22a2 (u : PFunctor.B (P F\u2082) (repr y).fst) \u00d7 PFunctor.B (P F\u2081) (repr (Sigma.snd (repr y) u)).fst \u2192 \u03b1\n[PROOFSTEP]\nintro x\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\ny : F\u2082 (F\u2081 \u03b1)\nx : (u : PFunctor.B (P F\u2082) (repr y).fst) \u00d7 PFunctor.B (P F\u2081) (repr (Sigma.snd (repr y) u)).fst\n\u22a2 \u03b1\n[PROOFSTEP]\nexact (repr ((repr y).2 x.1)).snd x.2\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 \u2200 (x : Functor.Comp F\u2082 F\u2081 \u03b1),\n (fun {\u03b1} =>\n id fun p =>\n abs\n { fst := p.fst.fst,\n snd := fun x => abs { fst := Sigma.snd p.fst x, snd := fun y => Sigma.snd p { fst := x, snd := y } } })\n ((fun {\u03b1} =>\n id fun y =>\n { fst := { fst := (repr y).fst, snd := fun u => (repr (Sigma.snd (repr y) u)).fst },\n snd := id fun x => Sigma.snd (repr (Sigma.snd (repr y) x.fst)) x.snd })\n x) =\n x\n[PROOFSTEP]\ndsimp [Functor.Comp]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\n\u22a2 \u2200 (x : F\u2082 (F\u2081 \u03b1)),\n abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n x\n[PROOFSTEP]\nintro x\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n\u22a2 abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n x\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 abs_repr x]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n| abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n x\n[PROOFSTEP]\n rhs\n rw [\u2190 abs_repr x]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n| abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n x\n[PROOFSTEP]\n rhs\n rw [\u2190 abs_repr x]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n| abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n x\n[PROOFSTEP]\nrhs\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n| x\n[PROOFSTEP]\nrw [\u2190 abs_repr x]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\n\u22a2 abs\n { fst := (repr x).fst,\n snd := fun x_1 =>\n abs\n { fst := (repr (Sigma.snd (repr x) x_1)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd (repr x) x_1)) y } } =\n abs (repr x)\n[PROOFSTEP]\ncases' h : repr x with a f\n[GOAL]\ncase mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\na : (P F\u2082).A\nf : PFunctor.B (P F\u2082) a \u2192 F\u2081 \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 abs\n { fst := { fst := a, snd := f }.fst,\n snd := fun x =>\n abs\n { fst := (repr (Sigma.snd { fst := a, snd := f } x)).fst,\n snd := fun y => Sigma.snd (repr (Sigma.snd { fst := a, snd := f } x)) y } } =\n abs { fst := a, snd := f }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx : F\u2082 (F\u2081 \u03b1)\na : (P F\u2082).A\nf : PFunctor.B (P F\u2082) a \u2192 F\u2081 \u03b1\nh : repr x = { fst := a, snd := f }\n\u22a2 abs { fst := a, snd := fun x => abs { fst := (repr (f x)).fst, snd := fun y => Sigma.snd (repr (f x)) y } } =\n abs { fst := a, snd := f }\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase mk.e_a.e_snd.h\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx\u271d : F\u2082 (F\u2081 \u03b1)\na : (P F\u2082).A\nf : PFunctor.B (P F\u2082) a \u2192 F\u2081 \u03b1\nh : repr x\u271d = { fst := a, snd := f }\nx : PFunctor.B (P F\u2082) a\n\u22a2 abs { fst := (repr (f x)).fst, snd := fun y => Sigma.snd (repr (f x)) y } = f x\n[PROOFSTEP]\ncases' h' : repr (f x) with b g\n[GOAL]\ncase mk.e_a.e_snd.h.mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx\u271d : F\u2082 (F\u2081 \u03b1)\na : (P F\u2082).A\nf : PFunctor.B (P F\u2082) a \u2192 F\u2081 \u03b1\nh : repr x\u271d = { fst := a, snd := f }\nx : PFunctor.B (P F\u2082) a\nb : (P F\u2081).A\ng : PFunctor.B (P F\u2081) b \u2192 \u03b1\nh' : repr (f x) = { fst := b, snd := g }\n\u22a2 abs { fst := { fst := b, snd := g }.fst, snd := fun y => Sigma.snd { fst := b, snd := g } y } = f x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.e_a.e_snd.h.mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 : Type u\nx\u271d : F\u2082 (F\u2081 \u03b1)\na : (P F\u2082).A\nf : PFunctor.B (P F\u2082) a \u2192 F\u2081 \u03b1\nh : repr x\u271d = { fst := a, snd := f }\nx : PFunctor.B (P F\u2082) a\nb : (P F\u2081).A\ng : PFunctor.B (P F\u2081) b \u2192 \u03b1\nh' : repr (f x) = { fst := b, snd := g }\n\u22a2 abs { fst := b, snd := fun y => g y } = f x\n[PROOFSTEP]\nrw [\u2190 h', abs_repr]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (p : PFunctor.Obj (PFunctor.comp (P F\u2082) (P F\u2081)) \u03b1),\n (fun {\u03b1} =>\n id fun p =>\n abs\n { fst := p.fst.fst,\n snd := fun x => abs { fst := Sigma.snd p.fst x, snd := fun y => Sigma.snd p { fst := x, snd := y } } })\n (f <$> p) =\n f <$>\n (fun {\u03b1} =>\n id fun p =>\n abs\n { fst := p.fst.fst,\n snd := fun x =>\n abs { fst := Sigma.snd p.fst x, snd := fun y => Sigma.snd p { fst := x, snd := y } } })\n p\n[PROOFSTEP]\ndsimp [Functor.Comp, PFunctor.comp]\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200\n (p :\n PFunctor.Obj\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n \u03b1),\n abs\n { fst := (f <$> p).fst.fst,\n snd := fun x =>\n abs { fst := Sigma.snd (f <$> p).fst x, snd := fun y => Sigma.snd (f <$> p) { fst := x, snd := y } } } =\n f <$>\n abs\n { fst := p.fst.fst,\n snd := fun x => abs { fst := Sigma.snd p.fst x, snd := fun y => Sigma.snd p { fst := x, snd := y } } }\n[PROOFSTEP]\nintro p\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\np :\n PFunctor.Obj\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n \u03b1\n\u22a2 abs\n { fst := (f <$> p).fst.fst,\n snd := fun x =>\n abs { fst := Sigma.snd (f <$> p).fst x, snd := fun y => Sigma.snd (f <$> p) { fst := x, snd := y } } } =\n f <$>\n abs\n { fst := p.fst.fst,\n snd := fun x => abs { fst := Sigma.snd p.fst x, snd := fun y => Sigma.snd p { fst := x, snd := y } } }\n[PROOFSTEP]\ncases' p with a g\n[GOAL]\ncase mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\na :\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }.A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n a \u2192\n \u03b1\n\u22a2 abs\n { fst := (f <$> { fst := a, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := a, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := a, snd := g }) { fst := x, snd := y } } } =\n f <$>\n abs\n { fst := { fst := a, snd := g }.fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd { fst := a, snd := g }.fst x,\n snd := fun y => Sigma.snd { fst := a, snd := g } { fst := x, snd := y } } }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\na :\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }.A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n a \u2192\n \u03b1\n\u22a2 abs\n { fst := (f <$> { fst := a, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := a, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := a, snd := g }) { fst := x, snd := y } } } =\n f <$> abs { fst := a.fst, snd := fun x => abs { fst := Sigma.snd a x, snd := fun y => g { fst := x, snd := y } } }\n[PROOFSTEP]\ncases' a with b h\n[GOAL]\ncase mk.mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } } =\n f <$>\n abs\n { fst := { fst := b, snd := h }.fst,\n snd := fun x => abs { fst := Sigma.snd { fst := b, snd := h } x, snd := fun y => g { fst := x, snd := y } } }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } } =\n f <$> abs { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } }\n[PROOFSTEP]\nsymm\n[GOAL]\ncase mk.mk\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 f <$> abs { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } } =\n abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\ntrans\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 f <$> abs { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } } = ?m.45227\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 ?m.45227 =\n abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 F\u2082 (F\u2081 \u03b2)\n[PROOFSTEP]\nsymm\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 ?a = f <$> abs { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } }\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 ?a =\n abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\napply abs_map\n[GOAL]\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 abs\n ((fun x x_1 => x <$> x_1) f <$>\n { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } }) =\n abs\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 (fun x x_1 => x <$> x_1) f <$>\n { fst := b, snd := fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } } =\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\nrw [PFunctor.map_eq]\n[GOAL]\ncase e_a\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 { fst := b,\n snd := (fun x x_1 => x <$> x_1) f \u2218 fun x => abs { fst := h x, snd := fun y => g { fst := x, snd := y } } } =\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\ndsimp [Function.comp]\n[GOAL]\ncase e_a\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 { fst := b, snd := fun x => f <$> abs { fst := h x, snd := fun y => g { fst := x, snd := y } } } =\n { fst := (f <$> { fst := { fst := b, snd := h }, snd := g }).fst.fst,\n snd := fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } } }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_snd\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\n\u22a2 (fun x => f <$> abs { fst := h x, snd := fun y => g { fst := x, snd := y } }) = fun x =>\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } }\n[PROOFSTEP]\next x\n[GOAL]\ncase e_a.e_snd.h\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\nx : PFunctor.B (P F\u2082) b\n\u22a2 f <$> abs { fst := h x, snd := fun y => g { fst := x, snd := y } } =\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } }\n[PROOFSTEP]\nrw [\u2190 abs_map]\n[GOAL]\ncase e_a.e_snd.h\nF\u2082 : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\u2082\nq\u2082 : Qpf F\u2082\nF\u2081 : Type u \u2192 Type u\ninst\u271d : Functor F\u2081\nq\u2081 : Qpf F\u2081\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nb : (P F\u2082).A\nh : PFunctor.B (P F\u2082) b \u2192 (P F\u2081).A\ng :\n PFunctor.B\n { A := (a\u2082 : (P F\u2082).A) \u00d7 (PFunctor.B (P F\u2082) a\u2082 \u2192 (P F\u2081).A),\n B := fun a\u2082a\u2081 => (u : PFunctor.B (P F\u2082) a\u2082a\u2081.fst) \u00d7 PFunctor.B (P F\u2081) (Sigma.snd a\u2082a\u2081 u) }\n { fst := b, snd := h } \u2192\n \u03b1\nx : PFunctor.B (P F\u2082) b\n\u22a2 abs (f <$> { fst := h x, snd := fun y => g { fst := x, snd := y } }) =\n abs\n { fst := Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }).fst x,\n snd := fun y => Sigma.snd (f <$> { fst := { fst := b, snd := h }, snd := g }) { fst := x, snd := y } }\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\nq : Qpf F\nG : Type u \u2192 Type u\ninst\u271d : Functor G\nFG_abs : {\u03b1 : Type u} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : Type u} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : Type u} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), FG_abs (f <$> x) = f <$> FG_abs x\n\u03b1 : Type u\nx : G \u03b1\n\u22a2 (fun {\u03b1} p => FG_abs (abs p)) ((fun {\u03b1} x => repr (FG_repr x)) x) = x\n[PROOFSTEP]\nsimp only\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\nq : Qpf F\nG : Type u \u2192 Type u\ninst\u271d : Functor G\nFG_abs : {\u03b1 : Type u} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : Type u} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : Type u} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), FG_abs (f <$> x) = f <$> FG_abs x\n\u03b1 : Type u\nx : G \u03b1\n\u22a2 FG_abs (abs (repr (FG_repr x))) = x\n[PROOFSTEP]\nrw [abs_repr, FG_abs_repr]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\nq : Qpf F\nG : Type u \u2192 Type u\ninst\u271d : Functor G\nFG_abs : {\u03b1 : Type u} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : Type u} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : Type u} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), FG_abs (f <$> x) = f <$> FG_abs x\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nx : PFunctor.Obj (P F) \u03b1\n\u22a2 (fun {\u03b1} p => FG_abs (abs p)) (f <$> x) = f <$> (fun {\u03b1} p => FG_abs (abs p)) x\n[PROOFSTEP]\nsimp only\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d\u00b9 : Functor F\nq : Qpf F\nG : Type u \u2192 Type u\ninst\u271d : Functor G\nFG_abs : {\u03b1 : Type u} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : Type u} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : Type u} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), FG_abs (f <$> x) = f <$> FG_abs x\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nx : PFunctor.Obj (P F) \u03b1\n\u22a2 FG_abs (abs (f <$> x)) = f <$> FG_abs (abs x)\n[PROOFSTEP]\nrw [abs_map, FG_abs_map]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\n\u22a2 u \u2208 supp x \u2194 \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\n[PROOFSTEP]\nrw [supp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\n\u22a2 u \u2208 {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y} \u2194\n \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\n[PROOFSTEP]\ndsimp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\n\u22a2 (\u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u) \u2194\n \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\n\u22a2 (\u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u) \u2192\n \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\n[PROOFSTEP]\nintro h a f haf\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nhaf : abs { fst := a, snd := f } = x\n\u22a2 u \u2208 f '' univ\n[PROOFSTEP]\nhave : Liftp (fun u => u \u2208 f '' univ) x := by\n rw [liftp_iff]\n refine' \u27e8a, f, haf.symm, fun i => mem_image_of_mem _ (mem_univ _)\u27e9\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nhaf : abs { fst := a, snd := f } = x\n\u22a2 Liftp (fun u => u \u2208 f '' univ) x\n[PROOFSTEP]\nrw [liftp_iff]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nhaf : abs { fst := a, snd := f } = x\n\u22a2 \u2203 a_1 f_1, x = abs { fst := a_1, snd := f_1 } \u2227 \u2200 (i : PFunctor.B (P F) a_1), f_1 i \u2208 f '' univ\n[PROOFSTEP]\nrefine' \u27e8a, f, haf.symm, fun i => mem_image_of_mem _ (mem_univ _)\u27e9\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nhaf : abs { fst := a, snd := f } = x\nthis : Liftp (fun u => u \u2208 f '' univ) x\n\u22a2 u \u2208 f '' univ\n[PROOFSTEP]\nexact h this\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\n\u22a2 (\u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ) \u2192\n \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p u\n[PROOFSTEP]\nintro h p\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\np : \u03b1 \u2192 Prop\n\u22a2 Liftp p x \u2192 p u\n[PROOFSTEP]\nrw [liftp_iff]\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)) \u2192 p u\n[PROOFSTEP]\nrintro \u27e8a, f, xeq, h'\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), p (f i)\n\u22a2 p u\n[PROOFSTEP]\nrcases h a f xeq.symm with \u27e8i, _, hi\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), p (f i)\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhi : f i = u\n\u22a2 p u\n[PROOFSTEP]\nrw [\u2190 hi]\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nu : \u03b1\nh : \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), p (f i)\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhi : f i = u\n\u22a2 p (f i)\n[PROOFSTEP]\napply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 supp x = {u | \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ}\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 supp x \u2194 x\u271d \u2208 {u | \u2200 (a : (P F).A) (f : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f } = x \u2192 u \u2208 f '' univ}\n[PROOFSTEP]\napply mem_supp\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 (\u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u) \u2194\n \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 (\u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u) \u2192\n \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n\u22a2 \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nhave : Liftp (supp x) x := by rw [h]; intro u; exact id\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n\u22a2 Liftp (supp x) x\n[PROOFSTEP]\nrw [h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n\u22a2 \u2200 (u : \u03b1), u \u2208 supp x \u2192 supp x u\n[PROOFSTEP]\nintro u\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nu : \u03b1\n\u22a2 u \u2208 supp x \u2192 supp x u\n[PROOFSTEP]\nexact id\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\n\u22a2 \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nrw [liftp_iff] at this \n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis\u271d : Liftp (supp x) x\nthis : \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), supp x (f i)\n\u22a2 \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nrcases this with \u27e8a, f, xeq, h'\u27e9\n[GOAL]\ncase mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\n\u22a2 \u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nrefine' \u27e8a, f, xeq.symm, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\n\u22a2 \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nintro a' f' h''\n[GOAL]\ncase mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh'' : abs { fst := a', snd := f' } = x\n\u22a2 f '' univ \u2286 f' '' univ\n[PROOFSTEP]\nrintro u \u27e8i, _, hfi\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh'' : abs { fst := a', snd := f' } = x\nu : \u03b1\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhfi : f i = u\n\u22a2 u \u2208 f' '' univ\n[PROOFSTEP]\nhave : u \u2208 supp x := by rw [\u2190 hfi]; apply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh'' : abs { fst := a', snd := f' } = x\nu : \u03b1\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhfi : f i = u\n\u22a2 u \u2208 supp x\n[PROOFSTEP]\nrw [\u2190 hfi]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh'' : abs { fst := a', snd := f' } = x\nu : \u03b1\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhfi : f i = u\n\u22a2 f i \u2208 supp x\n[PROOFSTEP]\napply h'\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\nh : \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\nthis\u271d : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : x = abs { fst := a, snd := f }\nh' : \u2200 (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh'' : abs { fst := a', snd := f' } = x\nu : \u03b1\ni : PFunctor.B (P F) a\nleft\u271d : i \u2208 univ\nhfi : f i = u\nthis : u \u2208 supp x\n\u22a2 u \u2208 f' '' univ\n[PROOFSTEP]\nexact (mem_supp x u).mp this _ _ h''\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\n\u22a2 (\u2203 a f,\n abs { fst := a, snd := f } = x \u2227\n \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ) \u2192\n \u2200 (p : \u03b1 \u2192 Prop), Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n[PROOFSTEP]\nrintro \u27e8a, f, xeq, h\u27e9 p\n[GOAL]\ncase mpr.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\n\u22a2 Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n[PROOFSTEP]\nrw [liftp_iff]\n[GOAL]\ncase mpr.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)) \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.intro.intro.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)) \u2192 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n[PROOFSTEP]\nrintro \u27e8a', f', xeq', h'\u27e9 u usuppx\n[GOAL]\ncase mpr.intro.intro.intro.mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nxeq' : x = abs { fst := a', snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\nusuppx : u \u2208 supp x\n\u22a2 p u\n[PROOFSTEP]\nrcases(mem_supp x u).mp usuppx a' f' xeq'.symm with \u27e8i, _, f'ieq\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nxeq' : x = abs { fst := a', snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\nusuppx : u \u2208 supp x\ni : PFunctor.B (P F) a'\nleft\u271d : i \u2208 univ\nf'ieq : f' i = u\n\u22a2 p u\n[PROOFSTEP]\nrw [\u2190 f'ieq]\n[GOAL]\ncase mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nxeq' : x = abs { fst := a', snd := f' }\nh' : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\nusuppx : u \u2208 supp x\ni : PFunctor.B (P F) a'\nleft\u271d : i \u2208 univ\nf'ieq : f' i = u\n\u22a2 p (f' i)\n[PROOFSTEP]\napply h'\n[GOAL]\ncase mpr.intro.intro.intro.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (u : \u03b1), u \u2208 supp x \u2192 p u) \u2192 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mpr.intro.intro.intro.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n\u22a2 \u2203 a f, x = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nrefine' \u27e8a, f, xeq.symm, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n\u22a2 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mpr.intro.intro.intro.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\ni : PFunctor.B (P F) a\n\u22a2 p (f i)\n[PROOFSTEP]\napply h'\n[GOAL]\ncase mpr.intro.intro.intro.mpr.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\ni : PFunctor.B (P F) a\n\u22a2 f i \u2208 supp x\n[PROOFSTEP]\nrw [mem_supp]\n[GOAL]\ncase mpr.intro.intro.intro.mpr.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\ni : PFunctor.B (P F) a\n\u22a2 \u2200 (a : (P F).A) (f_1 : PFunctor.B (P F) a \u2192 \u03b1), abs { fst := a, snd := f_1 } = x \u2192 f i \u2208 f_1 '' univ\n[PROOFSTEP]\nintro a' f' xeq'\n[GOAL]\ncase mpr.intro.intro.intro.mpr.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\ni : PFunctor.B (P F) a\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nxeq' : abs { fst := a', snd := f' } = x\n\u22a2 f i \u2208 f' '' univ\n[PROOFSTEP]\napply h a' f' xeq'\n[GOAL]\ncase mpr.intro.intro.intro.mpr.a.a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u03b1 : Type u\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nxeq : abs { fst := a, snd := f } = x\nh : \u2200 (a' : (P F).A) (f' : PFunctor.B (P F) a' \u2192 \u03b1), abs { fst := a', snd := f' } = x \u2192 f '' univ \u2286 f' '' univ\np : \u03b1 \u2192 Prop\nh' : \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\ni : PFunctor.B (P F) a\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nxeq' : abs { fst := a', snd := f' } = x\n\u22a2 f i \u2208 f '' univ\n[PROOFSTEP]\napply mem_image_of_mem _ (mem_univ _)\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = f '' univ\n[PROOFSTEP]\next u\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\n\u22a2 u \u2208 supp (abs { fst := a, snd := f }) \u2194 u \u2208 f '' univ\n[PROOFSTEP]\nrw [mem_supp]\n[GOAL]\ncase h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\n\u22a2 (\u2200 (a_1 : (P F).A) (f_1 : PFunctor.B (P F) a_1 \u2192 \u03b1),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } \u2192 u \u2208 f_1 '' univ) \u2194\n u \u2208 f '' univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\n\u22a2 (\u2200 (a_1 : (P F).A) (f_1 : PFunctor.B (P F) a_1 \u2192 \u03b1),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } \u2192 u \u2208 f_1 '' univ) \u2192\n u \u2208 f '' univ\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase h.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\nh' :\n \u2200 (a_1 : (P F).A) (f_1 : PFunctor.B (P F) a_1 \u2192 \u03b1),\n abs { fst := a_1, snd := f_1 } = abs { fst := a, snd := f } \u2192 u \u2208 f_1 '' univ\n\u22a2 u \u2208 f '' univ\n[PROOFSTEP]\napply h' _ _ rfl\n[GOAL]\ncase h.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\n\u22a2 u \u2208 f '' univ \u2192\n \u2200 (a_2 : (P F).A) (f_1 : PFunctor.B (P F) a_2 \u2192 \u03b1),\n abs { fst := a_2, snd := f_1 } = abs { fst := a, snd := f } \u2192 u \u2208 f_1 '' univ\n[PROOFSTEP]\nintro h' a' f' e\n[GOAL]\ncase h.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\nh' : u \u2208 f '' univ\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n\u22a2 u \u2208 f' '' univ\n[PROOFSTEP]\nrw [\u2190 h _ _ _ _ e.symm]\n[GOAL]\ncase h.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nu : \u03b1\nh' : u \u2208 f '' univ\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\ne : abs { fst := a', snd := f' } = abs { fst := a, snd := f }\n\u22a2 u \u2208 f '' univ\n[PROOFSTEP]\napply h'\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 Liftp p x \u2194 \u2200 (u : \u03b1), u \u2208 supp x \u2192 p u\n[PROOFSTEP]\nrw [liftp_iff, \u2190 abs_repr x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 a f, abs (repr x) = abs { fst := a, snd := f } \u2227 \u2200 (i : PFunctor.B (P F) a), p (f i)) \u2194\n \u2200 (u : \u03b1), u \u2208 supp (abs (repr x)) \u2192 p u\n[PROOFSTEP]\ncases' repr x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 (\u2203 a_1 f_1, abs { fst := a, snd := f } = abs { fst := a_1, snd := f_1 } \u2227 \u2200 (i : PFunctor.B (P F) a_1), p (f_1 i)) \u2194\n \u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 (\u2203 a_1 f_1, abs { fst := a, snd := f } = abs { fst := a_1, snd := f_1 } \u2227 \u2200 (i : PFunctor.B (P F) a_1), p (f_1 i)) \u2192\n \u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\n[PROOFSTEP]\nrintro \u27e8a', f', abseq, hf\u27e9 u\n[GOAL]\ncase mk.mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nabseq : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\nhf : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\n\u22a2 u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\n[PROOFSTEP]\nrw [supp_eq_of_isUniform h, h _ _ _ _ abseq]\n[GOAL]\ncase mk.mp.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nabseq : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\nhf : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\n\u22a2 u \u2208 f' '' univ \u2192 p u\n[PROOFSTEP]\nrintro \u27e8i, _, hi\u27e9\n[GOAL]\ncase mk.mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nabseq : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\nhf : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\ni : PFunctor.B (P F) a'\nleft\u271d : i \u2208 univ\nhi : f' i = u\n\u22a2 p u\n[PROOFSTEP]\nrw [\u2190 hi]\n[GOAL]\ncase mk.mp.intro.intro.intro.intro.intro\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\na' : (P F).A\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nabseq : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\nhf : \u2200 (i : PFunctor.B (P F) a'), p (f' i)\nu : \u03b1\ni : PFunctor.B (P F) a'\nleft\u271d : i \u2208 univ\nhi : f' i = u\n\u22a2 p (f' i)\n[PROOFSTEP]\napply hf\n[GOAL]\ncase mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 (\u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u) \u2192\n \u2203 a_2 f_1, abs { fst := a, snd := f } = abs { fst := a_2, snd := f_1 } \u2227 \u2200 (i : PFunctor.B (P F) a_2), p (f_1 i)\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : \u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\n\u22a2 \u2203 a_1 f_1, abs { fst := a, snd := f } = abs { fst := a_1, snd := f_1 } \u2227 \u2200 (i : PFunctor.B (P F) a_1), p (f_1 i)\n[PROOFSTEP]\nrefine' \u27e8a, f, rfl, fun i => h' _ _\u27e9\n[GOAL]\ncase mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : \u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\ni : PFunctor.B (P F) a\n\u22a2 f i \u2208 supp (abs { fst := a, snd := f })\n[PROOFSTEP]\nrw [supp_eq_of_isUniform h]\n[GOAL]\ncase mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\nx : F \u03b1\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : \u2200 (u : \u03b1), u \u2208 supp (abs { fst := a, snd := f }) \u2192 p u\ni : PFunctor.B (P F) a\n\u22a2 f i \u2208 f '' univ\n[PROOFSTEP]\nexact \u27e8i, mem_univ i, rfl\u27e9\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 \u03b2 : Type u\ng : \u03b1 \u2192 \u03b2\nx : F \u03b1\n\u22a2 supp (g <$> x) = g '' supp x\n[PROOFSTEP]\nrw [\u2190 abs_repr x]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 \u03b2 : Type u\ng : \u03b1 \u2192 \u03b2\nx : F \u03b1\n\u22a2 supp (g <$> abs (repr x)) = g '' supp (abs (repr x))\n[PROOFSTEP]\ncases' repr x with a f\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 \u03b2 : Type u\ng : \u03b1 \u2192 \u03b2\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (g <$> abs { fst := a, snd := f }) = g '' supp (abs { fst := a, snd := f })\n[PROOFSTEP]\nrw [\u2190 abs_map, PFunctor.map_eq]\n[GOAL]\ncase mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 \u03b2 : Type u\ng : \u03b1 \u2192 \u03b2\nx : F \u03b1\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (abs { fst := a, snd := g \u2218 f }) = g '' supp (abs { fst := a, snd := f })\n[PROOFSTEP]\nrw [supp_eq_of_isUniform h, supp_eq_of_isUniform h, image_comp]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 SuppPreservation \u2194 IsUniform\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 SuppPreservation \u2192 IsUniform\n[PROOFSTEP]\nintro h \u03b1 a a' f f' h'\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : SuppPreservation\n\u03b1 : Type u\na a' : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nf' : PFunctor.B (P F) a' \u2192 \u03b1\nh' : abs { fst := a, snd := f } = abs { fst := a', snd := f' }\n\u22a2 f '' univ = f' '' univ\n[PROOFSTEP]\nrw [\u2190 PFunctor.supp_eq, \u2190 PFunctor.supp_eq, \u2190 h, h', h]\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 IsUniform \u2192 SuppPreservation\n[PROOFSTEP]\nrintro h \u03b1 \u27e8a, f\u27e9\n[GOAL]\ncase mpr.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : IsUniform\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }\n[PROOFSTEP]\nrwa [supp_eq_of_isUniform, PFunctor.supp_eq]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 SuppPreservation \u2194 LiftpPreservation\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 SuppPreservation \u2192 LiftpPreservation\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 LiftpPreservation \u2192 SuppPreservation\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : SuppPreservation\n\u22a2 LiftpPreservation\n[PROOFSTEP]\nrintro \u03b1 p \u27e8a, f\u27e9\n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }\n[PROOFSTEP]\nhave h' := h\n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : SuppPreservation\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }\n[PROOFSTEP]\nrw [suppPreservation_iff_uniform] at h' \n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }\n[PROOFSTEP]\ndsimp only [SuppPreservation, supp] at h \n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs { fst := a, snd := f }) \u2194 Liftp p { fst := a, snd := f }\n[PROOFSTEP]\nrw [liftp_iff_of_isUniform h', supp_eq_of_isUniform h', PFunctor.liftp_iff']\n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1), u \u2208 f '' univ \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nsimp only [image_univ, mem_range, exists_imp]\n[GOAL]\ncase mp.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u) \u2194 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.mk.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u) \u2192 \u2200 (i : PFunctor.B (P F) a), p (f i)\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (i : PFunctor.B (P F) a), p (f i)) \u2192 \u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.mk.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\na\u271d : \u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u\ni\u271d : PFunctor.B (P F) a\n\u22a2 p (f i\u271d)\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase mp.mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\na\u271d\u00b9 : \u2200 (i : PFunctor.B (P F) a), p (f i)\nu\u271d : \u03b1\nx\u271d : PFunctor.B (P F) a\na\u271d : f x\u271d = u\u271d\n\u22a2 p u\u271d\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase mp.mk.mp\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\na\u271d : \u2200 (u : \u03b1) (x : PFunctor.B (P F) a), f x = u \u2192 p u\ni\u271d : PFunctor.B (P F) a\n\u22a2 p (f i\u271d)\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\ncase mp.mk.mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh :\n \u2200 \u2983\u03b1 : Type u\u2984 (x : PFunctor.Obj (P F) \u03b1),\n {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p (abs x) \u2192 p y} = {y | \u2200 \u2983p : \u03b1 \u2192 Prop\u2984, Liftp p x \u2192 p y}\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\nh' : IsUniform\na\u271d : \u2200 (i : PFunctor.B (P F) a), p (f i)\nx\u271d : PFunctor.B (P F) a\n\u22a2 p (f x\u271d)\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\ncase mpr\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : LiftpPreservation\n\u22a2 SuppPreservation\n[PROOFSTEP]\nrintro \u03b1 \u27e8a, f\u27e9\n[GOAL]\ncase mpr.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : LiftpPreservation\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }\n[PROOFSTEP]\nsimp only [LiftpPreservation] at h \n[GOAL]\ncase mpr.mk\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\nh : \u2200 \u2983\u03b1 : Type u\u2984 (p : \u03b1 \u2192 Prop) (x : PFunctor.Obj (P F) \u03b1), Liftp p (abs x) \u2194 Liftp p x\n\u03b1 : Type u\na : (P F).A\nf : PFunctor.B (P F) a \u2192 \u03b1\n\u22a2 supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }\n[PROOFSTEP]\nsimp only [supp, h]\n[GOAL]\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\n\u22a2 LiftpPreservation \u2194 IsUniform\n[PROOFSTEP]\nrw [\u2190 suppPreservation_iff_liftpPreservation, suppPreservation_iff_uniform]\n", "meta": {"mathlib_filename": "Mathlib.Data.QPF.Univariate.Basic", "llama_tokens": 74022, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.25730724104350405}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.725, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nj : \u2115\n\u22a2 \u00acComplexShape.Rel c 0 j\n[PROOFSTEP]\nintro hj\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.725, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nj : \u2115\nhj : ComplexShape.Rel c 0 j\n\u22a2 False\n[PROOFSTEP]\ndsimp at hj \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.725, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nj : \u2115\nhj : j + 1 = 0\n\u22a2 False\n[PROOFSTEP]\napply Nat.not_succ_le_zero j\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.725, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nj : \u2115\nhj : j + 1 = 0\n\u22a2 Nat.succ j \u2264 0\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, hj]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.35594, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\n\u22a2 X.obj (op [n + 1]) = HomologicalComplex.X K[X] m\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n m : \u2115\nhnq : n < q\nhnm : ComplexShape.Rel c m n\n\u22a2 h\u03c3' q n m hnm = 0\n[PROOFSTEP]\nsimp only [h\u03c3', h\u03c3]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n m : \u2115\nhnq : n < q\nhnm : ComplexShape.Rel c m n\n\u22a2 (if n < q then 0 else (-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) =\n 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n m : \u2115\nhnq : n < q\nhnm : ComplexShape.Rel c m n\n\u22a2 0 \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) = 0\n[PROOFSTEP]\nexact zero_comp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.71579, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\n\u22a2 X.obj (op [n + 1]) = HomologicalComplex.X K[X] m\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\n\u22a2 h\u03c3' q n m hnm =\n ((-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }) \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m]))\n[PROOFSTEP]\nsimp only [h\u03c3', h\u03c3]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\n\u22a2 (if n < q then 0 else (-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) =\n ((-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }) \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m]))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\nh\u271d : n < q\n\u22a2 0 \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) =\n ((-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }) \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m]))\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\nh\u271d : n < q\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\nh\u271d : \u00acn < q\n\u22a2 ((-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) =\n ((-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }) \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m]))\n[PROOFSTEP]\nhave h' := tsub_eq_of_eq_add ha\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a m : \u2115\nha : n = a + q\nhnm : ComplexShape.Rel c m n\nh\u271d : \u00acn < q\nh' : n - q = a\n\u22a2 ((-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])) =\n ((-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }) \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m]))\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a : \u2115\nha : n = a + q\n\u22a2 h\u03c3' q n (n + 1) (_ : n + 1 = n + 1) = (-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }\n[PROOFSTEP]\nrw [h\u03c3'_eq ha rfl, eqToHom_refl, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\n\u22a2 HomologicalComplex.Hom.f (H\u03c3 q) 0 = 0\n[PROOFSTEP]\nunfold H\u03c3\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) 0 = 0\n[PROOFSTEP]\nrw [nullHomotopicMap'_f_of_not_rel_left (c_mk 1 0 rfl) cs_down_0_not_rel_left]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\n\u22a2 h\u03c3' q 0 1 (_ : ComplexShape.Rel c 1 0) \u226b HomologicalComplex.d K[X] 1 0 = 0\n[PROOFSTEP]\nrcases q with (_ | q)\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 h\u03c3' Nat.zero 0 1 (_ : ComplexShape.Rel c 1 0) \u226b HomologicalComplex.d K[X] 1 0 = 0\n[PROOFSTEP]\nrw [h\u03c3'_eq (show 0 = 0 + 0 by rfl) (c_mk 1 0 rfl)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 0 = 0 + 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 (((-1) ^ 0 \u2022 \u03c3 X { val := 0, isLt := (_ : 0 < Nat.succ 0) }) \u226b eqToHom (_ : X.obj (op [0 + 1]) = X.obj (op [1]))) \u226b\n HomologicalComplex.d K[X] 1 0 =\n 0\n[PROOFSTEP]\nsimp only [pow_zero, Fin.mk_zero, one_zsmul, eqToHom_refl, Category.comp_id]\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 \u03c3 X 0 \u226b HomologicalComplex.d K[X] 1 0 = 0\n[PROOFSTEP]\nerw [ChainComplex.of_d]\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 \u03c3 X 0 \u226b AlternatingFaceMapComplex.objD X 0 = 0\n[PROOFSTEP]\nrw [AlternatingFaceMapComplex.objD, Fin.sum_univ_two, Fin.val_zero, Fin.val_one, pow_zero, pow_one, one_smul, neg_smul,\n one_smul, comp_add, comp_neg, add_neg_eq_zero]\n[GOAL]\ncase zero\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\n\u22a2 \u03c3 X 0 \u226b \u03b4 X 0 = \u03c3 X 0 \u226b \u03b4 X 1\n[PROOFSTEP]\nerw [\u03b4_comp_\u03c3_self, \u03b4_comp_\u03c3_succ]\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 : \u2115\n\u22a2 h\u03c3' (Nat.succ q) 0 1 (_ : ComplexShape.Rel c 1 0) \u226b HomologicalComplex.d K[X] 1 0 = 0\n[PROOFSTEP]\nrw [h\u03c3'_eq_zero (Nat.succ_pos q) (c_mk 1 0 rfl), zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\nX Y : SimplicialObject C\nf : X \u27f6 Y\n\u22a2 NatTrans.app f (op [n]) \u226b h\u03c3' q n m hnm = h\u03c3' q n m hnm \u226b NatTrans.app f (op [m])\n[PROOFSTEP]\nhave h : n + 1 = m := hnm\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\nX Y : SimplicialObject C\nf : X \u27f6 Y\nh : n + 1 = m\n\u22a2 NatTrans.app f (op [n]) \u226b h\u03c3' q n m hnm = h\u03c3' q n m hnm \u226b NatTrans.app f (op [m])\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\n\u22a2 NatTrans.app f (op [n]) \u226b h\u03c3' q n (n + 1) hnm = h\u03c3' q n (n + 1) hnm \u226b NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nsimp only [h\u03c3', eqToHom_refl, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\n\u22a2 NatTrans.app f (op [n]) \u226b h\u03c3 q n = h\u03c3 q n \u226b NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nunfold h\u03c3\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\n\u22a2 (NatTrans.app f (op [n]) \u226b\n if n < q then 0 else (-1) ^ (n - q) \u2022 \u03c3 Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n (if n < q then 0 else (-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\nh\u271d : n < q\n\u22a2 NatTrans.app f (op [n]) \u226b 0 = 0 \u226b NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nrw [zero_comp, comp_zero]\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\nh\u271d : \u00acn < q\n\u22a2 NatTrans.app f (op [n]) \u226b ((-1) ^ (n - q) \u2022 \u03c3 Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n ((-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nsimp only [zsmul_comp, comp_zsmul]\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\nh\u271d : \u00acn < q\n\u22a2 (-1) ^ (n - q) \u2022 NatTrans.app f (op [n]) \u226b \u03c3 Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) } \u226b NatTrans.app f (op [n + 1])\n[PROOFSTEP]\nerw [f.naturality]\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Preadditive C\nX\u271d : SimplicialObject C\nq n : \u2115\nX Y : SimplicialObject C\nf : X \u27f6 Y\nhnm : ComplexShape.Rel c (n + 1) n\nh\u271d : \u00acn < q\n\u22a2 (-1) ^ (n - q) \u2022 NatTrans.app f (op [n]) \u226b \u03c3 Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) \u2022\n NatTrans.app f (op [n]) \u226b Y.map (SimplexCategory.\u03c3 { val := n - q, isLt := (_ : n - q < Nat.succ n) }).op\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.259122, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nx\u271d\u00b9 x\u271d : SimplicialObject C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (alternatingFaceMapComplex C).map f \u226b (fun X => H\u03c3 q) x\u271d = (fun X => H\u03c3 q) x\u271d\u00b9 \u226b (alternatingFaceMapComplex C).map f\n[PROOFSTEP]\nunfold H\u03c3\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.259122, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nx\u271d\u00b9 x\u271d : SimplicialObject C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (alternatingFaceMapComplex C).map f \u226b (fun X => nullHomotopicMap' (h\u03c3' q)) x\u271d =\n (fun X => nullHomotopicMap' (h\u03c3' q)) x\u271d\u00b9 \u226b (alternatingFaceMapComplex C).map f\n[PROOFSTEP]\nrw [nullHomotopicMap'_comp, comp_nullHomotopicMap']\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.259122, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nx\u271d\u00b9 x\u271d : SimplicialObject C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (nullHomotopicMap' fun i j hij => HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) i \u226b h\u03c3' q i j hij) =\n nullHomotopicMap' fun i j hij => h\u03c3' q i j hij \u226b HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) j\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.259122, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nx\u271d\u00b9 x\u271d : SimplicialObject C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (fun i j hij => HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) i \u226b h\u03c3' q i j hij) = fun i j hij =>\n h\u03c3' q i j hij \u226b HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) j\n[PROOFSTEP]\next n m hnm\n[GOAL]\ncase e_h.h.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.259122, u_1} C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq : \u2115\nx\u271d\u00b9 x\u271d : SimplicialObject C\nf : x\u271d\u00b9 \u27f6 x\u271d\nn m : \u2115\nhnm : ComplexShape.Rel (ComplexShape.down \u2115) m n\n\u22a2 HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) n \u226b h\u03c3' q n m hnm =\n h\u03c3' q n m hnm \u226b HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map f) m\n[PROOFSTEP]\nsimp only [alternatingFaceMapComplex_map_f, h\u03c3'_naturality]\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\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\n\u22a2 h\u03c3' q n m hnm = G.map (h\u03c3' q n m hnm)\n[PROOFSTEP]\nunfold h\u03c3' h\u03c3\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\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\n\u22a2 (if n < q then 0\n else (-1) ^ (n - q) \u2022 \u03c3 (((whiskering C D).obj G).obj X) { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : (((whiskering C D).obj G).obj X).obj (op [n + 1]) = (((whiskering C D).obj G).obj X).obj (op [m])) =\n G.map\n ((if n < q then 0 else (-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\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\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\nh\u271d : n < q\n\u22a2 0 \u226b eqToHom (_ : (((whiskering C D).obj G).obj X).obj (op [n + 1]) = (((whiskering C D).obj G).obj X).obj (op [m])) =\n G.map (0 \u226b eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])))\n[PROOFSTEP]\nsimp only [Functor.map_zero, zero_comp]\n[GOAL]\ncase neg\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\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\nh\u271d : \u00acn < q\n\u22a2 ((-1) ^ (n - q) \u2022 \u03c3 (((whiskering C D).obj G).obj X) { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : (((whiskering C D).obj G).obj X).obj (op [n + 1]) = (((whiskering C D).obj G).obj X).obj (op [m])) =\n G.map\n (((-1) ^ (n - q) \u2022 \u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : X.obj (op [n + 1]) = X.obj (op [m])))\n[PROOFSTEP]\nsimp only [eqToHom_map, Functor.map_comp, Functor.map_zsmul]\n[GOAL]\ncase neg\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\nq n m : \u2115\nhnm : ComplexShape.Rel c m n\nh\u271d : \u00acn < q\n\u22a2 ((-1) ^ (n - q) \u2022 \u03c3 (((whiskering C D).obj G).obj X) { val := n - q, isLt := (_ : n - q < Nat.succ n) }) \u226b\n eqToHom (_ : (((whiskering C D).obj G).obj X).obj (op [n + 1]) = (((whiskering C D).obj G).obj X).obj (op [m])) =\n ((-1) ^ (n - q) \u2022 G.map (\u03c3 X { val := n - q, isLt := (_ : n - q < Nat.succ n) })) \u226b\n eqToHom (_ : G.obj (X.obj (op [n + 1])) = G.obj (HomologicalComplex.X K[X] m))\n[PROOFSTEP]\nrfl\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\nq n : \u2115\n\u22a2 HomologicalComplex.Hom.f (H\u03c3 q) n = G.map (HomologicalComplex.Hom.f (H\u03c3 q) n)\n[PROOFSTEP]\nunfold H\u03c3\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\nq n : \u2115\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n =\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n)\n[PROOFSTEP]\nhave eq := HomologicalComplex.congr_hom (map_nullHomotopicMap' G (@h\u03c3' _ _ _ X q)) n\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\nq n : \u2115\neq :\n HomologicalComplex.Hom.f ((Functor.mapHomologicalComplex G c).map (nullHomotopicMap' (h\u03c3' q))) n =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => G.map (h\u03c3' q i j hij)) n\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n =\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n)\n[PROOFSTEP]\nsimp only [Functor.mapHomologicalComplex_map_f, \u2190 map_h\u03c3'] at eq \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\nq n : \u2115\neq :\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n) =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => h\u03c3' q i j hij) n\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n =\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n)\n[PROOFSTEP]\nrw [eq]\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\nq n : \u2115\neq :\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n) =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => h\u03c3' q i j hij) n\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => h\u03c3' q i j hij) n\n[PROOFSTEP]\nlet h := (Functor.congr_obj (map_alternatingFaceMapComplex G) X).symm\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\nq n : \u2115\neq :\n G.map (HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n) =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => h\u03c3' q i j hij) n\nh : ((whiskering C D).obj G \u22d9 alternatingFaceMapComplex D).obj X =\n (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj X :=\n Eq.symm (Functor.congr_obj (map_alternatingFaceMapComplex G) X)\n\u22a2 HomologicalComplex.Hom.f (nullHomotopicMap' (h\u03c3' q)) n =\n HomologicalComplex.Hom.f (nullHomotopicMap' fun i j hij => h\u03c3' q i j hij) n\n[PROOFSTEP]\ncongr\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.Homotopies", "llama_tokens": 9332, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.25686843524451275}} {"text": "[GOAL]\nX : Scheme\ninst\u271d : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nh : Nonempty { x // x \u2208 U }\n\u22a2 Set.Nonempty (\u22a4 \u2229 \u2191U)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\n\u22a2 Field \u2191(Scheme.functionField X)\n[PROOFSTEP]\napply fieldOfIsUnitOrEqZero\n[GOAL]\ncase h\nX : Scheme\ninst\u271d : IsIntegral X\n\u22a2 \u2200 (a : \u2191(Scheme.functionField X)), IsUnit a \u2228 a = 0\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nX : Scheme\ninst\u271d : IsIntegral X\na : \u2191(Scheme.functionField X)\n\u22a2 IsUnit a \u2228 a = 0\n[PROOFSTEP]\nobtain \u27e8U, m, s, rfl\u27e9 := TopCat.Presheaf.germ_exist _ _ a\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\n\u22a2 IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s) \u2228\n \u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s = 0\n[PROOFSTEP]\nrw [or_iff_not_imp_right, \u2190 (X.presheaf.germ \u27e8_, m\u27e9).map_zero]\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\n\u22a2 \u00ac\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s =\n \u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) 0 \u2192\n IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s)\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha :\n \u00ac\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s =\n \u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) 0\n\u22a2 IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s)\n[PROOFSTEP]\nreplace ha := ne_of_apply_ne _ ha\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s)\n[PROOFSTEP]\nhave hs : genericPoint X.carrier \u2208 RingedSpace.basicOpen _ s :=\n by\n rw [\u2190 SetLike.mem_coe, (genericPoint_spec X.carrier).mem_open_set_iff, Set.top_eq_univ, Set.univ_inter,\n Set.nonempty_iff_ne_empty, Ne.def, \u2190 Opens.coe_bot, \u2190 SetLike.ext'_iff]\n erw [basicOpen_eq_bot_iff]\n exacts [ha, (RingedSpace.basicOpen _ _).isOpen]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 genericPoint \u2191\u2191X.toPresheafedSpace \u2208 RingedSpace.basicOpen X.toSheafedSpace s\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, (genericPoint_spec X.carrier).mem_open_set_iff, Set.top_eq_univ, Set.univ_inter,\n Set.nonempty_iff_ne_empty, Ne.def, \u2190 Opens.coe_bot, \u2190 SetLike.ext'_iff]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 \u00acRingedSpace.basicOpen X.toSheafedSpace s = \u22a5\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 IsOpen \u2191(RingedSpace.basicOpen X.toSheafedSpace s)\n[PROOFSTEP]\nerw [basicOpen_eq_bot_iff]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 \u00acs = 0\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\n\u22a2 IsOpen \u2191(RingedSpace.basicOpen X.toSheafedSpace s)\n[PROOFSTEP]\nexacts [ha, (RingedSpace.basicOpen _ _).isOpen]\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\nhs : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 RingedSpace.basicOpen X.toSheafedSpace s\n\u22a2 IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s)\n[PROOFSTEP]\nhave := (X.presheaf.germ \u27e8_, hs\u27e9).isUnit_map (RingedSpace.isUnit_res_basicOpen _ s)\n[GOAL]\ncase h.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nm : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 U\ns : (forget CommRingCat).obj (X.presheaf.obj (op U))\nha : s \u2260 0\nhs : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 RingedSpace.basicOpen X.toSheafedSpace s\nthis :\n IsUnit\n (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := hs })\n (\u2191(X.presheaf.map (homOfLE (_ : RingedSpace.basicOpen X.toSheafedSpace s \u2264 U)).op) s))\n\u22a2 IsUnit (\u2191(Presheaf.germ X.presheaf { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := m }) s)\n[PROOFSTEP]\nrwa [TopCat.Presheaf.germ_res_apply] at this \n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\n\u22a2 Function.Injective \u2191(Presheaf.germ X.presheaf x)\n[PROOFSTEP]\nrw [injective_iff_map_eq_zero]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\n\u22a2 \u2200 (a : (forget CommRingCat).obj (X.presheaf.obj (op U))), \u2191(Presheaf.germ X.presheaf x) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro y hy\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ny : (forget CommRingCat).obj (X.presheaf.obj (op U))\nhy : \u2191(Presheaf.germ X.presheaf x) y = 0\n\u22a2 y = 0\n[PROOFSTEP]\nrw [\u2190 (X.presheaf.germ x).map_zero] at hy \n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ny : (forget CommRingCat).obj (X.presheaf.obj (op U))\nhy : \u2191(Presheaf.germ X.presheaf x) y = \u2191(Presheaf.germ X.presheaf x) 0\n\u22a2 y = 0\n[PROOFSTEP]\nobtain \u27e8W, hW, iU, iV, e\u27e9 := X.presheaf.germ_eq _ x.prop x.prop _ _ hy\n[GOAL]\ncase intro.intro.intro.intro\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ny : (forget CommRingCat).obj (X.presheaf.obj (op U))\nhy : \u2191(Presheaf.germ X.presheaf x) y = \u2191(Presheaf.germ X.presheaf x) 0\nW : Opens \u2191\u2191X.toPresheafedSpace\nhW : \u2191x \u2208 W\niU iV : W \u27f6 U\ne : \u2191(X.presheaf.map iU.op) y = \u2191(X.presheaf.map iV.op) 0\n\u22a2 y = 0\n[PROOFSTEP]\ncases show iU = iV from Subsingleton.elim _ _\n[GOAL]\ncase intro.intro.intro.intro.refl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ny : (forget CommRingCat).obj (X.presheaf.obj (op U))\nhy : \u2191(Presheaf.germ X.presheaf x) y = \u2191(Presheaf.germ X.presheaf x) 0\nW : Opens \u2191\u2191X.toPresheafedSpace\nhW : \u2191x \u2208 W\niU : W \u27f6 U\ne : \u2191(X.presheaf.map iU.op) y = \u2191(X.presheaf.map iU.op) 0\n\u22a2 y = 0\n[PROOFSTEP]\nhaveI : Nonempty W := \u27e8\u27e8_, hW\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refl\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ny : (forget CommRingCat).obj (X.presheaf.obj (op U))\nhy : \u2191(Presheaf.germ X.presheaf x) y = \u2191(Presheaf.germ X.presheaf x) 0\nW : Opens \u2191\u2191X.toPresheafedSpace\nhW : \u2191x \u2208 W\niU : W \u27f6 U\ne : \u2191(X.presheaf.map iU.op) y = \u2191(X.presheaf.map iU.op) 0\nthis : Nonempty { x // x \u2208 W }\n\u22a2 y = 0\n[PROOFSTEP]\nexact map_injective_of_isIntegral X iU e\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base (genericPoint \u2191\u2191X.toPresheafedSpace) = genericPoint \u2191\u2191Y.toPresheafedSpace\n[PROOFSTEP]\napply ((genericPoint_spec Y).eq _).symm\n[GOAL]\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\n\u22a2 IsGenericPoint (\u2191f.val.base (genericPoint \u2191\u2191X.toPresheafedSpace)) \u22a4\n[PROOFSTEP]\nconvert (genericPoint_spec X.carrier).image (show Continuous f.1.base from ContinuousMap.continuous_toFun _)\n[GOAL]\ncase h.e'_4.h\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 \u22a4 = closure (\u2191f.val.base '' \u22a4)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_4.h\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 closure (\u2191f.val.base '' \u22a4) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, Set.top_eq_univ, Set.top_eq_univ]\n[GOAL]\ncase h.e'_4.h\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 Set.univ \u2264 closure (\u2191f.val.base '' Set.univ)\n[PROOFSTEP]\nconvert subset_closure_inter_of_isPreirreducible_of_isOpen _ H.base_open.open_range _\n[GOAL]\ncase h.e'_2.h.e'_3\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 \u2191f.val.base '' Set.univ = Set.univ \u2229 Set.range \u2191f.val.base\ncase h.e'_4.h.convert_2\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 IsPreirreducible Set.univ\ncase h.e'_4.h.convert_3\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 Set.Nonempty (Set.univ \u2229 Set.range \u2191f.val.base)\n[PROOFSTEP]\nrw [Set.univ_inter, Set.image_univ]\n[GOAL]\ncase h.e'_4.h.convert_2\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 IsPreirreducible Set.univ\ncase h.e'_4.h.convert_3\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 Set.Nonempty (Set.univ \u2229 Set.range \u2191f.val.base)\n[PROOFSTEP]\napply PreirreducibleSpace.isPreirreducible_univ (\u03b1 := Y.carrier)\n[GOAL]\ncase h.e'_4.h.convert_3\nX\u271d : Scheme\nX Y : Scheme\nf : X \u27f6 Y\nH : IsOpenImmersion f\nhX : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\ninst\u271d : IrreducibleSpace \u2191\u2191Y.toPresheafedSpace\ne_1\u271d : \u2191\u2191Y.toPresheafedSpace = (forget TopCat).obj \u2191Y.toPresheafedSpace\n\u22a2 Set.Nonempty (Set.univ \u2229 Set.range \u2191f.val.base)\n[PROOFSTEP]\nexact \u27e8_, trivial, Set.mem_range_self hX.2.some\u27e9\n[GOAL]\nX : Scheme\ninst\u271d : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 Algebra \u2191(Presheaf.stalk X.presheaf x) \u2191(Scheme.functionField X)\n[PROOFSTEP]\napply RingHom.toAlgebra\n[GOAL]\ncase i\nX : Scheme\ninst\u271d : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 \u2191(Presheaf.stalk X.presheaf x) \u2192+* \u2191(Scheme.functionField X)\n[PROOFSTEP]\nexact X.presheaf.stalkSpecializes ((genericPoint_spec X.carrier).specializes trivial)\n[GOAL]\nX : Scheme\ninst\u271d\u00b9 : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ninst\u271d : Nonempty { x // x \u2208 U }\n\u22a2 IsScalarTower \u2191(X.presheaf.obj (op U)) \u2191(Presheaf.stalk X.presheaf \u2191x) \u2191(Scheme.functionField X)\n[PROOFSTEP]\napply IsScalarTower.of_algebraMap_eq'\n[GOAL]\ncase h\nX : Scheme\ninst\u271d\u00b9 : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ninst\u271d : Nonempty { x // x \u2208 U }\n\u22a2 algebraMap \u2191(X.presheaf.obj (op U)) \u2191(Scheme.functionField X) =\n RingHom.comp (algebraMap \u2191(Presheaf.stalk X.presheaf \u2191x) \u2191(Scheme.functionField X))\n (algebraMap \u2191(X.presheaf.obj (op U)) \u2191(Presheaf.stalk X.presheaf \u2191x))\n[PROOFSTEP]\nsimp_rw [RingHom.algebraMap_toAlgebra]\n[GOAL]\ncase h\nX : Scheme\ninst\u271d\u00b9 : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ninst\u271d : Nonempty { x // x \u2208 U }\n\u22a2 Scheme.germToFunctionField X U =\n RingHom.comp (Presheaf.stalkSpecializes X.presheaf (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2933 \u2191x))\n (Presheaf.germ X.presheaf x)\n[PROOFSTEP]\nchange _ = X.presheaf.germ x \u226b _\n[GOAL]\ncase h\nX : Scheme\ninst\u271d\u00b9 : IrreducibleSpace \u2191\u2191X.toPresheafedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : { x // x \u2208 U }\ninst\u271d : Nonempty { x // x \u2208 U }\n\u22a2 Scheme.germToFunctionField X U =\n Presheaf.germ X.presheaf x \u226b Presheaf.stalkSpecializes X.presheaf (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2933 \u2191x)\n[PROOFSTEP]\nrw [X.presheaf.germ_stalkSpecializes]\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 \u2191R \u2192+* \u2191(Scheme.functionField (Scheme.Spec.obj (op R)))\n[PROOFSTEP]\nchange CommRingCat.of R \u27f6 _\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 CommRingCat.of \u2191R \u27f6 Scheme.functionField (Scheme.Spec.obj (op R))\n[PROOFSTEP]\napply StructureSheaf.toStalk\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace = { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }\n[PROOFSTEP]\napply (genericPoint_spec (Scheme.Spec.obj <| op R).carrier).eq\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 IsGenericPoint { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) } \u22a4\n[PROOFSTEP]\nrw [isGenericPoint_def]\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 closure {{ asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }} = \u22a4\n[PROOFSTEP]\nrw [\u2190 PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure, PrimeSpectrum.vanishingIdeal_singleton]\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 PrimeSpectrum.zeroLocus \u2191{ asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal = \u22a4\n[PROOFSTEP]\nrw [Set.top_eq_univ, \u2190 PrimeSpectrum.zeroLocus_singleton_zero]\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 PrimeSpectrum.zeroLocus \u2191{ asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal = PrimeSpectrum.zeroLocus {0}\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 IsFractionRing \u2191R \u2191(Scheme.functionField (Scheme.Spec.obj (op R)))\n[PROOFSTEP]\nconvert StructureSheaf.IsLocalization.to_stalk R (genericPoint _)\n[GOAL]\ncase a\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 IsFractionRing \u2191R \u2191(Scheme.functionField (Scheme.Spec.obj (op R))) \u2194\n IsLocalization.AtPrime\n (\u2191(Presheaf.stalk (Sheaf.presheaf (Spec.structureSheaf \u2191R))\n (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace)))\n (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace).asIdeal\n[PROOFSTEP]\ndelta IsFractionRing IsLocalization.AtPrime\n[GOAL]\ncase a\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 IsLocalization (nonZeroDivisors \u2191R) \u2191(Scheme.functionField (Scheme.Spec.obj (op R))) \u2194\n IsLocalization (Ideal.primeCompl (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace).asIdeal)\n \u2191(Presheaf.stalk (Sheaf.presheaf (Spec.structureSheaf \u2191R))\n (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace))\n[PROOFSTEP]\napply Eq.to_iff\n[GOAL]\ncase a.a\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 IsLocalization (nonZeroDivisors \u2191R) \u2191(Scheme.functionField (Scheme.Spec.obj (op R))) =\n IsLocalization (Ideal.primeCompl (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace).asIdeal)\n \u2191(Presheaf.stalk (Sheaf.presheaf (Spec.structureSheaf \u2191R))\n (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase a.a.e_M\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 nonZeroDivisors \u2191R = Ideal.primeCompl (genericPoint \u2191\u2191(Scheme.Spec.obj (op R)).toPresheafedSpace).asIdeal\n[PROOFSTEP]\nrw [genericPoint_eq_bot_of_affine]\n[GOAL]\ncase a.a.e_M\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\n\u22a2 nonZeroDivisors \u2191R = Ideal.primeCompl { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal\n[PROOFSTEP]\next\n[GOAL]\ncase a.a.e_M.h\nX : Scheme\nR : CommRingCat\ninst\u271d : IsDomain \u2191R\nx\u271d : \u2191R\n\u22a2 x\u271d \u2208 nonZeroDivisors \u2191R \u2194 x\u271d \u2208 Ideal.primeCompl { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal\n[PROOFSTEP]\nexact mem_nonZeroDivisors_iff_ne_zero\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\n\u22a2 Set.Nonempty (\u22a4 \u2229 \u2191U)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\n\u22a2 primeIdealOf hU\n { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n genericPoint \u2191\u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace\n[PROOFSTEP]\nhaveI : IsAffine _ := hU\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 primeIdealOf hU\n { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n genericPoint \u2191\u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace\n[PROOFSTEP]\nhave e : U.openEmbedding.isOpenMap.functor.obj \u22a4 = U := by ext1; exact Set.image_univ.trans Subtype.range_coe\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4) = \u2191U\n[PROOFSTEP]\nexact Set.image_univ.trans Subtype.range_coe\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\ne : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\n\u22a2 primeIdealOf hU\n { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n genericPoint \u2191\u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace\n[PROOFSTEP]\ndelta IsAffineOpen.primeIdealOf\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\ne : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\n\u22a2 \u2191(Scheme.Spec.map\n (X.presheaf.map\n (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op).op).val.base\n (\u2191(Scheme.isoSpec (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom.val.base\n { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) }) =\n genericPoint \u2191\u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace\n[PROOFSTEP]\nerw [\u2190 Scheme.comp_val_base_apply]\n[GOAL]\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\ne : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\n\u22a2 \u2191((Scheme.isoSpec (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))).hom \u226b\n Scheme.Spec.map\n (X.presheaf.map\n (eqToHom (_ : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U)).op).op).val.base\n { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n genericPoint \u2191\u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace\n[PROOFSTEP]\nconvert\n genericPoint_eq_of_isOpenImmersion\n ((X.restrict U.openEmbedding).isoSpec.hom \u226b Scheme.Spec.map (X.presheaf.map (eqToHom e).op).op)\n -- Porting note: this was `ext1`\n[GOAL]\ncase h.e'_2.h.h.e'_6\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\ne : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\ne_1\u271d :\n PrimeSpectrum \u2191(X.presheaf.obj (op U)) =\n (fun x => (forget TopCat).obj \u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace)\n (genericPoint\n \u2191\u2191(Scheme.restrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n\u22a2 { val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n genericPoint\n \u2191\u2191(Scheme.restrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase h.e'_2.h.h.e'_6.a\nX\u271d : Scheme\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nh : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\ne : (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4 = U\ne_1\u271d :\n PrimeSpectrum \u2191(X.presheaf.obj (op U)) =\n (fun x => (forget TopCat).obj \u2191(Scheme.Spec.obj (op (X.presheaf.obj (op U)))).toPresheafedSpace)\n (genericPoint\n \u2191\u2191(Scheme.restrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n\u22a2 \u2191{ val := genericPoint \u2191\u2191X.toPresheafedSpace, property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) } =\n \u2191(genericPoint\n \u2191\u2191(Scheme.restrict X\n (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace)\n[PROOFSTEP]\nexact (genericPoint_eq_of_isOpenImmersion (X.ofRestrict U.openEmbedding)).symm\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\n\u22a2 IsFractionRing \u2191(X.presheaf.obj (op U)) \u2191(Scheme.functionField X)\n[PROOFSTEP]\nhaveI : IsAffine _ := hU\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 IsFractionRing \u2191(X.presheaf.obj (op U)) \u2191(Scheme.functionField X)\n[PROOFSTEP]\nhaveI : Nonempty (X.restrict U.openEmbedding).carrier := hU'\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsFractionRing \u2191(X.presheaf.obj (op U)) \u2191(Scheme.functionField X)\n[PROOFSTEP]\nhaveI : IsIntegral (X.restrict U.openEmbedding) :=\n @isIntegralOfIsAffineIsDomain _ _ _ (by dsimp; rw [Opens.openEmbedding_obj_top]; infer_instance)\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsDomain\n \u2191((Scheme.restrict X\n (_ :\n OpenEmbedding\n \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.obj\n (op \u22a4))\n[PROOFSTEP]\ndsimp\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsDomain \u2191(X.presheaf.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion U))).obj \u22a4)))\n[PROOFSTEP]\nrw [Opens.openEmbedding_obj_top]\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\n\u22a2 IsDomain \u2191(X.presheaf.obj (op U))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d\u00b9 : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis\u271d :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nthis : IsIntegral (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 IsFractionRing \u2191(X.presheaf.obj (op U)) \u2191(Scheme.functionField X)\n[PROOFSTEP]\ndelta IsFractionRing Scheme.functionField\n[GOAL]\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d\u00b9 : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis\u271d :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nthis : IsIntegral (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 IsLocalization (nonZeroDivisors \u2191(X.presheaf.obj (op U)))\n \u2191(Presheaf.stalk X.presheaf (genericPoint \u2191\u2191X.toPresheafedSpace))\n[PROOFSTEP]\nconvert hU.isLocalization_stalk \u27e8genericPoint X.carrier, _\u27e9 using 1\n[GOAL]\ncase h.e'_3.h\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d\u00b9 : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis\u271d :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nthis : IsIntegral (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 nonZeroDivisors \u2191(X.presheaf.obj (op U)) =\n Ideal.primeCompl\n (IsAffineOpen.primeIdealOf hU\n { val := genericPoint \u2191\u2191X.toPresheafedSpace,\n property := (_ : genericPoint \u2191\u2191X.toPresheafedSpace \u2208 \u2191U) }).asIdeal\n[PROOFSTEP]\nrw [hU.primeIdealOf_genericPoint, genericPoint_eq_bot_of_affine]\n[GOAL]\ncase h.e'_3.h\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d\u00b9 : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis\u271d :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nthis : IsIntegral (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\n\u22a2 nonZeroDivisors \u2191(X.presheaf.obj (op U)) = Ideal.primeCompl { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h.h\nX : Scheme\ninst\u271d : IsIntegral X\nU : Opens \u2191\u2191X.toPresheafedSpace\nhU : IsAffineOpen U\nhU' : Nonempty { x // x \u2208 U }\nthis\u271d\u00b9 : IsAffine (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nthis\u271d :\n Nonempty\n \u2191\u2191(Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U))).toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace\nthis : IsIntegral (Scheme.restrict X (_ : OpenEmbedding \u2191(Opens.inclusion U)))\nx\u271d : \u2191(X.presheaf.obj (op U))\n\u22a2 x\u271d \u2208 nonZeroDivisors \u2191(X.presheaf.obj (op U)) \u2194\n x\u271d \u2208 Ideal.primeCompl { asIdeal := 0, IsPrime := (_ : Ideal.IsPrime \u22a5) }.asIdeal\n[PROOFSTEP]\nexact mem_nonZeroDivisors_iff_ne_zero\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.FunctionField", "llama_tokens": 12759, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.40733340004593027, "lm_q1q2_score": 0.2565282320966306}} {"text": "[GOAL]\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid A\ninst\u271d\u00b3 : Algebra A S\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : A\ns : { x // x \u2208 M }\nhr : \u2191(aeval (mk' S r s)) p = 0\n\u22a2 \u2191(aeval (\u2191(algebraMap A S) r)) (scaleRoots p \u2191s) = 0\n[PROOFSTEP]\nconvert scaleRoots_eval\u2082_eq_zero (algebraMap A S) hr\n[GOAL]\ncase h.e'_2.h.e\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid A\ninst\u271d\u00b3 : Algebra A S\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : A\ns : { x // x \u2208 M }\nhr : \u2191(aeval (mk' S r s)) p = 0\n\u22a2 \u2191(aeval (\u2191(algebraMap A S) r)) = eval\u2082 (algebraMap A S) (\u2191(algebraMap A S) \u2191s * mk' S r s)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h.e'_2.h.e.h\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\nM : Submonoid A\ninst\u271d\u00b3 : Algebra A S\ninst\u271d\u00b2 : IsLocalization M S\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : A\ns : { x // x \u2208 M }\nhr : \u2191(aeval (mk' S r s)) p = 0\nx\u271d : A[X]\n\u22a2 \u2191(aeval (\u2191(algebraMap A S) r)) x\u271d = eval\u2082 (algebraMap A S) (\u2191(algebraMap A S) \u2191s * mk' S r s) x\u271d\n[PROOFSTEP]\nrw [aeval_def, mk'_spec' _ r s]\n[GOAL]\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid A\ninst\u271d\u2075 : Algebra A S\ninst\u271d\u2074 : IsLocalization M S\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : \u2191(aeval x) p = 0\n\u22a2 IsRoot (scaleRoots p \u2191(den A x)) (num A x)\n[PROOFSTEP]\napply isRoot_of_eval\u2082_map_eq_zero (IsFractionRing.injective A K)\n[GOAL]\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid A\ninst\u271d\u2075 : Algebra A S\ninst\u271d\u2074 : IsLocalization M S\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : \u2191(aeval x) p = 0\n\u22a2 eval\u2082 (algebraMap A K) (\u2191(algebraMap A K) (num A x)) (scaleRoots p \u2191(den A x)) = 0\n[PROOFSTEP]\nrefine' scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero _\n[GOAL]\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid A\ninst\u271d\u2075 : Algebra A S\ninst\u271d\u2074 : IsLocalization M S\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : \u2191(aeval x) p = 0\n\u22a2 \u2191(aeval (mk' K (num A x) (den A x))) p = 0\n[PROOFSTEP]\nrw [mk'_num_den]\n[GOAL]\nA : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\nM : Submonoid A\ninst\u271d\u2075 : Algebra A S\ninst\u271d\u2074 : IsLocalization M S\ninst\u271d\u00b3 : Algebra A K\ninst\u271d\u00b2 : IsFractionRing A K\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : \u2191(aeval x) p = 0\n\u22a2 \u2191(aeval x) p = 0\n[PROOFSTEP]\nexact hr\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nsuffices num A r \u2223 (scaleRoots p (den A r)).coeff 0\n by\n simp only [coeff_scaleRoots, tsub_zero] at this \n haveI inst := Classical.propDecidable\n by_cases hr : num A r = 0\n \u00b7 obtain \u27e8u, hu\u27e9 := (isUnit_den_of_num_eq_zero hr).pow p.natDegree\n rw [\u2190 hu] at this \n exact Units.dvd_mul_right.mp this\n \u00b7 refine' dvd_of_dvd_mul_left_of_no_prime_factors hr _ this\n intro q dvd_num dvd_denom_pow hq\n apply hq.not_unit\n exact num_den_reduced A r dvd_num (hq.dvd_of_dvd_pow dvd_denom_pow)\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff (scaleRoots p \u2191(den A r)) 0\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nsimp only [coeff_scaleRoots, tsub_zero] at this \n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nhaveI inst := Classical.propDecidable\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nby_cases hr : num A r = 0\n[GOAL]\ncase pos\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : num A r = 0\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nobtain \u27e8u, hu\u27e9 := (isUnit_den_of_num_eq_zero hr).pow p.natDegree\n[GOAL]\ncase pos.intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : num A r = 0\nu : A\u02e3\nhu : \u2191u = \u2191(den A r) ^ natDegree p\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nrw [\u2190 hu] at this \n[GOAL]\ncase pos.intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\ninst : (a : Prop) \u2192 Decidable a\nhr : num A r = 0\nu : A\u02e3\nthis : num A r \u2223 coeff p 0 * \u2191u\nhu : \u2191u = \u2191(den A r) ^ natDegree p\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nexact Units.dvd_mul_right.mp this\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : \u00acnum A r = 0\n\u22a2 num A r \u2223 coeff p 0\n[PROOFSTEP]\nrefine' dvd_of_dvd_mul_left_of_no_prime_factors hr _ this\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : \u00acnum A r = 0\n\u22a2 \u2200 {d : A}, d \u2223 num A r \u2192 d \u2223 \u2191(den A r) ^ natDegree p \u2192 \u00acPrime d\n[PROOFSTEP]\nintro q dvd_num dvd_denom_pow hq\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : \u00acnum A r = 0\nq : A\ndvd_num : q \u2223 num A r\ndvd_denom_pow : q \u2223 \u2191(den A r) ^ natDegree p\nhq : Prime q\n\u22a2 False\n[PROOFSTEP]\napply hq.not_unit\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr\u271d : \u2191(aeval r) p = 0\nthis : num A r \u2223 coeff p 0 * \u2191(den A r) ^ natDegree p\ninst : (a : Prop) \u2192 Decidable a\nhr : \u00acnum A r = 0\nq : A\ndvd_num : q \u2223 num A r\ndvd_denom_pow : q \u2223 \u2191(den A r) ^ natDegree p\nhq : Prime q\n\u22a2 IsUnit q\n[PROOFSTEP]\nexact num_den_reduced A r dvd_num (hq.dvd_of_dvd_pow dvd_denom_pow)\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 num A r \u2223 coeff (scaleRoots p \u2191(den A r)) 0\n[PROOFSTEP]\nconvert dvd_term_of_isRoot_of_dvd_terms 0 (num_isRoot_scaleRoots_of_aeval_eq_zero hr) _\n[GOAL]\ncase h.e'_4\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 coeff (scaleRoots p \u2191(den A r)) 0 = coeff (scaleRoots p \u2191(den A r)) 0 * num A r ^ 0\n[PROOFSTEP]\nrw [pow_zero, mul_one]\n[GOAL]\ncase convert_2\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 \u2200 (j : \u2115), j \u2260 0 \u2192 num A r \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase convert_2\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 0\n\u22a2 num A r \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\napply dvd_mul_of_dvd_right\n[GOAL]\ncase convert_2.h\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 0\n\u22a2 num A r \u2223 num A r ^ j\n[PROOFSTEP]\nconvert pow_dvd_pow (num A r) (Nat.succ_le_of_lt (bot_lt_iff_ne_bot.mpr hj))\n[GOAL]\ncase h.e'_3\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 0\n\u22a2 num A r = num A r ^ Nat.succ \u22a5\n[PROOFSTEP]\nexact (pow_one _).symm\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 \u2191(den A r) \u2223 leadingCoeff p\n[PROOFSTEP]\nsuffices (den A r : A) \u2223 p.leadingCoeff * num A r ^ p.natDegree\n by\n refine' dvd_of_dvd_mul_left_of_no_prime_factors (mem_nonZeroDivisors_iff_ne_zero.mp (den A r).2) _ this\n intro q dvd_den dvd_num_pow hq\n apply hq.not_unit\n exact num_den_reduced A r (hq.dvd_of_dvd_pow dvd_num_pow) dvd_den\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : \u2191(den A r) \u2223 leadingCoeff p * num A r ^ natDegree p\n\u22a2 \u2191(den A r) \u2223 leadingCoeff p\n[PROOFSTEP]\nrefine' dvd_of_dvd_mul_left_of_no_prime_factors (mem_nonZeroDivisors_iff_ne_zero.mp (den A r).2) _ this\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : \u2191(den A r) \u2223 leadingCoeff p * num A r ^ natDegree p\n\u22a2 \u2200 {d : A}, d \u2223 \u2191(den A r) \u2192 d \u2223 num A r ^ natDegree p \u2192 \u00acPrime d\n[PROOFSTEP]\nintro q dvd_den dvd_num_pow hq\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : \u2191(den A r) \u2223 leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q \u2223 \u2191(den A r)\ndvd_num_pow : q \u2223 num A r ^ natDegree p\nhq : Prime q\n\u22a2 False\n[PROOFSTEP]\napply hq.not_unit\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nthis : \u2191(den A r) \u2223 leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q \u2223 \u2191(den A r)\ndvd_num_pow : q \u2223 num A r ^ natDegree p\nhq : Prime q\n\u22a2 IsUnit q\n[PROOFSTEP]\nexact num_den_reduced A r (hq.dvd_of_dvd_pow dvd_num_pow) dvd_den\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 \u2191(den A r) \u2223 leadingCoeff p * num A r ^ natDegree p\n[PROOFSTEP]\nrw [\u2190 coeff_scaleRoots_natDegree]\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 \u2191(den A r) \u2223 coeff (scaleRoots p ?s) (natDegree p) * num A r ^ natDegree p\ncase s\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 A\n[PROOFSTEP]\napply dvd_term_of_isRoot_of_dvd_terms _ (num_isRoot_scaleRoots_of_aeval_eq_zero hr)\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\n\u22a2 \u2200 (j : \u2115), j \u2260 natDegree p \u2192 \u2191(den A r) \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nintro j hj\n[GOAL]\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\n\u22a2 \u2191(den A r) \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nby_cases h : j < p.natDegree\n[GOAL]\ncase pos\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : j < natDegree p\n\u22a2 \u2191(den A r) \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nrw [coeff_scaleRoots]\n[GOAL]\ncase pos\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : j < natDegree p\n\u22a2 \u2191(den A r) \u2223 coeff p j * \u2191(den A r) ^ (natDegree p - j) * num A r ^ j\n[PROOFSTEP]\nrefine' (dvd_mul_of_dvd_right _ _).mul_right _\n[GOAL]\ncase pos\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : j < natDegree p\n\u22a2 \u2191(den A r) \u2223 \u2191(den A r) ^ (natDegree p - j)\n[PROOFSTEP]\nconvert pow_dvd_pow (den A r : A) (Nat.succ_le_iff.mpr (lt_tsub_iff_left.mpr _))\n[GOAL]\ncase h.e'_3\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : j < natDegree p\n\u22a2 \u2191(den A r) = \u2191(den A r) ^ Nat.succ ?pos.convert_1\u271d\n[PROOFSTEP]\nexact (pow_one _).symm\n[GOAL]\ncase pos.convert_4\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : j < natDegree p\n\u22a2 j + 0 < natDegree p\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree p\nh : \u00acj < natDegree p\n\u22a2 \u2191(den A r) \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nrw [\u2190 natDegree_scaleRoots p (den A r)] at *\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree (scaleRoots p \u2191(den A r))\nh : \u00acj < natDegree (scaleRoots p \u2191(den A r))\n\u22a2 \u2191(den A r) \u2223 coeff (scaleRoots p \u2191(den A r)) j * num A r ^ j\n[PROOFSTEP]\nrw [coeff_eq_zero_of_natDegree_lt (lt_of_le_of_ne (le_of_not_gt h) hj.symm), zero_mul]\n[GOAL]\ncase neg\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nr : K\nhr : \u2191(aeval r) p = 0\nj : \u2115\nhj : j \u2260 natDegree (scaleRoots p \u2191(den A r))\nh : \u00acj < natDegree (scaleRoots p \u2191(den A r))\n\u22a2 \u2191(den A r) \u2223 0\n[PROOFSTEP]\nexact dvd_zero _\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.RationalRoot", "llama_tokens": 8855, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.2563057208693416}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\na b : \u03b1\n\u22a2 a = b \u2192 r a b \u2227 r b a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsRefl \u03b1 r\ninst\u271d : IsAntisymm \u03b1 r\na : \u03b1\n\u22a2 r a a \u2227 r a a\n[PROOFSTEP]\nexact \u27e8refl _, refl _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrichotomous \u03b1 r\na b : \u03b1\n\u22a2 Function.swap r a b \u2228 a = b \u2228 Function.swap r b a\n[PROOFSTEP]\nsimpa [Function.swap, or_comm, or_left_comm] using trichotomous_of r a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsIrrefl \u03b1 r\ninst\u271d : Subsingleton \u03b1\n\u22a2 \u2200 (a b : \u03b1), r a b = EmptyRelation a b\n[PROOFSTEP]\nsimpa using not_rel_of_subsingleton r\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\n\u22a2 \u00acr b a \u2192 r b c \u2192 r a c\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : \u00acr b a\nh\u2082 : r b c\n\u22a2 r a c\n[PROOFSTEP]\nrcases trichotomous_of r a b with (h\u2083 | rfl | h\u2083)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : \u00acr b a\nh\u2082 : r b c\nh\u2083 : r a b\n\u22a2 r a c\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na c : \u03b1\nh\u2081 : \u00acr a a\nh\u2082 : r a c\n\u22a2 r a c\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : \u00acr b a\nh\u2082 : r b c\nh\u2083 : r b a\n\u22a2 r a c\n[PROOFSTEP]\nexacts [_root_.trans h\u2083 h\u2082, h\u2082, absurd h\u2083 h\u2081]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\n\u22a2 r a b \u2192 \u00acr c b \u2192 r a c\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : r a b\nh\u2082 : \u00acr c b\n\u22a2 r a c\n[PROOFSTEP]\nrcases trichotomous_of r b c with (h\u2083 | rfl | h\u2083)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : r a b\nh\u2082 : \u00acr c b\nh\u2083 : r b c\n\u22a2 r a c\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b : \u03b1\nh\u2081 : r a b\nh\u2082 : \u00acr b b\n\u22a2 r a b\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsTrichotomous \u03b1 r\na b c : \u03b1\nh\u2081 : r a b\nh\u2082 : \u00acr c b\nh\u2083 : r c b\n\u22a2 r a c\n[PROOFSTEP]\nexacts [_root_.trans h\u2081 h\u2083, h\u2081, absurd h\u2083 h\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\nx y : \u03b1\nh : x < y\ne : y = x\n\u22a2 False\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsStrictOrder \u03b1 r\nx y : \u03b1\nh : x < x\ne : y = x\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsOrderConnected \u03b1 r\na b c : \u03b1\nh\u2081 : \u00acr a b\nh\u2082 : \u00acr b c\n\u22a2 \u00ac(r a b \u2228 r b c)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\n\u03b1 : Type ?u.14854\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 IsTrichotomous \u03b1 r\n[PROOFSTEP]\ninfer_instance\n -- see Note [lower instance priority]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\n\u03b1 : Type ?u.14937\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 IsTrans \u03b1 r\n[PROOFSTEP]\ninfer_instance\n -- see Note [lower instance priority]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\n\u03b1 : Type ?u.15023\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 IsIrrefl \u03b1 r\n[PROOFSTEP]\ninfer_instance\n -- see Note [lower instance priority]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\n\u03b1 : Type ?u.15135\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\n\u22a2 IsAsymm \u03b1 r\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\na b c : \u03b1 \u00d7 \u03b2\nh\u2081 : Prod.Lex r s a b\nh\u2082 : Prod.Lex r s b c\n\u22a2 Prod.Lex r s a c\n[PROOFSTEP]\ncases' h\u2081 with a\u2081 a\u2082 b\u2081 b\u2082 ab a\u2081 b\u2081 b\u2082 ab\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nc : \u03b1 \u00d7 \u03b2\na\u2081 : \u03b1\na\u2082 : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nab : r a\u2081 b\u2081\nh\u2082 : Prod.Lex r s (b\u2081, b\u2082) c\n\u22a2 Prod.Lex r s (a\u2081, a\u2082) c\n[PROOFSTEP]\ncases' h\u2082 with _ _ c\u2081 c\u2082 bc _ _ c\u2082 bc\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\nc : \u03b1 \u00d7 \u03b2\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nab : s b\u2081 b\u2082\nh\u2082 : Prod.Lex r s (a\u2081, b\u2082) c\n\u22a2 Prod.Lex r s (a\u2081, b\u2081) c\n[PROOFSTEP]\ncases' h\u2082 with _ _ c\u2081 c\u2082 bc _ _ c\u2082 bc\n[GOAL]\ncase left.left\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\na\u2081 : \u03b1\na\u2082 : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nab : r a\u2081 b\u2081\nc\u2081 : \u03b1\nc\u2082 : \u03b2\nbc : r b\u2081 c\u2081\n\u22a2 Prod.Lex r s (a\u2081, a\u2082) (c\u2081, c\u2082)\ncase left.right\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\na\u2081 : \u03b1\na\u2082 : \u03b2\nb\u2081 : \u03b1\nb\u2082 : \u03b2\nab : r a\u2081 b\u2081\nc\u2082 : \u03b2\nbc : s b\u2082 c\u2082\n\u22a2 Prod.Lex r s (a\u2081, a\u2082) (b\u2081, c\u2082)\ncase right.left\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nab : s b\u2081 b\u2082\nc\u2081 : \u03b1\nc\u2082 : \u03b2\nbc : r a\u2081 c\u2081\n\u22a2 Prod.Lex r s (a\u2081, b\u2081) (c\u2081, c\u2082)\ncase right.right\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b9 : IsWellOrder \u03b1 r\ninst\u271d : IsWellOrder \u03b2 s\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nab : s b\u2081 b\u2082\nc\u2082 : \u03b2\nbc : s b\u2082 c\u2082\n\u22a2 Prod.Lex r s (a\u2081, b\u2081) (a\u2081, c\u2082)\n[PROOFSTEP]\nexacts [.left _ _ (_root_.trans ab bc), .left _ _ ab, .left _ _ bc, .right _ (_root_.trans ab bc)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\n\u22a2 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nrefine @Subrelation.wf (\u03b1 \u00d7 \u03b2) (Prod.Lex (\u00b7 < \u00b7) (\u00b7 < \u00b7)) (\u00b7 < \u00b7) ?_ IsWellFounded.wf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\n\u22a2 Subrelation (fun x x_1 => x < x_1) (Prod.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1)\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 w\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nw : (a\u2081, b\u2081) < (a\u2082, b\u2082)\n\u22a2 Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nsimp only [Prod.mk_lt_mk] at w \n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nw : a\u2081 < a\u2082 \u2227 b\u2081 \u2264 b\u2082 \u2228 a\u2081 \u2264 a\u2082 \u2227 b\u2081 < b\u2082\n\u22a2 Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nrcases eq_or_ne a\u2081 a\u2082 with rfl | ha\n[GOAL]\ncase mk.mk.inl\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nw : a\u2081 < a\u2081 \u2227 b\u2081 \u2264 b\u2082 \u2228 a\u2081 \u2264 a\u2081 \u2227 b\u2081 < b\u2082\n\u22a2 Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (a\u2081, b\u2081) (a\u2081, b\u2082)\n[PROOFSTEP]\nright\n[GOAL]\ncase mk.mk.inl.h\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nw : a\u2081 < a\u2081 \u2227 b\u2081 \u2264 b\u2082 \u2228 a\u2081 \u2264 a\u2081 \u2227 b\u2081 < b\u2082\n\u22a2 b\u2081 < b\u2082\n[PROOFSTEP]\nsimpa using w\n[GOAL]\ncase mk.mk.inr\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nw : a\u2081 < a\u2082 \u2227 b\u2081 \u2264 b\u2082 \u2228 a\u2081 \u2264 a\u2082 \u2227 b\u2081 < b\u2082\nha : a\u2081 \u2260 a\u2082\n\u22a2 Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (a\u2081, b\u2081) (a\u2082, b\u2082)\n[PROOFSTEP]\nleft\n[GOAL]\ncase mk.mk.inr.h\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nw : a\u2081 < a\u2082 \u2227 b\u2081 \u2264 b\u2082 \u2228 a\u2081 \u2264 a\u2082 \u2227 b\u2081 < b\u2082\nha : a\u2081 \u2260 a\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nrcases w with \u27e8a_lt, _\u27e9 | \u27e8a_le, _\u27e9\n[GOAL]\ncase mk.mk.inr.h.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha : a\u2081 \u2260 a\u2082\na_lt : a\u2081 < a\u2082\nright\u271d : b\u2081 \u2264 b\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.inr.h.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : WellFoundedLT \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : WellFoundedLT \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha : a\u2081 \u2260 a\u2082\na_le : a\u2081 \u2264 a\u2082\nright\u271d : b\u2081 < b\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nexact Ne.lt_of_le ha a_le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b1\n\u22a2 \u00acBounded r s \u2194 Unbounded r s\n[PROOFSTEP]\nsimp only [Bounded, Unbounded, not_forall, not_exists, exists_prop, not_and, not_not]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b1\n\u22a2 \u00acUnbounded r s \u2194 Bounded r s\n[PROOFSTEP]\nrw [not_iff_comm, not_bounded_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : HasSubset \u03b1\na b c : \u03b1\nhab : a = b\nhbc : b \u2286 c\n\u22a2 a \u2286 c\n[PROOFSTEP]\nrwa [hab]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : HasSubset \u03b1\na b c : \u03b1\nhab : a \u2286 b\nhbc : b = c\n\u22a2 a \u2286 c\n[PROOFSTEP]\nrwa [\u2190 hbc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : HasSSubset \u03b1\na b c : \u03b1\nhab : a = b\nhbc : b \u2282 c\n\u22a2 a \u2282 c\n[PROOFSTEP]\nrwa [hab]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : HasSSubset \u03b1\na b c : \u03b1\nhab : a \u2282 b\nhbc : b = c\n\u22a2 a \u2282 c\n[PROOFSTEP]\nrwa [\u2190 hbc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\n\u22a2 IsOrderConnected \u03b1 fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\n\u22a2 IsIncompTrans \u03b1 fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\n\u22a2 IsStrictWeakOrder \u03b1 fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Order.RelClasses", "llama_tokens": 5644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.3849121444839335, "lm_q1q2_score": 0.2561246051294886}} {"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 torsionOf R M 0 = \u22a4\n[PROOFSTEP]\nsimp [torsionOf]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : M\n\u22a2 torsionOf R M m = \u22a4 \u2194 m = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => by simp [h]\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : M\nh : m = 0\n\u22a2 torsionOf R M m = \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : M\nh : torsionOf R M m = \u22a4\n\u22a2 m = 0\n[PROOFSTEP]\nrw [\u2190 one_smul R m, \u2190 mem_torsionOf_iff m (1 : R), h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : M\nh : torsionOf R M m = \u22a4\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\nexact Submodule.mem_top\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroSMulDivisors R M\nm : M\n\u22a2 torsionOf R M m = \u22a5 \u2194 m \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun h contra => _, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroSMulDivisors R M\nm : M\nh : torsionOf R M m = \u22a5\ncontra : m = 0\n\u22a2 False\n[PROOFSTEP]\nrw [contra, torsionOf_zero] at h \n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroSMulDivisors R M\nm : M\nh : \u22a4 = \u22a5\ncontra : m = 0\n\u22a2 False\n[PROOFSTEP]\nexact bot_ne_top.symm h\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroSMulDivisors R M\nm : M\nh : m \u2260 0\nr : R\nhr : r \u2208 torsionOf R M m\n\u22a2 r = 0\n[PROOFSTEP]\nrw [mem_torsionOf_iff, smul_eq_zero] at hr \n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroSMulDivisors R M\nm : M\nh : m \u2260 0\nr : R\nhr : r = 0 \u2228 m = 0\n\u22a2 r = 0\n[PROOFSTEP]\ntauto\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nhv : CompleteLattice.Independent fun i => Submodule.span R {v i}\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\n\u22a2 LinearIndependent R v\n[PROOFSTEP]\nrefine' linearIndependent_iff_not_smul_mem_span.mpr fun i r hi => _\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nhv : CompleteLattice.Independent fun i => Submodule.span R {v i}\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\n\u22a2 r = 0\n[PROOFSTEP]\nreplace hv := CompleteLattice.independent_def.mp hv i\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Disjoint (Submodule.span R {v i}) (\u2a06 (j : \u03b9) (_ : j \u2260 i), Submodule.span R {v j})\n\u22a2 r = 0\n[PROOFSTEP]\nsimp only [iSup_subtype', \u2190 Submodule.span_range_eq_iSup, disjoint_iff] at hv \n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\n\u22a2 r = 0\n[PROOFSTEP]\nhave : r \u2022 v i \u2208 \u22a5 := by\n rw [\u2190 hv, Submodule.mem_inf]\n refine' \u27e8Submodule.mem_span_singleton.mpr \u27e8r, rfl\u27e9, _\u27e9\n convert hi\n ext\n simp\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\n\u22a2 r \u2022 v i \u2208 \u22a5\n[PROOFSTEP]\nrw [\u2190 hv, Submodule.mem_inf]\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\n\u22a2 r \u2022 v i \u2208 Submodule.span R {v i} \u2227 r \u2022 v i \u2208 Submodule.span R (Set.range fun i_1 => v \u2191i_1)\n[PROOFSTEP]\nrefine' \u27e8Submodule.mem_span_singleton.mpr \u27e8r, rfl\u27e9, _\u27e9\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\n\u22a2 r \u2022 v i \u2208 Submodule.span R (Set.range fun i_1 => v \u2191i_1)\n[PROOFSTEP]\nconvert hi\n[GOAL]\ncase h.e'_5.h.e'_6\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\n\u22a2 (Set.range fun i_1 => v \u2191i_1) = v '' (Set.univ \\ {i})\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h.e'_6.h\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\nx\u271d : M\n\u22a2 (x\u271d \u2208 Set.range fun i_1 => v \u2191i_1) \u2194 x\u271d \u2208 v '' (Set.univ \\ {i})\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\nthis : r \u2022 v i \u2208 \u22a5\n\u22a2 r = 0\n[PROOFSTEP]\nrw [\u2190 Submodule.mem_bot R, \u2190 h_ne_zero i]\n[GOAL]\nR\u271d : Type u_1\nM\u271d : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b3 : Module R\u271d M\u271d\n\u03b9 : Type u_3\nR : Type u_4\nM : Type u_5\nv : \u03b9 \u2192 M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nh_ne_zero : \u2200 (i : \u03b9), torsionOf R M (v i) = \u22a5\ni : \u03b9\nr : R\nhi : r \u2022 v i \u2208 Submodule.span R (v '' (Set.univ \\ {i}))\nhv : Submodule.span R {v i} \u2293 Submodule.span R (Set.range fun i_1 => v \u2191i_1) = \u22a5\nthis : r \u2022 v i \u2208 \u22a5\n\u22a2 r \u2208 torsionOf R M (v i)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nS : Type w\ninst\u271d\u00b9 : CommMonoid S\ninst\u271d : DistribMulAction S M\n\u22a2 \u2200 {a b : M}, a \u2208 {x | \u2203 a, a \u2022 x = 0} \u2192 b \u2208 {x | \u2203 a, a \u2022 x = 0} \u2192 a + b \u2208 {x | \u2203 a, a \u2022 x = 0}\n[PROOFSTEP]\nintro x y \u27e8a, hx\u27e9 \u27e8b, hy\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nS : Type w\ninst\u271d\u00b9 : CommMonoid S\ninst\u271d : DistribMulAction S M\nx y : M\na : S\nhx : a \u2022 x = 0\nb : S\nhy : b \u2022 y = 0\n\u22a2 x + y \u2208 {x | \u2203 a, a \u2022 x = 0}\n[PROOFSTEP]\nuse b * a\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nS : Type w\ninst\u271d\u00b9 : CommMonoid S\ninst\u271d : DistribMulAction S M\nx y : M\na : S\nhx : a \u2022 x = 0\nb : S\nhy : b \u2022 y = 0\n\u22a2 (b * a) \u2022 (x + y) = 0\n[PROOFSTEP]\nrw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type w\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\nsrc\u271d : AddSubmonoid M := torsion'AddSubMonoid M S\na : R\nx : M\nx\u271d : x \u2208 { toAddSubsemigroup := src\u271d.toAddSubsemigroup, zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\nb : S\nh : b \u2022 x = 0\n\u22a2 b \u2022 a \u2022 x = 0\n[PROOFSTEP]\nrw [smul_comm, h, smul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\n\u22a2 x \u2208 torsionBySet R M s \u2194 \u2200 (a : \u2191s), \u2191a \u2022 x = 0\n[PROOFSTEP]\nrefine' \u27e8fun h \u27e8a, ha\u27e9 => mem_sInf.mp h _ (Set.mem_image_of_mem _ ha), fun h => mem_sInf.mpr _\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nh : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\n\u22a2 \u2200 (p : Submodule R M), p \u2208 torsionBy R M '' s \u2192 x \u2208 p\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d : R\nx : M\nh : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\na : R\nha : a \u2208 s\n\u22a2 x \u2208 torsionBy R M a\n[PROOFSTEP]\nexact h \u27e8a, ha\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 torsionBySet R M {a} = torsionBy R M a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\n\u22a2 x \u2208 torsionBySet R M {a} \u2194 x \u2208 torsionBy R M a\n[PROOFSTEP]\nsimp only [mem_torsionBySet_iff, SetCoe.forall, Subtype.coe_mk, Set.mem_singleton_iff, forall_eq, mem_torsionBy_iff]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 torsionBySet R M s = torsionBySet R M \u2191(Ideal.span s)\n[PROOFSTEP]\nrefine le_antisymm (fun x hx => ?_) (torsionBySet_le_torsionBySet_of_subset subset_span)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nhx : x \u2208 torsionBySet R M s\n\u22a2 x \u2208 torsionBySet R M \u2191(Ideal.span s)\n[PROOFSTEP]\nrw [mem_torsionBySet_iff] at hx \u22a2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nhx : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\n\u22a2 \u2200 (a : \u2191\u2191(Ideal.span s)), \u2191a \u2022 x = 0\n[PROOFSTEP]\nsuffices Ideal.span s \u2264 Ideal.torsionOf R M x by\n rintro \u27e8a, ha\u27e9\n exact this ha\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nhx : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\nthis : Ideal.span s \u2264 Ideal.torsionOf R M x\n\u22a2 \u2200 (a : \u2191\u2191(Ideal.span s)), \u2191a \u2022 x = 0\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d : R\nx : M\nhx : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\nthis : Ideal.span s \u2264 Ideal.torsionOf R M x\na : R\nha : a \u2208 \u2191(Ideal.span s)\n\u22a2 \u2191{ val := a, property := ha } \u2022 x = 0\n[PROOFSTEP]\nexact this ha\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nhx : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\n\u22a2 Ideal.span s \u2264 Ideal.torsionOf R M x\n[PROOFSTEP]\nrw [Ideal.span_le]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx : M\nhx : \u2200 (a : \u2191s), \u2191a \u2022 x = 0\n\u22a2 s \u2286 \u2191(Ideal.torsionOf R M x)\n[PROOFSTEP]\nexact fun a ha => hx \u27e8a, ha\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d a b : R\ndvd : a \u2223 b\n\u22a2 torsionBy R M a \u2264 torsionBy R M b\n[PROOFSTEP]\nrw [\u2190 torsionBySet_span_singleton_eq, \u2190 torsionBySet_singleton_eq]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d a b : R\ndvd : a \u2223 b\n\u22a2 torsionBySet R M \u2191(span R {a}) \u2264 torsionBySet R M {b}\n[PROOFSTEP]\napply torsionBySet_le_torsionBySet_of_subset\n[GOAL]\ncase st\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d a b : R\ndvd : a \u2223 b\n\u22a2 {b} \u2286 \u2191(span R {a})\n[PROOFSTEP]\nrintro c (rfl : c = b)\n[GOAL]\ncase st\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na\u271d a c : R\ndvd : a \u2223 c\n\u22a2 c \u2208 \u2191(span R {a})\n[PROOFSTEP]\nexact Ideal.mem_span_singleton.mpr dvd\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx\u271d : M\nh : x\u271d \u2208 torsionBy R M 1\n\u22a2 x\u271d \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_torsionBy_iff, one_smul] at h \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nx\u271d : M\nh : x\u271d = 0\n\u22a2 x\u271d \u2208 \u22a5\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 torsionBySet R M Set.univ = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff, \u2190 torsionBy_one, \u2190 torsionBySet_singleton_eq]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 torsionBySet R M Set.univ \u2264 torsionBySet R M {1}\n[PROOFSTEP]\nexact torsionBySet_le_torsionBySet_of_subset fun _ _ => trivial\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 IsTorsionBySet R M {a} \u2194 IsTorsionBy R M a\n[PROOFSTEP]\nrefine' \u27e8fun h x => @h _ \u27e8_, Set.mem_singleton _\u27e9, fun h x => _\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nh : IsTorsionBy R M a\nx : M\n\u22a2 \u2200 \u2983a_1 : \u2191{a}\u2984, \u2191a_1 \u2022 x = 0\n[PROOFSTEP]\nrintro \u27e8b, rfl : b = a\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\nx : M\nb : R\nh : IsTorsionBy R M b\n\u22a2 \u2191{ val := b, property := (_ : b = b) } \u2022 x = 0\n[PROOFSTEP]\nexact @h _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nh : torsionBySet R M s = \u22a4\nx : M\n\u22a2 \u2200 \u2983a : \u2191s\u2984, \u2191a \u2022 x = 0\n[PROOFSTEP]\nrw [\u2190 mem_torsionBySet_iff, h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\nh : torsionBySet R M s = \u22a4\nx : M\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 IsTorsionBy R M a \u2194 torsionBy R M a = \u22a4\n[PROOFSTEP]\nrw [\u2190 torsionBySet_singleton_eq, \u2190 isTorsionBySet_singleton_iff, isTorsionBySet_iff_torsionBySet_eq_top]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ns : Set R\na : R\n\u22a2 IsTorsionBySet R M s \u2194 IsTorsionBySet R M \u2191(Ideal.span s)\n[PROOFSTEP]\nrw [isTorsionBySet_iff_torsionBySet_eq_top, isTorsionBySet_iff_torsionBySet_eq_top, torsionBySet_eq_torsionBySet_span]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i) = torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\ncases' S.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : S = \u2205\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i) = torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\nsimp only [h]\n -- Porting note: converts were not cooperating\n[GOAL]\ncase inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : S = \u2205\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 \u2205), torsionBySet R M \u2191(p i) = torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 \u2205), p i)\n[PROOFSTEP]\nconvert iSup_emptyset (f := fun i => torsionBySet R M (p i))\n[GOAL]\ncase h.e'_2.h.e'_4.h.pq.a.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : S = \u2205\nx\u271d : \u03b9\n\u22a2 x\u271d \u2208 \u2205 \u2194 x\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : S = \u2205\n\u22a2 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 \u2205), p i) = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i) = torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase inr.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i) \u2264 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\n\u22a2 \u2200 (i : \u03b9), \u2a06 (_ : i \u2208 S), torsionBySet R M \u2191(p i) \u2264 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\ni : \u03b9\n\u22a2 \u2a06 (_ : i \u2208 S), torsionBySet R M \u2191(p i) \u2264 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\napply iSup_le _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\ni : \u03b9\n\u22a2 i \u2208 S \u2192 torsionBySet R M \u2191(p i) \u2264 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\nintro is\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\ni : \u03b9\nis : i \u2208 S\n\u22a2 torsionBySet R M \u2191(p i) \u2264 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n[PROOFSTEP]\napply torsionBySet_le_torsionBySet_of_subset\n[GOAL]\ncase st\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\ni : \u03b9\nis : i \u2208 S\n\u22a2 \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i) \u2286 \u2191(p i)\n[PROOFSTEP]\nexact (iInf_le (fun i => \u2a05 _ : i \u2208 S, p i) i).trans (iInf_le _ is)\n[GOAL]\ncase inr.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\n\u22a2 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i) \u2264 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase inr.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : x \u2208 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u22a2 x \u2208 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(p i)\n[PROOFSTEP]\nrw [mem_iSup_finset_iff_exists_sum]\n[GOAL]\ncase inr.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : x \u2208 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u22a2 \u2203 \u03bc, \u2211 i in S, \u2191(\u03bc i) = x\n[PROOFSTEP]\nobtain \u27e8\u03bc, h\u03bc\u27e9 :=\n (mem_iSup_finset_iff_exists_sum _ _).mp\n ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)\n[GOAL]\ncase inr.a.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : x \u2208 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\n\u22a2 \u2203 \u03bc, \u2211 i in S, \u2191(\u03bc i) = x\n[PROOFSTEP]\nrefine' \u27e8fun i => \u27e8(\u03bc i : R) \u2022 x, _\u27e9, _\u27e9\n[GOAL]\ncase inr.a.intro.refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : x \u2208 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\n\u22a2 \u2191(\u03bc i) \u2022 x \u2208 torsionBySet R M \u2191(p i)\n[PROOFSTEP]\nrw [mem_torsionBySet_iff] at hx \u22a2\n[GOAL]\ncase inr.a.intro.refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\n\u22a2 \u2200 (a : \u2191\u2191(p i)), \u2191a \u2022 \u2191(\u03bc i) \u2022 x = 0\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase inr.a.intro.refine'_1.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\n\u22a2 \u2191{ val := a, property := ha } \u2022 \u2191(\u03bc i) \u2022 x = 0\n[PROOFSTEP]\nrw [smul_smul]\n[GOAL]\ncase inr.a.intro.refine'_1.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\n\u22a2 (\u2191{ val := a, property := ha } * \u2191(\u03bc i)) \u2022 x = 0\n[PROOFSTEP]\nsuffices : a * \u03bc i \u2208 \u2a05 i \u2208 S, p i\n[GOAL]\ncase inr.a.intro.refine'_1.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nthis : a * \u2191(\u03bc i) \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 S), p i\n\u22a2 (\u2191{ val := a, property := ha } * \u2191(\u03bc i)) \u2022 x = 0\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\n\u22a2 a * \u2191(\u03bc i) \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 S), p i\n[PROOFSTEP]\nexact hx \u27e8_, this\u27e9\n[GOAL]\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\n\u22a2 a * \u2191(\u03bc i) \u2208 \u2a05 (i : \u03b9) (_ : i \u2208 S), p i\n[PROOFSTEP]\nrw [mem_iInf]\n[GOAL]\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\n\u22a2 \u2200 (i_1 : \u03b9), a * \u2191(\u03bc i) \u2208 \u2a05 (_ : i_1 \u2208 S), p i_1\n[PROOFSTEP]\nintro j\n[GOAL]\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\n\u22a2 a * \u2191(\u03bc i) \u2208 \u2a05 (_ : j \u2208 S), p j\n[PROOFSTEP]\nrw [mem_iInf]\n[GOAL]\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\n\u22a2 j \u2208 S \u2192 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nintro hj\n[GOAL]\ncase this\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\n\u22a2 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nby_cases ij : j = i\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\nij : j = i\n\u22a2 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nrw [ij]\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\nij : j = i\n\u22a2 a * \u2191(\u03bc i) \u2208 p i\n[PROOFSTEP]\nexact Ideal.mul_mem_right _ _ ha\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\nij : \u00acj = i\n\u22a2 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nhave := coe_mem (\u03bc i)\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\nij : \u00acj = i\nthis : \u2191(\u03bc i) \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j\n\u22a2 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nsimp only [mem_iInf] at this \n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na\u271d : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : \u2200 (a : \u2191\u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)), \u2191a \u2022 x = 0\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\ni : \u03b9\na : R\nha : a \u2208 \u2191(p i)\nj : \u03b9\nhj : j \u2208 S\nij : \u00acj = i\nthis : \u2200 (i_1 : \u03b9), i_1 \u2208 S \u2192 i_1 \u2260 i \u2192 \u2191(\u03bc i) \u2208 p i_1\n\u22a2 a * \u2191(\u03bc i) \u2208 p j\n[PROOFSTEP]\nexact Ideal.mul_mem_left _ _ (this j hj ij)\n[GOAL]\ncase inr.a.intro.refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nh : Finset.Nonempty S\nx : M\nhx : x \u2208 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u03bc : (i : \u03b9) \u2192 { x // x \u2208 \u2a05 (j : \u03b9) (_ : j \u2208 S) (_ : j \u2260 i), p j }\nh\u03bc : \u2211 i in S, \u2191(\u03bc i) = 1\n\u22a2 \u2211 i in S, \u2191((fun i => { val := \u2191(\u03bc i) \u2022 x, property := (_ : \u2191(\u03bc i) \u2022 x \u2208 torsionBySet R M \u2191(p i)) }) i) = x\n[PROOFSTEP]\nrw [\u2190 Finset.sum_smul, h\u03bc, one_smul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\n\u22a2 Disjoint ((fun i => torsionBySet R M \u2191(p i)) i) (Finset.sup T fun i => torsionBySet R M \u2191(p i))\n[PROOFSTEP]\nrw [disjoint_iff, Finset.sup_eq_iSup,\n iSup_torsionBySet_ideal_eq_torsionBySet_iInf fun i hi j hj ij => hp (hT hi) (hT hj) ij]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\n\u22a2 (fun i => torsionBySet R M \u2191(p i)) i \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i) = \u22a5\n[PROOFSTEP]\nhave := GaloisConnection.u_inf (b\u2081 := OrderDual.toDual (p i)) (b\u2082 := OrderDual.toDual (\u2a05 i \u2208 T, p i)) (torsion_gc R M)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(\u2191OrderDual.ofDual (\u2191OrderDual.toDual (p i) \u2293 \u2191OrderDual.toDual (\u2a05 (i : \u03b9) (_ : i \u2208 T), p i))) =\n torsionBySet R M \u2191(\u2191OrderDual.ofDual (\u2191OrderDual.toDual (p i))) \u2293\n torsionBySet R M \u2191(\u2191OrderDual.ofDual (\u2191OrderDual.toDual (\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)))\n\u22a2 (fun i => torsionBySet R M \u2191(p i)) i \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i) = \u22a5\n[PROOFSTEP]\ndsimp at this \u22a2\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(p i \u2294 \u2a05 (i : \u03b9) (_ : i \u2208 T), p i) =\n torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)\n\u22a2 torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i) = \u22a5\n[PROOFSTEP]\nrw [\u2190 this, Ideal.sup_iInf_eq_top, top_coe, torsionBySet_univ]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(p i \u2294 \u2a05 (i : \u03b9) (_ : i \u2208 T), p i) =\n torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 T \u2192 p i \u2294 p i_1 = \u22a4\n[PROOFSTEP]\nintro j hj\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(p i \u2294 \u2a05 (i : \u03b9) (_ : i \u2208 T), p i) =\n torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)\nj : \u03b9\nhj : j \u2208 T\n\u22a2 p i \u2294 p j = \u22a4\n[PROOFSTEP]\napply hp hi (hT hj)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(p i \u2294 \u2a05 (i : \u03b9) (_ : i \u2208 T), p i) =\n torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)\nj : \u03b9\nhj : j \u2208 T\n\u22a2 i \u2260 j\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\ninst\u271d : DecidableEq \u03b9\nT : Finset \u03b9\nhT : T \u2286 S\ni : \u03b9\nhi : i \u2208 S\nhiT : \u00aci \u2208 T\nthis :\n torsionBySet R M \u2191(p i \u2294 \u2a05 (i : \u03b9) (_ : i \u2208 T), p i) =\n torsionBySet R M \u2191(p i) \u2293 torsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 T), p i)\nhj : i \u2208 T\n\u22a2 False\n[PROOFSTEP]\nexact hiT hj\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBy R M (q i) = torsionBy R M (\u220f i in S, q i)\n[PROOFSTEP]\nrw [\u2190 torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, \u2190 Ideal.finset_inf_span_singleton _ _ hq,\n Finset.inf_eq_iInf, \u2190 iSup_torsionBySet_ideal_eq_torsionBySet_iInf]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBy R M (q i) = \u2a06 (i : \u03b9) (_ : i \u2208 S), torsionBySet R M \u2191(Ideal.span {q i})\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 (fun i => \u2a06 (_ : i \u2208 S), torsionBy R M (q i)) = fun i => \u2a06 (_ : i \u2208 S), torsionBySet R M \u2191(Ideal.span {q i})\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\next : 1\n[GOAL]\ncase e_s.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\nx\u271d : \u03b9\n\u22a2 \u2a06 (_ : x\u271d \u2208 S), torsionBy R M (q x\u271d) = \u2a06 (_ : x\u271d \u2208 S), torsionBySet R M \u2191(Ideal.span {q x\u271d})\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s.h.e_s\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\nx\u271d : \u03b9\n\u22a2 (fun h => torsionBy R M (q x\u271d)) = fun h => torsionBySet R M \u2191(Ideal.span {q x\u271d})\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\next : 1\n[GOAL]\ncase e_s.h.e_s.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\nx\u271d\u00b9 : \u03b9\nx\u271d : x\u271d\u00b9 \u2208 S\n\u22a2 torsionBy R M (q x\u271d\u00b9) = torsionBySet R M \u2191(Ideal.span {q x\u271d\u00b9})\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\nexact (torsionBySet_span_singleton_eq _).symm\n[GOAL]\ncase hp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Set.Pairwise \u2191S fun i j => Ideal.span {q i} \u2294 Ideal.span {q j} = \u22a4\n[PROOFSTEP]\nexact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\n\u22a2 Finset.SupIndep S fun i => torsionBy R M (q i)\n[PROOFSTEP]\nconvert\n supIndep_torsionBySet_ideal (M := M) fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij\n[GOAL]\ncase h.e'_6.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ns : Set R\na : R\n\u03b9 : Type u_3\np : \u03b9 \u2192 Ideal R\nS : Finset \u03b9\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\ninst\u271d : DecidableEq \u03b9\nx\u271d : \u03b9\n\u22a2 torsionBy R M (q x\u271d) = torsionBySet R M \u2191(Ideal.span {q x\u271d})\n[PROOFSTEP]\nexact (torsionBySet_span_singleton_eq (R := R) (M := M) _).symm\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\n\u03b9 : Type u_3\ninst\u271d : DecidableEq \u03b9\nS : Finset \u03b9\np : \u03b9 \u2192 Ideal R\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nhM : Module.IsTorsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u22a2 \u2a06 (i : { x // x \u2208 S }), torsionBySet R M \u2191(p \u2191i) = \u22a4\n[PROOFSTEP]\napply (iSup_subtype'' \u2191S fun i => torsionBySet R M <| p i).trans\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\n\u03b9 : Type u_3\ninst\u271d : DecidableEq \u03b9\nS : Finset \u03b9\np : \u03b9 \u2192 Ideal R\nhp : Set.Pairwise \u2191S fun i j => p i \u2294 p j = \u22a4\nhM : Module.IsTorsionBySet R M \u2191(\u2a05 (i : \u03b9) (_ : i \u2208 S), p i)\n\u22a2 \u2a06 (t : \u03b9) (_ : t \u2208 \u2191S), torsionBySet R M \u2191(p t) = \u22a4\n[PROOFSTEP]\napply (iSup_torsionBySet_ideal_eq_torsionBySet_iInf hp).trans <| (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM\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\n\u03b9 : Type u_3\ninst\u271d : DecidableEq \u03b9\nS : Finset \u03b9\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\nhM : Module.IsTorsionBy R M (\u220f i in S, q i)\n\u22a2 DirectSum.IsInternal fun i => torsionBy R M (q \u2191i)\n[PROOFSTEP]\nrw [\u2190 Module.isTorsionBySet_span_singleton_iff, Ideal.submodule_span_eq, \u2190 Ideal.finset_inf_span_singleton _ _ hq,\n Finset.inf_eq_iInf] at hM \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\n\u03b9 : Type u_3\ninst\u271d : DecidableEq \u03b9\nS : Finset \u03b9\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\nhM : Module.IsTorsionBySet R M \u2191(\u2a05 (a : \u03b9) (_ : a \u2208 S), Ideal.span {q a})\n\u22a2 DirectSum.IsInternal fun i => torsionBy R M (q \u2191i)\n[PROOFSTEP]\nconvert torsionBySet_isInternal (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM\n[GOAL]\ncase h.e'_8.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : DecidableEq \u03b9\nS : Finset \u03b9\nq : \u03b9 \u2192 R\nhq : Set.Pairwise (\u2191S) (IsCoprime on q)\nhM : Module.IsTorsionBySet R M \u2191(\u2a05 (a : \u03b9) (_ : a \u2208 S), Ideal.span {q a})\nx\u271d : { x // x \u2208 S }\n\u22a2 torsionBy R M (q \u2191x\u271d) = torsionBySet R M \u2191(Ideal.span {q \u2191x\u271d})\n[PROOFSTEP]\nexact (torsionBySet_span_singleton_eq _ (R := R) (M := M)).symm\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 : Ideal R\nhM : IsTorsionBySet R M \u2191I\nb : R \u29f8 I\nx : M\nb\u2081 b\u2082 : R\nh : Setoid.r b\u2081 b\u2082\n\u22a2 b\u2081 \u2022 x = b\u2082 \u2022 x\n[PROOFSTEP]\nhave : (-b\u2081 + b\u2082) \u2022 x = 0 := @hM x \u27e8_, QuotientAddGroup.leftRel_apply.mp h\u27e9\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 : Ideal R\nhM : IsTorsionBySet R M \u2191I\nb : R \u29f8 I\nx : M\nb\u2081 b\u2082 : R\nh : Setoid.r b\u2081 b\u2082\nthis : (-b\u2081 + b\u2082) \u2022 x = 0\n\u22a2 b\u2081 \u2022 x = b\u2082 \u2022 x\n[PROOFSTEP]\nrw [add_smul, neg_smul, neg_add_eq_zero] at this \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 : Ideal R\nhM : IsTorsionBySet R M \u2191I\nb : R \u29f8 I\nx : M\nb\u2081 b\u2082 : R\nh : Setoid.r b\u2081 b\u2082\nthis : b\u2081 \u2022 x = b\u2082 \u2022 x\n\u22a2 b\u2081 \u2022 x = b\u2082 \u2022 x\n[PROOFSTEP]\nexact this\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 : Ideal R\nhM : IsTorsionBySet R M \u2191I\nx : M \u29f8 I \u2022 \u22a4\nr : \u2191\u2191I\n\u22a2 \u2191r \u2022 x = 0\n[PROOFSTEP]\ninduction x using Quotient.inductionOn\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhM : IsTorsionBySet R M \u2191I\nr : \u2191\u2191I\na\u271d : M\n\u22a2 \u2191r \u2022 Quotient.mk (Submodule.quotientRel (I \u2022 \u22a4)) a\u271d = 0\n[PROOFSTEP]\nrefine' (Submodule.Quotient.mk_eq_zero _).mpr (Submodule.smul_mem_smul r.prop _)\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhM : IsTorsionBySet R M \u2191I\nr : \u2191\u2191I\na\u271d : M\n\u22a2 a\u271d \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\ns : S\nx : { x // x \u2208 torsion' R M S }\n\u22a2 s \u2022 \u2191x \u2208 torsion' R M S\n[PROOFSTEP]\nobtain \u27e8x, a, h\u27e9 := x\n[GOAL]\ncase mk.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\ns : S\nx : M\na : S\nh : a \u2022 x = 0\n\u22a2 s \u2022 \u2191{ val := x, property := (_ : \u2203 a, a \u2022 x = 0) } \u2208 torsion' R M S\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\ns : S\nx : M\na : S\nh : a \u2022 x = 0\n\u22a2 a \u2022 s \u2022 \u2191{ val := x, property := (_ : \u2203 a, a \u2022 x = 0) } = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\ns : S\nx : M\na : S\nh : a \u2022 x = 0\n\u22a2 a \u2022 s \u2022 x = 0\n[PROOFSTEP]\nrw [smul_comm, h, smul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\nh : torsion' R M S = \u22a4\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nrw [\u2190 @mem_torsion'_iff R, h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nS : Type u_3\ninst\u271d\u00b2 : CommMonoid S\ninst\u271d\u00b9 : DistribMulAction S M\ninst\u271d : SMulCommClass S R M\nh : torsion' R M S = \u22a4\nx : M\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\n\u22a2 \u2191(annihilator \u22a4) \u2229 \u2191R\u2070 \u2260 \u2205\n[PROOFSTEP]\nobtain \u27e8S, hS\u27e9 := \u2039Module.Finite R M\u203a.out\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nS : Finset M\nhS : span R \u2191S = \u22a4\n\u22a2 \u2191(annihilator \u22a4) \u2229 \u2191R\u2070 \u2260 \u2205\n[PROOFSTEP]\nrefine' Set.Nonempty.ne_empty \u27e8_, _, (\u220f x in S, (@hM x).choose : R\u2070).prop\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nS : Finset M\nhS : span R \u2191S = \u22a4\n\u22a2 \u2191(\u220f x in S, Exists.choose (_ : \u2203 a, a \u2022 x = 0)) \u2208 \u2191(annihilator \u22a4)\n[PROOFSTEP]\nrw [Submonoid.coe_finset_prod, SetLike.mem_coe, \u2190 hS, mem_annihilator_span]\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nS : Finset M\nhS : span R \u2191S = \u22a4\n\u22a2 \u2200 (n : \u2191\u2191S), (\u220f i in S, \u2191(Exists.choose (_ : \u2203 a, a \u2022 i = 0))) \u2022 \u2191n = 0\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nS : Finset M\nhS : span R \u2191S = \u22a4\nn : \u2191\u2191S\n\u22a2 (\u220f i in S, \u2191(Exists.choose (_ : \u2203 a, a \u2022 i = 0))) \u2022 \u2191n = 0\n[PROOFSTEP]\nletI := Classical.decEq M\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module.Finite R M\nhM : Module.IsTorsion R M\nS : Finset M\nhS : span R \u2191S = \u22a4\nn : \u2191\u2191S\nthis : DecidableEq M := Classical.decEq M\n\u22a2 (\u220f i in S, \u2191(Exists.choose (_ : \u2203 a, a \u2022 i = 0))) \u2022 \u2191n = 0\n[PROOFSTEP]\nrw [\u2190 Finset.prod_erase_mul _ _ n.prop, mul_smul, \u2190 Submonoid.smul_def, (@hM n).choose_spec, smul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 \u2191(torsion R M) = {x | annihilator (span R {x}) \u2260 \u22a5}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nx : M\n\u22a2 x \u2208 \u2191(torsion R M) \u2194 x \u2208 {x | annihilator (span R {x}) \u2260 \u22a5}\n[PROOFSTEP]\nsimp_rw [Submodule.ne_bot_iff, mem_annihilator, mem_span_singleton]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nx : M\n\u22a2 x \u2208 \u2191(torsion R M) \u2194 x \u2208 {x | \u2203 x_1, (\u2200 (n : M), (\u2203 a, a \u2022 x = n) \u2192 x_1 \u2022 n = 0) \u2227 x_1 \u2260 0}\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8a, hax\u27e9 =>\n \u27e8a, fun _ \u27e8b, hb\u27e9 => by rw [\u2190 hb, smul_comm, \u2190 Submonoid.smul_def, hax, smul_zero], nonZeroDivisors.coe_ne_zero _\u27e9,\n fun \u27e8a, hax, ha\u27e9 => \u27e8\u27e8_, mem_nonZeroDivisors_of_ne_zero ha\u27e9, hax x \u27e81, one_smul _ _\u27e9\u27e9\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nx : M\nx\u271d\u00b2 : x \u2208 \u2191(torsion R M)\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x = 0\nx\u271d\u00b9 : M\nx\u271d : \u2203 a, a \u2022 x = x\u271d\u00b9\nb : R\nhb : b \u2022 x = x\u271d\u00b9\n\u22a2 \u2191a \u2022 x\u271d\u00b9 = 0\n[PROOFSTEP]\nrw [\u2190 hb, smul_comm, \u2190 Submonoid.smul_def, hax, smul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 NoZeroSMulDivisors R M \u2194 torsion R M = \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 NoZeroSMulDivisors R M \u2192 torsion R M = \u22a5\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 torsion R M = \u22a5 \u2192 NoZeroSMulDivisors R M\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5\n[PROOFSTEP]\nhaveI : NoZeroSMulDivisors R M := h\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M \u2264 \u22a5\n[PROOFSTEP]\nrintro x \u27e8a, hax\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nchange (a : R) \u2022 x = 0 at hax \n[GOAL]\ncase mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\ncases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0\n[GOAL]\ncase mp.intro.inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase mp.intro.inl.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nexact nonZeroDivisors.coe_ne_zero a h0\n[GOAL]\ncase mp.intro.inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : x = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexact h0\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\n\u22a2 NoZeroSMulDivisors R M\n[PROOFSTEP]\nexact\n {\n eq_zero_or_eq_zero_of_smul_eq_zero := fun {a} {x} hax =>\n by\n by_cases ha : a = 0\n \u00b7 left\n exact ha\n \u00b7 right\n rw [\u2190 mem_bot R, \u2190 h]\n exact \u27e8\u27e8a, mem_nonZeroDivisors_of_ne_zero ha\u27e9, hax\u27e9 }\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\n\u22a2 a = 0 \u2228 x = 0\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0 \u2228 x = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nexact ha\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 a = 0 \u2228 x = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 mem_bot R, \u2190 h]\n[GOAL]\ncase neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x \u2208 torsion R M\n[PROOFSTEP]\nexact \u27e8\u27e8a, mem_nonZeroDivisors_of_ne_zero ha\u27e9, hax\u27e9\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\nz : M \u29f8 torsion R M\nx : M\nx\u271d : Quotient.mk'' x \u2208 torsion R (M \u29f8 torsion R M)\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 Quotient.mk'' x = 0\n\u22a2 Quotient.mk'' x \u2208 \u22a5\n[PROOFSTEP]\nrw [Quotient.mk''_eq_mk, \u2190 Quotient.mk_smul, Quotient.mk_eq_zero] at hax \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\nz : M \u29f8 torsion R M\nx : M\nx\u271d : Quotient.mk'' x \u2208 torsion R (M \u29f8 torsion R M)\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x \u2208 torsion R M\n\u22a2 Quotient.mk'' x \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_bot, Quotient.mk''_eq_mk, Quotient.mk_eq_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\nz : M \u29f8 torsion R M\nx : M\nx\u271d : Quotient.mk'' x \u2208 torsion R (M \u29f8 torsion R M)\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x \u2208 torsion R M\n\u22a2 x \u2208 torsion R M\n[PROOFSTEP]\ncases' hax with b h\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nz : M \u29f8 torsion R M\nx : M\nx\u271d : Quotient.mk'' x \u2208 torsion R (M \u29f8 torsion R M)\na b : { x // x \u2208 R\u2070 }\nh : b \u2022 a \u2022 x = 0\n\u22a2 x \u2208 torsion R M\n[PROOFSTEP]\nexact \u27e8b * a, (mul_smul _ _ _).trans h\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\n\u22a2 IsTorsion' M { x // x \u2208 Submonoid.powers p } \u2194 \u2200 (x : M), \u2203 n, p ^ n \u2022 x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\n\u22a2 IsTorsion' M { x // x \u2208 Submonoid.powers p } \u2192 \u2200 (x : M), \u2203 n, p ^ n \u2022 x = 0\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nx : M\n\u22a2 \u2203 n, p ^ n \u2022 x = 0\n[PROOFSTEP]\nlet \u27e8\u27e8a, \u27e8n, hn\u27e9\u27e9, hx\u27e9 := @h x\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nx : M\na : R\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) p n = a\nhx : { val := a, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) p y = a) } \u2022 x = 0\n\u22a2 \u2203 n, p ^ n \u2022 x = 0\n[PROOFSTEP]\ndsimp at hn \n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nx : M\na : R\nn : \u2115\nhn : p ^ n = a\nhx : { val := a, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) p y = a) } \u2022 x = 0\n\u22a2 \u2203 n, p ^ n \u2022 x = 0\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nx : M\na : R\nn : \u2115\nhn : p ^ n = a\nhx : { val := a, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) p y = a) } \u2022 x = 0\n\u22a2 p ^ n \u2022 x = 0\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nx : M\na : R\nn : \u2115\nhn : p ^ n = a\nhx : { val := a, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) p y = a) } \u2022 x = 0\n\u22a2 a \u2022 x = 0\n[PROOFSTEP]\napply hx\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\n\u22a2 (\u2200 (x : M), \u2203 n, p ^ n \u2022 x = 0) \u2192 IsTorsion' M { x // x \u2208 Submonoid.powers p }\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : \u2200 (x : M), \u2203 n, p ^ n \u2022 x = 0\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nlet \u27e8n, hn\u27e9 := h x\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : DistribMulAction R M\np : R\nh : \u2200 (x : M), \u2203 n, p ^ n \u2022 x = 0\nx : M\nn : \u2115\nhn : p ^ n \u2022 x = 0\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nexact \u27e8\u27e8_, \u27e8n, rfl\u27e9\u27e9, hn\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\n\u22a2 \u2203 j, IsTorsionBy R M (p ^ pOrder hM (s j))\n[PROOFSTEP]\nlet oj := List.argmax (fun i => pOrder hM <| s i) (List.finRange d)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\n\u22a2 \u2203 j, IsTorsionBy R M (p ^ pOrder hM (s j))\n[PROOFSTEP]\nhave hoj : oj.isSome :=\n Option.ne_none_iff_isSome.mp fun eq_none => hd <| List.finRange_eq_nil.mp <| List.argmax_eq_none.mp eq_none\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\n\u22a2 \u2203 j, IsTorsionBy R M (p ^ pOrder hM (s j))\n[PROOFSTEP]\nuse Option.get _ hoj\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\n\u22a2 IsTorsionBy R M (p ^ pOrder hM (s (Option.get oj hoj)))\n[PROOFSTEP]\nrw [isTorsionBy_iff_torsionBy_eq_top, eq_top_iff, \u2190 hs, Submodule.span_le, Set.range_subset_iff]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\n\u22a2 \u2200 (y : Fin d), s y \u2208 \u2191(torsionBy R M (p ^ pOrder hM (s (Option.get oj hoj))))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\ni : Fin d\n\u22a2 s i \u2208 \u2191(torsionBy R M (p ^ pOrder hM (s (Option.get oj hoj))))\n[PROOFSTEP]\nchange (p ^ pOrder hM (s (Option.get oj hoj))) \u2022 s i = 0\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\ni : Fin d\n\u22a2 p ^ pOrder hM (s (Option.get oj hoj)) \u2022 s i = 0\n[PROOFSTEP]\nhave : pOrder hM (s i) \u2264 pOrder hM (s <| Option.get _ hoj) :=\n List.le_of_mem_argmax (List.mem_finRange i) (Option.get_mem hoj)\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : (x : M) \u2192 Decidable (x = 0)\np : R\nhM : IsTorsion' M { x // x \u2208 Submonoid.powers p }\nd : \u2115\nhd : d \u2260 0\ns : Fin d \u2192 M\nhs : span R (Set.range s) = \u22a4\noj : Option (Fin d) := List.argmax (fun i => pOrder hM (s i)) (List.finRange d)\nhoj : Option.isSome oj = true\ni : Fin d\nthis : pOrder hM (s i) \u2264 pOrder hM (s (Option.get oj hoj))\n\u22a2 p ^ pOrder hM (s (Option.get oj hoj)) \u2022 s i = 0\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel this, pow_add, mul_smul, pow_pOrder_smul, smul_zero]\n[GOAL]\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\n\u22a2 torsionBy R (R \u29f8 Submodule.span R {a * b}) a = Submodule.span R {\u2191(mk (Submodule.span R {a * b})) b}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R \u29f8 Submodule.span R {a * b}\n\u22a2 x \u2208 torsionBy R (R \u29f8 Submodule.span R {a * b}) a \u2194 x \u2208 Submodule.span R {\u2191(mk (Submodule.span R {a * b})) b}\n[PROOFSTEP]\nrw [mem_torsionBy_iff, Submodule.mem_span_singleton]\n[GOAL]\ncase h\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R \u29f8 Submodule.span R {a * b}\n\u22a2 a \u2022 x = 0 \u2194 \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = x\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := mk_surjective x\n[GOAL]\ncase h.intro\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\n\u22a2 a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0 \u2194\n \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.intro.mp\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\n\u22a2 a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0 \u2192\n \u2203 a_2, a_2 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.intro.mpr\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\n\u22a2 (\u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x) \u2192\n a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.intro.mp\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\nh : a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0\n\u22a2 \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nrw [\u2190 mk_eq_mk, \u2190 Quotient.mk_smul, Quotient.mk_eq_zero, Submodule.mem_span_singleton] at h \n[GOAL]\ncase h.intro.mp\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\nh : \u2203 a_1, a_1 \u2022 (a * b) = a \u2022 x\n\u22a2 \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := h\n[GOAL]\ncase h.intro.mp.intro\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx c : R\nh : c \u2022 (a * b) = a \u2022 x\n\u22a2 \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nrw [smul_eq_mul, smul_eq_mul, mul_comm, mul_assoc, mul_cancel_left_mem_nonZeroDivisors ha, mul_comm] at h \n[GOAL]\ncase h.intro.mp.intro\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx c : R\nh : c * b = x\n\u22a2 \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx c : R\nh : c * b = x\n\u22a2 c \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n[PROOFSTEP]\nrw [\u2190 h, \u2190 mk_eq_mk, \u2190 Quotient.mk_smul, smul_eq_mul, mk_eq_mk]\n[GOAL]\ncase h.intro.mpr\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx : R\nh : \u2203 a_1, a_1 \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n\u22a2 a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := h\n[GOAL]\ncase h.intro.mpr.intro\nR\u271d : Type u_1\nM : Type u_2\nR : Type w\ninst\u271d : CommRing R\na b : R\nha : a \u2208 R\u2070\nx c : R\nh : c \u2022 \u2191(mk (Submodule.span R {a * b})) b = \u2191(mk (Submodule.span R {a * b})) x\n\u22a2 a \u2022 \u2191(mk (Submodule.span R {a * b})) x = 0\n[PROOFSTEP]\nrw [\u2190 h, smul_comm, \u2190 mk_eq_mk, \u2190 Quotient.mk_smul, (Quotient.mk_eq_zero _).mpr <| mem_span_singleton_self _, smul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\n\u22a2 IsTorsion M \u2194 Module.IsTorsion \u2115 M\n[PROOFSTEP]\nrefine' \u27e8fun h x => _, fun h x => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nh : IsTorsion M\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8n, h0, hn\u27e9 := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)\n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nh : IsTorsion M\nx : M\nn : \u2115\nh0 : 0 < n\nhn : n \u2022 x = 0\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nexact \u27e8\u27e8n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt h0\u27e9, hn\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nh : Module.IsTorsion \u2115 M\nx : M\n\u22a2 IsOfFinAddOrder x\n[PROOFSTEP]\nrw [isOfFinAddOrder_iff_nsmul_eq_zero]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nh : Module.IsTorsion \u2115 M\nx : M\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := @h x\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommMonoid M\nh : Module.IsTorsion \u2115 M\nx : M\nn : { x // x \u2208 \u2115\u2070 }\nhn : n \u2022 x = 0\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0\n[PROOFSTEP]\nrefine' \u27e8n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn\u27e9\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\n\u22a2 IsTorsion M \u2194 Module.IsTorsion \u2124 M\n[PROOFSTEP]\nrefine' \u27e8fun h x => _, fun h x => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8n, h0, hn\u27e9 := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)\n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\nn : \u2115\nh0 : 0 < n\nhn : n \u2022 x = 0\n\u22a2 \u2203 a, a \u2022 x = 0\n[PROOFSTEP]\nexact \u27e8\u27e8n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.coe_nat_pos.mpr h0\u27e9, (coe_nat_zsmul _ _).trans hn\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 IsOfFinAddOrder x\n[PROOFSTEP]\nrw [isOfFinAddOrder_iff_nsmul_eq_zero]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := @h x\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nM : Type u_2\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\nn : { x // x \u2208 \u2124\u2070 }\nhn : n \u2022 x = 0\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0\n[PROOFSTEP]\nexact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Module.Torsion", "llama_tokens": 37402, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.4493926344647597, "lm_q1q2_score": 0.2560875908415096}} {"text": "[GOAL]\n\u03b1 \u03b2 : NonemptyFinLinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191e \u226b \u2191(OrderIso.symm e) = \ud835\udfd9 \u03b1\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : NonemptyFinLinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget NonemptyFinLinOrdCat).obj \u03b1\n\u22a2 \u2191(\u2191e \u226b \u2191(OrderIso.symm e)) x = \u2191(\ud835\udfd9 \u03b1) x\n[PROOFSTEP]\nexact e.symm_apply_apply x\n[GOAL]\n\u03b1 \u03b2 : NonemptyFinLinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191(OrderIso.symm e) \u226b \u2191e = \ud835\udfd9 \u03b2\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : NonemptyFinLinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget NonemptyFinLinOrdCat).obj \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm e) \u226b \u2191e) x = \u2191(\ud835\udfd9 \u03b2) x\n[PROOFSTEP]\nexact e.apply_symm_apply x\n[GOAL]\nA B : NonemptyFinLinOrdCat\nx\u271d\u00b9 x\u271d : A \u27f6 B\nh :\n (fun f =>\n \u2191(let_fun this := f;\n this))\n x\u271d\u00b9 =\n (fun f =>\n \u2191(let_fun this := f;\n this))\n x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nA B : NonemptyFinLinOrdCat\nx\u271d\u00b9 x\u271d : A \u27f6 B\nh :\n (fun f =>\n \u2191(let_fun this := f;\n this))\n x\u271d\u00b9 =\n (fun f =>\n \u2191(let_fun this := f;\n this))\n x\u271d\nx : (forget NonemptyFinLinOrdCat).obj A\n\u22a2 \u2191x\u271d\u00b9 x = \u2191x\u271d x\n[PROOFSTEP]\nexact congr_fun h x\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\n\u22a2 Mono f \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nrefine' \u27e8_, ConcreteCategory.mono_of_injective f\u27e9\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\n\u22a2 Mono f \u2192 Function.Injective \u2191f\n[PROOFSTEP]\nintro\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\n\u22a2 Function.Injective \u2191f\n[PROOFSTEP]\nintro a\u2081 a\u2082 h\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nlet X := NonemptyFinLinOrdCat.of (ULift (Fin 1))\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nlet g\u2081 : X \u27f6 A := \u27e8fun _ => a\u2081, fun _ _ _ => by rfl\u27e9\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\nx\u271d\u00b2 x\u271d\u00b9 : \u2191X\nx\u271d : x\u271d\u00b2 \u2264 x\u271d\u00b9\n\u22a2 (fun x => a\u2081) x\u271d\u00b2 \u2264 (fun x => a\u2081) x\u271d\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nlet g\u2082 : X \u27f6 A := \u27e8fun _ => a\u2082, fun _ _ _ => by rfl\u27e9\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\nx\u271d\u00b2 x\u271d\u00b9 : \u2191X\nx\u271d : x\u271d\u00b2 \u2264 x\u271d\u00b9\n\u22a2 (fun x => a\u2082) x\u271d\u00b2 \u2264 (fun x => a\u2082) x\u271d\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nchange g\u2081 (ULift.up (0 : Fin 1)) = g\u2082 (ULift.up (0 : Fin 1))\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\n\u22a2 \u2191g\u2081 { down := 0 } = \u2191g\u2082 { down := 0 }\n[PROOFSTEP]\nhave eq : g\u2081 \u226b f = g\u2082 \u226b f := by\n ext\n exact h\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\n\u22a2 g\u2081 \u226b f = g\u2082 \u226b f\n[PROOFSTEP]\next\n[GOAL]\ncase w\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\nx\u271d : (forget NonemptyFinLinOrdCat).obj X\n\u22a2 \u2191(g\u2081 \u226b f) x\u271d = \u2191(g\u2082 \u226b f) x\u271d\n[PROOFSTEP]\nexact h\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\neq : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 \u2191g\u2081 { down := 0 } = \u2191g\u2082 { down := 0 }\n[PROOFSTEP]\nrw [cancel_mono] at eq \n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : \u2191f a\u2081 = \u2191f a\u2082\nX : NonemptyFinLinOrdCat := of (ULift (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2081) x \u2264 (fun x => a\u2081) x) }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := (_ : \u2200 (x x_1 : \u2191X), x \u2264 x_1 \u2192 (fun x => a\u2082) x \u2264 (fun x => a\u2082) x) }\neq : g\u2081 = g\u2082\n\u22a2 \u2191g\u2081 { down := 0 } = \u2191g\u2082 { down := 0 }\n[PROOFSTEP]\nrw [eq]\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\n\u22a2 Epi f \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\n\u22a2 Epi f \u2192 Function.Surjective \u2191f\n[PROOFSTEP]\nintro\n[GOAL]\ncase mp\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\n\u22a2 Function.Surjective \u2191f\n[PROOFSTEP]\ndsimp only [Function.Surjective]\n[GOAL]\ncase mp\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\n\u22a2 \u2200 (b : \u2191B), \u2203 a, \u2191f a = b\n[PROOFSTEP]\nby_contra' hf'\n[GOAL]\ncase mp\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nhf' : \u2203 b, \u2200 (a : \u2191A), \u2191f a \u2260 b\n\u22a2 False\n[PROOFSTEP]\nrcases hf' with \u27e8m, hm\u27e9\n[GOAL]\ncase mp.intro\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\n\u22a2 False\n[PROOFSTEP]\nlet Y := NonemptyFinLinOrdCat.of (ULift (Fin 2))\n[GOAL]\ncase mp.intro\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\n\u22a2 False\n[PROOFSTEP]\nlet p\u2081 : B \u27f6 Y :=\n \u27e8fun b => if b < m then ULift.up 0 else ULift.up 1, fun x\u2081 x\u2082 h =>\n by\n simp only\n split_ifs with h\u2081 h\u2082 h\u2082\n any_goals apply Fin.zero_le\n \u00b7 exfalso\n exact h\u2081 (lt_of_le_of_lt h h\u2082)\n \u00b7 rfl\u27e9\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\n\u22a2 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082\n[PROOFSTEP]\nsimp only\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\n\u22a2 (if x\u2081 < m then { down := 0 } else { down := 1 }) \u2264 if x\u2082 < m then { down := 0 } else { down := 1 }\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 { down := 0 } \u2264 { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 < m\nh\u2082 : \u00acx\u2082 < m\n\u22a2 { down := 0 } \u2264 { down := 1 }\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : \u00acx\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\nany_goals apply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 { down := 0 } \u2264 { down := 0 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 < m\nh\u2082 : \u00acx\u2082 < m\n\u22a2 { down := 0 } \u2264 { down := 1 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 0 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : \u00acx\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 0 }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : x\u2082 < m\n\u22a2 False\n[PROOFSTEP]\nexact h\u2081 (lt_of_le_of_lt h h\u2082)\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 < m\nh\u2082 : \u00acx\u2082 < m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.intro\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\n\u22a2 False\n[PROOFSTEP]\nlet p\u2082 : B \u27f6 Y :=\n \u27e8fun b => if b \u2264 m then ULift.up 0 else ULift.up 1, fun x\u2081 x\u2082 h =>\n by\n simp only\n split_ifs with h\u2081 h\u2082 h\u2082\n any_goals apply Fin.zero_le\n \u00b7 exfalso\n exact h\u2081 (h.trans h\u2082)\n \u00b7 rfl\u27e9\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\n\u22a2 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082\n[PROOFSTEP]\nsimp only\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\n\u22a2 (if x\u2081 \u2264 m then { down := 0 } else { down := 1 }) \u2264 if x\u2082 \u2264 m then { down := 0 } else { down := 1 }\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 { down := 0 } \u2264 { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 \u2264 m\nh\u2082 : \u00acx\u2082 \u2264 m\n\u22a2 { down := 0 } \u2264 { down := 1 }\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : \u00acx\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\nany_goals apply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 { down := 0 } \u2264 { down := 0 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : x\u2081 \u2264 m\nh\u2082 : \u00acx\u2082 \u2264 m\n\u22a2 { down := 0 } \u2264 { down := 1 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 0 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : \u00acx\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\napply Fin.zero_le\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 0 }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : x\u2082 \u2264 m\n\u22a2 False\n[PROOFSTEP]\nexact h\u2081 (h.trans h\u2082)\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\nx\u2081 x\u2082 : \u2191B\nh : x\u2081 \u2264 x\u2082\nh\u2081 : \u00acx\u2081 \u2264 m\nh\u2082 : \u00acx\u2082 \u2264 m\n\u22a2 { down := 1 } \u2264 { down := 1 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.intro\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\n\u22a2 False\n[PROOFSTEP]\nhave h : p\u2081 m = p\u2082 m := by\n congr\n rw [\u2190 cancel_epi f]\n ext a\n simp only [coe_of, comp_apply]\n change ite _ _ _ = ite _ _ _\n split_ifs with h\u2081 h\u2082 h\u2082\n any_goals rfl\n \u00b7 exfalso\n exact h\u2082 (le_of_lt h\u2081)\n \u00b7 exfalso\n exact hm a (eq_of_le_of_not_lt h\u2082 h\u2081)\n[GOAL]\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\n\u22a2 \u2191p\u2081 m = \u2191p\u2082 m\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\n\u22a2 p\u2081 = p\u2082\n[PROOFSTEP]\nrw [\u2190 cancel_epi f]\n[GOAL]\ncase e_a\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\n\u22a2 f \u226b p\u2081 = f \u226b p\u2082\n[PROOFSTEP]\next a\n[GOAL]\ncase e_a.w\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\n\u22a2 \u2191(f \u226b p\u2081) a = \u2191(f \u226b p\u2082) a\n[PROOFSTEP]\nsimp only [coe_of, comp_apply]\n[GOAL]\ncase e_a.w\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\n\u22a2 \u2191{ toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\n (\u2191f a) =\n \u2191{ toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\n (\u2191f a)\n[PROOFSTEP]\nchange ite _ _ _ = ite _ _ _\n[GOAL]\ncase e_a.w\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\n\u22a2 (if \u2191f a < m then { down := 0 } else { down := 1 }) = if \u2191f a \u2264 m then { down := 0 } else { down := 1 }\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 { down := 0 } = { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 { down := 0 } = { down := 1 }\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 { down := 1 } = { down := 0 }\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 { down := 1 } = { down := 1 }\n[PROOFSTEP]\nany_goals rfl\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 { down := 0 } = { down := 0 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 { down := 0 } = { down := 1 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 { down := 1 } = { down := 0 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 { down := 1 } = { down := 1 }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 { down := 0 } = { down := 1 }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u2191f a < m\nh\u2082 : \u00ac\u2191f a \u2264 m\n\u22a2 False\n[PROOFSTEP]\nexact h\u2082 (le_of_lt h\u2081)\n[GOAL]\ncase pos\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 { down := 1 } = { down := 0 }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\na : (forget NonemptyFinLinOrdCat).obj A\nh\u2081 : \u00ac\u2191f a < m\nh\u2082 : \u2191f a \u2264 m\n\u22a2 False\n[PROOFSTEP]\nexact hm a (eq_of_le_of_not_lt h\u2082 h\u2081)\n[GOAL]\ncase mp.intro\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\na\u271d : Epi f\nm : \u2191B\nhm : \u2200 (a : \u2191A), \u2191f a \u2260 m\nY : NonemptyFinLinOrdCat := of (ULift (Fin 2))\np\u2081 : B \u27f6 Y :=\n { toFun := fun b => if b < m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) }\np\u2082 : B \u27f6 Y :=\n { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 },\n monotone' :=\n (_ :\n \u2200 (x\u2081 x\u2082 : \u2191B),\n x\u2081 \u2264 x\u2082 \u2192\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264\n (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) }\nh : \u2191p\u2081 m = \u2191p\u2082 m\n\u22a2 False\n[PROOFSTEP]\nsimp [FunLike.coe] at h \n[GOAL]\ncase mpr\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\n\u22a2 Function.Surjective \u2191f \u2192 Epi f\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nA B : NonemptyFinLinOrdCat\nf : A \u27f6 B\nh : Function.Surjective \u2191f\n\u22a2 Epi f\n[PROOFSTEP]\nexact ConcreteCategory.epi_of_surjective f h\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\n\u22a2 IsSplitEpi f\n[PROOFSTEP]\nhave H : \u2200 y : Y, Nonempty (f \u207b\u00b9' { y }) := by\n rw [epi_iff_surjective] at hf \n intro y\n exact Nonempty.intro \u27e8(hf y).choose, (hf y).choose_spec\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\n\u22a2 \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n[PROOFSTEP]\nrw [epi_iff_surjective] at hf \n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Function.Surjective \u2191f\n\u22a2 \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n[PROOFSTEP]\nintro y\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Function.Surjective \u2191f\ny : \u2191Y\n\u22a2 Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n[PROOFSTEP]\nexact Nonempty.intro \u27e8(hf y).choose, (hf y).choose_spec\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u22a2 IsSplitEpi f\n[PROOFSTEP]\nlet \u03c6 : Y \u2192 X := fun y => (H y).some.1\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\n\u22a2 IsSplitEpi f\n[PROOFSTEP]\nhave h\u03c6 : \u2200 y : Y, f (\u03c6 y) = y := fun y => (H y).some.2\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\n\u22a2 IsSplitEpi f\n[PROOFSTEP]\nrefine' IsSplitEpi.mk' \u27e8\u27e8\u03c6, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\n\u22a2 Monotone \u03c6\ncase refine'_2\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\n\u22a2 { toFun := \u03c6, monotone' := ?refine'_1 } \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_2\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\n\u22a2 { toFun := \u03c6, monotone' := ?refine'_1 } \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\next b\n[GOAL]\ncase refine'_2.w\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\nb : (forget NonemptyFinLinOrdCat).obj Y\n\u22a2 \u2191({ toFun := \u03c6, monotone' := ?refine'_1 } \u226b f) b = \u2191(\ud835\udfd9 Y) b\n[PROOFSTEP]\napply h\u03c6\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\n\u22a2 Monotone \u03c6\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\n\u22a2 a \u2264 b \u2192 \u03c6 a \u2264 \u03c6 b\n[PROOFSTEP]\ncontrapose\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\n\u22a2 \u00ac\u03c6 a \u2264 \u03c6 b \u2192 \u00aca \u2264 b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u00ac\u03c6 a \u2264 \u03c6 b\n\u22a2 \u00aca \u2264 b\n[PROOFSTEP]\nsimp only [not_le] at h \u22a2\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {a})))\n\u22a2 b < a\n[PROOFSTEP]\nsuffices b \u2264 a by\n apply lt_of_le_of_ne this\n rintro rfl\n exfalso\n simp at h \n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {a})))\nthis : b \u2264 a\n\u22a2 b < a\n[PROOFSTEP]\napply lt_of_le_of_ne this\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {a})))\nthis : b \u2264 a\n\u22a2 b \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\nb : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b})))\nthis : b \u2264 b\n\u22a2 False\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\nb : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b})))\nthis : b \u2264 b\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {a})))\n\u22a2 b \u2264 a\n[PROOFSTEP]\nhave H : f (\u03c6 b) \u2264 f (\u03c6 a) := f.monotone (le_of_lt h)\n[GOAL]\ncase refine'_1\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nhf : Epi f\nH\u271d : \u2200 (y : \u2191Y), Nonempty \u2191(\u2191f \u207b\u00b9' {y})\n\u03c6 : \u2191Y \u2192 \u2191X := fun y => \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {y})))\nh\u03c6 : \u2200 (y : \u2191Y), \u2191f (\u03c6 y) = y\na b : \u2191Y\nh : \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {b}))) < \u2191(Nonempty.some (_ : Nonempty \u2191(\u2191f \u207b\u00b9' {a})))\nH : \u2191f (\u03c6 b) \u2264 \u2191f (\u03c6 a)\n\u22a2 b \u2264 a\n[PROOFSTEP]\nsimpa only [h\u03c6] using H\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nletI : NonemptyFinLinOrd (Set.image f \u22a4) := \u27e8by infer_instance\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\n\u22a2 Nonempty \u2191(\u2191f '' \u22a4)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nlet I := NonemptyFinLinOrdCat.of (Set.image f \u22a4)\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nlet e : X \u27f6 I := \u27e8fun x => \u27e8f x, \u27e8x, by tauto\u27e9\u27e9, fun x\u2081 x\u2082 h => f.monotone h\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\nx : \u2191X\n\u22a2 x \u2208 \u22a4 \u2227 \u2191f x = \u2191f x\n[PROOFSTEP]\ntauto\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nlet m : I \u27f6 Y := \u27e8fun y => y.1, by tauto\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\n\u22a2 Monotone fun y => \u2191y\n[PROOFSTEP]\ntauto\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nhaveI : Epi e := by\n rw [epi_iff_surjective]\n rintro \u27e8_, y, h, rfl\u27e9\n exact \u27e8y, rfl\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\n\u22a2 Epi e\n[PROOFSTEP]\nrw [epi_iff_surjective]\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\n\u22a2 Function.Surjective \u2191e\n[PROOFSTEP]\nrintro \u27e8_, y, h, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\ny : \u2191X\nh : y \u2208 \u22a4\n\u22a2 \u2203 a, \u2191e a = { val := \u2191f y, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f y) }\n[PROOFSTEP]\nexact \u27e8y, rfl\u27e9\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis\u271d : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\nthis : Epi e\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nhaveI : StrongEpi e := strongEpi_of_epi e\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis\u271d\u00b9 : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\nthis\u271d : Epi e\nthis : StrongEpi e\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nhaveI : Mono m := ConcreteCategory.mono_of_injective _ (fun x y h => Subtype.ext h)\n[GOAL]\nX Y : NonemptyFinLinOrdCat\nf : X \u27f6 Y\nthis\u271d\u00b2 : NonemptyFinLinOrd \u2191(\u2191f '' \u22a4) := NonemptyFinLinOrd.mk\nI : NonemptyFinLinOrdCat := of \u2191(\u2191f '' \u22a4)\ne : X \u27f6 I :=\n { toFun := fun x => { val := \u2191f x, property := (_ : \u2203 a, a \u2208 \u22a4 \u2227 \u2191f a = \u2191f x) },\n monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191X), x\u2081 \u2264 x\u2082 \u2192 \u2191f x\u2081 \u2264 \u2191f x\u2082) }\nm : I \u27f6 Y := { toFun := fun y => \u2191y, monotone' := (_ : \u2200 \u2983a b : \u2191I\u2984, a \u2264 b \u2192 a \u2264 b) }\nthis\u271d\u00b9 : Epi e\nthis\u271d : StrongEpi e\nthis : Mono m\n\u22a2 Nonempty (StrongEpiMonoFactorisation f)\n[PROOFSTEP]\nexact \u27e8\u27e8I, m, e, rfl\u27e9\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.NonemptyFinLinOrdCat", "llama_tokens": 23409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.2549339603760972}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA A\u2081 A\u2082 : IndexSet \u0394\nh\u2081 : A\u2081.fst = A\u2082.fst\n\u22a2 A\u2081.fst.unop = A\u2082.fst.unop\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA A\u2081 A\u2082 : IndexSet \u0394\nh\u2081 : A\u2081.fst = A\u2082.fst\nh\u2082 : e A\u2081 \u226b eqToHom (_ : A\u2081.fst.unop = A\u2082.fst.unop) = e A\u2082\n\u22a2 A\u2081 = A\u2082\n[PROOFSTEP]\nrcases A\u2081 with \u27e8\u0394\u2081, \u27e8\u03b1\u2081, h\u03b1\u2081\u27e9\u27e9\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA A\u2082 : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2081 : Epi \u03b1\u2081\nh\u2081 : { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst = A\u2082.fst\nh\u2082 :\n e { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } \u226b\n eqToHom (_ : { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst.unop = A\u2082.fst.unop) =\n e A\u2082\n\u22a2 { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } = A\u2082\n[PROOFSTEP]\nrcases A\u2082 with \u27e8\u0394\u2082, \u27e8\u03b1\u2082, h\u03b1\u2082\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2081 : Epi \u03b1\u2081\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082 : \u0394.unop \u27f6 \u0394\u2082.unop\nh\u03b1\u2082 : Epi \u03b1\u2082\nh\u2081 : { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst = { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }.fst\nh\u2082 :\n e { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } \u226b\n eqToHom\n (_ :\n { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst.unop =\n { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }.fst.unop) =\n e { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n\u22a2 { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } = { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n[PROOFSTEP]\nsimp only at h\u2081 \n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2081 : Epi \u03b1\u2081\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082 : \u0394.unop \u27f6 \u0394\u2082.unop\nh\u03b1\u2082 : Epi \u03b1\u2082\nh\u2081 : \u0394\u2081 = \u0394\u2082\nh\u2082 :\n e { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } \u226b\n eqToHom\n (_ :\n { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst.unop =\n { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }.fst.unop) =\n e { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n\u22a2 { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } = { fst := \u0394\u2082, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n[PROOFSTEP]\nsubst h\u2081\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2081 : Epi \u03b1\u2081\n\u03b1\u2082 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2082 : Epi \u03b1\u2082\nh\u2082 :\n e { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } \u226b\n eqToHom\n (_ :\n { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } }.fst.unop =\n { fst := \u0394\u2081, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }.fst.unop) =\n e { fst := \u0394\u2081, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n\u22a2 { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } = { fst := \u0394\u2081, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n[PROOFSTEP]\nsimp only [eqToHom_refl, comp_id, IndexSet.e] at h\u2082 \n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.913, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2081 : Epi \u03b1\u2081\n\u03b1\u2082 : \u0394.unop \u27f6 \u0394\u2081.unop\nh\u03b1\u2082 : Epi \u03b1\u2082\nh\u2082 : \u03b1\u2081 = \u03b1\u2082\n\u22a2 { fst := \u0394\u2081, snd := { val := \u03b1\u2081, property := h\u03b1\u2081 } } = { fst := \u0394\u2081, snd := { val := \u03b1\u2082, property := h\u03b1\u2082 } }\n[PROOFSTEP]\nsimp only [h\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 Function.Injective fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) }\n[PROOFSTEP]\nrintro \u27e8\u0394\u2081, \u03b1\u2081\u27e9 \u27e8\u0394\u2082, \u03b1\u2082\u27e9 h\u2081\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081 : SimplexCategory\u1d52\u1d56\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082 }\n\u22a2 { fst := \u0394\u2081, snd := \u03b1\u2081 } = { fst := \u0394\u2082, snd := \u03b1\u2082 }\n[PROOFSTEP]\ninduction' \u0394\u2081 using Opposite.rec with \u0394\u2081\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082 }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082 }\n\u22a2 { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } = { fst := \u0394\u2082, snd := \u03b1\u2082 }\n[PROOFSTEP]\ninduction' \u0394\u2082 using Opposite.rec with \u0394\u2082\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2082 : SimplexCategory\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\nh\u2081 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\n\u22a2 { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } = { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\n[PROOFSTEP]\nsimp only [unop_op, Sigma.mk.inj_iff, Fin.mk.injEq] at h\u2081 \n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2082 : SimplexCategory\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2082 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }))\n\u22a2 { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } = { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\n[PROOFSTEP]\nhave h\u2082 : \u0394\u2081 = \u0394\u2082 := by\n ext1\n simpa only [Fin.mk_eq_mk] using h\u2081.1\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2082 : SimplexCategory\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2082 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }))\n\u22a2 \u0394\u2081 = \u0394\u2082\n[PROOFSTEP]\next1\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2082 : SimplexCategory\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2082 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }))\n\u22a2 len \u0394\u2081 = len \u0394\u2082\n[PROOFSTEP]\nsimpa only [Fin.mk_eq_mk] using h\u2081.1\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082\u271d, snd := \u03b1\u2082\u271d }\n\u0394\u2082 : SimplexCategory\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2082 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }))\nh\u2082 : \u0394\u2081 = \u0394\u2082\n\u22a2 { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } = { fst := { unop := \u0394\u2082 }, snd := \u03b1\u2082 }\n[PROOFSTEP]\nsubst h\u2082\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2081 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }))\n\u22a2 { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } = { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }\n[PROOFSTEP]\nrefine' ext _ _ rfl _\n[GOAL]\ncase mk.mk.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2081 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }))\n\u22a2 e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } \u226b\n eqToHom (_ : { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }.fst.unop = { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }.fst.unop) =\n e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }\n[PROOFSTEP]\next : 2\n[GOAL]\ncase mk.mk.mk.mk.a.h\nC : Type u_1\ninst\u271d : Category.{?u.1648, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u0394\u2081\u271d : SimplexCategory\u1d52\u1d56\n\u03b1\u2081\u271d : { \u03b1 // Epi \u03b1 }\n\u0394\u2082 : SimplexCategory\u1d52\u1d56\n\u03b1\u2082\u271d : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b2 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u0394\u2081 : SimplexCategory\n\u03b1\u2081 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d\u00b9 :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2082, snd := \u03b1\u2082\u271d }\n\u03b1\u2082 : { \u03b1 // Epi \u03b1 }\nh\u2081\u271d :\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := \u0394\u2081\u271d, snd := \u03b1\u2081\u271d } =\n (fun A =>\n { fst := { val := len A.fst.unop, isLt := (_ : len A.fst.unop < Nat.succ (len \u0394.unop)) },\n snd := \u2191(Hom.toOrderHom (e A)) })\n { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }\nh\u2081 :\n len \u0394\u2081 = len \u0394\u2081 \u2227\n HEq \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }))\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }))\n\u22a2 \u2191(Hom.toOrderHom\n (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 } \u226b\n eqToHom\n (_ : { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2081 }.fst.unop = { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }.fst.unop))) =\n \u2191(Hom.toOrderHom (e { fst := { unop := \u0394\u2081 }, snd := \u03b1\u2082 }))\n[PROOFSTEP]\nexact eq_of_heq h\u2081.2\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.3923, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 Epi (\ud835\udfd9 \u0394.unop)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2194 A.fst = \u0394\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2192 A.fst = \u0394\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : EqId A\n\u22a2 A.fst = \u0394\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : A = id \u0394\n\u22a2 A.fst = \u0394\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : A = id \u0394\n\u22a2 (id \u0394).fst = \u0394\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 A.fst = \u0394 \u2192 EqId A\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : A.fst = \u0394\n\u22a2 EqId A\n[PROOFSTEP]\nrcases A with \u27e8_, \u27e8f, hf\u27e9\u27e9\n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 fst\u271d : SimplexCategory\u1d52\u1d56\nf : \u0394.unop \u27f6 fst\u271d.unop\nhf : Epi f\nh : { fst := fst\u271d, snd := { val := f, property := hf } }.fst = \u0394\n\u22a2 EqId { fst := fst\u271d, snd := { val := f, property := hf } }\n[PROOFSTEP]\nsimp only at h \n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\n\u0394 fst\u271d : SimplexCategory\u1d52\u1d56\nf : \u0394.unop \u27f6 fst\u271d.unop\nhf : Epi f\nh : fst\u271d = \u0394\n\u22a2 EqId { fst := fst\u271d, snd := { val := f, property := hf } }\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\nfst\u271d : SimplexCategory\u1d52\u1d56\nf : fst\u271d.unop \u27f6 fst\u271d.unop\nhf : Epi f\n\u22a2 EqId { fst := fst\u271d, snd := { val := f, property := hf } }\n[PROOFSTEP]\nrefine' ext _ _ rfl _\n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\nfst\u271d : SimplexCategory\u1d52\u1d56\nf : fst\u271d.unop \u27f6 fst\u271d.unop\nhf : Epi f\n\u22a2 e { fst := fst\u271d, snd := { val := f, property := hf } } \u226b\n eqToHom (_ : { fst := fst\u271d, snd := { val := f, property := hf } }.fst.unop = (id fst\u271d).fst.unop) =\n e (id fst\u271d)\n[PROOFSTEP]\nhaveI := hf\n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\nfst\u271d : SimplexCategory\u1d52\u1d56\nf : fst\u271d.unop \u27f6 fst\u271d.unop\nhf this : Epi f\n\u22a2 e { fst := fst\u271d, snd := { val := f, property := hf } } \u226b\n eqToHom (_ : { fst := fst\u271d, snd := { val := f, property := hf } }.fst.unop = (id fst\u271d).fst.unop) =\n e (id fst\u271d)\n[PROOFSTEP]\nsimp only [eqToHom_refl, comp_id]\n[GOAL]\ncase mpr.mk.mk\nC : Type u_1\ninst\u271d : Category.{?u.4321, u_1} C\nfst\u271d : SimplexCategory\u1d52\u1d56\nf : fst\u271d.unop \u27f6 fst\u271d.unop\nhf this : Epi f\n\u22a2 e { fst := fst\u271d, snd := { val := f, property := hf } } = e (id fst\u271d)\n[PROOFSTEP]\nexact eq_id_of_epi f\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2194 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nrw [eqId_iff_eq]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 A.fst = \u0394 \u2194 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 A.fst = \u0394 \u2192 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : A.fst = \u0394\n\u22a2 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 len A.fst.unop = len \u0394.unop \u2192 A.fst = \u0394\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : len A.fst.unop = len \u0394.unop\n\u22a2 A.fst = \u0394\n[PROOFSTEP]\nrw [\u2190 unop_inj_iff]\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : len A.fst.unop = len \u0394.unop\n\u22a2 A.fst.unop = \u0394.unop\n[PROOFSTEP]\next\n[GOAL]\ncase mpr.a\nC : Type u_1\ninst\u271d : Category.{?u.4752, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : len A.fst.unop = len \u0394.unop\n\u22a2 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4880, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2194 len \u0394.unop \u2264 len A.fst.unop\n[PROOFSTEP]\nrw [eqId_iff_len_eq]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4880, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 len A.fst.unop = len \u0394.unop \u2194 len \u0394.unop \u2264 len A.fst.unop\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4880, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 len A.fst.unop = len \u0394.unop \u2192 len \u0394.unop \u2264 len A.fst.unop\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.4880, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : len A.fst.unop = len \u0394.unop\n\u22a2 len \u0394.unop \u2264 len A.fst.unop\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.4880, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 len \u0394.unop \u2264 len A.fst.unop \u2192 len A.fst.unop = len \u0394.unop\n[PROOFSTEP]\nexact le_antisymm (len_le_of_epi (inferInstance : Epi A.e))\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2194 Mono (e A)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 EqId A \u2192 Mono (e A)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : EqId A\n\u22a2 Mono (e A)\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : A = id \u0394\n\u22a2 Mono (e A)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 Mono (e (id \u0394))\n[PROOFSTEP]\ndsimp only [id, e]\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 Mono (\ud835\udfd9 \u0394.unop)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 Mono (e A) \u2192 EqId A\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : Mono (e A)\n\u22a2 EqId A\n[PROOFSTEP]\nrw [eqId_iff_len_le]\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d : Category.{?u.5425, u_1} C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\nh : Mono (e A)\n\u22a2 len \u0394.unop \u2264 len A.fst.unop\n[PROOFSTEP]\nexact len_le_of_mono h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A = \u03b9 s (len A.fst.unop) \u226b X.map (IndexSet.e A).op\n[PROOFSTEP]\ndsimp only [\u03b9Summand, Iso.hom]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Coprod s.N A \u226b (iso s \u0394).hom = \u03b9 s (len A.fst.unop) \u226b X.map (IndexSet.e A).op\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, Cofan.mk_\u03b9_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nn : \u2115\n\u22a2 \u03b9Summand s (IndexSet.id (op [n])) = \u03b9 s n\n[PROOFSTEP]\nerw [\u03b9Summand_eq, X.map_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nn : \u2115\n\u22a2 \u03b9 s (len (IndexSet.id (op [n])).fst.unop) = \u03b9 s n\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf : X \u27f6 Y\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A \u226b NatTrans.app f \u0394 = \u03c6 s f (len A.fst.unop) \u226b Y.map (IndexSet.e A).op\n[PROOFSTEP]\nsimp only [\u03b9Summand_eq_assoc, \u03c6, assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf : X \u27f6 Y\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9 s (len A.fst.unop) \u226b X.map (IndexSet.e A).op \u226b NatTrans.app f \u0394 =\n \u03b9 s (len A.fst.unop) \u226b NatTrans.app f (op [len A.fst.unop]) \u226b Y.map (IndexSet.e A).op\n[PROOFSTEP]\nerw [NatTrans.naturality]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nf g : X.obj \u0394 \u27f6 Z\nh : \u2200 (A : IndexSet \u0394), \u03b9Summand s A \u226b f = \u03b9Summand s A \u226b g\n\u22a2 f = g\n[PROOFSTEP]\nrw [\u2190 cancel_epi (s.iso \u0394).hom]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nf g : X.obj \u0394 \u27f6 Z\nh : \u2200 (A : IndexSet \u0394), \u03b9Summand s A \u226b f = \u03b9Summand s A \u226b g\n\u22a2 (iso s \u0394).hom \u226b f = (iso s \u0394).hom \u226b g\n[PROOFSTEP]\next A\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nf g : X.obj \u0394 \u27f6 Z\nh : \u2200 (A : IndexSet \u0394), \u03b9Summand s A \u226b f = \u03b9Summand s A \u226b g\nA : IndexSet \u0394\n\u22a2 Sigma.\u03b9 (summand s.N \u0394) A \u226b (iso s \u0394).hom \u226b f = Sigma.\u03b9 (summand s.N \u0394) A \u226b (iso s \u0394).hom \u226b g\n[PROOFSTEP]\nsimpa only [\u03b9Summand_eq, iso_hom, map, colimit.\u03b9_desc_assoc, Cofan.mk_\u03b9_app] using h A\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u22a2 f = g\n[PROOFSTEP]\next \u0394\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app f \u0394 = NatTrans.app g \u0394\n[PROOFSTEP]\napply s.hom_ext'\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 \u2200 (A : IndexSet \u0394), \u03b9Summand s A \u226b NatTrans.app f \u0394 = \u03b9Summand s A \u226b NatTrans.app g \u0394\n[PROOFSTEP]\nintro A\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A \u226b NatTrans.app f \u0394 = \u03b9Summand s A \u226b NatTrans.app g \u0394\n[PROOFSTEP]\ninduction' \u0394 using Opposite.rec with \u0394\n[GOAL]\ncase h.h.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394\u271d : SimplexCategory\u1d52\u1d56\nA\u271d : IndexSet \u0394\u271d\n\u0394 : SimplexCategory\nA : IndexSet { unop := \u0394 }\n\u22a2 \u03b9Summand s A \u226b NatTrans.app f { unop := \u0394 } = \u03b9Summand s A \u226b NatTrans.app g { unop := \u0394 }\n[PROOFSTEP]\ninduction' \u0394 using SimplexCategory.rec with n\n[GOAL]\ncase h.h.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394\u271d : SimplexCategory\u1d52\u1d56\nA\u271d\u00b9 : IndexSet \u0394\u271d\n\u0394 : SimplexCategory\nA\u271d : IndexSet { unop := \u0394 }\nn : \u2115\nA : IndexSet { unop := [n] }\n\u22a2 \u03b9Summand s A \u226b NatTrans.app f { unop := [n] } = \u03b9Summand s A \u226b NatTrans.app g { unop := [n] }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X \u27f6 Y\nh : \u2200 (n : \u2115), \u03c6 s f n = \u03c6 s g n\n\u0394\u271d : SimplexCategory\u1d52\u1d56\nA\u271d\u00b9 : IndexSet \u0394\u271d\n\u0394 : SimplexCategory\nA\u271d : IndexSet { unop := \u0394 }\nn : \u2115\nA : IndexSet { unop := [n] }\n\u22a2 \u03b9Summand s A \u226b NatTrans.app f { unop := [n] } = \u03b9Summand s A \u226b NatTrans.app g { unop := [n] }\n[PROOFSTEP]\nsimp only [s.\u03b9Summand_comp_app, h]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nF : (A : IndexSet \u0394) \u2192 N s (len A.fst.unop) \u27f6 Z\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A \u226b desc s \u0394 F = F A\n[PROOFSTEP]\ndsimp only [\u03b9Summand, desc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nF : (A : IndexSet \u0394) \u2192 N s (len A.fst.unop) \u27f6 Z\nA : IndexSet \u0394\n\u22a2 (\u03b9Coprod s.N A \u226b (iso s \u0394).hom) \u226b (iso s \u0394).inv \u226b Sigma.desc F = F A\n[PROOFSTEP]\nsimp only [assoc, Iso.hom_inv_id_assoc, \u03b9Coprod]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nZ : C\n\u0394 : SimplexCategory\u1d52\u1d56\nF : (A : IndexSet \u0394) \u2192 N s (len A.fst.unop) \u27f6 Z\nA : IndexSet \u0394\n\u22a2 Sigma.\u03b9 (summand s.N \u0394) A \u226b Sigma.desc F = F A\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, Cofan.mk_\u03b9_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.147648, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\ne : X \u2245 Y\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 IsIso (map Y (fun n => \u03b9 s n \u226b NatTrans.app e.hom (op [n])) \u0394)\n[PROOFSTEP]\nconvert (inferInstance : IsIso ((s.iso \u0394).hom \u226b e.hom.app \u0394))\n[GOAL]\ncase h.e'_5\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.147648, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\ne : X \u2245 Y\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 map Y (fun n => \u03b9 s n \u226b NatTrans.app e.hom (op [n])) \u0394 = (iso s \u0394).hom \u226b NatTrans.app e.hom \u0394\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.147648, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\ne : X \u2245 Y\n\u0394 : SimplexCategory\u1d52\u1d56\nb\u271d : IndexSet \u0394\n\u22a2 Sigma.\u03b9 (summand (fun n => N s n) \u0394) b\u271d \u226b map Y (fun n => \u03b9 s n \u226b NatTrans.app e.hom (op [n])) \u0394 =\n Sigma.\u03b9 (summand (fun n => N s n) \u0394) b\u271d \u226b (iso s \u0394).hom \u226b NatTrans.app e.hom \u0394\n[PROOFSTEP]\nsimp [map]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394\u2081 \u0394\u2082 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\u2081\np : \u0394\u2081 \u27f6 \u0394\u2082\ninst\u271d : Epi p.unop\n\u22a2 \u03b9Summand s A \u226b X.map p = \u03b9Summand s (IndexSet.epiComp A p)\n[PROOFSTEP]\ndsimp [\u03b9Summand]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394\u2081 \u0394\u2082 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\u2081\np : \u0394\u2081 \u27f6 \u0394\u2082\ninst\u271d : Epi p.unop\n\u22a2 (Sigma.\u03b9 (fun A => N s (len A.fst.unop)) A \u226b Sigma.desc fun A => \u03b9 s (len A.fst.unop) \u226b X.map (IndexSet.e A).op) \u226b\n X.map p =\n Sigma.\u03b9 (fun A => N s (len A.fst.unop)) (IndexSet.epiComp A p) \u226b\n Sigma.desc fun A => \u03b9 s (len A.fst.unop) \u226b X.map (IndexSet.e A).op\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, colimit.\u03b9_desc, Cofan.mk_\u03b9_app, Cofan.mk_\u03b9_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394\u2081 \u0394\u2082 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\u2081\np : \u0394\u2081 \u27f6 \u0394\u2082\ninst\u271d : Epi p.unop\n\u22a2 (\u03b9 s (len { as := A }.as.fst.unop) \u226b X.map (IndexSet.e { as := A }.as).op) \u226b X.map p =\n \u03b9 s (len { as := IndexSet.epiComp A p }.as.fst.unop) \u226b X.map (IndexSet.e { as := IndexSet.epiComp A p }.as).op\n[PROOFSTEP]\ndsimp only [IndexSet.epiComp, IndexSet.e]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\n\u0394\u2081 \u0394\u2082 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\u2081\np : \u0394\u2081 \u27f6 \u0394\u2082\ninst\u271d : Epi p.unop\n\u22a2 (\u03b9 s (len A.fst.unop) \u226b X.map (\u2191A.snd).op) \u226b X.map p = \u03b9 s (len A.fst.unop) \u226b X.map (p.unop \u226b \u2191A.snd).op\n[PROOFSTEP]\nrw [op_comp, X.map_comp, assoc, Quiver.Hom.op_unop]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2081 \u03a6\u2082 : Hom S\u2081 S\u2082\nh : \u2200 (n : \u2115), f \u03a6\u2081 n = f \u03a6\u2082 n\n\u22a2 \u03a6\u2081 = \u03a6\u2082\n[PROOFSTEP]\nrcases \u03a6\u2081 with \u27e8F\u2081, f\u2081, c\u2081\u27e9\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2082 : Hom S\u2081 S\u2082\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f \u03a6\u2082 n\n\u22a2 mk F\u2081 f\u2081 = \u03a6\u2082\n[PROOFSTEP]\nrcases \u03a6\u2082 with \u27e8F\u2082, f\u2082, c\u2082\u27e9\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082\n[PROOFSTEP]\nhave h' : f\u2081 = f\u2082 := by\n ext\n apply h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nx\u271d : \u2115\n\u22a2 f\u2081 x\u271d = f\u2082 x\u271d\n[PROOFSTEP]\napply h\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nh' : f\u2081 = f\u2082\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082\n[PROOFSTEP]\nsubst h'\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2081\n[PROOFSTEP]\nsimp only [mk.injEq, and_true]\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 F\u2081 = F\u2082\n[PROOFSTEP]\napply S\u2081.s.hom_ext\n[GOAL]\ncase mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 \u2200 (n : \u2115), Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n]) = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2081 (op [n]) = Splitting.\u03b9 S\u2081.s n \u226b NatTrans.app F\u2082 (op [n])\n[PROOFSTEP]\nrw [c\u2081, c\u2082]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.183784, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d Z\u271d : Split C\n\u03a6\u2081\u2082 : X\u271d \u27f6 Y\u271d\n\u03a6\u2082\u2083 : Y\u271d \u27f6 Z\u271d\nn : \u2115\n\u22a2 Splitting.\u03b9 X\u271d.s n \u226b NatTrans.app (\u03a6\u2081\u2082.F \u226b \u03a6\u2082\u2083.F) (op [n]) =\n (fun n => Split.Hom.f \u03a6\u2081\u2082 n \u226b Split.Hom.f \u03a6\u2082\u2083 n) n \u226b Splitting.\u03b9 Z\u271d.s n\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.183784, u_1} C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d Z\u271d : Split C\n\u03a6\u2081\u2082 : X\u271d \u27f6 Y\u271d\n\u03a6\u2082\u2083 : Y\u271d \u27f6 Z\u271d\nn : \u2115\n\u22a2 Splitting.\u03b9 X\u271d.s n \u226b NatTrans.app \u03a6\u2081\u2082.F (op [n]) \u226b NatTrans.app \u03a6\u2082\u2083.F (op [n]) =\n (Split.Hom.f \u03a6\u2081\u2082 n \u226b Split.Hom.f \u03a6\u2082\u2083 n) \u226b Splitting.\u03b9 Z\u271d.s n\n[PROOFSTEP]\nsimp only [assoc, Split.Hom.comm_assoc, Split.Hom.comm]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2081 \u03a6\u2082 : S\u2081 \u27f6 S\u2082\nh : \u03a6\u2081 = \u03a6\u2082\n\u22a2 \u03a6\u2081.f = \u03a6\u2082.f\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2081 \u03a6\u2082 : S\u2081 \u27f6 S\u2082\nh : \u03a6\u2081 = \u03a6\u2082\nn : \u2115\n\u22a2 Hom.f \u03a6\u2081 n = Hom.f \u03a6\u2082 n\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\n\u0394 : SimplexCategory\u1d52\u1d56\nA : Splitting.IndexSet \u0394\n\u22a2 Splitting.\u03b9Summand S\u2081.s A \u226b NatTrans.app \u03a6.F \u0394 = Hom.f \u03a6 (len A.fst.unop) \u226b Splitting.\u03b9Summand S\u2082.s A\n[PROOFSTEP]\nerw [S\u2081.s.\u03b9Summand_eq, S\u2082.s.\u03b9Summand_eq, assoc, \u03a6.F.naturality, \u2190 \u03a6.comm_assoc]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.SplitSimplicialObject", "llama_tokens": 22154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.661922862511608, "lm_q2_score": 0.3849121444839335, "lm_q1q2_score": 0.2547821484922869}} {"text": "[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\nf : \u03b1\u271d \u2243 \u03b2\u271d\n\u03b1 : Sort u_1\n\u03b2 : Type u_2\np : \u03b2 \u2192 Prop\ne : \u03b1 \u2243 Subtype p\nx : \u03b2\nhs : x \u2208 setOf p\n\u22a2 \u2191(asEmbedding e) (\u2191e.symm { val := x, property := hs }) = x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nf : Option \u03b1 \u21aa \u03b2\n\u22a2 \u2200 (x : Option \u03b1),\n \u2191((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n ((fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd :=\n { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n f))\n x =\n \u2191f x\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase none\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nf : Option \u03b1 \u21aa \u03b2\n\u22a2 \u2191((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n ((fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd :=\n { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n f))\n none =\n \u2191f none\n[PROOFSTEP]\nsimp [Option.coe_def]\n[GOAL]\ncase some\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nf : Option \u03b1 \u21aa \u03b2\nval\u271d : \u03b1\n\u22a2 \u2191((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n ((fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd :=\n { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n f))\n (some val\u271d) =\n \u2191f (some val\u271d)\n[PROOFSTEP]\nsimp [Option.coe_def]\n[GOAL]\ncase some\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nf : Option \u03b1 \u21aa \u03b2\nval\u271d : \u03b1\n\u22a2 \u2191val\u271d = some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nx\u271d : (f : \u03b1 \u21aa \u03b2) \u00d7 \u2191(Set.range \u2191f)\u1d9c\nf : \u03b1 \u21aa \u03b2\ny : \u03b2\nhy : y \u2208 (Set.range \u2191f)\u1d9c\n\u22a2 (fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd := { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n ((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n { fst := f, snd := { val := y, property := hy } }) =\n { fst := f, snd := { val := y, property := hy } }\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nx\u271d\u00b9 : (f : \u03b1 \u21aa \u03b2) \u00d7 \u2191(Set.range \u2191f)\u1d9c\nf : \u03b1 \u21aa \u03b2\ny : \u03b2\nhy : y \u2208 (Set.range \u2191f)\u1d9c\nx\u271d : \u03b1\n\u22a2 \u2191((fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd :=\n { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n ((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n { fst := f, snd := { val := y, property := hy } })).fst\n x\u271d =\n \u2191{ fst := f, snd := { val := y, property := hy } }.fst x\u271d\n[PROOFSTEP]\nsimp [Option.coe_def]\n[GOAL]\ncase a\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nx\u271d : (f : \u03b1 \u21aa \u03b2) \u00d7 \u2191(Set.range \u2191f)\u1d9c\nf : \u03b1 \u21aa \u03b2\ny : \u03b2\nhy : y \u2208 (Set.range \u2191f)\u1d9c\n\u22a2 \u2191((fun f =>\n { fst := Embedding.trans coeWithTop f,\n snd :=\n { val := \u2191f none, property := (_ : \u2191f none \u2208 Set.range \u2191(Embedding.trans coeWithTop f) \u2192 False) } })\n ((fun f => optionElim f.fst \u2191f.snd (_ : \u2191f.snd \u2208 (Set.range \u2191f.fst)\u1d9c))\n { fst := f, snd := { val := y, property := hy } })).snd =\n \u2191{ fst := f, snd := { val := y, property := hy } }.snd\n[PROOFSTEP]\nsimp [Option.coe_def]\n[GOAL]\ncase a.h\n\u03b1 : Type ?u.1349\n\u03b2 : Type ?u.1350\nx\u271d\u00b9 : (f : \u03b1 \u21aa \u03b2) \u00d7 \u2191(Set.range \u2191f)\u1d9c\nf : \u03b1 \u21aa \u03b2\ny : \u03b2\nhy : y \u2208 (Set.range \u2191f)\u1d9c\nx\u271d : \u03b1\n\u22a2 \u2191(optionElim f y\n (_ :\n \u2191{ fst := f, snd := { val := y, property := hy } }.snd \u2208\n (Set.range \u2191{ fst := f, snd := { val := y, property := hy } }.fst)\u1d9c))\n \u2191x\u271d =\n \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.4882\ns t : Set \u03b1\nh\u271d : s \u2286 t\nx\u271d\u00b9 x\u271d : \u2191s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh :\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 t) }) { val := x, property := hx } =\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 t) }) { val := y, property := hy }\n\u22a2 { val := x, property := hx } = { val := y, property := hy }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_val\n\u03b1 : Type ?u.4882\ns t : Set \u03b1\nh\u271d : s \u2286 t\nx\u271d\u00b9 x\u271d : \u2191s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh :\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 t) }) { val := x, property := hx } =\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 t) }) { val := y, property := hy }\n\u22a2 x = y\n[PROOFSTEP]\ninjection h\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x \u2228 q x }\n\u22a2 Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)))\n (\u2191(subtypeOrLeftEmbedding p q) x) =\n x\n[PROOFSTEP]\nby_cases hx : p x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x \u2228 q x }\nhx : p \u2191x\n\u22a2 Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)))\n (\u2191(subtypeOrLeftEmbedding p q) x) =\n x\n[PROOFSTEP]\nrw [subtypeOrLeftEmbedding_apply_left _ hx]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x \u2228 q x }\nhx : p \u2191x\n\u22a2 Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)))\n (Sum.inl { val := \u2191x, property := hx }) =\n x\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x \u2228 q x }\nhx : \u00acp \u2191x\n\u22a2 Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)))\n (\u2191(subtypeOrLeftEmbedding p q) x) =\n x\n[PROOFSTEP]\nrw [subtypeOrLeftEmbedding_apply_right _ hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x \u2228 q x }\nhx : \u00acp \u2191x\n\u22a2 Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)))\n (Sum.inr { val := \u2191x, property := (_ : q \u2191x) }) =\n x\n[PROOFSTEP]\nsimp [Subtype.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x } \u2295 { x // q x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) x) =\n x\n[PROOFSTEP]\ncases x with\n| inl x =>\n simp only [Sum.elim_inl]\n rw [subtypeOrLeftEmbedding_apply_left]\n \u00b7 simp\n \u00b7 simpa using x.prop\n| inr x =>\n simp only [Sum.elim_inr]\n rw [subtypeOrLeftEmbedding_apply_right]\n \u00b7 simp\n \u00b7 suffices \u00acp x by simpa\n intro hp\n simpa using h.le_bot x \u27e8hp, x.prop\u27e9\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x } \u2295 { x // q x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) x) =\n x\n[PROOFSTEP]\ncases x with\n| inl x =>\n simp only [Sum.elim_inl]\n rw [subtypeOrLeftEmbedding_apply_left]\n \u00b7 simp\n \u00b7 simpa using x.prop\n| inr x =>\n simp only [Sum.elim_inr]\n rw [subtypeOrLeftEmbedding_apply_right]\n \u00b7 simp\n \u00b7 suffices \u00acp x by simpa\n intro hp\n simpa using h.le_bot x \u27e8hp, x.prop\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) (Sum.inl x)) =\n Sum.inl x\n[PROOFSTEP]\n\n| inl x =>\n simp only [Sum.elim_inl]\n rw [subtypeOrLeftEmbedding_apply_left]\n \u00b7 simp\n \u00b7 simpa using x.prop\n[GOAL]\ncase inl\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) (Sum.inl x)) =\n Sum.inl x\n[PROOFSTEP]\nsimp only [Sum.elim_inl]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)) x) =\n Sum.inl x\n[PROOFSTEP]\nrw [subtypeOrLeftEmbedding_apply_left]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x }\n\u22a2 Sum.inl\n { val := \u2191(\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)) x),\n property := ?inl.hx } =\n Sum.inl x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.hx\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // p x }\n\u22a2 p \u2191(\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)) x)\n[PROOFSTEP]\nsimpa using x.prop\n[GOAL]\ncase inr\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) (Sum.inr x)) =\n Sum.inr x\n[PROOFSTEP]\n\n| inr x =>\n simp only [Sum.elim_inr]\n rw [subtypeOrLeftEmbedding_apply_right]\n \u00b7 simp\n \u00b7 suffices \u00acp x by simpa\n intro hp\n simpa using h.le_bot x \u27e8hp, x.prop\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (Sum.elim (\u2191(Subtype.impEmbedding (fun x => p x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), p x \u2192 p x \u2228 q x)))\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x))) (Sum.inr x)) =\n Sum.inr x\n[PROOFSTEP]\nsimp only [Sum.elim_inr]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 \u2191(subtypeOrLeftEmbedding p q)\n (\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)) x) =\n Sum.inr x\n[PROOFSTEP]\nrw [subtypeOrLeftEmbedding_apply_right]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 Sum.inr\n { val := \u2191(\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)) x),\n property :=\n (_ : q \u2191(\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)) x)) } =\n Sum.inr x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.hx\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 \u00acp \u2191(\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)) x)\n[PROOFSTEP]\nsuffices \u00acp x by simpa\n[GOAL]\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\nthis : \u00acp \u2191x\n\u22a2 \u00acp \u2191(\u2191(Subtype.impEmbedding (fun x => q x) (fun x => p x \u2228 q x) (_ : \u2200 (x : \u03b1), q x \u2192 p x \u2228 q x)) x)\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr.hx\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n\u22a2 \u00acp \u2191x\n[PROOFSTEP]\nintro hp\n[GOAL]\ncase inr.hx\n\u03b1 : Type u_1\np q : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\nhp : p \u2191x\n\u22a2 False\n[PROOFSTEP]\nsimpa using h.le_bot x \u27e8hp, x.prop\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Logic.Embedding.Set", "llama_tokens": 5856, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.2547684695954492}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nnodup\u271d\u00b9 : Nodup s\nt : Multiset \u03b1\nnodup\u271d : Nodup t\nh : { val := s, nodup := nodup\u271d\u00b9 }.val = { val := t, nodup := nodup\u271d }.val\n\u22a2 { val := s, nodup := nodup\u271d\u00b9 } = { val := t, nodup := nodup\u271d }\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Multiset \u03b1\nnodup\u271d\u00b9 nodup\u271d : Nodup s\n\u22a2 { val := s, nodup := nodup\u271d\u00b9 } = { val := s, nodup := nodup\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsRefl (Finset \u03b1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsTrans (Finset \u03b1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsAntisymm (Finset \u03b1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsIrrefl (Finset \u03b1) fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsTrans (Finset \u03b1) fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 IsAsymm (Finset \u03b1) fun x x_1 => x < x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\n\u22a2 \u00acs \u2286 t \u2194 \u2203 x, x \u2208 s \u2227 \u00acx \u2208 t\n[PROOFSTEP]\nsimp only [\u2190 coe_subset, Set.not_subset, mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t s\u2081 s\u2082 : Finset \u03b1\n\u22a2 \u2191s\u2081 \u2282 \u2191s\u2082 \u2194 s\u2081 \u2286 s\u2082 \u2227 \u00acs\u2082 \u2286 s\u2081\n[PROOFSTEP]\nsimp only [Set.ssubset_def, Finset.coe_subset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\nl : List \u03b1\n\u22a2 \u00ac\u2203 a, a \u2208 Quotient.mk (List.isSetoid \u03b1) []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 a \u2208 Quotient.mk (List.isSetoid \u03b1) (a :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u00aca \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_def, empty_val, not_mem_zero, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s : Finset \u03b1\n\u22a2 s = \u2205 \u2192 \u2200 (x : \u03b1), \u00acx \u2208 s\n[PROOFSTEP]\nrintro rfl x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\nx : \u03b1\n\u22a2 \u00acx \u2208 \u2205\n[PROOFSTEP]\napply not_mem_empty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 \u2191\u2205 \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s : Finset \u03b1\n\u22a2 \u2191s = \u2205 \u2194 s = \u2205\n[PROOFSTEP]\nrw [\u2190 coe_empty, coe_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s : Finset \u03b1\n\u22a2 IsEmpty { x // x \u2208 s } \u2194 s = \u2205\n[PROOFSTEP]\nsimpa using @Set.isEmpty_coe_sort \u03b1 s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s.val = {a} \u2194 s = {a}\n[PROOFSTEP]\nrw [\u2190 val_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s.val = {a} \u2194 s.val = {a}.val\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na\u271d b a : \u03b1\n\u22a2 \u2191{a} = {a}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na\u271d b a x\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191{a} \u2194 x\u271d \u2208 {a}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 \u2191s = {a} \u2194 s = {a}\n[PROOFSTEP]\nrw [\u2190 coe_singleton, coe_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2194 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2192 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 (a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a) \u2192 s = {a}\n[PROOFSTEP]\nintro t\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nrw [t]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a\n[PROOFSTEP]\nexact \u27e8Finset.mem_singleton_self _, fun _ => Finset.mem_singleton.1\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}\n[PROOFSTEP]\next\n[GOAL]\ncase mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 {a}\n[PROOFSTEP]\nrw [Finset.mem_singleton]\n[GOAL]\ncase mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d = a\n[PROOFSTEP]\nexact \u27e8t.right _, fun r => r.symm \u25b8 t.left\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2194 Finset.Nonempty s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2192 Finset.Nonempty s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na\u271d b a : \u03b1\n\u22a2 Finset.Nonempty {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 (Finset.Nonempty s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a) \u2192 s = {a}\n[PROOFSTEP]\nrintro \u27e8hne, h_uniq\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nhne : Finset.Nonempty s\nh_uniq : \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}\n[PROOFSTEP]\nrw [eq_singleton_iff_unique_mem]\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nhne : Finset.Nonempty s\nh_uniq : \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n[PROOFSTEP]\nrefine' \u27e8_, h_uniq\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nhne : Finset.Nonempty s\nh_uniq : \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 h_uniq hne.choose hne.choose_spec]\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nhne : Finset.Nonempty s\nh_uniq : \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 Exists.choose hne \u2208 s\n[PROOFSTEP]\nexact hne.choose_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na b : \u03b1\ninst\u271d : Unique \u03b1\ns : Finset \u03b1\n\u22a2 Finset.Nonempty s \u2194 s = {default}\n[PROOFSTEP]\nsimp [eq_singleton_iff_nonempty_unique_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2203 a, s = {a}) \u2194 \u2203! a, a \u2208 s\n[PROOFSTEP]\nsimp only [eq_singleton_iff_unique_mem, ExistsUnique]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191{a} \u2286 s \u2194 a \u2208 s\n[PROOFSTEP]\nrw [coe_singleton, Set.singleton_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s \u2286 {a} \u2194 s = \u2205 \u2228 s = {a}\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_singleton, Set.subset_singleton_iff_eq, coe_eq_empty, coe_eq_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\n\u22a2 {a} \u2286 {b} \u2194 a = b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s \u2282 {a} \u2194 s = \u2205\n[PROOFSTEP]\nrw [\u2190 coe_ssubset, coe_singleton, Set.ssubset_singleton_iff, coe_eq_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 s = {a} \u2228 Set.Nontrivial \u2191s\n[PROOFSTEP]\nrw [\u2190 coe_eq_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 \u2191s = {a} \u2228 Set.Nontrivial \u2191s\n[PROOFSTEP]\nexact Set.eq_singleton_or_nontrivial ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nh : \u00aca \u2208 s\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : \u03b1), x \u2208 cons a s h \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp only [mem_cons, or_imp, forall_and, forall_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nm : Multiset \u03b1\nhm : Nodup m\n\u22a2 Finset.Nonempty { val := m, nodup := hm } \u2194 m \u2260 0\n[PROOFSTEP]\ninduction m using Multiset.induction_on\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nhm : Nodup 0\n\u22a2 Finset.Nonempty { val := 0, nodup := hm } \u2194 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b a\u271d\u00b9 : \u03b1\ns\u271d : Multiset \u03b1\na\u271d : \u2200 {hm : Nodup s\u271d}, Finset.Nonempty { val := s\u271d, nodup := hm } \u2194 s\u271d \u2260 0\nhm : Nodup (a\u271d\u00b9 ::\u2098 s\u271d)\n\u22a2 Finset.Nonempty { val := a\u271d\u00b9 ::\u2098 s\u271d, nodup := hm } \u2194 a\u271d\u00b9 ::\u2098 s\u271d \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191(cons a s h) = insert a \u2191s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(cons a s h) \u2194 x\u271d \u2208 insert a \u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nhs : \u00aca \u2208 s\nht : \u00aca \u2208 t\n\u22a2 cons a s hs \u2286 cons a t ht \u2194 s \u2286 t\n[PROOFSTEP]\nrwa [\u2190 coe_subset, coe_cons, coe_cons, Set.insert_subset_insert_iff, coe_subset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 s \u2282 t \u2194 \u2203 a h, cons a s h \u2286 t\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_ssubset_of_subset (ssubset_cons _) h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na b : \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a h, cons a s h \u2286 t\n[PROOFSTEP]\nobtain \u27e8a, hs, ht\u27e9 := not_subset.1 h.2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Finset \u03b1\na\u271d b : \u03b1\nh : s \u2282 t\na : \u03b1\nhs : a \u2208 t\nht : \u00aca \u2208 s\n\u22a2 \u2203 a h, cons a s h \u2286 t\n[PROOFSTEP]\nexact \u27e8a, ht, cons_subset.2 \u27e8hs, h.subset\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint s t \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 t \u2192 \u00aca \u2208 s\n[PROOFSTEP]\nrw [_root_.disjoint_comm, disjoint_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint s t \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b\n[PROOFSTEP]\nsimp only [disjoint_left, imp_not_comm, forall_eq']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b x\u271d : \u03b1\n\u22a2 \u00ac(x\u271d \u2208 s \u2192 \u00acx\u271d \u2208 t) \u2194 x\u271d \u2208 s \u2227 x\u271d \u2208 t\n[PROOFSTEP]\nrw [not_imp, not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint {a} s \u2194 \u00aca \u2208 s\n[PROOFSTEP]\nsimp only [disjoint_left, mem_singleton, forall_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint {a} {b} \u2194 a \u2260 b\n[PROOFSTEP]\nrw [disjoint_singleton_left, mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ns t u : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint \u2191s \u2191t \u2194 _root_.Disjoint s t\n[PROOFSTEP]\nsimp only [Finset.disjoint_left, Set.disjoint_left, mem_coe]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\ns t : Finset \u03b1\nh : _root_.Disjoint s t\na : \u03b1\n\u22a2 a \u2208 disjUnion s t h \u2194 a \u2208 s \u2228 a \u2208 t\n[PROOFSTEP]\nrcases s with \u27e8\u27e8s\u27e9\u27e9\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\nt : Finset \u03b1\na : \u03b1\nval\u271d : Multiset \u03b1\ns : List \u03b1\nnodup\u271d : Nodup (Quot.mk Setoid.r s)\nh : _root_.Disjoint { val := Quot.mk Setoid.r s, nodup := nodup\u271d } t\n\u22a2 a \u2208 disjUnion { val := Quot.mk Setoid.r s, nodup := nodup\u271d } t h \u2194\n a \u2208 { val := Quot.mk Setoid.r s, nodup := nodup\u271d } \u2228 a \u2208 t\n[PROOFSTEP]\nrcases t with \u27e8\u27e8t\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\na : \u03b1\nval\u271d\u00b9 : Multiset \u03b1\ns : List \u03b1\nnodup\u271d\u00b9 : Nodup (Quot.mk Setoid.r s)\nval\u271d : Multiset \u03b1\nt : List \u03b1\nnodup\u271d : Nodup (Quot.mk Setoid.r t)\nh : _root_.Disjoint { val := Quot.mk Setoid.r s, nodup := nodup\u271d\u00b9 } { val := Quot.mk Setoid.r t, nodup := nodup\u271d }\n\u22a2 a \u2208 disjUnion { val := Quot.mk Setoid.r s, nodup := nodup\u271d\u00b9 } { val := Quot.mk Setoid.r t, nodup := nodup\u271d } h \u2194\n a \u2208 { val := Quot.mk Setoid.r s, nodup := nodup\u271d\u00b9 } \u2228 a \u2208 { val := Quot.mk Setoid.r t, nodup := nodup\u271d }\n[PROOFSTEP]\napply List.mem_append\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nh : _root_.Disjoint s {a}\n\u22a2 disjUnion s {a} h = cons a s (_ : \u00aca \u2208 s)\n[PROOFSTEP]\nrw [disjUnion_comm, singleton_disjUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 (insert a s).val = dedup (a ::\u2098 s.val)\n[PROOFSTEP]\nrw [dedup_cons, dedup_eq_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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 (insert a s).val = ndinsert a s.val\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 (insert a s).val = a ::\u2098 s.val\n[PROOFSTEP]\nrw [insert_val, ndinsert_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 u v : Finset \u03b1\na\u271d\u00b9 b a\u271d : \u03b1\ns : Finset \u03b1\nh : \u00aca\u271d \u2208 s\na : \u03b1\n\u22a2 a \u2208 cons a\u271d s h \u2194 a \u2208 insert a\u271d 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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 \u2191(insert a s) \u2194 x \u2208 insert a \u2191s\n[PROOFSTEP]\nsimp only [mem_coe, mem_insert, Set.mem_insert_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx y : \u03b1\n\u22a2 x \u2208 insert y s \u2194 x \u2208 insert y \u2191s\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 v : Finset \u03b1\na\u271d b a : \u03b1\n\u22a2 insert a \u2205 = {a}\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 u v : Finset \u03b1\na\u271d\u00b9 b a a\u271d : \u03b1\n\u22a2 a\u271d \u2208 insert a \u2205 \u2194 a\u271d \u2208 {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\ns\u271d t u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 insert a (insert b s) \u2194 x \u2208 insert b (insert a s)\n[PROOFSTEP]\nsimp only [mem_insert, or_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 \u2191{a, b} = {a, b}\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na\u271d b\u271d a b x\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191{a, b} \u2194 x\u271d \u2208 {a, b}\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 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b1\na b : \u03b1\n\u22a2 \u2191s = {a, b} \u2194 s = {a, b}\n[PROOFSTEP]\nrw [\u2190 coe_pair, coe_inj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 insert a (insert a s) \u2194 x \u2208 insert a s\n[PROOFSTEP]\nsimp only [mem_insert, \u2190 or_assoc, or_self_iff]\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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2260 insert a t\n[PROOFSTEP]\ncontrapose! 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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\nh : s = insert a t\n\u22a2 a \u2208 s\n[PROOFSTEP]\nsimp [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 v : Finset \u03b1\na b : \u03b1\n\u22a2 insert a s \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t\n[PROOFSTEP]\nsimp only [subset_iff, mem_insert, forall_eq, 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 v : Finset \u03b1\na b : \u03b1\n\u22a2 s \u2282 t \u2194 \u2203 a x, insert a s \u2286 t\n[PROOFSTEP]\nexact_mod_cast @Set.ssubset_iff_insert \u03b1 s t\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\ns : Multiset \u03b1\nnd : Nodup s\n\u22a2 p { val := s, nodup := nd }\n[PROOFSTEP]\ninduction s using Multiset.induction with\n| empty => exact empty\n| @cons a s IH =>\n cases' nodup_cons.1 nd with m nd'\n rw [\u2190 (eq_of_veq _ : Finset.cons a \u27e8s, _\u27e9 m = \u27e8a ::\u2098 s, nd\u27e9)]\n \u00b7 exact cons m (IH nd')\n \u00b7 rw [cons_val]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\ns : Multiset \u03b1\nnd : Nodup s\n\u22a2 p { val := s, nodup := nd }\n[PROOFSTEP]\ninduction s using Multiset.induction with\n| empty => exact empty\n| @cons a s IH =>\n cases' nodup_cons.1 nd with m nd'\n rw [\u2190 (eq_of_veq _ : Finset.cons a \u27e8s, _\u27e9 m = \u27e8a ::\u2098 s, nd\u27e9)]\n \u00b7 exact cons m (IH nd')\n \u00b7 rw [cons_val]\n[GOAL]\ncase empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\nnd : Nodup 0\n\u22a2 p { val := 0, nodup := nd }\n[PROOFSTEP]\n\n| empty => exact empty\n[GOAL]\ncase empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\nnd : Nodup 0\n\u22a2 p { val := 0, nodup := nd }\n[PROOFSTEP]\nexact empty\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (nd : Nodup s), p { val := s, nodup := nd }\nnd : Nodup (a ::\u2098 s)\n\u22a2 p { val := a ::\u2098 s, nodup := nd }\n[PROOFSTEP]\n\n| @cons a s IH =>\n cases' nodup_cons.1 nd with m nd'\n rw [\u2190 (eq_of_veq _ : Finset.cons a \u27e8s, _\u27e9 m = \u27e8a ::\u2098 s, nd\u27e9)]\n \u00b7 exact cons m (IH nd')\n \u00b7 rw [cons_val]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (nd : Nodup s), p { val := s, nodup := nd }\nnd : Nodup (a ::\u2098 s)\n\u22a2 p { val := a ::\u2098 s, nodup := nd }\n[PROOFSTEP]\ncases' nodup_cons.1 nd with m nd'\n[GOAL]\ncase cons.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (nd : Nodup s), p { val := s, nodup := nd }\nnd : Nodup (a ::\u2098 s)\nm : \u00aca \u2208 s\nnd' : Nodup s\n\u22a2 p { val := a ::\u2098 s, nodup := nd }\n[PROOFSTEP]\nrw [\u2190 (eq_of_veq _ : Finset.cons a \u27e8s, _\u27e9 m = \u27e8a ::\u2098 s, nd\u27e9)]\n[GOAL]\ncase cons.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (nd : Nodup s), p { val := s, nodup := nd }\nnd : Nodup (a ::\u2098 s)\nm : \u00aca \u2208 s\nnd' : Nodup s\n\u22a2 p (Finset.cons a { val := s, nodup := ?m.116533 } m)\n[PROOFSTEP]\nexact cons m (IH nd')\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : Finset \u03b1 \u2192 Prop\nempty : p \u2205\ncons : \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1} (h : \u00aca \u2208 s), p s \u2192 p (Finset.cons a s h)\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (nd : Nodup s), p { val := s, nodup := nd }\nnd : Nodup (a ::\u2098 s)\nm : \u00aca \u2208 s\nnd' : Nodup s\n\u22a2 (Finset.cons a { val := s, nodup := nd' } m).val = { val := a ::\u2098 s, nodup := nd }.val\n[PROOFSTEP]\nrw [cons_val]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 p s hs\n[PROOFSTEP]\ninduction' s using Finset.cons_induction with a t ha h\n[GOAL]\ncase empty\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\nhs : Finset.Nonempty \u2205\n\u22a2 p \u2205 hs\n[PROOFSTEP]\nexact (not_nonempty_empty hs).elim\n[GOAL]\ncase cons\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\n\u22a2 p (cons a t ha) hs\n[PROOFSTEP]\nobtain rfl | ht := t.eq_empty_or_nonempty\n[GOAL]\ncase cons.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nha : \u00aca \u2208 \u2205\nh : \u2200 (hs : Finset.Nonempty \u2205), p \u2205 hs\nhs : Finset.Nonempty (cons a \u2205 ha)\n\u22a2 p (cons a \u2205 ha) hs\n[PROOFSTEP]\nexact h\u2080 a\n[GOAL]\ncase cons.inr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d t\u271d u v : Finset \u03b1\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_4\np : (s : Finset \u03b1) \u2192 Finset.Nonempty s \u2192 Prop\nh\u2080 : \u2200 (a : \u03b1), p {a} (_ : Finset.Nonempty {a})\nh\u2081 :\n \u2200 \u2983a : \u03b1\u2984 (s : Finset \u03b1) (h : \u00aca \u2208 s) (hs : Finset.Nonempty s),\n p s hs \u2192 p (cons a s h) (_ : Finset.Nonempty (cons a s h))\ns : Finset \u03b1\nhs\u271d : Finset.Nonempty s\na : \u03b1\nt : Finset \u03b1\nha : \u00aca \u2208 t\nh : \u2200 (hs : Finset.Nonempty t), p t hs\nhs : Finset.Nonempty (cons a t ha)\nht : Finset.Nonempty t\n\u22a2 p (cons a t ha) hs\n[PROOFSTEP]\nexact h\u2081 t ha ht (h ht)\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 v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 { i // i \u2208 insert x t } \u2243 Option { i // i \u2208 t }\n[PROOFSTEP]\nrefine'\n { toFun := fun y => if h : \u2191y = x then none else some \u27e8y, (mem_insert.mp y.2).resolve_left h\u27e9\n invFun := fun y => (y.elim \u27e8x, mem_insert_self _ _\u27e9) fun z => \u27e8z, mem_insert_of_mem z.2\u27e9 .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 LeftInverse\n (fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) }\n[PROOFSTEP]\nintro y\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\ny : { i // i \u2208 insert x t }\n\u22a2 (fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n ((fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) }) y) =\n y\n[PROOFSTEP]\nby_cases h : \u2191y = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh\u271d : \u00acx \u2208 t\ny : { i // i \u2208 insert x t }\nh : \u2191y = x\n\u22a2 (fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n ((fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) }) y) =\n y\n[PROOFSTEP]\nsimp only [Subtype.ext_iff, h, Option.elim, dif_pos, Subtype.coe_mk]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh\u271d : \u00acx \u2208 t\ny : { i // i \u2208 insert x t }\nh : \u00ac\u2191y = x\n\u22a2 (fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n ((fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) }) y) =\n y\n[PROOFSTEP]\nsimp only [h, Option.elim, dif_neg, not_false_iff, Subtype.coe_eta, Subtype.coe_mk]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 Function.RightInverse\n (fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) }\n[PROOFSTEP]\nrintro (_ | y)\n[GOAL]\ncase refine'_2.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 (fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) })\n ((fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n none) =\n none\n[PROOFSTEP]\nsimp only [Option.elim, dif_pos]\n[GOAL]\ncase refine'_2.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\ny : { i // i \u2208 t }\n\u22a2 (fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) })\n ((fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n (some y)) =\n some y\n[PROOFSTEP]\nhave : \u2191y \u2260 x := by\n rintro \u27e8\u27e9\n exact h y.2\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 v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\ny : { i // i \u2208 t }\n\u22a2 \u2191y \u2260 x\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\ny : { i // i \u2208 t }\nh : \u00ac\u2191y \u2208 t\n\u22a2 False\n[PROOFSTEP]\nexact h y.2\n[GOAL]\ncase refine'_2.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b : \u03b1\nt : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\ny : { i // i \u2208 t }\nthis : \u2191y \u2260 x\n\u22a2 (fun y => if h : \u2191y = x then none else some { val := \u2191y, property := (_ : \u2191y \u2208 t) })\n ((fun y =>\n Option.elim y { val := x, property := (_ : x \u2208 insert x t) } fun z =>\n { val := \u2191z, property := (_ : \u2191z \u2208 insert x t) })\n (some y)) =\n some y\n[PROOFSTEP]\nsimp only [this, Option.elim, Subtype.eta, dif_neg, not_false_iff, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (insert a s) t \u2194 \u00aca \u2208 t \u2227 _root_.Disjoint s t\n[PROOFSTEP]\nsimp only [disjoint_left, mem_insert, or_imp, forall_and, forall_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (insert a t) s \u2194 \u00aca \u2208 s \u2227 _root_.Disjoint s t\n[PROOFSTEP]\nrw [disjoint_insert_left, _root_.disjoint_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\nh : _root_.Disjoint s t\na : \u03b1\n\u22a2 a \u2208 disjUnion s t h \u2194 a \u2208 s \u222a 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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\n\u22a2 \u00aca \u2208 s \u222a t \u2194 \u00aca \u2208 s \u2227 \u00aca \u2208 t\n[PROOFSTEP]\nrw [mem_union, not_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u\u271d v : Finset \u03b1\na b : \u03b1\ns t u : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s \u222a (t \u222a u) \u2194 x\u271d \u2208 t \u222a (s \u222a u)\n[PROOFSTEP]\nsimp only [mem_union, or_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u\u271d v : Finset \u03b1\na b : \u03b1\ns t u : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 s \u222a t \u222a u \u2194 x \u2208 s \u222a u \u222a t\n[PROOFSTEP]\nsimp only [mem_union, or_assoc, @or_comm (x \u2208 t)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 s \u2228 x \u2208 \u2205 \u2194 x \u2208 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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 \u2205 \u2228 x \u2208 s \u2194 x \u2208 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 s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 insert a s \u222a t = insert a (s \u222a t)\n[PROOFSTEP]\nsimp only [insert_eq, union_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u222a insert a t = insert a (s \u222a t)\n[PROOFSTEP]\nsimp only [insert_eq, union_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 insert a (s \u222a t) = insert a s \u222a insert a t\n[PROOFSTEP]\nsimp only [insert_union, union_insert, insert_idem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s = s \u222a t \u2194 t \u2286 s\n[PROOFSTEP]\nrw [\u2190 union_eq_left_iff_subset, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s = t \u222a s \u2194 t \u2286 s\n[PROOFSTEP]\nrw [\u2190 union_eq_right_iff_subset, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (s \u222a t) u \u2194 _root_.Disjoint s u \u2227 _root_.Disjoint t u\n[PROOFSTEP]\nsimp only [disjoint_left, mem_union, 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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint s (t \u222a u) \u2194 _root_.Disjoint s t \u2227 _root_.Disjoint s u\n[PROOFSTEP]\nsimp only [disjoint_right, mem_union, 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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\n\u22a2 \u2200 (a b : Finset \u03b1), P a b\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na b : Finset \u03b1\n\u22a2 P a b\n[PROOFSTEP]\nrefine' Finset.induction_on b empty_right fun x s _xs hi => symm _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na b : Finset \u03b1\nx : \u03b1\ns : Finset \u03b1\n_xs : \u00acx \u2208 s\nhi : P a s\n\u22a2 P (insert x s) a\n[PROOFSTEP]\nrw [Finset.insert_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na b : Finset \u03b1\nx : \u03b1\ns : Finset \u03b1\n_xs : \u00acx \u2208 s\nhi : P a s\n\u22a2 P ({x} \u222a s) a\n[PROOFSTEP]\napply union_of _ (symm hi)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na b : Finset \u03b1\nx : \u03b1\ns : Finset \u03b1\n_xs : \u00acx \u2208 s\nhi : P a s\n\u22a2 P {x} a\n[PROOFSTEP]\nrefine' Finset.induction_on a empty_right fun a t _ta hi => symm _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d\u00b9 b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na\u271d b : Finset \u03b1\nx : \u03b1\ns : Finset \u03b1\n_xs : \u00acx \u2208 s\nhi\u271d : P a\u271d s\na : \u03b1\nt : Finset \u03b1\n_ta : \u00aca \u2208 t\nhi : P {x} t\n\u22a2 P (insert a t) {x}\n[PROOFSTEP]\nrw [Finset.insert_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d\u00b9 b\u271d : \u03b1\nP : Finset \u03b1 \u2192 Finset \u03b1 \u2192 Prop\nsymm : \u2200 {a b : Finset \u03b1}, P a b \u2192 P b a\nempty_right : \u2200 {a : Finset \u03b1}, P a \u2205\nsingletons : \u2200 {a b : \u03b1}, P {a} {b}\nunion_of : \u2200 {a b c : Finset \u03b1}, P a c \u2192 P b c \u2192 P (a \u222a b) c\na\u271d b : Finset \u03b1\nx : \u03b1\ns : Finset \u03b1\n_xs : \u00acx \u2208 s\nhi\u271d : P a\u271d s\na : \u03b1\nt : Finset \u03b1\n_ta : \u00aca \u2208 t\nhi : P {x} t\n\u22a2 P ({a} \u222a t) {x}\n[PROOFSTEP]\nexact union_of singletons (symm hi)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 \u2203 i, \u2191s \u2286 f i\n[PROOFSTEP]\nclassical\nrevert hs\nrefine' s.induction_on _ _\n\u00b7 refine' fun _ => \u27e8hn.some, _\u27e9\n simp only [coe_empty, Set.empty_subset]\n\u00b7 intro b t _hbt htc hbtc\n obtain \u27e8i : \u03b9, hti : (t : Set \u03b1) \u2286 f i\u27e9 := htc (Set.Subset.trans (t.subset_insert b) hbtc)\n obtain \u27e8j, hbj\u27e9 : \u2203 j, b \u2208 f j := by simpa [Set.mem_iUnion\u2082] using hbtc (t.mem_insert_self b)\n rcases h j i with \u27e8k, hk, hk'\u27e9\n use k\n rw [coe_insert, Set.insert_subset_iff]\n exact \u27e8hk hbj, _root_.trans hti hk'\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 \u2203 i, \u2191s \u2286 f i\n[PROOFSTEP]\nrevert hs\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\n\u22a2 \u2191s \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191s \u2286 f i\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\n\u22a2 \u2191\u2205 \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191\u2205 \u2286 f i\n[PROOFSTEP]\nrefine' fun _ => \u27e8hn.some, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nx\u271d : \u2191\u2205 \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 \u2191\u2205 \u2286 f (Nonempty.some hn)\n[PROOFSTEP]\nsimp only [coe_empty, Set.empty_subset]\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1},\n \u00aca \u2208 s \u2192 (\u2191s \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191s \u2286 f i) \u2192 \u2191(insert a s) \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191(insert a s) \u2286 f i\n[PROOFSTEP]\nintro b t _hbt htc hbtc\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 \u2203 i, \u2191(insert b t) \u2286 f i\n[PROOFSTEP]\nobtain \u27e8i : \u03b9, hti : (t : Set \u03b1) \u2286 f i\u27e9 := htc (Set.Subset.trans (t.subset_insert b) hbtc)\n[GOAL]\ncase refine'_2.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\n\u22a2 \u2203 i, \u2191(insert b t) \u2286 f i\n[PROOFSTEP]\nobtain \u27e8j, hbj\u27e9 : \u2203 j, b \u2208 f j := by simpa [Set.mem_iUnion\u2082] using hbtc (t.mem_insert_self b)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\n\u22a2 \u2203 j, b \u2208 f j\n[PROOFSTEP]\nsimpa [Set.mem_iUnion\u2082] using hbtc (t.mem_insert_self b)\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\nj : \u03b9\nhbj : b \u2208 f j\n\u22a2 \u2203 i, \u2191(insert b t) \u2286 f i\n[PROOFSTEP]\nrcases h j i with \u27e8k, hk, hk'\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\nj : \u03b9\nhbj : b \u2208 f j\nk : \u03b9\nhk : f j \u2286 f k\nhk' : f i \u2286 f k\n\u22a2 \u2203 i, \u2191(insert b t) \u2286 f i\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\nj : \u03b9\nhbj : b \u2208 f j\nk : \u03b9\nhk : f j \u2286 f k\nhk' : f i \u2286 f k\n\u22a2 \u2191(insert b t) \u2286 f k\n[PROOFSTEP]\nrw [coe_insert, Set.insert_subset_iff]\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b\u271d : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nhn : Nonempty \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : Directed (fun x x_1 => x \u2286 x_1) f\ns : Finset \u03b1\nb : \u03b1\nt : Finset \u03b1\n_hbt : \u00acb \u2208 t\nhtc : \u2191t \u2286 \u22c3 (i : \u03b9), f i \u2192 \u2203 i, \u2191t \u2286 f i\nhbtc : \u2191(insert b t) \u2286 \u22c3 (i : \u03b9), f i\ni : \u03b9\nhti : \u2191t \u2286 f i\nj : \u03b9\nhbj : b \u2208 f j\nk : \u03b9\nhk : f j \u2286 f k\nhk' : f i \u2286 f k\n\u22a2 b \u2208 f k \u2227 \u2191t \u2286 f k\n[PROOFSTEP]\nexact \u27e8hk hbj, _root_.trans hti hk'\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nf : \u03b9 \u2192 Set \u03b1\nc : Set \u03b9\nhn : Set.Nonempty c\nhc : DirectedOn (fun i j => f i \u2286 f j) c\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 c), f i\n\u22a2 \u2203 i, i \u2208 c \u2227 \u2191s \u2286 f i\n[PROOFSTEP]\nrw [Set.biUnion_eq_iUnion] at hs \n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nf : \u03b9 \u2192 Set \u03b1\nc : Set \u03b9\nhn : Set.Nonempty c\nhc : DirectedOn (fun i j => f i \u2286 f j) c\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (x : \u2191c), f \u2191x\n\u22a2 \u2203 i, i \u2208 c \u2227 \u2191s \u2286 f i\n[PROOFSTEP]\nhaveI := hn.coe_sort\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nf : \u03b9 \u2192 Set \u03b1\nc : Set \u03b9\nhn : Set.Nonempty c\nhc : DirectedOn (fun i j => f i \u2286 f j) c\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (x : \u2191c), f \u2191x\nthis : Nonempty \u2191c\n\u22a2 \u2203 i, i \u2208 c \u2227 \u2191s \u2286 f i\n[PROOFSTEP]\nobtain \u27e8\u27e8i, hic\u27e9, hi\u27e9 := (directed_comp.2 hc.directed_val).exists_mem_subset_of_finset_subset_biUnion hs\n[GOAL]\ncase intro.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\u271d\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_4\n\u03b9 : Type u_5\nf : \u03b9 \u2192 Set \u03b1\nc : Set \u03b9\nhn : Set.Nonempty c\nhc : DirectedOn (fun i j => f i \u2286 f j) c\ns : Finset \u03b1\nhs : \u2191s \u2286 \u22c3 (x : \u2191c), f \u2191x\nthis : Nonempty \u2191c\ni : \u03b9\nhic : i \u2208 c\nhi : \u2191s \u2286 ((fun j => f j) \u2218 Subtype.val) { val := i, property := hic }\n\u22a2 \u2203 i, i \u2208 c \u2227 \u2191s \u2286 f i\n[PROOFSTEP]\nexact \u27e8i, hic, hi\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u\u271d v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 u : Finset \u03b1\n\u22a2 s\u2081 \u2286 s\u2082 \u2192 s\u2081 \u2286 u \u2192 s\u2081 \u2286 s\u2082 \u2229 u\n[PROOFSTEP]\nsimp (config := { contextual := true }) [subset_iff, mem_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 (s \u222a t) \u2229 s = s\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_inter, coe_union, Set.union_inter_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 (s \u222a t) \u2229 t = t\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_inter, coe_union, Set.union_inter_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \u2229 s\u2082 \u2194 x\u271d \u2208 s\u2082 \u2229 s\u2081\n[PROOFSTEP]\nsimp only [mem_inter, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 s\u2083 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \u2229 s\u2082 \u2229 s\u2083 \u2194 x\u271d \u2208 s\u2081 \u2229 (s\u2082 \u2229 s\u2083)\n[PROOFSTEP]\nsimp only [mem_inter, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 s\u2083 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \u2229 (s\u2082 \u2229 s\u2083) \u2194 x\u271d \u2208 s\u2082 \u2229 (s\u2081 \u2229 s\u2083)\n[PROOFSTEP]\nsimp only [mem_inter, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 s\u2083 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \u2229 s\u2082 \u2229 s\u2083 \u2194 x\u271d \u2208 s\u2081 \u2229 s\u2083 \u2229 s\u2082\n[PROOFSTEP]\nsimp only [mem_inter, and_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s \u2227 x\u271d \u2208 \u2205 \u2194 x\u271d \u2208 \u2205\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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2205 \u2227 x\u271d \u2208 s \u2194 x\u271d \u2208 \u2205\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 s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2229 (t \u222a s) = s\n[PROOFSTEP]\nrw [inter_comm, union_inter_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : a \u2208 s\u2082\nx : \u03b1\n\u22a2 x \u2208 insert a s\u2081 \u2229 s\u2082 \u2194 x \u2208 insert a (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nhave : x = a \u2228 x \u2208 s\u2082 \u2194 x \u2208 s\u2082 := or_iff_right_of_imp <| by rintro rfl; exact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : a \u2208 s\u2082\nx : \u03b1\n\u22a2 x = a \u2192 x \u2208 s\u2082\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx : \u03b1\nh : x \u2208 s\u2082\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : a \u2208 s\u2082\nx : \u03b1\nthis : x = a \u2228 x \u2208 s\u2082 \u2194 x \u2208 s\u2082\n\u22a2 x \u2208 insert a s\u2081 \u2229 s\u2082 \u2194 x \u2208 insert a (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nsimp only [mem_inter, mem_insert, or_and_left, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : a \u2208 s\u2081\n\u22a2 s\u2081 \u2229 insert a s\u2082 = insert a (s\u2081 \u2229 s\u2082)\n[PROOFSTEP]\nrw [inter_comm, insert_inter_of_mem h, inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\u2082\nx : \u03b1\n\u22a2 x \u2208 insert a s\u2081 \u2229 s\u2082 \u2194 x \u2208 s\u2081 \u2229 s\u2082\n[PROOFSTEP]\nhave : \u00ac(x = a \u2227 x \u2208 s\u2082) := by rintro \u27e8rfl, H\u27e9; exact h H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\u2082\nx : \u03b1\n\u22a2 \u00ac(x = a \u2227 x \u2208 s\u2082)\n[PROOFSTEP]\nrintro \u27e8rfl, H\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx : \u03b1\nH : x \u2208 s\u2082\nh : \u00acx \u2208 s\u2082\n\u22a2 False\n[PROOFSTEP]\nexact h H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\u2082\nx : \u03b1\nthis : \u00ac(x = a \u2227 x \u2208 s\u2082)\n\u22a2 x \u2208 insert a s\u2081 \u2229 s\u2082 \u2194 x \u2208 s\u2081 \u2229 s\u2082\n[PROOFSTEP]\nsimp only [mem_inter, mem_insert, or_and_right, this, false_or_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\u2081\n\u22a2 s\u2081 \u2229 insert a s\u2082 = s\u2081 \u2229 s\u2082\n[PROOFSTEP]\nrw [inter_comm, insert_inter_of_not_mem h, 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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nH : a \u2208 s\n\u22a2 insert a \u2205 \u2229 s = insert a \u2205\n[PROOFSTEP]\nrw [insert_inter_of_mem H, empty_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nH : \u00aca \u2208 s\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 {a} \u2229 s\n[PROOFSTEP]\nsimp only [mem_inter, mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nH : \u00aca \u2208 s\n\u22a2 \u2200 (x : \u03b1), \u00ac(x = a \u2227 x \u2208 s)\n[PROOFSTEP]\nrintro x \u27e8rfl, h\u27e9\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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : x \u2208 s\nH : \u00acx \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact H h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\n\u22a2 s \u2229 {a} = {a}\n[PROOFSTEP]\nrw [inter_comm, singleton_inter_of_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 s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2229 {a} = \u2205\n[PROOFSTEP]\nrw [inter_comm, singleton_inter_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 s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\nx y s t : Finset \u03b1\nh : x \u2286 y\nh' : s \u2286 t\n\u22a2 x \u2229 s \u2286 y \u2229 t\n[PROOFSTEP]\nintro a a_in\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\nx y s t : Finset \u03b1\nh : x \u2286 y\nh' : s \u2286 t\na : \u03b1\na_in : a \u2208 x \u2229 s\n\u22a2 a \u2208 y \u2229 t\n[PROOFSTEP]\nrw [Finset.mem_inter] at a_in \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b : \u03b1\nx y s t : Finset \u03b1\nh : x \u2286 y\nh' : s \u2286 t\na : \u03b1\na_in : a \u2208 x \u2227 a \u2208 s\n\u22a2 a \u2208 y \u2227 a \u2208 t\n[PROOFSTEP]\nexact \u27e8h a_in.1, h' a_in.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b\u271d : \u03b1\na b c : Finset \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) \u2264 a \u2294 b \u2293 c\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [sup_eq_union, inf_eq_inter, le_eq_subset, subset_iff, mem_inter,\n mem_union, and_imp, or_imp, true_or_iff, imp_true_iff, true_and_iff, or_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\n\u22a2 (\u2203 a, a \u2208 s \u2227 a \u2208 t) \u2194 Finset.Nonempty (s \u2229 t)\n[PROOFSTEP]\nsimp [Finset.Nonempty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 _root_.Disjoint s t \u2228 Finset.Nonempty (s \u2229 t)\n[PROOFSTEP]\nrw [\u2190 not_disjoint_iff_nonempty_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 _root_.Disjoint s t \u2228 \u00ac_root_.Disjoint s t\n[PROOFSTEP]\nexact em _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 IsDirected (Finset \u03b1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nclassical infer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 IsDirected (Finset \u03b1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na\u271d b a : \u03b1\n\u22a2 erase {a} a = \u2205\n[PROOFSTEP]\next x\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 u v : Finset \u03b1\na\u271d b a x : \u03b1\n\u22a2 x \u2208 erase {a} a \u2194 x \u2208 \u2205\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 v : Finset \u03b1\na b : \u03b1\n\u22a2 a \u2260 b \u2192 a \u2208 s \u2192 a \u2208 erase s b\n[PROOFSTEP]\nsimp only [mem_erase]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 a \u2260 b \u2192 a \u2208 s \u2192 a \u2260 b \u2227 a \u2208 s\n[PROOFSTEP]\nexact And.intro\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nhs : b \u2208 s\nhsa : \u00acb \u2208 erase s a\n\u22a2 b = a\n[PROOFSTEP]\nrw [mem_erase, not_and] at hsa \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nhs : b \u2208 s\nhsa : b \u2260 a \u2192 \u00acb \u2208 s\n\u22a2 b = a\n[PROOFSTEP]\nexact not_imp_not.mp hsa hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na x : \u03b1\n\u22a2 x \u2208 erase (insert a s) a \u2194 x \u2208 erase s a\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_erase, mem_insert, and_congr_right_iff, false_or_iff, iff_self_iff,\n imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 erase (insert a s) a = s\n[PROOFSTEP]\nrw [erase_insert_eq_erase, erase_eq_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 u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nh : a \u2260 b\nx : \u03b1\n\u22a2 x \u2208 erase (insert a s) b \u2194 x \u2208 insert a (erase s b)\n[PROOFSTEP]\nhave : x \u2260 b \u2227 x = a \u2194 x = a := and_iff_right_of_imp fun hx => hx.symm \u25b8 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 u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nh : a \u2260 b\nx : \u03b1\nthis : x \u2260 b \u2227 x = a \u2194 x = a\n\u22a2 x \u2208 erase (insert a s) b \u2194 x \u2208 insert a (erase s b)\n[PROOFSTEP]\nsimp only [mem_erase, mem_insert, and_or_left, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nhb : a \u2260 b\n\u22a2 erase (cons a s ha) b = cons a (erase s b) (_ : a \u2208 erase s b \u2192 False)\n[PROOFSTEP]\nsimp only [cons_eq_insert, erase_insert_of_ne hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\nx : \u03b1\n\u22a2 x \u2208 insert a (erase s a) \u2194 x \u2208 s\n[PROOFSTEP]\nsimp only [mem_insert, mem_erase, or_and_left, dec_em, true_and_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\nx : \u03b1\n\u22a2 x = a \u2228 x \u2208 s \u2194 x \u2208 s\n[PROOFSTEP]\napply or_iff_right_of_imp\n[GOAL]\ncase ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : a \u2208 s\nx : \u03b1\n\u22a2 x = a \u2192 x \u2208 s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase ha\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : x \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact 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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2260 a \u2227 x\u271d \u2208 s \u2194 x\u271d \u2208 \u2191s \\ {a}\n[PROOFSTEP]\nrw [and_comm, Set.mem_diff, Set.mem_singleton_iff, mem_coe]\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 v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2282 t \u2194 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_subset_of_ssubset h <| erase_ssubset ha\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\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a\n[PROOFSTEP]\nobtain \u27e8a, ht, hs\u27e9 := not_subset.1 h.2\n[GOAL]\ncase intro.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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\na : \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a\n[PROOFSTEP]\nexact \u27e8a, ht, subset_erase.2 \u27e8h.1, hs\u27e9\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 u v : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 erase (cons a s h) a = s\n[PROOFSTEP]\nrw [cons_eq_insert, erase_insert_eq_erase, erase_eq_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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 erase (erase s a) a = erase s 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\ns\u271d t u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\n\u22a2 erase (erase s a) b = erase (erase s b) a\n[PROOFSTEP]\next x\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 u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 erase (erase s a) b \u2194 x \u2208 erase (erase s b) a\n[PROOFSTEP]\nsimp only [mem_erase, \u2190 and_assoc]\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 u v : Finset \u03b1\na\u271d b\u271d a b : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 (x \u2260 b \u2227 x \u2260 a) \u2227 x \u2208 s \u2194 (x \u2260 a \u2227 x \u2260 b) \u2227 x \u2208 s\n[PROOFSTEP]\nrw [@and_comm (x \u2260 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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2286 insert a t \u2194 erase s a \u2286 t\n[PROOFSTEP]\nsimp only [subset_iff, or_iff_not_imp_left, mem_erase, mem_insert, and_imp]\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 v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 t) \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2260 a \u2192 x \u2208 s \u2192 x \u2208 t\n[PROOFSTEP]\nexact forall_congr' fun x => forall_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2286 insert a t \u2194 s \u2286 t\n[PROOFSTEP]\nrw [subset_insert_iff, erase_eq_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 t u v : Finset \u03b1\na b : \u03b1\nh : a \u2208 t\n\u22a2 erase s a \u2286 t \u2194 s \u2286 t\n[PROOFSTEP]\nrw [\u2190 subset_insert_iff, insert_eq_of_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 u v : Finset \u03b1\na b x y : \u03b1\ns : Finset \u03b1\nhx : x \u2208 s\n\u22a2 erase s x = erase s y \u2194 x = y\n[PROOFSTEP]\nrefine \u27e8fun h => eq_of_mem_of_not_mem_erase hx ?_, congr_arg _\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 u v : Finset \u03b1\na b x y : \u03b1\ns : Finset \u03b1\nhx : x \u2208 s\nh : erase s x = erase s y\n\u22a2 \u00acx \u2208 erase s y\n[PROOFSTEP]\nrw [\u2190 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 u v : Finset \u03b1\na b x y : \u03b1\ns : Finset \u03b1\nhx : x \u2208 s\nh : erase s x = erase s y\n\u22a2 \u00acx \u2208 erase s 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\u271d t\u271d u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nhs : s \u2208 {s | a \u2208 s}\nt : Finset \u03b1\nht : t \u2208 {s | a \u2208 s}\nh : erase s a = (fun s => erase s a) t\n\u22a2 s = t\n[PROOFSTEP]\nrw [\u2190 insert_erase hs, \u2190 insert_erase ht, 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 v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 s\u2081 \u2229 (s\u2082 \\ s\u2081)\n[PROOFSTEP]\nsimp only [mem_inter, mem_sdiff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\n\u22a2 \u2200 (x : \u03b1), \u00ac(x \u2208 s\u2081 \u2227 x \u2208 s\u2082 \u2227 \u00acx \u2208 s\u2081)\n[PROOFSTEP]\nrintro x \u27e8h, _, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx : \u03b1\nh : x \u2208 s\u2081\nleft\u271d : x \u2208 s\u2082\nhn : \u00acx \u2208 s\u2081\n\u22a2 False\n[PROOFSTEP]\nexact hn 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 v : Finset \u03b1\na b : \u03b1\nx y : Finset \u03b1\n\u22a2 x \u2293 y \u2294 x \\ y = x\n[PROOFSTEP]\nsimp only [ext_iff, mem_union, mem_sdiff, inf_eq_inter, sup_eq_union, mem_inter, \u2190 and_or_left, em, and_true,\n implies_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nx y : Finset \u03b1\n\u22a2 x \u2293 y \u2293 x \\ y = \u22a5\n[PROOFSTEP]\nsimp only [ext_iff, inter_sdiff_self, inter_empty, inter_assoc, false_iff_iff, inf_eq_inter, not_mem_empty,\n bot_eq_empty, not_false_iff, implies_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nh : a \u2208 t\n\u22a2 \u00aca \u2208 s \\ t\n[PROOFSTEP]\nsimp only [mem_sdiff, h, not_true, not_false_iff, and_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 s \\ t\n[PROOFSTEP]\nsimp [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 v : Finset \u03b1\na b : \u03b1\ns t u : Finset \u03b1\n\u22a2 s \u2229 (t \\ u) = (s \u2229 t) \\ u\n[PROOFSTEP]\next x\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\u271d v : Finset \u03b1\na b : \u03b1\ns t u : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 s \u2229 (t \\ u) \u2194 x \u2208 (s \u2229 t) \\ u\n[PROOFSTEP]\nsimp [and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 s \u222a t \\ s = t \u222a s \\ t\n[PROOFSTEP]\nsimp [union_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 v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 insert x s \\ t = insert x (s \\ t)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_insert, coe_sdiff, coe_sdiff, coe_insert]\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 v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\nh : \u00acx \u2208 t\n\u22a2 insert x \u2191s \\ \u2191t = insert x (\u2191s \\ \u2191t)\n[PROOFSTEP]\nexact Set.insert_diff_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 u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : x \u2208 t\n\u22a2 insert x s \\ t = s \\ t\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_sdiff, coe_sdiff, coe_insert]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na b : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : x \u2208 t\n\u22a2 insert x \u2191s \\ \u2191t = \u2191s \\ \u2191t\n[PROOFSTEP]\nexact Set.insert_diff_of_mem _ h\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 v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\n\u22a2 s \\ insert x t = s \\ t\n[PROOFSTEP]\nrefine' Subset.antisymm (sdiff_subset_sdiff (Subset.refl _) (subset_insert _ _)) fun y hy => _\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 v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \\ t\n\u22a2 y \u2208 s \\ insert x t\n[PROOFSTEP]\nsimp only [mem_sdiff, mem_insert, not_or] at hy \u22a2\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 v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \u2227 \u00acy \u2208 t\n\u22a2 y \u2208 s \u2227 \u00acy = x \u2227 \u00acy \u2208 t\n[PROOFSTEP]\nexact \u27e8hy.1, fun hxy => h <| hxy \u25b8 hy.1, hy.2\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 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\n\u22a2 s \\ {a} = erase s a\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\u271d t u v : Finset \u03b1\na\u271d\u00b9 b a : \u03b1\ns : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \\ {a} \u2194 a\u271d \u2208 erase s a\n[PROOFSTEP]\nrw [mem_erase, mem_sdiff, mem_singleton, and_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (erase s a) t \u2194 _root_.Disjoint s (erase t a)\n[PROOFSTEP]\nsimp_rw [erase_eq, disjoint_sdiff_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : \u00aca \u2208 t\nhst : _root_.Disjoint (erase s a) t\n\u22a2 _root_.Disjoint s t\n[PROOFSTEP]\nrw [\u2190 erase_insert ha, \u2190 disjoint_erase_comm, disjoint_insert_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : \u00aca \u2208 t\nhst : _root_.Disjoint (erase s a) t\n\u22a2 \u00aca \u2208 erase s a \u2227 _root_.Disjoint (erase s a) t\n[PROOFSTEP]\nexact \u27e8not_mem_erase _ _, hst\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhst : _root_.Disjoint s (erase t a)\n\u22a2 _root_.Disjoint s t\n[PROOFSTEP]\nrw [\u2190 erase_insert ha, disjoint_erase_comm, disjoint_insert_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 v : Finset \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\nhst : _root_.Disjoint s (erase t a)\n\u22a2 \u00aca \u2208 erase t a \u2227 _root_.Disjoint s (erase t a)\n[PROOFSTEP]\nexact \u27e8not_mem_erase _ _, hst\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\u271d u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2229 erase t a = erase (s \u2229 t) a\n[PROOFSTEP]\nsimp only [erase_eq, inter_sdiff]\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 v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 erase s a \u2229 t = erase (s \u2229 t) a\n[PROOFSTEP]\nsimpa only [inter_comm t] using inter_erase a t 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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 erase s a \\ t = erase (s \\ t) a\n[PROOFSTEP]\nsimp_rw [erase_eq, sdiff_right_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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 insert a s \u222a t = s \u222a insert a t\n[PROOFSTEP]\nrw [insert_union, union_insert]\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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 erase s a \u2229 t = s \u2229 erase t a\n[PROOFSTEP]\nrw [erase_inter, inter_erase]\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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 erase (s \u222a t) a = erase s a \u222a erase t a\n[PROOFSTEP]\nsimp_rw [erase_eq, union_sdiff_distrib]\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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 insert a (s \u2229 t) = insert a s \u2229 insert a t\n[PROOFSTEP]\nsimp_rw [insert_eq, union_distrib_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 v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\na : \u03b1\n\u22a2 erase (s \\ t) a = erase s a \\ erase t a\n[PROOFSTEP]\nsimp_rw [erase_eq, sdiff_sdiff, sup_sdiff_eq_sup le_rfl, sup_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 u v : Finset \u03b1\na b : \u03b1\nha : a \u2208 t\ns : Finset \u03b1\n\u22a2 erase s a \u222a t = s \u222a t\n[PROOFSTEP]\nrw [\u2190 insert_erase (mem_union_right s ha), erase_union_distrib, \u2190 union_insert, insert_erase ha]\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 v : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\nt : Finset \u03b1\n\u22a2 s \u222a erase t a = s \u222a t\n[PROOFSTEP]\nrw [\u2190 insert_erase (mem_union_left t ha), erase_union_distrib, \u2190 insert_union, insert_erase ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : \u00aca \u2208 s\n\u22a2 _root_.Disjoint {a} s\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nhts : t \u2286 s\nha : a \u2208 t\n\u22a2 s \\ t \u222a erase t a = erase s a\n[PROOFSTEP]\nsimp_rw [erase_eq, sdiff_union_sdiff_cancel hts (singleton_subset_iff.2 ha)]\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 v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\n\u22a2 s \\ insert x t = erase (s \\ t) x\n[PROOFSTEP]\nsimp_rw [\u2190 sdiff_singleton_eq_erase, insert_eq, sdiff_sdiff_left', sdiff_union_distrib, 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 v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nx : \u03b1\nhxs : x \u2208 s\nhxt : \u00acx \u2208 t\n\u22a2 insert x (s \\ insert x t) = s \\ t\n[PROOFSTEP]\nrw [sdiff_insert, insert_erase (mem_sdiff.mpr \u27e8hxs, hxt\u27e9)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nh : a \u2208 s\n\u22a2 s \\ erase t a = insert a (s \\ t)\n[PROOFSTEP]\nrw [\u2190 sdiff_singleton_eq_erase, sdiff_sdiff_eq_sdiff_union (singleton_subset_iff.2 h), insert_eq, union_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t u v : Finset \u03b1\na b : \u03b1\nha : a \u2208 s\n\u22a2 s \\ erase s a = {a}\n[PROOFSTEP]\nrw [sdiff_erase ha, sdiff_self, insert_emptyc_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 erase s a = \u2205 \u2194 s = \u2205 \u2228 s = {a}\n[PROOFSTEP]\nrw [\u2190 sdiff_singleton_eq_erase, sdiff_eq_empty_iff_subset, subset_singleton_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\n\u22a2 a \u2208 s \u2206 t \u2194 a \u2208 s \u2227 \u00aca \u2208 t \u2228 a \u2208 t \u2227 \u00aca \u2208 s\n[PROOFSTEP]\nsimp_rw [symmDiff, sup_eq_union, mem_union, mem_sdiff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na b x : \u03b1\n\u22a2 x \u2208 \u2191(s \u2206 t) \u2194 x \u2208 \u2191s \u2206 \u2191t\n[PROOFSTEP]\nsimp [mem_symmDiff, Set.mem_symmDiff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\ns : Finset \u03b1\nhx : x \u2208 s\n\u22a2 sizeOf x < sizeOf s\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < sizeOf { val := val\u271d, nodup := nodup\u271d }\n[PROOFSTEP]\ndsimp [SizeOf.sizeOf, SizeOf.sizeOf, Multiset.sizeOf]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < 1 + Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1) + 1\n[PROOFSTEP]\nrefine' lt_trans _ (Nat.lt_succ_self _)\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : SizeOf \u03b1\nx : \u03b1\nval\u271d : Multiset \u03b1\nnodup\u271d : Nodup val\u271d\nhx : x \u2208 { val := val\u271d, nodup := nodup\u271d }\n\u22a2 sizeOf x < Quot.liftOn val\u271d (fun m => List._sizeOf_1 m) (_ : \u2200 (x x_1 : List \u03b1), x ~ x_1 \u2192 sizeOf x = sizeOf x_1)\n[PROOFSTEP]\nexact Multiset.sizeOf_lt_sizeOf_of_mem hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\n\u22a2 Finset.Nonempty (attach s) \u2194 Finset.Nonempty s\n[PROOFSTEP]\nsimp [Finset.Nonempty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\n\u22a2 attach s = \u2205 \u2194 s = \u2205\n[PROOFSTEP]\nsimp [eq_empty_iff_forall_not_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 piecewise (insert j s) f g j = f j\n[PROOFSTEP]\nsimp [piecewise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 \u2205)\n\u22a2 piecewise \u2205 f g = g\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 \u2205)\ni : \u03b1\n\u22a2 piecewise \u2205 f g i = g i\n[PROOFSTEP]\nsimp [piecewise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 \u2191s)\n\u22a2 Set.piecewise (\u2191s) f g = piecewise s f g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 \u2191s)\nx\u271d : \u03b1\n\u22a2 Set.piecewise (\u2191s) f g x\u271d = piecewise s f g x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ni : \u03b1\nhi : i \u2208 s\n\u22a2 piecewise s f g i = f i\n[PROOFSTEP]\nsimp [piecewise, hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ni : \u03b1\nhi : \u00aci \u2208 s\n\u22a2 piecewise s f g i = g i\n[PROOFSTEP]\nsimp [piecewise, hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\ni j : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nh : i \u2260 j\n\u22a2 piecewise (insert j s) f g i = piecewise s f g i\n[PROOFSTEP]\nsimp [piecewise, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j)\n[PROOFSTEP]\nclassical simp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 piecewise (insert j s) f g = update (piecewise s f g) j (f j)\n[PROOFSTEP]\nsimp only [\u2190 piecewise_coe, coe_insert, \u2190 Set.piecewise_insert]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\n\u22a2 Set.piecewise (\u2191(insert j s)) f g = Set.piecewise (insert j \u2191s) f g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 Set.piecewise (\u2191(insert j s)) f g x\u271d = Set.piecewise (insert j \u2191s) f g x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_s\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b2 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : DecidableEq \u03b1\nj : \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 insert j s)\nx\u271d : \u03b1\n\u22a2 \u2191(insert j s) = insert j \u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ni : \u03b1\np : \u03b4 i \u2192 Prop\nhf : p (f i)\nhg : p (g i)\n\u22a2 p (piecewise s f g i)\n[PROOFSTEP]\nby_cases hi : i \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ni : \u03b1\np : \u03b4 i \u2192 Prop\nhf : p (f i)\nhg : p (g i)\nhi : i \u2208 s\n\u22a2 p (piecewise s f g i)\n[PROOFSTEP]\nsimpa [hi]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ni : \u03b1\np : \u03b4 i \u2192 Prop\nhf : p (f i)\nhg : p (g i)\nhi : \u00aci \u2208 s\n\u22a2 p (piecewise s f g i)\n[PROOFSTEP]\nsimpa [hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'\n[PROOFSTEP]\nclassical\nrw [\u2190 piecewise_coe]\nexact Set.piecewise_mem_pi (\u2191s) hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'\n[PROOFSTEP]\nrw [\u2190 piecewise_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t'\n[PROOFSTEP]\nexact Set.piecewise_mem_pi (\u2191s) hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\n\u22a2 piecewise {i} f g = update g i (f i)\n[PROOFSTEP]\nrw [\u2190 insert_emptyc_eq, piecewise_insert, piecewise_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nv : \u03b4 i\n\u22a2 update (piecewise s f g) i v = piecewise s (update f i v) (update g i v)\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nv : \u03b4 i\nj : \u03b1\n\u22a2 update (piecewise s f g) i v j = piecewise s (update f i v) (update g i v) j\n[PROOFSTEP]\nrcases em (j = i) with (rfl | hj)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\nj : \u03b1\nv : \u03b4 j\n\u22a2 update (piecewise s f g) j v j = piecewise s (update f j v) (update g j v) j\n[PROOFSTEP]\nby_cases hs : j \u2208 s\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nv : \u03b4 i\nj : \u03b1\nhj : \u00acj = i\n\u22a2 update (piecewise s f g) i v j = piecewise s (update f i v) (update g i v) j\n[PROOFSTEP]\nby_cases hs : j \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\nj : \u03b1\nv : \u03b4 j\nhs : j \u2208 s\n\u22a2 update (piecewise s f g) j v j = piecewise s (update f j v) (update g j v) j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\nj : \u03b1\nv : \u03b4 j\nhs : \u00acj \u2208 s\n\u22a2 update (piecewise s f g) j v j = piecewise s (update f j v) (update g j v) j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nv : \u03b4 i\nj : \u03b1\nhj : \u00acj = i\nhs : j \u2208 s\n\u22a2 update (piecewise s f g) i v j = piecewise s (update f i v) (update g i v) j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nv : \u03b4 i\nj : \u03b1\nhj : \u00acj = i\nhs : \u00acj \u2208 s\n\u22a2 update (piecewise s f g) i v j = piecewise s (update f i v) (update g i v) j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : i \u2208 s\nv : \u03b4 i\n\u22a2 update (piecewise s f g) i v = piecewise s (update f i v) g\n[PROOFSTEP]\nrw [update_piecewise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : i \u2208 s\nv : \u03b4 i\n\u22a2 piecewise s (update f i v) (update g i v) = piecewise s (update f i v) g\n[PROOFSTEP]\nrefine' s.piecewise_congr (fun _ _ => rfl) fun j hj => update_noteq _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : i \u2208 s\nv : \u03b4 i\nj : \u03b1\nhj : \u00acj \u2208 s\n\u22a2 j \u2260 i\n[PROOFSTEP]\nexact fun h => hj (h.symm \u25b8 hi)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : \u00aci \u2208 s\nv : \u03b4 i\n\u22a2 update (piecewise s f g) i v = piecewise s f (update g i v)\n[PROOFSTEP]\nrw [update_piecewise]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : \u00aci \u2208 s\nv : \u03b4 i\n\u22a2 piecewise s (update f i v) (update g i v) = piecewise s f (update g i v)\n[PROOFSTEP]\nrefine' s.piecewise_congr (fun j hj => update_noteq _ _ _) fun _ _ => rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\ninst\u271d : DecidableEq \u03b1\ni : \u03b1\nhi : \u00aci \u2208 s\nv : \u03b4 i\nj : \u03b1\nhj : j \u2208 s\n\u22a2 j \u2260 i\n[PROOFSTEP]\nexact fun h => hi (h \u25b8 hj)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\ninst\u271d : (i : \u03b1) \u2192 Preorder (\u03b4 i)\nf g f' g' : (i : \u03b1) \u2192 \u03b4 i\nHf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2264 f' x\nHg : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 g x \u2264 g' x\nx : \u03b1\n\u22a2 piecewise s f g x \u2264 piecewise s f' g' 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\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\ninst\u271d : (i : \u03b1) \u2192 Preorder (\u03b4 i)\nf g f' g' : (i : \u03b1) \u2192 \u03b4 i\nHf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2264 f' x\nHg : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 g x \u2264 g' x\nx : \u03b1\nhx : x \u2208 s\n\u22a2 piecewise s f g x \u2264 piecewise s f' g' x\n[PROOFSTEP]\nsimp [hx, *]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4\u271d : \u03b1 \u2192 Sort u_4\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_5\ninst\u271d : (i : \u03b1) \u2192 Preorder (\u03b4 i)\nf g f' g' : (i : \u03b1) \u2192 \u03b4 i\nHf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2264 f' x\nHg : \u2200 (x : \u03b1), \u00acx \u2208 s \u2192 g x \u2264 g' x\nx : \u03b1\nhx : \u00acx \u2208 s\n\u22a2 piecewise s f g x \u2264 piecewise s f' g' x\n[PROOFSTEP]\nsimp [hx, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 Decidable (s \u2282 t)\n[PROOFSTEP]\nrw [ssubset_iff_subset_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 Decidable (s \u2286 t \u2227 s \u2260 t)\n[PROOFSTEP]\nhave h\u2081 : Decidable (s \u2286 t) := decidableSubsetFinset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh\u2081 : Decidable (s \u2286 t)\n\u22a2 Decidable (s \u2286 t \u2227 s \u2260 t)\n[PROOFSTEP]\nhave h\u2082 : Decidable (s \u2260 t) := instDecidableNot\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nh\u2081 : Decidable (s \u2286 t)\nh\u2082 : Decidable (s \u2260 t)\n\u22a2 Decidable (s \u2286 t \u2227 s \u2260 t)\n[PROOFSTEP]\nexact instDecidableAnd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\np : \u03b1 \u2192 Prop\n_hp : (a : \u03b1) \u2192 Decidable (p a)\n\u22a2 (\u2203 a x, p a) \u2194 \u2203 a, a \u2208 s \u2227 p a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\na : \u03b1\n\u22a2 a \u2208 filter q (filter p s) \u2194 a \u2208 filter (fun a => p a \u2227 q a) s\n[PROOFSTEP]\nsimp only [mem_filter, and_assoc, Bool.decide_and, Bool.decide_coe, Bool.and_eq_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\n\u22a2 filter (fun x => True) s = s\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun x => True) s \u2194 a\u271d \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\na : \u03b1\n\u22a2 a \u2208 filter (fun x => False) s \u2194 a \u2208 \u2205\n[PROOFSTEP]\nsimp [mem_filter, and_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\n\u22a2 filter p s = s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp [Finset.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acp x\n\u22a2 \u2200 (x : \u03b1), \u00acx \u2208 filter p s\n[PROOFSTEP]\nsimpa only [eq_empty_iff_forall_not_mem, mem_filter, not_and] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\n\u22a2 filter p s = \u2205 \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acp x\n[PROOFSTEP]\nrefine' \u27e8_, filter_false_of_mem\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\n\u22a2 filter p s = \u2205 \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acp x\n[PROOFSTEP]\nintro hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\nhs : filter p s = \u2205\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acp x\n[PROOFSTEP]\ninjection hs with hs'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\nhs' : Multiset.filter p s.val = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acp x\n[PROOFSTEP]\nrwa [filter_eq_nil] at hs' \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Finset \u03b1\n\u22a2 Finset.Nonempty (filter p s) \u2194 \u2203 a, a \u2208 s \u2227 p a\n[PROOFSTEP]\nsimp [nonempty_iff_ne_empty, Ne.def, filter_eq_empty_iff, not_not, not_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\n\u22a2 filter p {a} = if p a then {a} else \u2205\n[PROOFSTEP]\nclassical\next x\nsimp\nsplit_ifs with h <;> by_cases h' : x = a <;> simp [h, h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\n\u22a2 filter p {a} = if p a then {a} else \u2205\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\n\u22a2 x \u2208 filter p {a} \u2194 x \u2208 if p a then {a} else \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\n\u22a2 x = a \u2227 p x \u2194 x \u2208 if p a then {a} 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\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : p a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 {a}\n[PROOFSTEP]\nby_cases h' : x = a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : \u00acp a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 \u2205\n[PROOFSTEP]\nby_cases h' : x = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : p a\nh' : x = a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 {a}\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : p a\nh' : \u00acx = a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 {a}\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : \u00acp a\nh' : x = a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na x : \u03b1\nh : \u00acp a\nh' : \u00acx = a\n\u22a2 x = a \u2227 p x \u2194 x \u2208 \u2205\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\n\u22a2 _root_.Disjoint (filter p s) (filter q s) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 p x \u2192 \u00acq x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\n\u22a2 _root_.Disjoint (filter p s) (filter q s) \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 p x \u2192 \u00acq x\n[PROOFSTEP]\nsimp (config := { contextual := true }) [disjoint_left]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 p x \u2192 \u00acq x) \u2192 _root_.Disjoint (filter p s) (filter q s)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [disjoint_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns t : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nh : _root_.Disjoint p q\n\u22a2 _root_.Disjoint (filter p s) (filter q t)\n[PROOFSTEP]\nsimp_rw [disjoint_left, mem_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns t : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nh : _root_.Disjoint p q\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2227 p a \u2192 \u00ac(a \u2208 t \u2227 q a)\n[PROOFSTEP]\nrintro a \u27e8_, hp\u27e9 \u27e8_, hq\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns t : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nh : _root_.Disjoint p q\na : \u03b1\nleft\u271d\u00b9 : a \u2208 s\nhp : p a\nleft\u271d : a \u2208 t\nhq : q a\n\u22a2 False\n[PROOFSTEP]\nrw [Pi.disjoint_iff] at h \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\u271d\ninst\u271d\u00b2 : DecidablePred q\u271d\ns t : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\nh : \u2200 (i : \u03b1), _root_.Disjoint (p i) (q i)\na : \u03b1\nleft\u271d\u00b9 : a \u2208 s\nhp : p a\nleft\u271d : a \u2208 t\nhq : q a\n\u22a2 False\n[PROOFSTEP]\nsimpa [hp, hq] using h a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\n\u22a2 _root_.Disjoint (if p a then {a} else \u2205) (filter p s)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 _root_.Disjoint {a} (filter p s)\n[PROOFSTEP]\nrw [disjoint_singleton_left]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : p a\n\u22a2 \u00aca \u2208 filter p s\n[PROOFSTEP]\nexact mem_filter.not.mpr <| mt And.left ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh\u271d : \u00acp a\n\u22a2 _root_.Disjoint \u2205 (filter p s)\n[PROOFSTEP]\nexact disjoint_empty_left _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\n\u22a2 filter p (cons a s ha) =\n disjUnion (if p a then {a} else \u2205) (filter p s) (_ : _root_.Disjoint (if p a then {a} else \u2205) (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 q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : p a\n\u22a2 filter p (cons a s ha) = disjUnion {a} (filter p s) (_ : _root_.Disjoint {a} (filter p s))\n[PROOFSTEP]\nrw [filter_cons_of_pos _ _ _ ha h, singleton_disjUnion]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nh : \u00acp a\n\u22a2 filter p (cons a s ha) = disjUnion \u2205 (filter p s) (_ : _root_.Disjoint \u2205 (filter p s))\n[PROOFSTEP]\nrw [filter_cons_of_neg _ _ _ ha h, empty_disjUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 filter p (s\u2081 \u222a s\u2082) \u2194 x\u271d \u2208 filter p s\u2081 \u222a filter p s\u2082\n[PROOFSTEP]\nsimp only [mem_filter, mem_union, or_and_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 filter p s \u222a filter q s \u2194 x \u2208 filter (fun x => p x \u2228 q x) s\n[PROOFSTEP]\nsimp [mem_filter, mem_union, \u2190 and_or_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 t)\ni : \u03b1\n\u22a2 i \u2208 filter (fun i => i \u2208 t) s \u2194 i \u2208 s \u2229 t\n[PROOFSTEP]\nsimp [mem_filter, mem_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 filter p (s \u2229 t) = filter p s \u2229 filter p t\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter p (s \u2229 t) \u2194 a\u271d \u2208 filter p s \u2229 filter p t\n[PROOFSTEP]\nsimp [mem_filter, mem_inter, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 filter p s \u2229 t = filter p (s \u2229 t)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter p s \u2229 t \u2194 a\u271d \u2208 filter p (s \u2229 t)\n[PROOFSTEP]\nsimp only [mem_inter, mem_filter, and_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2229 filter p t = filter p (s \u2229 t)\n[PROOFSTEP]\nrw [inter_comm, filter_inter, inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\n\u22a2 filter p (insert a s) = if p a then insert a (filter p s) else filter p s\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 if p a then insert a (filter p s) else 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 q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : p a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 insert a (filter p s)\n[PROOFSTEP]\nby_cases h' : x = a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : \u00acp a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 filter p s\n[PROOFSTEP]\nby_cases h' : x = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : p a\nh' : x = a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 insert a (filter p s)\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : p a\nh' : \u00acx = a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 insert a (filter p s)\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : \u00acp a\nh' : x = a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 filter p s\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\nh : \u00acp a\nh' : \u00acx = a\n\u22a2 x \u2208 filter p (insert a s) \u2194 x \u2208 filter p s\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\n\u22a2 filter p (erase s a) = erase (filter p s) a\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\nx : \u03b1\n\u22a2 x \u2208 filter p (erase s a) \u2194 x \u2208 erase (filter p s) a\n[PROOFSTEP]\nsimp only [and_assoc, mem_filter, iff_self_iff, mem_erase]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 filter (fun a => p a \u2228 q a) s \u2194 x\u271d \u2208 filter p s \u222a filter q s\n[PROOFSTEP]\nsimp [mem_filter, mem_union, and_or_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 filter (fun a => p a \u2227 q a) s \u2194 x\u271d \u2208 filter p s \u2229 filter q s\n[PROOFSTEP]\nsimp [mem_filter, mem_inter, and_comm, and_left_comm, and_self_iff, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 a \u2208 filter (fun a => \u00acp a) s \u2194 a \u2208 s \\ filter p s\n[PROOFSTEP]\nsimp only [Bool.decide_coe, Bool.not_eq_true', mem_filter, and_comm, mem_sdiff, not_and_or, Bool.not_eq_true,\n and_or_left, and_not_self, or_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \\ s\u2082 \u2194 x\u271d \u2208 filter (fun x => \u00acx \u2208 s\u2082) s\u2081\n[PROOFSTEP]\nsimp [mem_sdiff, mem_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Finset \u03b1\n\u22a2 s\u2081 \\ s\u2082 = s\u2081 \u2194 s\u2081 \u2229 s\u2082 \u2286 \u2205\n[PROOFSTEP]\nsimp [Subset.antisymm_iff, disjoint_iff_inter_eq_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\n\u22a2 \u2203 s\u2081 s\u2082, s\u2081 \u222a s\u2082 = s \u2227 \u2191s\u2081 \u2286 t\u2081 \u2227 \u2191s\u2082 \u2286 t\u2082 \\ t\u2081\n[PROOFSTEP]\nclassical\nrefine' \u27e8s.filter (\u00b7 \u2208 t\u2081), s.filter (\u00b7 \u2209 t\u2081), _, _, _\u27e9\n\u00b7 simp [filter_union_right, em]\n\u00b7 intro x\n simp\n\u00b7 intro x\n simp\n intro hx hx\u2082\n refine' \u27e8Or.resolve_left (h hx) hx\u2082, hx\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\n\u22a2 \u2203 s\u2081 s\u2082, s\u2081 \u222a s\u2082 = s \u2227 \u2191s\u2081 \u2286 t\u2081 \u2227 \u2191s\u2082 \u2286 t\u2082 \\ t\u2081\n[PROOFSTEP]\nrefine' \u27e8s.filter (\u00b7 \u2208 t\u2081), s.filter (\u00b7 \u2209 t\u2081), _, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\n\u22a2 filter (fun x => x \u2208 t\u2081) s \u222a filter (fun x => \u00acx \u2208 t\u2081) s = s\n[PROOFSTEP]\nsimp [filter_union_right, em]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\n\u22a2 \u2191(filter (fun x => x \u2208 t\u2081) s) \u2286 t\u2081\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\nx : \u03b1\n\u22a2 x \u2208 \u2191(filter (fun x => x \u2208 t\u2081) s) \u2192 x \u2208 t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\n\u22a2 \u2191(filter (fun x => \u00acx \u2208 t\u2081) s) \u2286 t\u2082 \\ t\u2081\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\nx : \u03b1\n\u22a2 x \u2208 \u2191(filter (fun x => \u00acx \u2208 t\u2081) s) \u2192 x \u2208 t\u2082 \\ t\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\nx : \u03b1\n\u22a2 x \u2208 s \u2192 \u00acx \u2208 t\u2081 \u2192 x \u2208 t\u2082 \u2227 \u00acx \u2208 t\u2081\n[PROOFSTEP]\nintro hx hx\u2082\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt\u2081 t\u2082 : Set \u03b1\nh : \u2191s \u2286 t\u2081 \u222a t\u2082\nx : \u03b1\nhx : x \u2208 s\nhx\u2082 : \u00acx \u2208 t\u2081\n\u22a2 x \u2208 t\u2082 \u2227 \u00acx \u2208 t\u2081\n[PROOFSTEP]\nrefine' \u27e8Or.resolve_left (h hx) hx\u2082, hx\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\n\u22a2 filter (Eq b) s = if b \u2208 s then {b} 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\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : b \u2208 s\n\u22a2 filter (Eq b) s = {b}\n[PROOFSTEP]\next\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : b \u2208 s\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 filter (Eq b) s \u2194 a\u271d \u2208 {b}\n[PROOFSTEP]\nsimp only [mem_filter, mem_singleton, decide_eq_true_eq]\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : b \u2208 s\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 s \u2227 b = a\u271d \u2194 a\u271d = b\n[PROOFSTEP]\nrefine \u27e8fun h => h.2.symm, ?_\u27e9\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : b \u2208 s\na\u271d : \u03b2\n\u22a2 a\u271d = b \u2192 a\u271d \u2208 s \u2227 b = a\u271d\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase pos.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\na\u271d : \u03b2\nh : a\u271d \u2208 s\n\u22a2 a\u271d \u2208 s \u2227 a\u271d = a\u271d\n[PROOFSTEP]\nexact \u27e8h, rfl\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\n\u22a2 filter (Eq b) s = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 filter (Eq b) s \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp only [mem_filter, not_and, iff_false_iff, not_mem_empty, decide_eq_true_eq]\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 s \u2192 \u00acb = a\u271d\n[PROOFSTEP]\nrintro m rfl\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\nh : \u00acb \u2208 s\nm : b \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact h m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb x\u271d\u00b9 : \u03b2\nx\u271d : x\u271d\u00b9 \u2208 s\n\u22a2 x\u271d\u00b9 = b \u2194 b = x\u271d\u00b9\n[PROOFSTEP]\nsimp_rw [@eq_comm _ b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb : \u03b2\n\u22a2 filter (fun a => b \u2260 a) s = erase s b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb a\u271d : \u03b2\n\u22a2 a\u271d \u2208 filter (fun a => b \u2260 a) s \u2194 a\u271d \u2208 erase s b\n[PROOFSTEP]\nsimp only [mem_filter, mem_erase, Ne.def, decide_not, Bool.not_eq_true', decide_eq_false_iff_not]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb a\u271d : \u03b2\n\u22a2 a\u271d \u2208 s \u2227 \u00acb = a\u271d \u2194 \u00aca\u271d = b \u2227 a\u271d \u2208 s\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidablePred p\ninst\u271d\u00b2 : DecidablePred q\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b2\nb x\u271d\u00b9 : \u03b2\nx\u271d : x\u271d\u00b9 \u2208 s\n\u22a2 x\u271d\u00b9 \u2260 b \u2194 b \u2260 x\u271d\u00b9\n[PROOFSTEP]\nsimp_rw [@ne_comm _ b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\ninst\u271d\u00b9 : DecidablePred q\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\n\u22a2 filter p s \u2229 filter (fun a => \u00acp a) t = \u2205\n[PROOFSTEP]\nsimpa using (disjoint_filter_filter_neg s t p).eq_bot\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn m l : \u2115\n\u22a2 range n = \u2205 \u2194 n = 0\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty, nonempty_range_iff, not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn\u271d m\u271d l n m : \u2115\n\u22a2 filter (fun x => x = m) (range n) = if m < n then {m} else \u2205\n[PROOFSTEP]\nconvert filter_eq (range n) m using 2\n[GOAL]\ncase h.e'_2.h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn\u271d m\u271d l n m : \u2115\n\u22a2 (fun x => x = m) = Eq m\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn\u271d m\u271d l n m x\u271d : \u2115\n\u22a2 x\u271d = m \u2194 m = x\u271d\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase h.e'_3.h\u2081.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn\u271d m\u271d l n m : \u2115\n\u22a2 m < n \u2194 m \u2208 range n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 x, x \u2208 \u2205 \u2227 p x) \u2194 False\n[PROOFSTEP]\nsimp only [not_mem_empty, false_and_iff, exists_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2203 x, x \u2208 insert a s \u2227 p x) \u2194 p a \u2228 \u2203 x, x \u2208 s \u2227 p x\n[PROOFSTEP]\nsimp only [mem_insert, or_and_right, exists_or, exists_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Finset \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : \u03b1), x \u2208 insert a s \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n[PROOFSTEP]\nsimp only [mem_insert, or_imp, forall_and, forall_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nk j : \u2115\n\u22a2 \u00acj + k \u2208 range k\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nk : \u2115\nj : { n // \u00acn \u2208 range k }\n\u22a2 (fun j => { val := j + k, property := (_ : \u00acj + k \u2208 range k) }) ((fun i => \u2191i - k) j) = j\n[PROOFSTEP]\nrw [Subtype.ext_iff_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nk : \u2115\nj : { n // \u00acn \u2208 range k }\n\u22a2 \u2191((fun j => { val := j + k, property := (_ : \u00acj + k \u2208 range k) }) ((fun i => \u2191i - k) j)) = \u2191j\n[PROOFSTEP]\napply tsub_add_cancel_of_le\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nk : \u2115\nj : { n // \u00acn \u2208 range k }\n\u22a2 k \u2264 \u2191j\n[PROOFSTEP]\nsimpa using j.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nk j : \u2115\n\u22a2 \u00acj + k \u2208 range k\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 l l' : Multiset \u03b1\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 l = l'\n[PROOFSTEP]\nsimpa [\u2190 toFinset_eq hl, \u2190 toFinset_eq hl'] using 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\na : \u03b1\n\u22a2 toFinset {a} = {a}\n[PROOFSTEP]\nrw [\u2190 cons_zero, toFinset_cons, toFinset_zero, IsLawfulSingleton.insert_emptyc_eq]\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 s t : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 toFinset (s + t) \u2194 a \u2208 toFinset s \u222a toFinset 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 s : Multiset \u03b1\nh : 0 \u2260 0\n\u22a2 toFinset (0 \u2022 s) = toFinset s\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\u271d t s : Multiset \u03b1\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 toFinset ((n + 1) \u2022 s) = toFinset s\n[PROOFSTEP]\nby_cases h : n = 0\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 s : Multiset \u03b1\nn : \u2115\nx\u271d : n + 1 \u2260 0\nh : n = 0\n\u22a2 toFinset ((n + 1) \u2022 s) = toFinset s\n[PROOFSTEP]\nrw [h, zero_add, one_nsmul]\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 s : Multiset \u03b1\nn : \u2115\nx\u271d : n + 1 \u2260 0\nh : \u00acn = 0\n\u22a2 toFinset ((n + 1) \u2022 s) = toFinset s\n[PROOFSTEP]\nrw [add_nsmul, toFinset_add, one_nsmul, toFinset_nsmul s n h, Finset.union_idempotent]\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 s t : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 toFinset (s \u2229 t) \u2194 a \u2208 toFinset s \u2229 toFinset 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 s t : Multiset \u03b1\n\u22a2 toFinset (s \u222a t) = toFinset s \u222a toFinset 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\u271d t\u271d s t : Multiset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 toFinset (s \u222a t) \u2194 a\u271d \u2208 toFinset s \u222a toFinset 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\n\u22a2 Finset.Nonempty (toFinset s) \u2194 s \u2260 0\n[PROOFSTEP]\nsimp only [toFinset_eq_empty, Ne.def, Finset.nonempty_iff_ne_empty]\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 toFinset s \u2286 toFinset t \u2194 s \u2286 t\n[PROOFSTEP]\nsimp only [Finset.subset_iff, Multiset.subset_iff, Multiset.mem_toFinset]\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 toFinset s \u2282 toFinset t \u2194 s \u2282 t\n[PROOFSTEP]\nsimp_rw [Finset.ssubset_def, toFinset_subset]\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 \u2286 t \u2227 \u00act \u2286 s \u2194 s \u2282 t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t m : Multiset \u03b1\n\u22a2 toFinset (dedup m) = toFinset m\n[PROOFSTEP]\nsimp_rw [toFinset, dedup_idempotent]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Multiset \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\n\u22a2 toFinset (bind (dedup m) f) = toFinset (bind m f)\n[PROOFSTEP]\nsimp_rw [toFinset, dedup_bind_dedup]\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 IsWellFounded (Multiset \u03b2) fun x x_1 => x \u2282 x_1\n[PROOFSTEP]\nclassical exact Subrelation.isWellFounded (InvImage _ toFinset) toFinset_ssubset.2\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 IsWellFounded (Multiset \u03b2) fun x x_1 => x \u2282 x_1\n[PROOFSTEP]\nexact Subrelation.isWellFounded (InvImage _ toFinset) toFinset_ssubset.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 toFinset s.val = s\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 : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 toFinset s.val \u2194 a\u271d \u2208 s\n[PROOFSTEP]\nrw [Multiset.mem_toFinset, \u2190 mem_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nn : Nodup l\n\u22a2 Multiset.Nodup \u2191l\n[PROOFSTEP]\nrwa [Multiset.coe_nodup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 (toFinset (a :: l)).val = (insert a (toFinset l)).val\n[PROOFSTEP]\nby_cases h : a \u2208 l\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nh : a \u2208 l\n\u22a2 (toFinset (a :: l)).val = (insert a (toFinset l)).val\n[PROOFSTEP]\nsimp [Finset.insert_val', Multiset.dedup_cons, h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nh : \u00aca \u2208 l\n\u22a2 (toFinset (a :: l)).val = (insert a (toFinset l)).val\n[PROOFSTEP]\nsimp [Finset.insert_val', Multiset.dedup_cons, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 Set.SurjOn toFinset {l | Nodup l} Set.univ\n[PROOFSTEP]\nrintro \u27e8\u27e8l\u27e9, hl\u27e9 _\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l' : List \u03b1\na : \u03b1\nval\u271d : Multiset \u03b1\nl : List \u03b1\nhl : Multiset.Nodup (Quot.mk Setoid.r l)\na\u271d : { val := Quot.mk Setoid.r l, nodup := hl } \u2208 Set.univ\n\u22a2 { val := Quot.mk Setoid.r l, nodup := hl } \u2208 toFinset '' {l | Nodup l}\n[PROOFSTEP]\nexact \u27e8l, hl, (toFinset_eq hl).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 toFinset l = toFinset l' \u2194 dedup l ~ dedup l'\n[PROOFSTEP]\nsimp [Finset.ext_iff, perm_ext (nodup_dedup _) (nodup_dedup _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na\u271d : \u03b1\na b : List \u03b1\n\u22a2 toFinset a = toFinset b \u2194 \u2200 (x : \u03b1), x \u2208 a \u2194 x \u2208 b\n[PROOFSTEP]\nsimp only [Finset.ext_iff, mem_toFinset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 l ~ l'\n[PROOFSTEP]\nrw [\u2190 Multiset.coe_eq_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 \u2191l = \u2191l'\n[PROOFSTEP]\nexact Multiset.Nodup.toFinset_inj hl hl' h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 toFinset (l ++ l') = toFinset l \u222a toFinset l'\n[PROOFSTEP]\ninduction' l with hd tl hl\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 toFinset ([] ++ l') = toFinset [] \u222a toFinset l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na hd : \u03b1\ntl : List \u03b1\nhl : toFinset (tl ++ l') = toFinset tl \u222a toFinset l'\n\u22a2 toFinset (hd :: tl ++ l') = toFinset (hd :: tl) \u222a toFinset l'\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\n\u22a2 toFinset (replicate n a) = {a}\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\nx : \u03b1\n\u22a2 x \u2208 toFinset (replicate n a) \u2194 x \u2208 {a}\n[PROOFSTEP]\nsimp [hn, List.mem_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l'\u271d : List \u03b1\na : \u03b1\nl l' : List \u03b1\n\u22a2 toFinset (l \u222a l') = toFinset l \u222a toFinset l'\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\nl\u271d l'\u271d : List \u03b1\na : \u03b1\nl l' : List \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 toFinset (l \u222a l') \u2194 a\u271d \u2208 toFinset l \u222a toFinset l'\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l'\u271d : List \u03b1\na : \u03b1\nl l' : List \u03b1\n\u22a2 toFinset (l \u2229 l') = toFinset l \u2229 toFinset l'\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\nl\u271d l'\u271d : List \u03b1\na : \u03b1\nl l' : List \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 toFinset (l \u2229 l') \u2194 a\u271d \u2208 toFinset l \u2229 toFinset l'\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l' : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 toFinset l = \u2205 \u2194 l = []\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na : \u03b1\n\u22a2 toFinset [] = \u2205 \u2194 [] = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\na head\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 toFinset (head\u271d :: tail\u271d) = \u2205 \u2194 head\u271d :: tail\u271d = []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l' : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 Finset.Nonempty (toFinset l) \u2194 l \u2260 []\n[PROOFSTEP]\nsimp [Finset.nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\n\u22a2 List.Nodup (toList s)\n[PROOFSTEP]\nrw [toList, \u2190 Multiset.coe_nodup, Multiset.coe_toList]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\n\u22a2 Nodup s.val\n[PROOFSTEP]\nexact s.nodup\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 List.toFinset (toList s) = s\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 : Finset \u03b1\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 List.toFinset (toList s) \u2194 a\u271d \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Finset \u03b1\n\u22a2 toList s = [a] \u2194 s = {a}\n[PROOFSTEP]\nrw [toList, Multiset.toList_eq_singleton_iff, val_eq_singleton_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 List.Nodup (a :: toList s)\n[PROOFSTEP]\nsimp [h, nodup_toList s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\nx : \u03b1\n\u22a2 x \u2208 toList (cons a s h) \u2194 x \u2208 a :: toList s\n[PROOFSTEP]\nsimp only [List.mem_cons, Finset.mem_toList, Finset.mem_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 t : \u03b1 \u2192 Finset \u03b2\n\u22a2 Set.PairwiseDisjoint (\u2191\u2205) t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nb : \u03b2\nh : Set.PairwiseDisjoint (\u2191s) t\n\u22a2 b \u2208 disjiUnion s t h \u2194 \u2203 a, a \u2208 s \u2227 b \u2208 t a\n[PROOFSTEP]\nsimp only [mem_def, disjiUnion_val, mem_bind, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nh : Set.PairwiseDisjoint (\u2191s) t\n\u22a2 \u2191(disjiUnion s t h) = \u22c3 (x : \u03b1) (_ : x \u2208 \u2191s), \u2191(t x)\n[PROOFSTEP]\nsimp [Set.ext_iff, mem_disjiUnion, Set.mem_iUnion, iff_self_iff, mem_coe, imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\nh1 : Set.PairwiseDisjoint (\u2191s) f\nh2 : Set.PairwiseDisjoint (\u2191(disjiUnion s f h1)) g\na : { x // x \u2208 s }\nx\u271d\u00b9 : a \u2208 \u2191(attach s)\nb : { x // x \u2208 s }\nx\u271d : b \u2208 \u2191(attach s)\nhab : a \u2260 b\nx : \u03b3\nhxa :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n a\nhxb :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n b\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8xa, hfa, hga\u27e9 := mem_disjiUnion.mp hxa\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\nh1 : Set.PairwiseDisjoint (\u2191s) f\nh2 : Set.PairwiseDisjoint (\u2191(disjiUnion s f h1)) g\na : { x // x \u2208 s }\nx\u271d\u00b9 : a \u2208 \u2191(attach s)\nb : { x // x \u2208 s }\nx\u271d : b \u2208 \u2191(attach s)\nhab : a \u2260 b\nx : \u03b3\nhxa :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n a\nhxb :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n b\nxa : \u03b2\nhfa : xa \u2208 f \u2191a\nhga : x \u2208 g xa\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8xb, hfb, hgb\u27e9 := mem_disjiUnion.mp hxb\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\nh1 : Set.PairwiseDisjoint (\u2191s) f\nh2 : Set.PairwiseDisjoint (\u2191(disjiUnion s f h1)) g\na : { x // x \u2208 s }\nx\u271d\u00b9 : a \u2208 \u2191(attach s)\nb : { x // x \u2208 s }\nx\u271d : b \u2208 \u2191(attach s)\nhab : a \u2260 b\nx : \u03b3\nhxa :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n a\nhxb :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n b\nxa : \u03b2\nhfa : xa \u2208 f \u2191a\nhga : x \u2208 g xa\nxb : \u03b2\nhfb : xb \u2208 f \u2191b\nhgb : x \u2208 g xb\n\u22a2 False\n[PROOFSTEP]\nrefine' disjoint_left.mp (h2 (mem_disjiUnion.mpr \u27e8_, a.prop, hfa\u27e9) (mem_disjiUnion.mpr \u27e8_, b.prop, hfb\u27e9) _) hga hgb\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\nh1 : Set.PairwiseDisjoint (\u2191s) f\nh2 : Set.PairwiseDisjoint (\u2191(disjiUnion s f h1)) g\na : { x // x \u2208 s }\nx\u271d\u00b9 : a \u2208 \u2191(attach s)\nb : { x // x \u2208 s }\nx\u271d : b \u2208 \u2191(attach s)\nhab : a \u2260 b\nx : \u03b3\nhxa :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n a\nhxb :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n b\nxa : \u03b2\nhfa : xa \u2208 f \u2191a\nhga : x \u2208 g xa\nxb : \u03b2\nhfb : xb \u2208 f \u2191b\nhgb : x \u2208 g xb\n\u22a2 xa \u2260 xb\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\nh1 : Set.PairwiseDisjoint (\u2191s) f\nh2 : Set.PairwiseDisjoint (\u2191(disjiUnion s f h1)) g\na : { x // x \u2208 s }\nx\u271d\u00b9 : a \u2208 \u2191(attach s)\nb : { x // x \u2208 s }\nx\u271d : b \u2208 \u2191(attach s)\nhab : a \u2260 b\nx : \u03b3\nhxa :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n a\nhxb :\n x \u2208\n (fun a =>\n disjiUnion (f \u2191a) g (_ : \u2200 (b : \u03b2), b \u2208 \u2191(f \u2191a) \u2192 \u2200 (c : \u03b2), c \u2208 \u2191(f \u2191a) \u2192 b \u2260 c \u2192 (_root_.Disjoint on g) b c))\n b\nxa : \u03b2\nhfa : xa \u2208 f \u2191a\nhga : x \u2208 g xa\nhfb : xa \u2208 f \u2191b\nhgb : x \u2208 g xa\n\u22a2 False\n[PROOFSTEP]\nexact disjoint_left.mp (h1 a.prop b.prop <| Subtype.coe_injective.ne hab) hfa hfb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t\nx' : \u03b2\nhx : x' \u2208 \u2191t\ny' : \u03b2\nhy : y' \u2208 \u2191t\nhne : x' \u2260 y'\n\u22a2 (_root_.Disjoint on fun a => filter (fun c => f c = a) s) x' y'\n[PROOFSTEP]\nsimp_rw [disjoint_left, mem_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t\nx' : \u03b2\nhx : x' \u2208 \u2191t\ny' : \u03b2\nhy : y' \u2208 \u2191t\nhne : x' \u2260 y'\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2227 f a = x' \u2192 \u00ac(a \u2208 s \u2227 f a = y')\n[PROOFSTEP]\nrintro i \u27e8_, rfl\u27e9 \u27e8_, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t\ni : \u03b1\nleft\u271d\u00b9 : i \u2208 s\nhx : f i \u2208 \u2191t\nleft\u271d : i \u2208 s\nhy : f i \u2208 \u2191t\nhne : f i \u2260 f i\n\u22a2 False\n[PROOFSTEP]\nexact hne rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t\nb : \u03b1\n\u22a2 b \u2208\n disjiUnion t (fun a => filter (fun c => f c = a) s)\n (_ :\n \u2200 (x' : \u03b2),\n x' \u2208 \u2191t \u2192 \u2200 (y' : \u03b2), y' \u2208 \u2191t \u2192 x' \u2260 y' \u2192 (_root_.Disjoint on fun a => filter (fun c => f c = a) s) x' y') \u2194\n b \u2208 s\n[PROOFSTEP]\nsimpa using h b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nb : \u03b2\n\u22a2 b \u2208 Finset.biUnion s t \u2194 \u2203 a, a \u2208 s \u2227 b \u2208 t a\n[PROOFSTEP]\nsimp only [mem_def, biUnion_val, mem_dedup, mem_bind, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\n\u22a2 \u2191(Finset.biUnion s t) = \u22c3 (x : \u03b1) (_ : x \u2208 \u2191s), \u2191(t x)\n[PROOFSTEP]\nsimp [Set.ext_iff, mem_biUnion, Set.mem_iUnion, iff_self_iff, mem_coe, imp_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : \u03b2\n\u22a2 x \u2208 Finset.biUnion (insert a s) t \u2194 x \u2208 t a \u222a Finset.biUnion s t\n[PROOFSTEP]\nsimp only [mem_biUnion, exists_prop, mem_union, mem_insert, or_and_right, exists_or, exists_eq_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nhs : s\u2081 = s\u2082\nht : \u2200 (a : \u03b1), a \u2208 s\u2081 \u2192 t\u2081 a = t\u2082 a\nx : \u03b2\n\u22a2 x \u2208 Finset.biUnion s\u2081 t\u2081 \u2194 x \u2208 Finset.biUnion s\u2082 t\u2082\n[PROOFSTEP]\nsimp_rw [mem_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nhs : s\u2081 = s\u2082\nht : \u2200 (a : \u03b1), a \u2208 s\u2081 \u2192 t\u2081 a = t\u2082 a\nx : \u03b2\n\u22a2 (\u2203 a, a \u2208 s\u2081 \u2227 x \u2208 t\u2081 a) \u2194 \u2203 a, a \u2208 s\u2082 \u2227 x \u2208 t\u2082 a\n[PROOFSTEP]\napply exists_congr\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nhs : s\u2081 = s\u2082\nht : \u2200 (a : \u03b1), a \u2208 s\u2081 \u2192 t\u2081 a = t\u2082 a\nx : \u03b2\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u2227 x \u2208 t\u2081 a \u2194 a \u2208 s\u2082 \u2227 x \u2208 t\u2082 a\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [hs, and_congr_right_iff, ht, implies_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns' : Finset \u03b2\n\u22a2 Finset.biUnion s t \u2286 s' \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 t x \u2286 s'\n[PROOFSTEP]\nsimp only [subset_iff, mem_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns' : Finset \u03b2\n\u22a2 (\u2200 \u2983x : \u03b2\u2984, (\u2203 a, a \u2208 s \u2227 x \u2208 t a) \u2192 x \u2208 s') \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 \u2983x_1 : \u03b2\u2984, x_1 \u2208 t x \u2192 x_1 \u2208 s'\n[PROOFSTEP]\nexact \u27e8fun H a ha b hb => H \u27e8a, ha, hb\u27e9, fun H b \u27e8a, ha, hb\u27e9 => H a ha hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\na : \u03b1\n\u22a2 Finset.biUnion {a} t = t a\n[PROOFSTEP]\nclassical rw [\u2190 insert_emptyc_eq, biUnion_insert, biUnion_empty, union_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\na : \u03b1\n\u22a2 Finset.biUnion {a} t = t a\n[PROOFSTEP]\nrw [\u2190 insert_emptyc_eq, biUnion_insert, biUnion_empty, union_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\n\u22a2 Finset.biUnion s f \u2229 t = Finset.biUnion s fun x => f x \u2229 t\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\nx : \u03b2\n\u22a2 x \u2208 Finset.biUnion s f \u2229 t \u2194 x \u2208 Finset.biUnion s fun x => f x \u2229 t\n[PROOFSTEP]\nsimp only [mem_biUnion, mem_inter]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\nx : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 x \u2208 f a) \u2227 x \u2208 t \u2194 \u2203 a, a \u2208 s \u2227 x \u2208 f a \u2227 x \u2208 t\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\n\u22a2 t \u2229 Finset.biUnion s f = Finset.biUnion s fun x => t \u2229 f x\n[PROOFSTEP]\nrw [inter_comm, biUnion_inter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\n\u22a2 (Finset.biUnion s fun x => f x \u2229 t) = Finset.biUnion s fun x => t \u2229 f x\n[PROOFSTEP]\nsimp [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\n\u22a2 Finset.biUnion (Finset.biUnion s f) g = Finset.biUnion s fun a => Finset.biUnion (f a) g\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Finset.biUnion (Finset.biUnion s f) g \u2194 a\u271d \u2208 Finset.biUnion s fun a => Finset.biUnion (f a) g\n[PROOFSTEP]\nsimp only [Finset.mem_biUnion, exists_prop]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\na\u271d : \u03b3\n\u22a2 (\u2203 a, (\u2203 a_1, a_1 \u2208 s \u2227 a \u2208 f a_1) \u2227 a\u271d \u2208 g a) \u2194 \u2203 a, a \u2208 s \u2227 \u2203 a_1, a_1 \u2208 f a \u2227 a\u271d \u2208 g a_1\n[PROOFSTEP]\nsimp_rw [\u2190 exists_and_right, \u2190 exists_and_left, and_assoc]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b3\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\ng : \u03b2 \u2192 Finset \u03b3\na\u271d : \u03b3\n\u22a2 (\u2203 a x, x \u2208 s \u2227 a \u2208 f x \u2227 a\u271d \u2208 g a) \u2194 \u2203 a x, a \u2208 s \u2227 x \u2208 f a \u2227 a\u271d \u2208 g x\n[PROOFSTEP]\nrw [exists_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nt : \u03b1 \u2192 Multiset \u03b2\nx : \u03b2\n\u22a2 x \u2208 toFinset (Multiset.bind s t) \u2194 x \u2208 Finset.biUnion (toFinset s) fun a => toFinset (t a)\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, mem_biUnion, Multiset.mem_bind, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 t\u2081 a \u2286 t\u2082 a\n\u22a2 Finset.biUnion s t\u2081 \u2286 Finset.biUnion s t\u2082\n[PROOFSTEP]\nhave : \u2200 b a, a \u2208 s \u2192 b \u2208 t\u2081 a \u2192 \u2203 a : \u03b1, a \u2208 s \u2227 b \u2208 t\u2082 a := fun b a ha hb => \u27e8a, ha, Finset.mem_of_subset (h a ha) hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 t\u2081 a \u2286 t\u2082 a\nthis : \u2200 (b : \u03b2) (a : \u03b1), a \u2208 s \u2192 b \u2208 t\u2081 a \u2192 \u2203 a, a \u2208 s \u2227 b \u2208 t\u2082 a\n\u22a2 Finset.biUnion s t\u2081 \u2286 Finset.biUnion s t\u2082\n[PROOFSTEP]\nsimpa only [subset_iff, mem_biUnion, exists_imp, and_imp, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 t : \u03b1 \u2192 Finset \u03b2\nh : s\u2081 \u2286 s\u2082\n\u22a2 Finset.biUnion s\u2081 t \u2286 Finset.biUnion s\u2082 t\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 t : \u03b1 \u2192 Finset \u03b2\nh : s\u2081 \u2286 s\u2082\nx : \u03b2\n\u22a2 x \u2208 Finset.biUnion s\u2081 t \u2192 x \u2208 Finset.biUnion s\u2082 t\n[PROOFSTEP]\nsimp only [and_imp, mem_biUnion, exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 t : \u03b1 \u2192 Finset \u03b2\nh : s\u2081 \u2286 s\u2082\nx : \u03b2\n\u22a2 (\u2203 a, a \u2208 s\u2081 \u2227 x \u2208 t a) \u2192 \u2203 a, a \u2208 s\u2082 \u2227 x \u2208 t a\n[PROOFSTEP]\nexact Exists.imp fun a ha => \u27e8h ha.1, ha.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 x \u2208 Finset.biUnion s singleton \u2194 x \u2208 s\n[PROOFSTEP]\nsimp only [mem_biUnion, mem_singleton, exists_prop, exists_eq_right']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 filter p (Finset.biUnion s f) = Finset.biUnion s fun a => filter p (f a)\n[PROOFSTEP]\next b\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\n\u22a2 b \u2208 filter p (Finset.biUnion s f) \u2194 b \u2208 Finset.biUnion s fun a => filter p (f a)\n[PROOFSTEP]\nsimp only [mem_biUnion, exists_prop, mem_filter]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 b \u2208 f a) \u2227 p b \u2194 \u2203 a, a \u2208 s \u2227 b \u2208 f a \u2227 p b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 b \u2208 f a) \u2227 p b \u2192 \u2203 a, a \u2208 s \u2227 b \u2208 f a \u2227 p b\n[PROOFSTEP]\nrintro \u27e8\u27e8a, ha, hba\u27e9, hb\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\nhb : p b\na : \u03b1\nha : a \u2208 s\nhba : b \u2208 f a\n\u22a2 \u2203 a, a \u2208 s \u2227 b \u2208 f a \u2227 p b\n[PROOFSTEP]\nexact \u27e8a, ha, hba, hb\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 b \u2208 f a \u2227 p b) \u2192 (\u2203 a, a \u2208 s \u2227 b \u2208 f a) \u2227 p b\n[PROOFSTEP]\nrintro \u27e8a, ha, hba, hb\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nb : \u03b2\na : \u03b1\nha : a \u2208 s\nhba : b \u2208 f a\nhb : p b\n\u22a2 (\u2203 a, a \u2208 s \u2227 b \u2208 f a) \u2227 p b\n[PROOFSTEP]\nexact \u27e8\u27e8a, ha, hba\u27e9, hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 t\n\u22a2 (Finset.biUnion t fun a => filter (fun c => f c = a) s) = s\n[PROOFSTEP]\nsimpa only [disjiUnion_eq_biUnion] using disjiUnion_filter_eq_of_maps_to h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 f : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nb : \u03b2\n\u22a2 erase (Finset.biUnion s f) b = Finset.biUnion s fun x => erase (f x) b\n[PROOFSTEP]\next a\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 f : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nb a : \u03b2\n\u22a2 a \u2208 erase (Finset.biUnion s f) b \u2194 a \u2208 Finset.biUnion s fun x => erase (f x) b\n[PROOFSTEP]\nsimp [Finset.mem_biUnion, iff_self_iff, exists_and_left, Finset.mem_erase]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 f : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nb a : \u03b2\n\u22a2 (\u00aca = b \u2227 \u2203 a_1, a_1 \u2208 s \u2227 a \u2208 f a_1) \u2194 \u2203 a_1, a_1 \u2208 s \u2227 \u00aca = b \u2227 a \u2208 f a_1\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\n\u22a2 Finset.Nonempty (Finset.biUnion s t) \u2194 \u2203 x, x \u2208 s \u2227 Finset.Nonempty (t x)\n[PROOFSTEP]\nsimp only [Finset.Nonempty, mem_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\n\u22a2 (\u2203 x a, a \u2208 s \u2227 x \u2208 t a) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 x_1, x_1 \u2208 t x\n[PROOFSTEP]\nrw [exists_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns s\u2081 s\u2082 : Finset \u03b1\nt t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\n\u22a2 (\u2203 y x, y \u2208 s \u2227 x \u2208 t y) \u2194 \u2203 x, x \u2208 s \u2227 \u2203 x_1, x_1 \u2208 t x\n[PROOFSTEP]\nsimp [exists_and_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\n\u22a2 _root_.Disjoint (Finset.biUnion s f) t \u2194 \u2200 (i : \u03b1), i \u2208 s \u2192 _root_.Disjoint (f i) t\n[PROOFSTEP]\nclassical\nrefine' s.induction _ _\n\u00b7 simp only [forall_mem_empty_iff, biUnion_empty, disjoint_empty_left]\n\u00b7 intro i s his ih\n simp only [disjoint_union_left, biUnion_insert, his, forall_mem_insert, ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\n\u22a2 _root_.Disjoint (Finset.biUnion s f) t \u2194 \u2200 (i : \u03b1), i \u2208 s \u2192 _root_.Disjoint (f i) t\n[PROOFSTEP]\nrefine' s.induction _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\n\u22a2 _root_.Disjoint (Finset.biUnion \u2205 f) t \u2194 \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 _root_.Disjoint (f i) t\n[PROOFSTEP]\nsimp only [forall_mem_empty_iff, biUnion_empty, disjoint_empty_left]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\n\u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Finset \u03b1},\n \u00aca \u2208 s \u2192\n (_root_.Disjoint (Finset.biUnion s f) t \u2194 \u2200 (i : \u03b1), i \u2208 s \u2192 _root_.Disjoint (f i) t) \u2192\n (_root_.Disjoint (Finset.biUnion (insert a s) f) t \u2194 \u2200 (i : \u03b1), i \u2208 insert a s \u2192 _root_.Disjoint (f i) t)\n[PROOFSTEP]\nintro i s his ih\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\nt : Finset \u03b2\ni : \u03b1\ns : Finset \u03b1\nhis : \u00aci \u2208 s\nih : _root_.Disjoint (Finset.biUnion s f) t \u2194 \u2200 (i : \u03b1), i \u2208 s \u2192 _root_.Disjoint (f i) t\n\u22a2 _root_.Disjoint (Finset.biUnion (insert i s) f) t \u2194 \u2200 (i_1 : \u03b1), i_1 \u2208 insert i s \u2192 _root_.Disjoint (f i_1) t\n[PROOFSTEP]\nsimp only [disjoint_union_left, biUnion_insert, his, forall_mem_insert, ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : \u03b1 \u2192 Finset \u03b2\ns : Finset \u03b2\nt : Finset \u03b1\nf : \u03b1 \u2192 Finset \u03b2\n\u22a2 _root_.Disjoint s (Finset.biUnion t f) \u2194 \u2200 (i : \u03b1), i \u2208 t \u2192 _root_.Disjoint s (f i)\n[PROOFSTEP]\nsimpa only [_root_.disjoint_comm] using disjoint_biUnion_left t f s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\n\u22a2 _root_.Pairwise (r on fun a_1 => f \u2191a_1) \u2194\n _root_.Pairwise (r on fun a => f \u2191a) \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 r (f a) (f b) \u2227 r (f b) (f a)\n[PROOFSTEP]\nsimp only [pairwise_subtype_iff_pairwise_finset', Finset.coe_cons, Set.pairwise_insert, Finset.mem_coe,\n and_congr_right_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\n\u22a2 Set.Pairwise (\u2191s) (r on f) \u2192\n ((\u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 (r on f) a b \u2227 (r on f) b a) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 r (f a) (f b) \u2227 r (f b) (f a))\n[PROOFSTEP]\nexact fun _ =>\n \u27e8fun h b hb =>\n h b hb <| by\n rintro rfl\n contradiction,\n fun h b hb _ => h b hb\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nx\u271d : Set.Pairwise (\u2191s) (r on f)\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 (r on f) a b \u2227 (r on f) b a\nb : \u03b1\nhb : b \u2208 s\n\u22a2 a \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : Finset \u03b1\na : \u03b1\nha : \u00aca \u2208 s\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nx\u271d : Set.Pairwise (\u2191s) (r on f)\nh : \u2200 (b : \u03b1), b \u2208 s \u2192 a \u2260 b \u2192 (r on f) a b \u2227 (r on f) b a\nhb : a \u2208 s\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 Option.elim' default val ((fun x => if h : x = default then none else some { val := x, property := h }) x) = x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\n\u22a2 Option.elim' default val (if h : x = default then none else some { val := x, property := h }) = x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh\u271d : x = default\n\u22a2 Option.elim' default val none = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh\u271d : \u00acx = default\n\u22a2 Option.elim' default val (some { val := x, property := h\u271d }) = x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 Function.RightInverse (Option.elim' default val) fun x =>\n if h : x = default then none else some { val := x, property := h }\n[PROOFSTEP]\nrintro (_ | \u27e8x, h\u27e9)\n[GOAL]\ncase none\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (fun x => if h : x = default then none else some { val := x, property := h }) (Option.elim' default val none) = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2260 default\n\u22a2 (fun x => if h : x = default then none else some { val := x, property := h })\n (Option.elim' default val (some { val := x, property := h })) =\n some { val := x, property := h }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase some.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2260 default\n\u22a2 (if h : Option.elim' default val (some { val := x, property := h }) = default then none\n else some { val := Option.elim' default val (some { val := x, property := h }), property := h }) =\n some { val := x, property := h }\n[PROOFSTEP]\nsplit_ifs with hi\n[GOAL]\ncase pos\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2260 default\nhi : Option.elim' default val (some { val := x, property := h }) = default\n\u22a2 False\n[PROOFSTEP]\nsimp [h] at hi \n[GOAL]\ncase neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2260 default\nhi : \u00acOption.elim' default val (some { val := x, property := h }) = default\n\u22a2 some { val := Option.elim' default val (some { val := x, property := h }), property := hi } =\n some { val := x, property := h }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\n\u22a2 _root_.Disjoint (toFinset m1) (toFinset m2) \u2194 Disjoint m1 m2\n[PROOFSTEP]\nrw [Finset.disjoint_iff_ne]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\n\u22a2 (\u2200 (a : \u03b1), a \u2208 toFinset m1 \u2192 \u2200 (b : \u03b1), b \u2208 toFinset m2 \u2192 a \u2260 b) \u2194 Disjoint m1 m2\n[PROOFSTEP]\nrefine' \u27e8fun h a ha1 ha2 => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\nh : \u2200 (a : \u03b1), a \u2208 toFinset m1 \u2192 \u2200 (b : \u03b1), b \u2208 toFinset m2 \u2192 a \u2260 b\na : \u03b1\nha1 : a \u2208 m1\nha2 : a \u2208 m2\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Multiset.mem_toFinset] at ha1 ha2 \n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\nh : \u2200 (a : \u03b1), a \u2208 toFinset m1 \u2192 \u2200 (b : \u03b1), b \u2208 toFinset m2 \u2192 a \u2260 b\na : \u03b1\nha1\u271d : a \u2208 m1\nha1 : a \u2208 toFinset m1\nha2\u271d : a \u2208 m2\nha2 : a \u2208 toFinset m2\n\u22a2 False\n[PROOFSTEP]\nexact h _ ha1 _ ha2 rfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\n\u22a2 Disjoint m1 m2 \u2192 \u2200 (a : \u03b1), a \u2208 toFinset m1 \u2192 \u2200 (b : \u03b1), b \u2208 toFinset m2 \u2192 a \u2260 b\n[PROOFSTEP]\nrintro h a ha b hb rfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\nh : Disjoint m1 m2\na : \u03b1\nha : a \u2208 toFinset m1\nhb : a \u2208 toFinset m2\n\u22a2 False\n[PROOFSTEP]\nrw [Multiset.mem_toFinset] at ha hb \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nm1 m2 : Multiset \u03b1\nh : Disjoint m1 m2\na : \u03b1\nha : a \u2208 m1\nhb : a \u2208 m2\n\u22a2 False\n[PROOFSTEP]\nexact h ha hb\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Basic", "llama_tokens": 72804, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5234203638047913, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.2535343999320546}} {"text": "[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nh : sel s = none\n\u22a2 enumerate sel s 0 = none\n[PROOFSTEP]\nsimp [h, enumerate]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nh : sel s = none\nn : \u2115\n\u22a2 enumerate sel s (n + 1) = none\n[PROOFSTEP]\nsimp [h, enumerate]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nh : sel s = none\nn : \u2115\n\u22a2 (do\n let a \u2190 none\n enumerate sel (s \\ {a}) n) =\n none\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\n\u22a2 enumerate sel s m = none\n[PROOFSTEP]\ncases hs : sel s\n[GOAL]\ncase none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nhs : sel s = none\n\u22a2 enumerate sel s m = none\n[PROOFSTEP]\nexact enumerate_eq_none_of_sel sel hs\n[GOAL]\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nval\u271d : \u03b1\nhs : sel s = some val\u271d\n\u22a2 enumerate sel s m = none\n[PROOFSTEP]\ncases m\n[GOAL]\ncase some.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nhm : n + 1 \u2264 Nat.zero\n\u22a2 enumerate sel s Nat.zero = none\ncase some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nn\u271d : \u2115\nhm : n + 1 \u2264 Nat.succ n\u271d\n\u22a2 enumerate sel s (Nat.succ n\u271d) = none\n[PROOFSTEP]\ncase zero => contradiction\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nhm : n + 1 \u2264 Nat.zero\n\u22a2 enumerate sel s Nat.zero = none\n[PROOFSTEP]\ncase zero => contradiction\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nhm : n + 1 \u2264 Nat.zero\n\u22a2 enumerate sel s Nat.zero = none\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nn\u271d : \u2115\nhm : n + 1 \u2264 Nat.succ n\u271d\n\u22a2 enumerate sel s (Nat.succ n\u271d) = none\n[PROOFSTEP]\ncase succ m' =>\n simp [hs, enumerate] at h \u22a2\n have hm : n \u2264 m' := Nat.le_of_succ_le_succ hm\n exact enumerate_eq_none h hm\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 Nat.succ m'\n\u22a2 enumerate sel s (Nat.succ m') = none\n[PROOFSTEP]\ncase succ m' =>\n simp [hs, enumerate] at h \u22a2\n have hm : n \u2264 m' := Nat.le_of_succ_le_succ hm\n exact enumerate_eq_none h hm\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 Nat.succ m'\n\u22a2 enumerate sel s (Nat.succ m') = none\n[PROOFSTEP]\nsimp [hs, enumerate] at h \u22a2\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 Nat.succ m'\nh :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) n) =\n none\n\u22a2 (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) m') =\n none\n[PROOFSTEP]\nhave hm : n \u2264 m' := Nat.le_of_succ_le_succ hm\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm\u271d : n + 1 \u2264 Nat.succ m'\nh :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) n) =\n none\nhm : n \u2264 m'\n\u22a2 (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) m') =\n none\n[PROOFSTEP]\nexact enumerate_eq_none h hm\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\ncases h : sel s\n[GOAL]\ncase none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\nh : sel s = none\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na val\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\ncase none => simp [enumerate_eq_none_of_sel, h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\nh : sel s = none\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\ncase none => simp [enumerate_eq_none_of_sel, h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\nh : sel s = none\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\nsimp [enumerate_eq_none_of_sel, h]\n[GOAL]\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na val\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\ncase some a' =>\n simp [enumerate, h]\n exact fun h' : enumerate sel (s \\ { a' }) n = some a \u21a6\n have : a \u2208 s \\ { a' } := enumerate_mem h_sel h'\n this.left\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\ncase some a' =>\n simp [enumerate, h]\n exact fun h' : enumerate sel (s \\ { a' }) n = some a \u21a6\n have : a \u2208 s \\ { a' } := enumerate_mem h_sel h'\n this.left\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s\n[PROOFSTEP]\nsimp [enumerate, h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 (do\n let a \u2190 some a'\n enumerate sel (s \\ {a}) n) =\n some a \u2192\n a \u2208 s\n[PROOFSTEP]\nexact fun h' : enumerate sel (s \\ { a' }) n = some a \u21a6\n have : a \u2208 s \\ { a' } := enumerate_mem h_sel h'\n this.left\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\nrcases le_total n\u2081 n\u2082 with (hn | hn)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\non_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\non_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\nswap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081\n[PROOFSTEP]\nall_goals\n rcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n clear hn\n induction n\u2081 generalizing s\n case zero =>\n cases m\n case zero => rfl\n case succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n case succ k ih =>\n cases h : sel s\n case none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n case some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\nrcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 = n\u2081 + m\n[PROOFSTEP]\nclear hn\n[GOAL]\ncase inl.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 = n\u2081 + m\n[PROOFSTEP]\ninduction n\u2081 generalizing s\n[GOAL]\ncase inl.intro.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero = Nat.zero + m\ncase inl.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d = n\u271d + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d = Nat.succ n\u271d + m\n[PROOFSTEP]\ncase zero =>\n cases m\n case zero => rfl\n case succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero = Nat.zero + m\n[PROOFSTEP]\ncase zero =>\n cases m\n case zero => rfl\n case succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero = Nat.zero + m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.zero\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.succ n\u271d\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.zero\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.succ n\u271d\n[PROOFSTEP]\ncase succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.succ m\n[PROOFSTEP]\ncase succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 Nat.zero = Nat.zero + Nat.succ m\n[PROOFSTEP]\nhave h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 enumerate sel (s \\ {a}) m = some a\n[PROOFSTEP]\nsimp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n (do\n let a \u2190 some a\n enumerate sel (s \\ {a}) m) =\n some a\n\u22a2 enumerate sel (s \\ {a}) m = some a\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 Nat.zero = Nat.zero + Nat.succ m\n[PROOFSTEP]\nhave : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 Nat.zero = Nat.zero + Nat.succ m\n[PROOFSTEP]\nsimp_all [Set.mem_diff_singleton]\n[GOAL]\ncase inl.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d = n\u271d + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d = Nat.succ n\u271d + m\n[PROOFSTEP]\ncase succ k ih =>\n cases h : sel s\n case none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n case some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncase succ k ih =>\n cases h : sel s\n case none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n case some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncases h : sel s\n[GOAL]\ncase none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k = Nat.succ k + m\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncase none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncase none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\nsimp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncase some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\ncase some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k = k + m\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k = Nat.succ k + m\n[PROOFSTEP]\nsimp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m = 0\nh\u2081 :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) k) =\n some a\nh\u2082 :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) (m + k)) =\n some a\nh : sel s = some val\u271d\n\u22a2 m = 0\n[PROOFSTEP]\nexact ih h\u2081 h\u2082\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081\n[PROOFSTEP]\nrcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 + m = n\u2081\n[PROOFSTEP]\nclear hn\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 + m = n\u2081\n[PROOFSTEP]\ninduction n\u2081 generalizing s\n[GOAL]\ncase inr.intro.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero\ncase inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d\n[PROOFSTEP]\ncase zero =>\n cases m\n case zero => rfl\n case succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero\n[PROOFSTEP]\ncase zero =>\n cases m\n case zero => rfl\n case succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + m) = some a\n\u22a2 Nat.zero + m = Nat.zero\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nh\u2082 : enumerate sel s (Nat.zero + Nat.zero) = some a\n\u22a2 Nat.zero + Nat.zero = Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nn\u271d : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ n\u271d) = some a\n\u22a2 Nat.zero + Nat.succ n\u271d = Nat.zero\n[PROOFSTEP]\ncase succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero\n[PROOFSTEP]\ncase succ\n m =>\n have h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n have : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n simp_all [Set.mem_diff_singleton]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero\n[PROOFSTEP]\nhave h' : enumerate sel (s \\ { a }) m = some a := by simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add];\n exact h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\n\u22a2 enumerate sel (s \\ {a}) m = some a\n[PROOFSTEP]\nsimp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n (do\n let a \u2190 some a\n enumerate sel (s \\ {a}) m) =\n some a\n\u22a2 enumerate sel (s \\ {a}) m = some a\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 Nat.zero + Nat.succ m = Nat.zero\n[PROOFSTEP]\nhave : a \u2208 s \\ { a } := enumerate_mem sel h_sel h'\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s Nat.zero = some a\nm : \u2115\nh\u2082 : enumerate sel s (Nat.zero + Nat.succ m) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 Nat.zero + Nat.succ m = Nat.zero\n[PROOFSTEP]\nsimp_all [Set.mem_diff_singleton]\n[GOAL]\ncase inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm n\u271d : \u2115\nn_ih\u271d : \u2200 {s : Set \u03b1}, enumerate sel s n\u271d = some a \u2192 enumerate sel s (n\u271d + m) = some a \u2192 n\u271d + m = n\u271d\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ n\u271d) = some a\nh\u2082 : enumerate sel s (Nat.succ n\u271d + m) = some a\n\u22a2 Nat.succ n\u271d + m = Nat.succ n\u271d\n[PROOFSTEP]\ncase succ k ih =>\n cases h : sel s\n case none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n case some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncase succ k ih =>\n cases h : sel s\n case none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n case some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncases h : sel s\n[GOAL]\ncase none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncase none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncase none => simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nh : sel s = none\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\nsimp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n[GOAL]\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncase some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\ncase some\n _ =>\n simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2,\n Nat.succ.injEq]\n exact ih h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (Nat.succ k) = some a\nh\u2082 : enumerate sel s (Nat.succ k + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 Nat.succ k + m = Nat.succ k\n[PROOFSTEP]\nsimp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq, Nat.add_succ, enumerate._eq_2, Nat.succ.injEq]\n[GOAL]\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k\nh\u2081 :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) k) =\n some a\nh\u2082 :\n (do\n let a \u2190 some val\u271d\n enumerate sel (s \\ {a}) (m + k)) =\n some a\nh : sel s = some val\u271d\n\u22a2 m + k = k\n[PROOFSTEP]\nexact ih h\u2081 h\u2082\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Enumerate", "llama_tokens": 14573, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.43014734858584286, "lm_q1q2_score": 0.2533090982108046}} {"text": "[GOAL]\nP : PartENat \u2192 Prop\n\u22a2 \u2200 (a : PartENat), P \u22a4 \u2192 (\u2200 (n : \u2115), P \u2191n) \u2192 P a\n[PROOFSTEP]\nexact PartENat.casesOn'\n[GOAL]\nx : PartENat\n\u22a2 x + \u22a4 = \u22a4\n[PROOFSTEP]\nrw [add_comm, top_add]\n[GOAL]\nx : PartENat\nh : x.Dom\n\u22a2 \u2191(Part.get x h) = x\n[PROOFSTEP]\nexact Part.ext' (iff_of_true trivial h) fun _ _ => rfl\n[GOAL]\nx : \u2115\nh : (\u2191x).Dom\n\u22a2 Part.get (\u2191x) h = x\n[PROOFSTEP]\nrw [\u2190 natCast_inj, natCast_get]\n[GOAL]\nx : \u2115\ny : PartENat\nh : (\u2191x + y).Dom\n\u22a2 Part.get (\u2191x + y) h = x + Part.get y (_ : y.Dom)\n[PROOFSTEP]\nrfl\n[GOAL]\na : PartENat\nha : a.Dom\nb : \u2115\n\u22a2 Part.get a ha = b \u2194 a = \u2191b\n[PROOFSTEP]\nrw [get_eq_iff_eq_some]\n[GOAL]\na : PartENat\nha : a.Dom\nb : \u2115\n\u22a2 a = \u2191b \u2194 a = \u2191b\n[PROOFSTEP]\nrfl\n[GOAL]\nx : PartENat\ny : \u2115\nh : x \u2264 \u2191y\n\u22a2 x.Dom\n[PROOFSTEP]\nexact dom_of_le_some h\n[GOAL]\nx y : PartENat\ninst\u271d\u00b9 : Decidable x.Dom\ninst\u271d : Decidable y.Dom\nhx : x.Dom\n\u22a2 ?m.35650 x y hx \u2194 x \u2264 y\n[PROOFSTEP]\nrw [le_def]\n[GOAL]\nx y : PartENat\n\u22a2 x < y \u2194 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nrw [lt_iff_le_not_le, le_def, le_def, not_exists]\n[GOAL]\nx y : PartENat\n\u22a2 ((\u2203 h, \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy) \u2227\n \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy) \u2194\n \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx y : PartENat\n\u22a2 ((\u2203 h, \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy) \u2227\n \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy) \u2192\n \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nrintro \u27e8\u27e8hyx, H\u27e9, h\u27e9\n[GOAL]\ncase mp.intro.intro\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nby_cases hx : x.Dom\n[GOAL]\ncase pos\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : x.Dom\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nuse hx\n[GOAL]\ncase h\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : x.Dom\n\u22a2 \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase h\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : x.Dom\nhy : y.Dom\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\nspecialize H hy\n[GOAL]\ncase h\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nhx : x.Dom\nhy : y.Dom\nH : Part.get x (_ : x.Dom) \u2264 Part.get y hy\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\nspecialize h fun _ => hy\n[GOAL]\ncase h\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nhx : x.Dom\nhy : y.Dom\nH : Part.get x (_ : x.Dom) \u2264 Part.get y hy\nh : \u00ac\u2200 (hy_1 : x.Dom), Part.get y hy \u2264 Part.get x hy_1\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\nrw [not_forall] at h \n[GOAL]\ncase h\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nhx : x.Dom\nhy : y.Dom\nH : Part.get x (_ : x.Dom) \u2264 Part.get y hy\nh : \u2203 x_1, \u00acPart.get y hy \u2264 Part.get x x_1\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\ncases' h with hx' h\n[GOAL]\ncase h.intro\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nhx : x.Dom\nhy : y.Dom\nH : Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx' : x.Dom\nh : \u00acPart.get y hy \u2264 Part.get x hx'\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\nrw [not_le] at h \n[GOAL]\ncase h.intro\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nhx : x.Dom\nhy : y.Dom\nH : Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx' : x.Dom\nh : Part.get x hx' < Part.get y hy\n\u22a2 Part.get x hx < Part.get y hy\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\nx y : PartENat\nh : \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : \u00acx.Dom\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nspecialize h fun hx' => (hx hx').elim\n[GOAL]\ncase neg\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : \u00acx.Dom\nh : \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nrw [not_forall] at h \n[GOAL]\ncase neg\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : \u00acx.Dom\nh : \u2203 x_1, \u00acPart.get y (_ : y.Dom) \u2264 Part.get x x_1\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\ncases' h with hx' h\n[GOAL]\ncase neg.intro\nx y : PartENat\nhyx : y.Dom \u2192 x.Dom\nH : \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy\nhx : \u00acx.Dom\nhx' : x.Dom\nh : \u00acPart.get y (_ : y.Dom) \u2264 Part.get x hx'\n\u22a2 \u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n[PROOFSTEP]\nexact (hx hx').elim\n[GOAL]\ncase mpr\nx y : PartENat\n\u22a2 (\u2203 hx, \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy) \u2192\n (\u2203 h, \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy) \u2227\n \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\n[PROOFSTEP]\nrintro \u27e8hx, H\u27e9\n[GOAL]\ncase mpr.intro\nx y : PartENat\nhx : x.Dom\nH : \u2200 (hy : y.Dom), Part.get x hx < Part.get y hy\n\u22a2 (\u2203 h, \u2200 (hy : y.Dom), Part.get x (_ : x.Dom) \u2264 Part.get y hy) \u2227\n \u2200 (x_1 : x.Dom \u2192 y.Dom), \u00ac\u2200 (hy : x.Dom), Part.get y (_ : y.Dom) \u2264 Part.get x hy\n[PROOFSTEP]\nexact \u27e8\u27e8fun _ => hx, fun hy => (H hy).le\u27e9, fun hxy h => not_lt_of_le (h _) (H _)\u27e9\n[GOAL]\nx y : \u2115\n\u22a2 \u2191x \u2264 \u2191y \u2194 x \u2264 y\n[PROOFSTEP]\nexact \u27e8fun \u27e8_, h\u27e9 => h trivial, fun h => \u27e8fun _ => trivial, fun _ => h\u27e9\u27e9\n[GOAL]\nx y : \u2115\n\u22a2 \u2191x < \u2191y \u2194 x < y\n[PROOFSTEP]\nrw [lt_iff_le_not_le, lt_iff_le_not_le, coe_le_coe, coe_le_coe]\n[GOAL]\nx y : PartENat\nhx : x.Dom\nhy : y.Dom\n\u22a2 Part.get x hx \u2264 Part.get y hy \u2194 x \u2264 y\n[PROOFSTEP]\nconv =>\n lhs\n rw [\u2190 coe_le_coe, natCast_get, natCast_get]\n[GOAL]\nx y : PartENat\nhx : x.Dom\nhy : y.Dom\n| Part.get x hx \u2264 Part.get y hy \u2194 x \u2264 y\n[PROOFSTEP]\n lhs\n rw [\u2190 coe_le_coe, natCast_get, natCast_get]\n[GOAL]\nx y : PartENat\nhx : x.Dom\nhy : y.Dom\n| Part.get x hx \u2264 Part.get y hy \u2194 x \u2264 y\n[PROOFSTEP]\n lhs\n rw [\u2190 coe_le_coe, natCast_get, natCast_get]\n[GOAL]\nx y : PartENat\nhx : x.Dom\nhy : y.Dom\n| Part.get x hx \u2264 Part.get y hy \u2194 x \u2264 y\n[PROOFSTEP]\nlhs\n[GOAL]\nx y : PartENat\nhx : x.Dom\nhy : y.Dom\n| Part.get x hx \u2264 Part.get y hy\n[PROOFSTEP]\nrw [\u2190 coe_le_coe, natCast_get, natCast_get]\n[GOAL]\nx : PartENat\nn : \u2115\n\u22a2 x \u2264 \u2191n \u2194 \u2203 h, Part.get x h \u2264 n\n[PROOFSTEP]\nshow (\u2203 h : True \u2192 x.Dom, _) \u2194 \u2203 h : x.Dom, x.get h \u2264 n\n[GOAL]\nx : PartENat\nn : \u2115\n\u22a2 (\u2203 h, \u2200 (hy : (\u2191n).Dom), Part.get x (_ : x.Dom) \u2264 Part.get (\u2191n) hy) \u2194 \u2203 h, Part.get x h \u2264 n\n[PROOFSTEP]\nsimp only [forall_prop_of_true, dom_natCast, get_natCast']\n[GOAL]\nx : PartENat\nn : \u2115\n\u22a2 x < \u2191n \u2194 \u2203 h, Part.get x h < n\n[PROOFSTEP]\nsimp only [lt_def, forall_prop_of_true, get_natCast', dom_natCast]\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 \u2191n \u2264 x \u2194 \u2200 (h : x.Dom), n \u2264 Part.get x h\n[PROOFSTEP]\nrw [\u2190 some_eq_natCast]\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 \u2191n \u2264 x \u2194 \u2200 (h : x.Dom), n \u2264 Part.get x h\n[PROOFSTEP]\nsimp only [le_def, exists_prop_of_true, dom_some, forall_true_iff]\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 (\u2200 (hy : x.Dom), Part.get \u2191n (_ : (\u2191n).Dom) \u2264 Part.get x hy) \u2194 \u2200 (h : x.Dom), n \u2264 Part.get x h\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 \u2191n < x \u2194 \u2200 (h : x.Dom), n < Part.get x h\n[PROOFSTEP]\nrw [\u2190 some_eq_natCast]\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 \u2191n < x \u2194 \u2200 (h : x.Dom), n < Part.get x h\n[PROOFSTEP]\nsimp only [lt_def, exists_prop_of_true, dom_some, forall_true_iff]\n[GOAL]\nn : \u2115\nx : PartENat\n\u22a2 (\u2200 (hy : x.Dom), Part.get \u2191n (_ : (\u2191n).Dom) < Part.get x hy) \u2194 \u2200 (h : x.Dom), n < Part.get x h\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u00ac1 = 0\n[PROOFSTEP]\ndecide\n[GOAL]\nx : \u2115\nh : \u2191x = \u22a4\n\u22a2 \u00ac(\u2191x).Dom = \u22a4.Dom\n[PROOFSTEP]\nsimp only [dom_natCast]\n[GOAL]\nx : \u2115\nh : \u2191x = \u22a4\n\u22a2 \u00acTrue = \u22a4.Dom\n[PROOFSTEP]\nexact true_ne_false\n[GOAL]\nx : PartENat\n\u22a2 x \u2260 \u22a4 \u2194 \u2203 n, x = \u2191n\n[PROOFSTEP]\nsimpa only [\u2190 some_eq_natCast] using Part.ne_none_iff\n[GOAL]\nx : PartENat\n\u22a2 x \u2260 \u22a4 \u2194 x.Dom\n[PROOFSTEP]\nclassical exact not_iff_comm.1 Part.eq_none_iff'.symm\n[GOAL]\nx : PartENat\n\u22a2 x \u2260 \u22a4 \u2194 x.Dom\n[PROOFSTEP]\nexact not_iff_comm.1 Part.eq_none_iff'.symm\n[GOAL]\nx : PartENat\n\u22a2 x = \u22a4 \u2194 \u2200 (n : \u2115), \u2191n < x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx : PartENat\n\u22a2 x = \u22a4 \u2192 \u2200 (n : \u2115), \u2191n < x\n[PROOFSTEP]\nrintro rfl n\n[GOAL]\ncase mp\nn : \u2115\n\u22a2 \u2191n < \u22a4\n[PROOFSTEP]\nexact natCast_lt_top _\n[GOAL]\ncase mpr\nx : PartENat\n\u22a2 (\u2200 (n : \u2115), \u2191n < x) \u2192 x = \u22a4\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mpr\nx : PartENat\n\u22a2 x \u2260 \u22a4 \u2192 \u2203 n, \u00ac\u2191n < x\n[PROOFSTEP]\nrw [ne_top_iff]\n[GOAL]\ncase mpr\nx : PartENat\n\u22a2 (\u2203 n, x = \u2191n) \u2192 \u2203 n, \u00ac\u2191n < x\n[PROOFSTEP]\nrintro \u27e8n, rfl\u27e9\n[GOAL]\ncase mpr.intro\nn : \u2115\n\u22a2 \u2203 n_1, \u00ac\u2191n_1 < \u2191n\n[PROOFSTEP]\nexact \u27e8n, irrefl _\u27e9\n[GOAL]\nx : PartENat\n\u22a2 0 < \u22a4 \u2194 1 \u2264 \u22a4\n[PROOFSTEP]\nsimp only [iff_true_iff, le_top, natCast_lt_top, \u2190 @Nat.cast_zero PartENat]\n[GOAL]\nx : PartENat\nn : \u2115\n\u22a2 0 < \u2191n \u2194 1 \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, \u2190 Nat.cast_one, PartENat.coe_lt_coe, PartENat.coe_le_coe]\n[GOAL]\nx : PartENat\nn : \u2115\n\u22a2 0 < n \u2194 1 \u2264 n\n[PROOFSTEP]\nrfl\n[GOAL]\nsrc\u271d : PartialOrder PartENat := partialOrder\na b : PartENat\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nchange (fun a b => a \u2294 b) a b = _\n[GOAL]\nsrc\u271d : PartialOrder PartENat := partialOrder\na b : PartENat\n\u22a2 (fun a b => a \u2294 b) a b = if a \u2264 b then b else a\n[PROOFSTEP]\nrw [@sup_eq_maxDefault PartENat _ (id _) _]\n[GOAL]\nsrc\u271d : PartialOrder PartENat := partialOrder\na b : PartENat\n\u22a2 maxDefault a b = if a \u2264 b then b else a\nsrc\u271d : PartialOrder PartENat := partialOrder a b : PartENat \u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrfl\n[GOAL]\nsrc\u271d\u00b9 : LinearOrder PartENat := linearOrder\nsrc\u271d : AddCommMonoid PartENat := addCommMonoid\na b : PartENat\nx\u271d : a \u2264 b\nc : PartENat\nh\u2081 : b.Dom \u2192 a.Dom\nh\u2082 : \u2200 (hy : b.Dom), Part.get a (_ : a.Dom) \u2264 Part.get b hy\n\u22a2 \u22a4 + a \u2264 \u22a4 + b\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d\u00b9 : LinearOrder PartENat := linearOrder\nsrc\u271d : AddCommMonoid PartENat := addCommMonoid\na b : PartENat\nx\u271d : a \u2264 b\nc\u271d : PartENat\nh\u2081 : b.Dom \u2192 a.Dom\nh\u2082 : \u2200 (hy : b.Dom), Part.get a (_ : a.Dom) \u2264 Part.get b hy\nc : \u2115\nh : (\u2191c + b).Dom\n\u22a2 Part.get (\u2191c + a) (_ : (\u2191c).Dom \u2227 a.Dom) \u2264 Part.get (\u2191c + b) h\n[PROOFSTEP]\nsimpa only [coe_add_get] using add_le_add_left (h\u2082 _) c\n[GOAL]\nsrc\u271d\u00b2 : SemilatticeSup PartENat := semilatticeSup\nsrc\u271d\u00b9 : OrderBot PartENat := orderBot\nsrc\u271d : OrderedAddCommMonoid PartENat := orderedAddCommMonoid\na\u271d b\u271d : PartENat\nb a : \u2115\nh : \u2191a \u2264 \u2191b\n\u22a2 \u2191b = \u2191a + \u2191(b - a)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add, natCast_inj, add_comm, tsub_add_cancel_of_le (coe_le_coe.1 h)]\n[GOAL]\nx y : PartENat\nn : \u2115\nh : x + y = \u2191n\n\u22a2 x = \u2191(n - Part.get y (_ : y.Dom))\n[PROOFSTEP]\nlift x to \u2115 using dom_of_le_natCast ((le_add_right le_rfl).trans_eq h)\n[GOAL]\ncase intro\ny : PartENat\nn x : \u2115\nh : \u2191x + y = \u2191n\n\u22a2 \u2191x = \u2191(n - Part.get y (_ : y.Dom))\n[PROOFSTEP]\nlift y to \u2115 using dom_of_le_natCast ((le_add_left le_rfl).trans_eq h)\n[GOAL]\ncase intro.intro\nn x y : \u2115\nh : \u2191x + \u2191y = \u2191n\n\u22a2 \u2191x = \u2191(n - Part.get \u2191y (_ : (\u2191y).Dom))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add, natCast_inj] at h \n[GOAL]\ncase intro.intro\nn x y : \u2115\nh\u271d : \u2191x + \u2191y = \u2191n\nh : x + y = n\n\u22a2 \u2191x = \u2191(n - Part.get \u2191y (_ : (\u2191y).Dom))\n[PROOFSTEP]\nrw [get_natCast, natCast_inj, eq_tsub_of_add_eq h]\n[GOAL]\nx y z : PartENat\nh : x < y\nhz : z \u2260 \u22a4\n\u22a2 x + z < y + z\n[PROOFSTEP]\nrcases ne_top_iff.mp (ne_top_of_lt h) with \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\ny z : PartENat\nhz : z \u2260 \u22a4\nm : \u2115\nh : \u2191m < y\n\u22a2 \u2191m + z < y + z\n[PROOFSTEP]\nrcases ne_top_iff.mp hz with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro.intro\ny : PartENat\nm : \u2115\nh : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\n\u22a2 \u2191m + \u2191k < y + \u2191k\n[PROOFSTEP]\ninduction' y using PartENat.casesOn with n\n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nh : \u2191m < \u22a4\n\u22a2 \u2191m + \u2191k < \u22a4 + \u2191k\n[PROOFSTEP]\nrw [top_add]\n -- Porting note: was apply_mod_cast natCast_lt_top\n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nh : \u2191m < \u22a4\n\u22a2 \u2191m + \u2191k < \u22a4\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nh : \u2191m < \u22a4\n\u22a2 \u2191(m + k) < \u22a4\n[PROOFSTEP]\napply natCast_lt_top\n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nn : \u2115\nh : \u2191m < \u2191n\n\u22a2 \u2191m + \u2191k < \u2191n + \u2191k\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nn : \u2115\nh : m < n\n\u22a2 \u2191m + \u2191k < \u2191n + \u2191k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro.a\ny : PartENat\nm : \u2115\nh\u271d : \u2191m < y\nk : \u2115\nhz : \u2191k \u2260 \u22a4\nn : \u2115\nh : m < n\n\u22a2 m + k < n + k\n[PROOFSTEP]\napply add_lt_add_right h\n[GOAL]\nx y z : PartENat\nhz : z \u2260 \u22a4\n\u22a2 z + x < z + y \u2194 x < y\n[PROOFSTEP]\nrw [add_comm z, add_comm z, PartENat.add_lt_add_iff_right hz]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n\u22a2 x < x + y \u2194 0 < y\n[PROOFSTEP]\nconv_rhs => rw [\u2190 PartENat.add_lt_add_iff_left hx]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n| 0 < y\n[PROOFSTEP]\nrw [\u2190 PartENat.add_lt_add_iff_left hx]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n| 0 < y\n[PROOFSTEP]\nrw [\u2190 PartENat.add_lt_add_iff_left hx]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n| 0 < y\n[PROOFSTEP]\nrw [\u2190 PartENat.add_lt_add_iff_left hx]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n\u22a2 x < x + y \u2194 x + 0 < x + y\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nx : PartENat\nhx : x \u2260 \u22a4\n\u22a2 x < x + 1\n[PROOFSTEP]\nrw [PartENat.lt_add_iff_pos_right hx]\n[GOAL]\nx : PartENat\nhx : x \u2260 \u22a4\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nx y : PartENat\nh : x < y + 1\n\u22a2 x \u2264 y\n[PROOFSTEP]\ninduction' y using PartENat.casesOn with n\n[GOAL]\ncase a\nx y : PartENat\nh\u271d : x < y + 1\nh : x < \u22a4 + 1\n\u22a2 x \u2264 \u22a4\n[PROOFSTEP]\napply le_top\n[GOAL]\ncase a\nx y : PartENat\nh\u271d : x < y + 1\nn : \u2115\nh : x < \u2191n + 1\n\u22a2 x \u2264 \u2191n\n[PROOFSTEP]\nrcases ne_top_iff.mp (ne_top_of_lt h) with\n \u27e8m, rfl\u27e9\n -- Porting note: was `apply_mod_cast Nat.le_of_lt_succ; apply_mod_cast h`\n[GOAL]\ncase a.intro\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y + 1\nh : \u2191m < \u2191n + 1\n\u22a2 \u2191m \u2264 \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase a.intro\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y + 1\nh : \u2191m < \u2191n + 1\n\u22a2 m \u2264 n\n[PROOFSTEP]\napply Nat.le_of_lt_succ\n[GOAL]\ncase a.intro.a\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y + 1\nh : \u2191m < \u2191n + 1\n\u22a2 m < Nat.succ n\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\nx y : PartENat\nh : x < y\n\u22a2 x + 1 \u2264 y\n[PROOFSTEP]\ninduction' y using PartENat.casesOn with n\n[GOAL]\ncase a\nx y : PartENat\nh\u271d : x < y\nh : x < \u22a4\n\u22a2 x + 1 \u2264 \u22a4\n[PROOFSTEP]\napply le_top\n[GOAL]\ncase a\nx y : PartENat\nh\u271d : x < y\nn : \u2115\nh : x < \u2191n\n\u22a2 x + 1 \u2264 \u2191n\n[PROOFSTEP]\nrcases ne_top_iff.mp (ne_top_of_lt h) with\n \u27e8m, rfl\u27e9\n -- Porting note: was `apply_mod_cast Nat.succ_le_of_lt; apply_mod_cast h`\n[GOAL]\ncase a.intro\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y\nh : \u2191m < \u2191n\n\u22a2 \u2191m + 1 \u2264 \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase a.intro\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y\nh : \u2191m < \u2191n\n\u22a2 m + 1 \u2264 n\n[PROOFSTEP]\napply Nat.succ_le_of_lt\n[GOAL]\ncase a.intro.h\ny : PartENat\nn m : \u2115\nh\u271d : \u2191m < y\nh : \u2191m < \u2191n\n\u22a2 m < n\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n\u22a2 x + 1 \u2264 y \u2194 x < y\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, add_one_le_of_lt\u27e9\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\nh : x + 1 \u2264 y\n\u22a2 x < y\n[PROOFSTEP]\nrcases ne_top_iff.mp hx with \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh : \u2191m + 1 \u2264 y\n\u22a2 \u2191m < y\n[PROOFSTEP]\ninduction' y using PartENat.casesOn with n\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m + 1 \u2264 y\nh : \u2191m + 1 \u2264 \u22a4\n\u22a2 \u2191m < \u22a4\n[PROOFSTEP]\napply natCast_lt_top\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m + 1 \u2264 y\nn : \u2115\nh : \u2191m + 1 \u2264 \u2191n\n\u22a2 \u2191m < \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m + 1 \u2264 y\nn : \u2115\nh : \u2191m + 1 \u2264 \u2191n\n\u22a2 m < n\n[PROOFSTEP]\napply Nat.lt_of_succ_le\n[GOAL]\ncase intro.a.h\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m + 1 \u2264 y\nn : \u2115\nh : \u2191m + 1 \u2264 \u2191n\n\u22a2 Nat.succ m \u2264 n\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\nn : \u2115\ne : PartENat\n\u22a2 \u2191(Nat.succ n) \u2264 e \u2194 \u2191n < e\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one n, Nat.cast_add, Nat.cast_one, add_one_le_iff_lt (natCast_ne_top n)]\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\n\u22a2 x < y + 1 \u2194 x \u2264 y\n[PROOFSTEP]\nrefine \u27e8le_of_lt_add_one, fun h => ?_\u27e9\n[GOAL]\nx y : PartENat\nhx : x \u2260 \u22a4\nh : x \u2264 y\n\u22a2 x < y + 1\n[PROOFSTEP]\nrcases ne_top_iff.mp hx with \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh : \u2191m \u2264 y\n\u22a2 \u2191m < y + 1\n[PROOFSTEP]\ninduction' y using PartENat.casesOn with n\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m \u2264 y\nh : \u2191m \u2264 \u22a4\n\u22a2 \u2191m < \u22a4 + 1\n[PROOFSTEP]\nrw [top_add]\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m \u2264 y\nh : \u2191m \u2264 \u22a4\n\u22a2 \u2191m < \u22a4\n[PROOFSTEP]\napply natCast_lt_top\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m \u2264 y\nn : \u2115\nh : \u2191m \u2264 \u2191n\n\u22a2 \u2191m < \u2191n + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m \u2264 y\nn : \u2115\nh : \u2191m \u2264 \u2191n\n\u22a2 m < n + 1\n[PROOFSTEP]\napply Nat.lt_succ_of_le\n[GOAL]\ncase intro.a.a\ny : PartENat\nm : \u2115\nhx : \u2191m \u2260 \u22a4\nh\u271d : \u2191m \u2264 y\nn : \u2115\nh : \u2191m \u2264 \u2191n\n\u22a2 m \u2264 n\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\nx : PartENat\nn : \u2115\nhx : x \u2260 \u22a4\n\u22a2 x < \u2191(Nat.succ n) \u2194 x \u2264 \u2191n\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one n, Nat.cast_add, Nat.cast_one, lt_add_one_iff_lt hx]\n[GOAL]\na b : PartENat\n\u22a2 a + b = \u22a4 \u2194 a = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\nrefine PartENat.casesOn a ?_ ?_\n[GOAL]\ncase refine_1\na b : PartENat\n\u22a2 \u22a4 + b = \u22a4 \u2194 \u22a4 = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\nrefine PartENat.casesOn b ?_ ?_\n[GOAL]\ncase refine_2\na b : PartENat\n\u22a2 \u2200 (n : \u2115), \u2191n + b = \u22a4 \u2194 \u2191n = \u22a4 \u2228 b = \u22a4\n[PROOFSTEP]\nrefine PartENat.casesOn b ?_ ?_\n[GOAL]\ncase refine_1.refine_1\na b : PartENat\n\u22a2 \u22a4 + \u22a4 = \u22a4 \u2194 \u22a4 = \u22a4 \u2228 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1.refine_2\na b : PartENat\n\u22a2 \u2200 (n : \u2115), \u22a4 + \u2191n = \u22a4 \u2194 \u22a4 = \u22a4 \u2228 \u2191n = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2.refine_1\na b : PartENat\n\u22a2 \u2200 (n : \u2115), \u2191n + \u22a4 = \u22a4 \u2194 \u2191n = \u22a4 \u2228 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2.refine_2\na b : PartENat\n\u22a2 \u2200 (n n_1 : \u2115), \u2191n_1 + \u2191n = \u22a4 \u2194 \u2191n_1 = \u22a4 \u2228 \u2191n = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_2.refine_2\na b : PartENat\n\u22a2 \u2200 (n n_1 : \u2115), \u00ac\u2191n_1 + \u2191n = \u22a4\n[PROOFSTEP]\nsimp only [\u2190 Nat.cast_add, PartENat.natCast_ne_top, forall_const]\n[GOAL]\na b c : PartENat\nhc : c \u2260 \u22a4\n\u22a2 a + c = b + c \u2194 a = b\n[PROOFSTEP]\nrcases ne_top_iff.1 hc with \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 a + \u2191c = b + \u2191c \u2194 a = b\n[PROOFSTEP]\nrefine PartENat.casesOn a ?_ ?_\n[GOAL]\ncase intro.refine_1\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u22a4 + \u2191c = b + \u2191c \u2194 \u22a4 = b\n[PROOFSTEP]\nrefine PartENat.casesOn b ?_ ?_\n[GOAL]\ncase intro.refine_2\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), \u2191n + \u2191c = b + \u2191c \u2194 \u2191n = b\n[PROOFSTEP]\nrefine PartENat.casesOn b ?_ ?_\n[GOAL]\ncase intro.refine_1.refine_1\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u22a4 + \u2191c = \u22a4 + \u2191c \u2194 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp [add_eq_top_iff, natCast_ne_top, @eq_comm _ (\u22a4 : PartENat)]\n[GOAL]\ncase intro.refine_1.refine_2\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), \u22a4 + \u2191c = \u2191n + \u2191c \u2194 \u22a4 = \u2191n\n[PROOFSTEP]\nsimp [add_eq_top_iff, natCast_ne_top, @eq_comm _ (\u22a4 : PartENat)]\n[GOAL]\ncase intro.refine_2.refine_1\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u2200 (n : \u2115), \u2191n + \u2191c = \u22a4 + \u2191c \u2194 \u2191n = \u22a4\n[PROOFSTEP]\nsimp [add_eq_top_iff, natCast_ne_top, @eq_comm _ (\u22a4 : PartENat)]\n[GOAL]\ncase intro.refine_2.refine_2\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u2200 (n n_1 : \u2115), \u2191n_1 + \u2191c = \u2191n + \u2191c \u2194 \u2191n_1 = \u2191n\n[PROOFSTEP]\nsimp [add_eq_top_iff, natCast_ne_top, @eq_comm _ (\u22a4 : PartENat)]\n[GOAL]\ncase intro.refine_2.refine_2\na b : PartENat\nc : \u2115\nhc : \u2191c \u2260 \u22a4\n\u22a2 \u2200 (n n_1 : \u2115), \u2191n_1 + \u2191c = \u2191n + \u2191c \u2194 n_1 = n\n[PROOFSTEP]\nsimp only [\u2190 Nat.cast_add, add_left_cancel_iff, PartENat.natCast_inj, add_comm, forall_const]\n[GOAL]\na b c : PartENat\nha : a \u2260 \u22a4\n\u22a2 a + b = a + c \u2194 b = c\n[PROOFSTEP]\nrw [add_comm a, add_comm a, PartENat.add_right_cancel_iff ha]\n[GOAL]\nh : Decidable \u22a4.Dom\n\u22a2 toWithTop \u22a4 = \u22a4\n[PROOFSTEP]\nconvert toWithTop_top\n[GOAL]\nh : Decidable 0.Dom\n\u22a2 toWithTop 0 = 0\n[PROOFSTEP]\nconvert toWithTop_zero\n[GOAL]\nn : \u2115\nx\u271d : Decidable (\u2191n).Dom\n\u22a2 toWithTop \u2191n = \u2191n\n[PROOFSTEP]\nsimp only [\u2190 toWithTop_some]\n[GOAL]\nn : \u2115\nx\u271d : Decidable (\u2191n).Dom\n\u22a2 toWithTop \u2191n = toWithTop \u2191n\n[PROOFSTEP]\ncongr\n[GOAL]\nn : \u2115\nh : Decidable (\u2191n).Dom\n\u22a2 toWithTop \u2191n = \u2191n\n[PROOFSTEP]\nrw [toWithTop_natCast n]\n[GOAL]\nx y : PartENat\nhx : Decidable x.Dom\nhy : Decidable y.Dom\n\u22a2 toWithTop x \u2264 toWithTop y \u2194 x \u2264 y\n[PROOFSTEP]\ninduction y using PartENat.casesOn generalizing hy\n[GOAL]\ncase a\nx : PartENat\nhx : Decidable x.Dom\nhy : Decidable \u22a4.Dom\n\u22a2 toWithTop x \u2264 toWithTop \u22a4 \u2194 x \u2264 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nx : PartENat\nhx : Decidable x.Dom\nn\u271d : \u2115\nhy : Decidable (\u2191n\u271d).Dom\n\u22a2 toWithTop x \u2264 toWithTop \u2191n\u271d \u2194 x \u2264 \u2191n\u271d\n[PROOFSTEP]\ninduction x using PartENat.casesOn generalizing hx\n[GOAL]\ncase a.a\nn\u271d : \u2115\nhy : Decidable (\u2191n\u271d).Dom\nhx : Decidable \u22a4.Dom\n\u22a2 toWithTop \u22a4 \u2264 toWithTop \u2191n\u271d \u2194 \u22a4 \u2264 \u2191n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.a\nn\u271d\u00b9 : \u2115\nhy : Decidable (\u2191n\u271d\u00b9).Dom\nn\u271d : \u2115\nhx : Decidable (\u2191n\u271d).Dom\n\u22a2 toWithTop \u2191n\u271d \u2264 toWithTop \u2191n\u271d\u00b9 \u2194 \u2191n\u271d \u2264 \u2191n\u271d\u00b9\n[PROOFSTEP]\nsimp\n -- Porting note: this takes too long.\n[GOAL]\n\u22a2 \u2191Option.none = \u22a4\n[PROOFSTEP]\nrfl\n -- Porting note : new\n[GOAL]\nn : \u2115\n\u22a2 \u2191(Option.some n) = \u2191n\n[PROOFSTEP]\nrfl\n -- Porting note : new\n[GOAL]\nn : \u2115\u221e\nx\u271d : Decidable (\u2191n).Dom\n\u22a2 toWithTop \u2191n = n\n[PROOFSTEP]\ninduction n with\n| none => simp\n| some n =>\n simp only [toWithTop_natCast', ofENat_some]\n rfl\n[GOAL]\nn : \u2115\u221e\nx\u271d : Decidable (\u2191n).Dom\n\u22a2 toWithTop \u2191n = n\n[PROOFSTEP]\ninduction n with\n| none => simp\n| some n =>\n simp only [toWithTop_natCast', ofENat_some]\n rfl\n[GOAL]\ncase none\nx\u271d : Decidable (\u2191Option.none).Dom\n\u22a2 toWithTop \u2191Option.none = Option.none\n[PROOFSTEP]\n\n| none => simp\n[GOAL]\ncase none\nx\u271d : Decidable (\u2191Option.none).Dom\n\u22a2 toWithTop \u2191Option.none = Option.none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\nn : \u2115\nx\u271d : Decidable (\u2191(Option.some n)).Dom\n\u22a2 toWithTop \u2191(Option.some n) = Option.some n\n[PROOFSTEP]\n\n| some n =>\n simp only [toWithTop_natCast', ofENat_some]\n rfl\n[GOAL]\ncase some\nn : \u2115\nx\u271d : Decidable (\u2191(Option.some n)).Dom\n\u22a2 toWithTop \u2191(Option.some n) = Option.some n\n[PROOFSTEP]\nsimp only [toWithTop_natCast', ofENat_some]\n[GOAL]\ncase some\nn : \u2115\nx\u271d : Decidable (\u2191(Option.some n)).Dom\n\u22a2 \u2191n = Option.some n\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : PartENat\n\u22a2 toWithTop (x + y) = toWithTop x + toWithTop y\n[PROOFSTEP]\nrefine PartENat.casesOn y ?_ ?_\n[GOAL]\ncase refine_1\nx y : PartENat\n\u22a2 toWithTop (x + \u22a4) = toWithTop x + toWithTop \u22a4\n[PROOFSTEP]\nrefine\n PartENat.casesOn x ?_\n ?_\n --Porting note: was `simp [\u2190 Nat.cast_add, \u2190 ENat.coe_add]`\n[GOAL]\ncase refine_2\nx y : PartENat\n\u22a2 \u2200 (n : \u2115), toWithTop (x + \u2191n) = toWithTop x + toWithTop \u2191n\n[PROOFSTEP]\nrefine\n PartENat.casesOn x ?_\n ?_\n --Porting note: was `simp [\u2190 Nat.cast_add, \u2190 ENat.coe_add]`\n[GOAL]\ncase refine_1.refine_1\nx y : PartENat\n\u22a2 toWithTop (\u22a4 + \u22a4) = toWithTop \u22a4 + toWithTop \u22a4\n[PROOFSTEP]\nsimp only [add_top, toWithTop_top', _root_.add_top]\n[GOAL]\ncase refine_1.refine_2\nx y : PartENat\n\u22a2 \u2200 (n : \u2115), toWithTop (\u2191n + \u22a4) = toWithTop \u2191n + toWithTop \u22a4\n[PROOFSTEP]\nsimp only [add_top, toWithTop_top', toWithTop_natCast', _root_.add_top, forall_const]\n[GOAL]\ncase refine_2.refine_1\nx y : PartENat\n\u22a2 \u2200 (n : \u2115), toWithTop (\u22a4 + \u2191n) = toWithTop \u22a4 + toWithTop \u2191n\n[PROOFSTEP]\nsimp only [top_add, toWithTop_top', toWithTop_natCast', _root_.top_add, forall_const]\n[GOAL]\ncase refine_2.refine_2\nx y : PartENat\n\u22a2 \u2200 (n n_1 : \u2115), toWithTop (\u2191n + \u2191n_1) = toWithTop \u2191n + toWithTop \u2191n_1\n[PROOFSTEP]\nsimp_rw [toWithTop_natCast', \u2190 Nat.cast_add, toWithTop_natCast', forall_const]\n[GOAL]\nx : PartENat\n\u22a2 (fun x => \u2191x) ((fun x => toWithTop x) x) = x\n[PROOFSTEP]\ninduction x using PartENat.casesOn\n[GOAL]\ncase a\n\u22a2 (fun x => \u2191x) ((fun x => toWithTop x) \u22a4) = \u22a4\n[PROOFSTEP]\nintros\n[GOAL]\ncase a\nn\u271d : \u2115\n\u22a2 (fun x => \u2191x) ((fun x => toWithTop x) \u2191n\u271d) = \u2191n\u271d\n[PROOFSTEP]\nintros\n[GOAL]\ncase a\n\u22a2 (fun x => \u2191x) ((fun x => toWithTop x) \u22a4) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nn\u271d : \u2115\n\u22a2 (fun x => \u2191x) ((fun x => toWithTop x) \u2191n\u271d) = \u2191n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u22a2 \u2191\u22a4 = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nn\u271d : \u2115\n\u22a2 \u2191\u2191n\u271d = \u2191n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nx : \u2115\u221e\n\u22a2 (fun x => toWithTop x) ((fun x => \u2191x) x) = x\n[PROOFSTEP]\nsimp [toWithTop_ofENat]\n[GOAL]\n\u22a2 \u2191withTopEquiv 0 = 0\n[PROOFSTEP]\nsimpa only [Nat.cast_zero] using withTopEquiv_natCast 0\n[GOAL]\nx y : \u2115\u221e\n\u22a2 \u2191withTopEquiv.symm x \u2264 \u2191withTopEquiv.symm y \u2194 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 withTopEquiv_le]\n[GOAL]\nx y : \u2115\u221e\n\u22a2 \u2191withTopEquiv (\u2191withTopEquiv.symm x) \u2264 \u2191withTopEquiv (\u2191withTopEquiv.symm y) \u2194 x \u2264 y\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : \u2115\u221e\n\u22a2 \u2191withTopEquiv.symm x < \u2191withTopEquiv.symm y \u2194 x < y\n[PROOFSTEP]\nrw [\u2190 withTopEquiv_lt]\n[GOAL]\nx y : \u2115\u221e\n\u22a2 \u2191withTopEquiv (\u2191withTopEquiv.symm x) < \u2191withTopEquiv (\u2191withTopEquiv.symm y) \u2194 x < y\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d : PartENat \u2243 \u2115\u221e := withTopEquiv\nx y : PartENat\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 (x + y) =\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 x +\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 y\n[PROOFSTEP]\nsimp only [withTopEquiv]\n[GOAL]\nsrc\u271d : PartENat \u2243 \u2115\u221e := withTopEquiv\nx y : PartENat\n\u22a2 toWithTop (x + y) = toWithTop x + toWithTop y\n[PROOFSTEP]\nexact toWithTop_add\n[GOAL]\n\u22a2 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nclassical\nchange WellFounded fun a b : PartENat => a < b\nsimp_rw [\u2190 withTopEquiv_lt]\nexact InvImage.wf _ (WithTop.wellFounded_lt Nat.lt_wfRel.wf)\n[GOAL]\n\u22a2 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\nchange WellFounded fun a b : PartENat => a < b\n[GOAL]\n\u22a2 WellFounded fun a b => a < b\n[PROOFSTEP]\nsimp_rw [\u2190 withTopEquiv_lt]\n[GOAL]\n\u22a2 WellFounded fun a b => \u2191withTopEquiv a < \u2191withTopEquiv b\n[PROOFSTEP]\nexact InvImage.wf _ (WithTop.wellFounded_lt Nat.lt_wfRel.wf)\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n\u22a2 \u2191n < find P\n[PROOFSTEP]\nrw [coe_lt_iff]\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n\u22a2 \u2200 (h : (find P).Dom), n < Part.get (find P) h\n[PROOFSTEP]\nintro h\u2081\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Part.get (find P) h\u2081\n[PROOFSTEP]\nrw [find_get]\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Nat.find h\u2081\n[PROOFSTEP]\nhave h\u2082 := @Nat.find_spec P _ h\u2081\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : P (Nat.find h\u2081)\n\u22a2 n < Nat.find h\u2081\n[PROOFSTEP]\nrevert h\u2082\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 P (Nat.find h\u2081) \u2192 n < Nat.find h\u2081\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 Nat.find h\u2081 \u2264 n \u2192 \u00acP (Nat.find h\u2081)\n[PROOFSTEP]\nexact h _\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\n\u22a2 \u2191n < find P \u2194 \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n[PROOFSTEP]\nrefine' \u27e8_, lt_find P n\u27e9\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\n\u22a2 \u2191n < find P \u2192 \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n[PROOFSTEP]\nintro h m hm\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2191n < find P\nm : \u2115\nhm : m \u2264 n\n\u22a2 \u00acP m\n[PROOFSTEP]\nby_cases H : (find P).Dom\n[GOAL]\ncase pos\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2191n < find P\nm : \u2115\nhm : m \u2264 n\nH : (find P).Dom\n\u22a2 \u00acP m\n[PROOFSTEP]\napply Nat.find_min H\n[GOAL]\ncase pos\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2191n < find P\nm : \u2115\nhm : m \u2264 n\nH : (find P).Dom\n\u22a2 m < Nat.find H\n[PROOFSTEP]\nrw [coe_lt_iff] at h \n[GOAL]\ncase pos\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (h : (find P).Dom), n < Part.get (find P) h\nm : \u2115\nhm : m \u2264 n\nH : (find P).Dom\n\u22a2 m < Nat.find H\n[PROOFSTEP]\nspecialize h H\n[GOAL]\ncase pos\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn m : \u2115\nhm : m \u2264 n\nH : (find P).Dom\nh : n < Part.get (find P) H\n\u22a2 m < Nat.find H\n[PROOFSTEP]\nexact lt_of_le_of_lt hm h\n[GOAL]\ncase neg\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2191n < find P\nm : \u2115\nhm : m \u2264 n\nH : \u00ac(find P).Dom\n\u22a2 \u00acP m\n[PROOFSTEP]\nexact not_exists.mp H m\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : P n\n\u22a2 find P \u2264 \u2191n\n[PROOFSTEP]\nrw [le_coe_iff]\n[GOAL]\nP : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : P n\n\u22a2 \u2203 h, Part.get (find P) h \u2264 n\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, h\u27e9, @Nat.find_min' P _ _ _ h\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.PartENat", "llama_tokens": 15332, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.25306831858711115}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 Compatibility.\u03c4\u2080 =\n Compatibility.\u03c4\u2081\n (eqToIso (_ : (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081))\n (eqToIso\n (_ :\n (toKaroubiEquivalence (ChainComplex C \u2115)).functor \u22d9 Preadditive.DoldKan.equivalence.inverse =\n \u0393 \u22d9 (toKaroubiEquivalence (SimplicialObject C)).functor))\n N\u2081\u0393\u2080\n[PROOFSTEP]\next K : 3\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app Compatibility.\u03c4\u2080.hom K =\n NatTrans.app\n (Compatibility.\u03c4\u2081\n (eqToIso\n (_ : (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081))\n (eqToIso\n (_ :\n (toKaroubiEquivalence (ChainComplex C \u2115)).functor \u22d9 Preadditive.DoldKan.equivalence.inverse =\n \u0393 \u22d9 (toKaroubiEquivalence (SimplicialObject C)).functor))\n N\u2081\u0393\u2080).hom\n K\n[PROOFSTEP]\nsimp only [Compatibility.\u03c4\u2080_hom_app, Compatibility.\u03c4\u2081_hom_app, eqToIso.hom]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app Preadditive.DoldKan.equivalence.counitIso.hom ((toKaroubiEquivalence (ChainComplex C \u2115)).functor.obj K) =\n Preadditive.DoldKan.equivalence.functor.map\n (NatTrans.app\n (eqToHom\n (_ :\n (toKaroubiEquivalence (ChainComplex C \u2115)).functor \u22d9 Preadditive.DoldKan.equivalence.inverse =\n \u0393 \u22d9 (toKaroubiEquivalence (SimplicialObject C)).functor))\n K) \u226b\n NatTrans.app\n (eqToHom\n (_ : (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081))\n (\u0393.obj K) \u226b\n NatTrans.app N\u2081\u0393\u2080.hom K\n[PROOFSTEP]\nrefine' (N\u2082\u0393\u2082_compatible_with_N\u2081\u0393\u2080 K).trans _\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nK : ChainComplex C \u2115\n\u22a2 NatTrans.app N\u2082\u0393\u2082ToKaroubiIso.hom K \u226b NatTrans.app N\u2081\u0393\u2080.hom K =\n Preadditive.DoldKan.equivalence.functor.map\n (NatTrans.app\n (eqToHom\n (_ :\n (toKaroubiEquivalence (ChainComplex C \u2115)).functor \u22d9 Preadditive.DoldKan.equivalence.inverse =\n \u0393 \u22d9 (toKaroubiEquivalence (SimplicialObject C)).functor))\n K) \u226b\n NatTrans.app\n (eqToHom\n (_ : (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081))\n (\u0393.obj K) \u226b\n NatTrans.app N\u2081\u0393\u2080.hom K\n[PROOFSTEP]\nsimp only [N\u2082\u0393\u2082ToKaroubiIso_hom, eqToHom_map, eqToHom_app, eqToHom_trans_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 Compatibility.\u03c5\n (eqToIso\n (_ : (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081)) =\n \u0393\u2082N\u2081\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 (Compatibility.\u03c5\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor = N\u2081))).hom =\n \u0393\u2082N\u2081.hom\n[PROOFSTEP]\nrw [\u2190 cancel_epi \u0393\u2082N\u2081.inv, Iso.inv_hom_id]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 \u0393\u2082N\u2081.inv \u226b\n (Compatibility.\u03c5\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081))).hom =\n \ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)\n[PROOFSTEP]\next X : 2\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 NatTrans.app\n (\u0393\u2082N\u2081.inv \u226b\n (Compatibility.\u03c5\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081))).hom)\n X =\n NatTrans.app (\ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)) X\n[PROOFSTEP]\nrw [NatTrans.comp_app]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 NatTrans.app \u0393\u2082N\u2081.inv X \u226b\n NatTrans.app\n (Compatibility.\u03c5\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081))).hom\n X =\n NatTrans.app (\ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)) X\n[PROOFSTEP]\nerw [compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_natTrans X]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 ((compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).inv \u226b NatTrans.app \u0393\u2082N\u2082.natTrans ((toKaroubi (SimplicialObject C)).obj X)) \u226b\n NatTrans.app\n (Compatibility.\u03c5\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081))).hom\n X =\n NatTrans.app (\ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)) X\n[PROOFSTEP]\nrw [Compatibility.\u03c5_hom_app, Preadditive.DoldKan.equivalence_unitIso, Iso.app_inv, assoc]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 NatTrans.app compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.inv X \u226b\n NatTrans.app \u0393\u2082N\u2082.natTrans ((toKaroubi (SimplicialObject C)).obj X) \u226b\n NatTrans.app \u0393\u2082N\u2082.hom ((toKaroubiEquivalence (SimplicialObject C)).functor.obj X) \u226b\n Preadditive.DoldKan.equivalence.inverse.map\n (NatTrans.app\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081)).hom\n X) =\n NatTrans.app (\ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)) X\n[PROOFSTEP]\nerw [\u2190 NatTrans.comp_app_assoc, IsIso.hom_inv_id]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 NatTrans.app compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.inv X \u226b\n NatTrans.app (\ud835\udfd9 (N\u2082 \u22d9 \u0393\u2082)) ((toKaroubi (SimplicialObject C)).obj X) \u226b\n Preadditive.DoldKan.equivalence.inverse.map\n (NatTrans.app\n (eqToIso\n (_ :\n (toKaroubiEquivalence (SimplicialObject C)).functor \u22d9 Preadditive.DoldKan.equivalence.functor =\n N\u2081)).hom\n X) =\n NatTrans.app (\ud835\udfd9 (N\u2081 \u22d9 \u0393\u2082)) X\n[PROOFSTEP]\nrw [NatTrans.id_app, id_comp, NatTrans.id_app, eqToIso.hom, eqToHom_app, eqToHom_map]\n[GOAL]\ncase w.w.h\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : IsIdempotentComplete C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 NatTrans.app compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.inv X \u226b\n eqToHom\n (_ :\n Preadditive.DoldKan.equivalence.inverse.obj\n (Preadditive.DoldKan.equivalence.functor.obj\n ((toKaroubiEquivalence (SimplicialObject C)).functor.obj X)) =\n Preadditive.DoldKan.equivalence.inverse.obj (N\u2081.obj X)) =\n \ud835\udfd9 ((N\u2081 \u22d9 \u0393\u2082).obj X)\n[PROOFSTEP]\nrw [compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_inv_app, eqToHom_trans, eqToHom_refl]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian", "llama_tokens": 3418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.25284104150788594}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\nh : L \u22a3 F\nA : C\ninst\u271d\u00b2 : PreservesLimitsOfShape (Discrete WalkingPair) L\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nB : D\n\u22a2 IsIso (NatTrans.app (frobeniusMorphism F h A) B)\n[PROOFSTEP]\ndsimp [frobeniusMorphism]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\nh : L \u22a3 F\nA : C\ninst\u271d\u00b2 : PreservesLimitsOfShape (Discrete WalkingPair) L\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nB : D\n\u22a2 IsIso (prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\n\u22a2 prod.map (\ud835\udfd9 (F.obj A)) (NatTrans.app (expComparison F A) B) \u226b NatTrans.app (exp.ev (F.obj A)) (F.obj B) =\n inv (prodComparison F A (A \u27f9 B)) \u226b F.map (NatTrans.app (exp.ev A) B)\n[PROOFSTEP]\nconvert transferNatTrans_counit _ _ (prodComparisonNatIso F A).inv B using 2\n[GOAL]\ncase h.e'_3.h.h.e'_6.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\ne_1\u271d :\n (F.obj A \u2a2f (exp A \u22d9 F).obj B \u27f6 (\ud835\udfed D).obj (F.obj B)) =\n ((prod.functor.obj (F.obj A)).obj ((exp A \u22d9 F).obj B) \u27f6 (\ud835\udfed D).obj (F.obj B))\ne_3\u271d : (F.obj A \u2a2f F.obj (A \u27f9 B)) = (F \u22d9 prod.functor.obj (F.obj A)).obj (A \u27f9 B)\ne_4\u271d : F.obj (A \u2a2f A \u27f9 B) = (prod.functor.obj A \u22d9 F).obj (A \u27f9 B)\n\u22a2 inv (prodComparison F A (A \u27f9 B)) = NatTrans.app (prodComparisonNatIso F A).inv (A \u27f9 B)\n[PROOFSTEP]\napply IsIso.inv_eq_of_hom_inv_id\n[GOAL]\ncase h.e'_3.h.h.e'_6.h.hom_inv_id\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\ne_1\u271d :\n (F.obj A \u2a2f (exp A \u22d9 F).obj B \u27f6 (\ud835\udfed D).obj (F.obj B)) =\n ((prod.functor.obj (F.obj A)).obj ((exp A \u22d9 F).obj B) \u27f6 (\ud835\udfed D).obj (F.obj B))\ne_3\u271d : (F.obj A \u2a2f F.obj (A \u27f9 B)) = (F \u22d9 prod.functor.obj (F.obj A)).obj (A \u27f9 B)\ne_4\u271d : F.obj (A \u2a2f A \u27f9 B) = (prod.functor.obj A \u22d9 F).obj (A \u27f9 B)\n\u22a2 prodComparison F A (A \u27f9 B) \u226b NatTrans.app (prodComparisonNatIso F A).inv (A \u27f9 B) = \ud835\udfd9 (F.obj (A \u2a2f A \u27f9 B))\n[PROOFSTEP]\nsimp only [Limits.prodComparisonNatIso_inv, asIso_inv, NatIso.isIso_inv_app, IsIso.hom_inv_id]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\n\u22a2 F.map (NatTrans.app (exp.coev A) B) \u226b NatTrans.app (expComparison F A) (A \u2a2f B) =\n NatTrans.app (exp.coev (F.obj A)) (F.obj B) \u226b (exp (F.obj A)).map (inv (prodComparison F A B))\n[PROOFSTEP]\nconvert unit_transferNatTrans _ _ (prodComparisonNatIso F A).inv B using 3\n[GOAL]\ncase h.e'_3.h.h.e'_7.h.h.e'_8.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\ne_1\u271d :\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj (A \u2a2f B)) =\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj ((prod.functor.obj A).obj B))\ne_4\u271d :\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj B) =\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_5\u271d : (F.obj A \u27f9 F.obj (A \u2a2f B)) = (F.obj A \u27f9 (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B))\ne_6\u271d : (prod.functor.obj (F.obj A)).obj (F.obj B) = (prod.functor.obj (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_7\u271d : F.obj (A \u2a2f B) = (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B)\n\u22a2 inv (prodComparison F A B) = NatTrans.app (prodComparisonNatIso F A).inv ((\ud835\udfed C).obj B)\n[PROOFSTEP]\napply IsIso.inv_eq_of_hom_inv_id\n[GOAL]\ncase h.e'_3.h.h.e'_7.h.h.e'_8.h.hom_inv_id\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\ne_1\u271d :\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj (A \u2a2f B)) =\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj ((prod.functor.obj A).obj B))\ne_4\u271d :\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj B) =\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_5\u271d : (F.obj A \u27f9 F.obj (A \u2a2f B)) = (F.obj A \u27f9 (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B))\ne_6\u271d : (prod.functor.obj (F.obj A)).obj (F.obj B) = (prod.functor.obj (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_7\u271d : F.obj (A \u2a2f B) = (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B)\n\u22a2 prodComparison F A B \u226b NatTrans.app (prodComparisonNatIso F A).inv ((\ud835\udfed C).obj B) = \ud835\udfd9 (F.obj (A \u2a2f B))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.h.e'_7.h.h.e'_8.h.hom_inv_id\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\ne_1\u271d :\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj (A \u2a2f B)) =\n (F.obj ((\ud835\udfed C).obj B) \u27f6 (F \u22d9 exp (F.obj A)).obj ((prod.functor.obj A).obj B))\ne_4\u271d :\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj B) =\n (prod.functor.obj (F.obj A) \u22d9 exp (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_5\u271d : (F.obj A \u27f9 F.obj (A \u2a2f B)) = (F.obj A \u27f9 (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B))\ne_6\u271d : (prod.functor.obj (F.obj A)).obj (F.obj B) = (prod.functor.obj (F.obj A)).obj (F.obj ((\ud835\udfed C).obj B))\ne_7\u271d : F.obj (A \u2a2f B) = (prod.functor.obj A \u22d9 F).obj ((\ud835\udfed C).obj B)\n\u22a2 prodComparison F A B \u226b NatTrans.app (inv (NatTrans.mk fun B => prodComparison F A B)) B = \ud835\udfd9 (F.obj (A \u2a2f B))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA B : C\n\u22a2 CartesianClosed.uncurry (NatTrans.app (expComparison F A) B) =\n inv (prodComparison F A (A \u27f9 B)) \u226b F.map (NatTrans.app (exp.ev A) B)\n[PROOFSTEP]\nrw [uncurry_eq, expComparison_ev]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA A' : C\nf : A' \u27f6 A\n\u22a2 expComparison F A \u226b whiskerLeft F (pre (F.map f)) = whiskerRight (pre f) F \u226b expComparison F A'\n[PROOFSTEP]\next B\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA A' : C\nf : A' \u27f6 A\nB : C\n\u22a2 NatTrans.app (expComparison F A \u226b whiskerLeft F (pre (F.map f))) B =\n NatTrans.app (whiskerRight (pre f) F \u226b expComparison F A') B\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA A' : C\nf : A' \u27f6 A\nB : C\n\u22a2 NatTrans.app (expComparison F A) B \u226b NatTrans.app (pre (F.map f)) (F.obj B) =\n F.map (NatTrans.app (pre f) B) \u226b NatTrans.app (expComparison F A') B\n[PROOFSTEP]\napply uncurry_injective\n[GOAL]\ncase w.h.a\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nA A' : C\nf : A' \u27f6 A\nB : C\n\u22a2 CartesianClosed.uncurry (NatTrans.app (expComparison F A) B \u226b NatTrans.app (pre (F.map f)) (F.obj B)) =\n CartesianClosed.uncurry (F.map (NatTrans.app (pre f) B) \u226b NatTrans.app (expComparison F A') B)\n[PROOFSTEP]\nrw [uncurry_natural_left, uncurry_natural_left, uncurry_expComparison, uncurry_pre, prod.map_swap_assoc, \u2190 F.map_id,\n expComparison_ev, \u2190 F.map_id, \u2190 prodComparison_inv_natural_assoc, \u2190 prodComparison_inv_natural_assoc, \u2190 F.map_comp, \u2190\n F.map_comp, prod_map_pre_app_comp_ev]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\n\u22a2 \u2191(transferNatTransSelf (Adjunction.comp h (exp.adjunction A)) (Adjunction.comp (exp.adjunction (F.obj A)) h))\n (frobeniusMorphism F h A) =\n expComparison F A\n[PROOFSTEP]\nrw [\u2190 Equiv.eq_symm_apply]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\n\u22a2 frobeniusMorphism F h A =\n \u2191(transferNatTransSelf (Adjunction.comp h (exp.adjunction A)) (Adjunction.comp (exp.adjunction (F.obj A)) h)).symm\n (expComparison F A)\n[PROOFSTEP]\next B : 2\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 NatTrans.app (frobeniusMorphism F h A) B =\n NatTrans.app\n (\u2191(transferNatTransSelf (Adjunction.comp h (exp.adjunction A))\n (Adjunction.comp (exp.adjunction (F.obj A)) h)).symm\n (expComparison F A))\n B\n[PROOFSTEP]\ndsimp [frobeniusMorphism, transferNatTransSelf, transferNatTrans, Adjunction.comp]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n \ud835\udfd9 (L.obj (F.obj A \u2a2f B)) \u226b\n (L.map\n (prod.map (\ud835\udfd9 (F.obj A))\n ((NatTrans.app h.unit B \u226b\n F.map (NatTrans.app (exp.adjunction A).unit (L.obj B)) \u226b \ud835\udfd9 (F.obj (A \u27f9 A \u2a2f L.obj B))) \u226b\n \ud835\udfd9 (F.obj (A \u27f9 A \u2a2f L.obj B)) \u226b\n NatTrans.app (expComparison F A) (A \u2a2f L.obj B) \u226b \ud835\udfd9 (F.obj A \u27f9 F.obj (A \u2a2f L.obj B)))) \u226b\n \ud835\udfd9 (L.obj (F.obj A \u2a2f F.obj A \u27f9 F.obj (A \u2a2f L.obj B))) \u226b\n L.map (NatTrans.app (exp.adjunction (F.obj A)).counit (F.obj (A \u2a2f L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)) \u226b\n \ud835\udfd9 (A \u2a2f L.obj B)\n[PROOFSTEP]\nsimp only [id_comp, comp_id]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map\n (prod.map (\ud835\udfd9 (F.obj A))\n ((NatTrans.app h.unit B \u226b F.map (NatTrans.app (exp.adjunction A).unit (L.obj B))) \u226b\n NatTrans.app (expComparison F A) (A \u2a2f L.obj B))) \u226b\n L.map (NatTrans.app (exp.adjunction (F.obj A)).counit (F.obj (A \u2a2f L.obj B))) \u226b NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\nrw [\u2190 L.map_comp_assoc, prod.map_id_comp, assoc]\n -- Porting note: need to use `erw` here.\n -- https://github.com/leanprover-community/mathlib4/issues/5164\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map\n (prod.map (\ud835\udfd9 (F.obj A)) (NatTrans.app h.unit B \u226b F.map (NatTrans.app (exp.adjunction A).unit (L.obj B))) \u226b\n prod.map (\ud835\udfd9 (F.obj A)) (NatTrans.app (expComparison F A) (A \u2a2f L.obj B)) \u226b\n NatTrans.app (exp.adjunction (F.obj A)).counit (F.obj (A \u2a2f L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\nerw [expComparison_ev]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map\n (prod.map (\ud835\udfd9 (F.obj A)) (NatTrans.app h.unit B \u226b F.map (NatTrans.app (exp.adjunction A).unit (L.obj B))) \u226b\n inv (prodComparison F A (A \u27f9 A \u2a2f L.obj B)) \u226b F.map (NatTrans.app (exp.ev A) (A \u2a2f L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\nrw [prod.map_id_comp, assoc, \u2190 F.map_id, \u2190 prodComparison_inv_natural_assoc, \u2190 F.map_comp]\n -- Porting note: need to use `erw` here.\n -- https://github.com/leanprover-community/mathlib4/issues/5164\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map\n (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b\n inv (prodComparison F A (L.obj B)) \u226b\n F.map\n (prod.map (\ud835\udfd9 A) (NatTrans.app (exp.adjunction A).unit (L.obj B)) \u226b\n NatTrans.app (exp.ev A) (A \u2a2f L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\nerw [exp.ev_coev]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map\n (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b\n inv (prodComparison F A (L.obj B)) \u226b F.map (\ud835\udfd9 (A \u2a2f L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\nrw [F.map_id (A \u2a2f L.obj B), comp_id]\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) =\n L.map (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b inv (prodComparison F A (L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.h\u2081\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 (prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B))) \u226b prod.fst =\n (L.map (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b inv (prodComparison F A (L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)) \u226b\n prod.fst\n[PROOFSTEP]\nrw [assoc, assoc, \u2190 h.counit_naturality, \u2190 L.map_comp_assoc, assoc, inv_prodComparison_map_fst]\n[GOAL]\ncase w.h.h\u2081\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) \u226b prod.fst =\n L.map (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b prod.fst) \u226b NatTrans.app h.counit A\n[PROOFSTEP]\nsimp\n[GOAL]\ncase w.h.h\u2082\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 (prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B))) \u226b prod.snd =\n (L.map (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b inv (prodComparison F A (L.obj B))) \u226b\n NatTrans.app h.counit (A \u2a2f L.obj B)) \u226b\n prod.snd\n[PROOFSTEP]\nrw [assoc, assoc, \u2190 h.counit_naturality, \u2190 L.map_comp_assoc, assoc, inv_prodComparison_map_snd]\n[GOAL]\ncase w.h.h\u2082\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\nB : D\n\u22a2 prodComparison L (F.obj A) B \u226b prod.map (NatTrans.app h.counit A) (\ud835\udfd9 (L.obj B)) \u226b prod.snd =\n L.map (prod.map (F.map (\ud835\udfd9 A)) (NatTrans.app h.unit B) \u226b prod.snd) \u226b NatTrans.app h.counit (L.obj B)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\ni : IsIso (expComparison F A)\n\u22a2 IsIso (frobeniusMorphism F h A)\n[PROOFSTEP]\nrw [\u2190 frobeniusMorphism_mate F h] at i \n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\ni :\n IsIso\n (\u2191(transferNatTransSelf (Adjunction.comp h (exp.adjunction A)) (Adjunction.comp (exp.adjunction (F.obj A)) h))\n (frobeniusMorphism F h A))\n\u22a2 IsIso (frobeniusMorphism F h A)\n[PROOFSTEP]\nexact @transferNatTransSelf_of_iso _ _ _ _ _ _ _ _ _ _ _ i\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\ni : IsIso (frobeniusMorphism F h A)\n\u22a2 IsIso (expComparison F A)\n[PROOFSTEP]\nrw [\u2190 frobeniusMorphism_mate F h]\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2075 : Category.{v, u'} D\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasFiniteProducts D\nF : C \u2964 D\nL : D \u2964 C\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : CartesianClosed D\ninst\u271d : PreservesLimitsOfShape (Discrete WalkingPair) F\nh : L \u22a3 F\nA : C\ni : IsIso (frobeniusMorphism F h A)\n\u22a2 IsIso\n (\u2191(transferNatTransSelf (Adjunction.comp h (exp.adjunction A)) (Adjunction.comp (exp.adjunction (F.obj A)) h))\n (frobeniusMorphism F h A))\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Closed.Functor", "llama_tokens": 10181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5428632979641571, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.25237800026447665}} {"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\nn : \u2115\na : \u03b1\nh : Stream'.cons none (\u2191c) n = some a\n\u22a2 Stream'.cons none (\u2191c) (n + 1) = some a\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\na : \u03b1\nh : Stream'.cons none (\u2191c) Nat.zero = some a\n\u22a2 Stream'.cons none (\u2191c) (Nat.zero + 1) = some a\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Stream'.cons none (\u2191c) (Nat.succ n) = some a\n\u22a2 Stream'.cons none (\u2191c) (Nat.succ n + 1) = some a\n[PROOFSTEP]\nexact c.2 h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\n\u22a2 destruct s = Sum.inl a \u2192 s = pure a\n[PROOFSTEP]\ndsimp [destruct]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\n\u22a2 (match \u2191s 0 with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a \u2192\n s = pure a\n[PROOFSTEP]\ninduction' f0 : s.1 0 with _\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\n\u22a2 (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a \u2192\n s = pure a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\n\u22a2 (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a \u2192\n s = pure a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\nh :\n (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n\u22a2 s = pure a\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh :\n (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n\u22a2 s = pure a\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase some.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh :\n (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n\u22a2 \u2191s = \u2191(pure a)\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase some.a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh :\n (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : \u2115\n\u22a2 \u2191s n = \u2191(pure a) n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase some.a.h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh :\n (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n\u22a2 \u2191s Nat.zero = \u2191(pure a) Nat.zero\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase some.a.h.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh' : val\u271d = a\n\u22a2 \u2191s Nat.zero = \u2191(pure a) Nat.zero\n[PROOFSTEP]\nrwa [h'] at f0 \n[GOAL]\ncase some.a.h.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nval\u271d : \u03b1\nf0 : \u2191s 0 = some val\u271d\nh :\n (match some val\u271d with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : \u2115\nIH : \u2191s n = \u2191(pure a) n\n\u22a2 \u2191s (Nat.succ n) = \u2191(pure a) (Nat.succ n)\n[PROOFSTEP]\nexact s.2 IH\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\n\u22a2 destruct s = Sum.inr s' \u2192 s = think s'\n[PROOFSTEP]\ndsimp [destruct]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\n\u22a2 (match \u2191s 0 with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inr s' \u2192\n s = think s'\n[PROOFSTEP]\ninduction' f0 : s.1 0 with a'\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\n\u22a2 (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inr s' \u2192\n s = think s'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\na' : \u03b1\nf0 : \u2191s 0 = some a'\n\u22a2 (match some a' with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inr s' \u2192\n s = think s'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\nh :\n (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inr s'\n\u22a2 s = think s'\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\nh' : tail s = s'\n\u22a2 s = think s'\n[PROOFSTEP]\nrw [\u2190 h']\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\nf0 : \u2191s 0 = none\nh' : tail s = s'\n\u22a2 s = think (tail s)\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase none.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns' : Computation \u03b1\nx\u271d : Option \u03b1\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nf0\u271d : \u2191{ val := f, property := al } 0 = x\u271d\nf0 : \u2191{ val := f, property := al } 0 = none\nh' : tail { val := f, property := al } = s'\n\u22a2 { val := f, property := al } = think (tail { val := f, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase none.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns' : Computation \u03b1\nx\u271d : Option \u03b1\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nf0\u271d : \u2191{ val := f, property := al } 0 = x\u271d\nf0 : \u2191{ val := f, property := al } 0 = none\nh' : tail { val := f, property := al } = s'\n\u22a2 \u2191{ val := f, property := al } = \u2191(think (tail { val := f, property := al }))\n[PROOFSTEP]\ndsimp [think, tail]\n[GOAL]\ncase none.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns' : Computation \u03b1\nx\u271d : Option \u03b1\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nf0\u271d : \u2191{ val := f, property := al } 0 = x\u271d\nf0 : \u2191{ val := f, property := al } 0 = none\nh' : tail { val := f, property := al } = s'\n\u22a2 f = Stream'.cons none (Stream'.tail f)\n[PROOFSTEP]\nrw [\u2190 f0]\n[GOAL]\ncase none.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns' : Computation \u03b1\nx\u271d : Option \u03b1\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nf0\u271d : \u2191{ val := f, property := al } 0 = x\u271d\nf0 : \u2191{ val := f, property := al } 0 = none\nh' : tail { val := f, property := al } = s'\n\u22a2 f = Stream'.cons (\u2191{ val := f, property := al } 0) (Stream'.tail f)\n[PROOFSTEP]\nexact (Stream'.eta f).symm\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s' : Computation \u03b1\nx\u271d : Option \u03b1\nf0\u271d : \u2191s 0 = x\u271d\na' : \u03b1\nf0 : \u2191s 0 = some a'\nh :\n (match some a' with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inr s'\n\u22a2 s = think s'\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\n\u22a2 tail (think s) = s\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 tail (think { val := f, property := al }) = { val := f, property := al }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 \u2191(tail (think { val := f, property := al })) = \u2191{ val := f, property := al }\n[PROOFSTEP]\ndsimp [tail, think]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\ns : Computation \u03b1\nh1 : (a : \u03b1) \u2192 C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C (think s)\nv : \u03b1\nH : destruct s = Sum.inl v\n\u22a2 C s\n[PROOFSTEP]\nrw [destruct_eq_pure H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\ns : Computation \u03b1\nh1 : (a : \u03b1) \u2192 C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C (think s)\nv : \u03b1\nH : destruct s = Sum.inl v\n\u22a2 C (pure v)\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\ns : Computation \u03b1\nh1 : (a : \u03b1) \u2192 C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C (think s)\nv : Computation \u03b1\na : Stream' (Option \u03b1)\ns' : \u2200 \u2983n : \u2115\u2984 \u2983a_1 : \u03b1\u2984, a n = some a_1 \u2192 a (n + 1) = some a_1\nH : destruct s = Sum.inr { val := a, property := s' }\n\u22a2 C s\n[PROOFSTEP]\nrw [destruct_eq_think H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\ns : Computation \u03b1\nh1 : (a : \u03b1) \u2192 C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C (think s)\nv : Computation \u03b1\na : Stream' (Option \u03b1)\ns' : \u2200 \u2983n : \u2115\u2984 \u2983a_1 : \u03b1\u2984, a n = some a_1 \u2192 a (n + 1) = some a_1\nH : destruct s = Sum.inr { val := a, property := s' }\n\u22a2 C (think { val := a, property := s' })\n[PROOFSTEP]\napply h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 Computation \u03b1\n[PROOFSTEP]\nrefine' \u27e8Stream'.corec' (Corec.f f) (Sum.inr b), fun n a' h => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\nh : Stream'.corec' (Corec.f f) (Sum.inr b) n = some a'\n\u22a2 Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a'\n[PROOFSTEP]\nrw [Stream'.corec'_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\nh : Stream'.corec' (Corec.f f) (Sum.inr b) n = some a'\n\u22a2 Stream'.cons (Corec.f f (Sum.inr b)).fst (Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).snd) (n + 1) = some a'\n[PROOFSTEP]\nchange Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).2 n = some a'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\nh : Stream'.corec' (Corec.f f) (Sum.inr b) n = some a'\n\u22a2 Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).snd n = some a'\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\n\u22a2 Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).snd n = some a'\n[PROOFSTEP]\ngeneralize Sum.inr b = o\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\no : \u03b1 \u2295 \u03b2\n\u22a2 Stream'.corec' (Corec.f f) o n = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd n = some a'\n[PROOFSTEP]\nrevert o\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nn : \u2115\na' : \u03b1\n\u22a2 \u2200 (o : \u03b1 \u2295 \u03b2), Stream'.corec' (Corec.f f) o n = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd n = some a'\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\n\u22a2 \u2200 (o : \u03b1 \u2295 \u03b2),\n Stream'.corec' (Corec.f f) o Nat.zero = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd Nat.zero = some a'\n[PROOFSTEP]\nintro o\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\nn : \u2115\nIH : \u2200 (o : \u03b1 \u2295 \u03b2), Stream'.corec' (Corec.f f) o n = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd n = some a'\n\u22a2 \u2200 (o : \u03b1 \u2295 \u03b2),\n Stream'.corec' (Corec.f f) o (Nat.succ n) = some a' \u2192\n Stream'.corec' (Corec.f f) (Corec.f f o).snd (Nat.succ n) = some a'\n[PROOFSTEP]\nintro o\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\no : \u03b1 \u2295 \u03b2\n\u22a2 Stream'.corec' (Corec.f f) o Nat.zero = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd Nat.zero = some a'\n[PROOFSTEP]\nchange (Corec.f f o).1 = some a' \u2192 (Corec.f f (Corec.f f o).2).1 = some a'\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\no : \u03b1 \u2295 \u03b2\n\u22a2 (Corec.f f o).fst = some a' \u2192 (Corec.f f (Corec.f f o).snd).fst = some a'\n[PROOFSTEP]\ncases' o with _ b\n[GOAL]\ncase zero.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' val\u271d : \u03b1\n\u22a2 (Corec.f f (Sum.inl val\u271d)).fst = some a' \u2192 (Corec.f f (Corec.f f (Sum.inl val\u271d)).snd).fst = some a'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb\u271d : \u03b2\na' : \u03b1\nb : \u03b2\n\u22a2 (Corec.f f (Sum.inr b)).fst = some a' \u2192 (Corec.f f (Corec.f f (Sum.inr b)).snd).fst = some a'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' val\u271d : \u03b1\nh : (Corec.f f (Sum.inl val\u271d)).fst = some a'\n\u22a2 (Corec.f f (Corec.f f (Sum.inl val\u271d)).snd).fst = some a'\n[PROOFSTEP]\nexact h\n[GOAL]\ncase zero.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb\u271d : \u03b2\na' : \u03b1\nb : \u03b2\nh : (Corec.f f (Sum.inr b)).fst = some a'\n\u22a2 (Corec.f f (Corec.f f (Sum.inr b)).snd).fst = some a'\n[PROOFSTEP]\nunfold Corec.f at *\n[GOAL]\ncase zero.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb\u271d : \u03b2\na' : \u03b1\nb : \u03b2\nh :\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).fst =\n some a'\n\u22a2 (match\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).snd with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).fst =\n some a'\n[PROOFSTEP]\nsplit\n[GOAL]\ncase zero.inr.h_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb\u271d : \u03b2\na' : \u03b1\nb : \u03b2\nh :\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).fst =\n some a'\nx\u271d : \u03b1 \u2295 \u03b2\na\u271d : \u03b1\nheq\u271d :\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).snd =\n Sum.inl a\u271d\n\u22a2 (some a\u271d, Sum.inl a\u271d).fst = some a'\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase zero.inr.h_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb\u271d : \u03b2\na' : \u03b1\nb : \u03b2\nh :\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).fst =\n some a'\nx\u271d : \u03b1 \u2295 \u03b2\nval\u271d : \u03b2\nheq\u271d :\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).snd =\n Sum.inr val\u271d\n\u22a2 (match f val\u271d with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f val\u271d).fst =\n some a'\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\nn : \u2115\nIH : \u2200 (o : \u03b1 \u2295 \u03b2), Stream'.corec' (Corec.f f) o n = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd n = some a'\no : \u03b1 \u2295 \u03b2\n\u22a2 Stream'.corec' (Corec.f f) o (Nat.succ n) = some a' \u2192\n Stream'.corec' (Corec.f f) (Corec.f f o).snd (Nat.succ n) = some a'\n[PROOFSTEP]\nrw [Stream'.corec'_eq (Corec.f f) (Corec.f f o).2, Stream'.corec'_eq (Corec.f f) o]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\na' : \u03b1\nn : \u2115\nIH : \u2200 (o : \u03b1 \u2295 \u03b2), Stream'.corec' (Corec.f f) o n = some a' \u2192 Stream'.corec' (Corec.f f) (Corec.f f o).snd n = some a'\no : \u03b1 \u2295 \u03b2\n\u22a2 Stream'.cons (Corec.f f o).fst (Stream'.corec' (Corec.f f) (Corec.f f o).snd) (Nat.succ n) = some a' \u2192\n Stream'.cons (Corec.f f (Corec.f f o).snd).fst (Stream'.corec' (Corec.f f) (Corec.f f (Corec.f f o).snd).snd)\n (Nat.succ n) =\n some a'\n[PROOFSTEP]\nexact IH (Corec.f f o).2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 destruct (corec f b) = rmap (corec f) (f b)\n[PROOFSTEP]\ndsimp [corec, destruct]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 (match Stream'.corec' (Corec.f f) (Sum.inr b) 0 with\n | none =>\n Sum.inr\n (tail\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') })\n | some a => Sum.inl a) =\n match f b with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }\n[PROOFSTEP]\nrw [show Stream'.corec' (Corec.f f) (Sum.inr b) 0 = Sum.rec Option.some (\u03bb _ => none) (f b)\n by\n dsimp [Corec.f, Stream'.corec', Stream'.corec, Stream'.map, Stream'.nth, Stream'.iterate]\n match (f b) with\n | Sum.inl x => rfl\n | Sum.inr x => rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 Stream'.corec' (Corec.f f) (Sum.inr b) 0 = Sum.rec some (fun x => none) (f b)\n[PROOFSTEP]\ndsimp [Corec.f, Stream'.corec', Stream'.corec, Stream'.map, Stream'.nth, Stream'.iterate]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none) =\n Sum.rec some (fun x => none) (f b)\n[PROOFSTEP]\nmatch (f b) with\n| Sum.inl x => rfl\n| Sum.inr x => rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx : \u03b1\n\u22a2 (match Sum.inl x with\n | Sum.inl a => some a\n | Sum.inr val => none) =\n Sum.rec some (fun x => none) (Sum.inl x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb x : \u03b2\n\u22a2 (match Sum.inr x with\n | Sum.inl a => some a\n | Sum.inr val => none) =\n Sum.rec some (fun x => none) (Sum.inr x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\n\u22a2 (match Sum.rec some (fun x => none) (f b) with\n | none =>\n Sum.inr\n (tail\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') })\n | some a => Sum.inl a) =\n match f b with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }\n[PROOFSTEP]\ninduction' h : f b with a b'\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\na : \u03b1\nh : f b = Sum.inl a\n\u22a2 (match Sum.rec some (fun x => none) (Sum.inl a) with\n | none =>\n Sum.inr\n (tail\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') })\n | some a => Sum.inl a) =\n match Sum.inl a with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 (match Sum.rec some (fun x => none) (Sum.inr b') with\n | none =>\n Sum.inr\n (tail\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') })\n | some a => Sum.inl a) =\n match Sum.inr b' with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n { val := Stream'.corec' (Corec.f f) (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }\n[PROOFSTEP]\ndsimp [Corec.f, destruct]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 Sum.inr\n (tail\n {\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }) =\n Sum.inr\n {\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b'),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b') n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b') (n + 1) = some a') }\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 tail\n {\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') } =\n {\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b'),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b') n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b') (n + 1) = some a') }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 \u2191(tail\n {\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b) n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b) (n + 1) = some a') }) =\n \u2191{\n val :=\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b'),\n property :=\n (_ :\n \u2200 (n : \u2115) (a' : \u03b1),\n Stream'.corec' (Corec.f f) (Sum.inr b') n = some a' \u2192\n Stream'.corec' (Corec.f f) (Sum.inr b') (n + 1) = some a') }\n[PROOFSTEP]\ndsimp [corec, tail]\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 Stream'.tail\n (Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b)) =\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b')\n[PROOFSTEP]\nrw [Stream'.corec'_eq, Stream'.tail_cons]\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (match Sum.inr b with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b)).snd =\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b')\n[PROOFSTEP]\ndsimp [Corec.f]\n[GOAL]\ncase inr.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b2 \u2192 \u03b1 \u2295 \u03b2\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nh\u271d : f b = x\u271d\nb' : \u03b2\nh : f b = Sum.inr b'\n\u22a2 Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (f b) =\n Stream'.corec'\n (fun x =>\n match x with\n | Sum.inl a => (some a, Sum.inl a)\n | Sum.inr b =>\n (match f b with\n | Sum.inl a => some a\n | Sum.inr val => none,\n f b))\n (Sum.inr b')\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 s\u2081 = s\u2082\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2191s\u2081 = \u2191s\u2082\n[PROOFSTEP]\napply Stream'.eq_of_bisim fun x y => \u2203 s s' : Computation \u03b1, s.1 = x \u2227 s'.1 = y \u2227 R s s'\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 Stream'.IsBisimulation fun x y => \u2203 s s', \u2191s = x \u2227 \u2191s' = y \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\ndsimp [Stream'.IsBisimulation]\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2200 \u2983s\u2081 s\u2082 : Stream' (Option \u03b1)\u2984,\n (\u2203 s s', \u2191s = s\u2081 \u2227 \u2191s' = s\u2082 \u2227 R s s') \u2192\n Stream'.head s\u2081 = Stream'.head s\u2082 \u2227 \u2203 s s', \u2191s = Stream'.tail s\u2081 \u2227 \u2191s' = Stream'.tail s\u2082 \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nintro t\u2081 t\u2082 e\n[GOAL]\ncase a.bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\n\u22a2 Stream'.head t\u2081 = Stream'.head t\u2082 \u2227 \u2203 s s', \u2191s = Stream'.tail t\u2081 \u2227 \u2191s' = Stream'.tail t\u2082 \u2227 R s s'\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nexact\n match t\u2081, t\u2082, e with\n | _, _, \u27e8s, s', rfl, rfl, r\u27e9 =>\n by\n suffices head s = head s' \u2227 R (tail s) (tail s') from\n And.imp id (fun r => \u27e8tail s, tail s', by cases s; rfl, by cases s'; rfl, r\u27e9) this\n have h := bisim r; revert r h\n apply recOn s _ _ <;> intro r' <;> apply recOn s' _ _ <;> intro a' r h\n \u00b7 constructor <;> dsimp at h \n \u00b7 rw [h]\n \u00b7 rw [h] at r \n rw [tail_pure, tail_pure, h]\n assumption\n \u00b7 rw [destruct_pure, destruct_think] at h \n exact False.elim h\n \u00b7 rw [destruct_pure, destruct_think] at h \n exact False.elim h\n \u00b7 simp at h \n simp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr : R s s'\n\u22a2 Stream'.head \u2191s = Stream'.head \u2191s' \u2227 \u2203 s_1 s'_1, \u2191s_1 = Stream'.tail \u2191s \u2227 \u2191s'_1 = Stream'.tail \u2191s' \u2227 R s_1 s'_1\n[PROOFSTEP]\nsuffices head s = head s' \u2227 R (tail s) (tail s') from\n And.imp id (fun r => \u27e8tail s, tail s', by cases s; rfl, by cases s'; rfl, r\u27e9) this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr\u271d : R s s'\nthis : head s = head s' \u2227 R (tail s) (tail s')\nr : R (tail s) (tail s')\n\u22a2 \u2191(tail s) = Stream'.tail \u2191s\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns' : Computation \u03b1\nval\u271d : Stream' (Option \u03b1)\nproperty\u271d : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, val\u271d n = some a \u2192 val\u271d (n + 1) = some a\nr\u271d : R { val := val\u271d, property := property\u271d } s'\nthis : head { val := val\u271d, property := property\u271d } = head s' \u2227 R (tail { val := val\u271d, property := property\u271d }) (tail s')\nr : R (tail { val := val\u271d, property := property\u271d }) (tail s')\n\u22a2 \u2191(tail { val := val\u271d, property := property\u271d }) = Stream'.tail \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr\u271d : R s s'\nthis : head s = head s' \u2227 R (tail s) (tail s')\nr : R (tail s) (tail s')\n\u22a2 \u2191(tail s') = Stream'.tail \u2191s'\n[PROOFSTEP]\ncases s'\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d\u00b9 : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns : Computation \u03b1\nval\u271d : Stream' (Option \u03b1)\nproperty\u271d : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, val\u271d n = some a \u2192 val\u271d (n + 1) = some a\nr\u271d : R s { val := val\u271d, property := property\u271d }\nthis : head s = head { val := val\u271d, property := property\u271d } \u2227 R (tail s) (tail { val := val\u271d, property := property\u271d })\nr : R (tail s) (tail { val := val\u271d, property := property\u271d })\n\u22a2 \u2191(tail { val := val\u271d, property := property\u271d }) = Stream'.tail \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr : R s s'\n\u22a2 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\nhave h := bisim r\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr : R s s'\nh : BisimO R (destruct s) (destruct s')\n\u22a2 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\nrevert r h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\n\u22a2 R s s' \u2192 BisimO R (destruct s) (destruct s') \u2192 head s = head s' \u2227 R (tail s) (tail s')\n[PROOFSTEP]\napply recOn s _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n R (pure a) s' \u2192 BisimO R (destruct (pure a)) (destruct s') \u2192 head (pure a) = head s' \u2227 R (tail (pure a)) (tail s')\n[PROOFSTEP]\nintro r'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n R (think s) s' \u2192\n BisimO R (destruct (think s)) (destruct s') \u2192 head (think s) = head s' \u2227 R (tail (think s)) (tail s')\n[PROOFSTEP]\nintro r'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' : \u03b1\n\u22a2 R (pure r') s' \u2192 BisimO R (destruct (pure r')) (destruct s') \u2192 head (pure r') = head s' \u2227 R (tail (pure r')) (tail s')\n[PROOFSTEP]\napply recOn s' _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' : Computation \u03b1\n\u22a2 R (think r') s' \u2192\n BisimO R (destruct (think r')) (destruct s') \u2192 head (think r') = head s' \u2227 R (tail (think r')) (tail s')\n[PROOFSTEP]\napply recOn s' _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' : \u03b1\n\u22a2 \u2200 (a : \u03b1),\n R (pure r') (pure a) \u2192\n BisimO R (destruct (pure r')) (destruct (pure a)) \u2192\n head (pure r') = head (pure a) \u2227 R (tail (pure r')) (tail (pure a))\n[PROOFSTEP]\nintro a' r h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' : \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n R (pure r') (think s) \u2192\n BisimO R (destruct (pure r')) (destruct (think s)) \u2192\n head (pure r') = head (think s) \u2227 R (tail (pure r')) (tail (think s))\n[PROOFSTEP]\nintro a' r h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n R (think r') (pure a) \u2192\n BisimO R (destruct (think r')) (destruct (pure a)) \u2192\n head (think r') = head (pure a) \u2227 R (tail (think r')) (tail (pure a))\n[PROOFSTEP]\nintro a' r h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n R (think r') (think s) \u2192\n BisimO R (destruct (think r')) (destruct (think s)) \u2192\n head (think r') = head (think s) \u2227 R (tail (think r')) (tail (think s))\n[PROOFSTEP]\nintro a' r h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure r') (pure a')\nh : BisimO R (destruct (pure r')) (destruct (pure a'))\n\u22a2 head (pure r') = head (pure a') \u2227 R (tail (pure r')) (tail (pure a'))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure r') (pure a')\nh : BisimO R (destruct (pure r')) (destruct (pure a'))\n\u22a2 head (pure r') = head (pure a')\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure r') (pure a')\nh : BisimO R (destruct (pure r')) (destruct (pure a'))\n\u22a2 R (tail (pure r')) (tail (pure a'))\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure r') (pure a')\nh : r' = a'\n\u22a2 head (pure r') = head (pure a')\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure r') (pure a')\nh : r' = a'\n\u22a2 R (tail (pure r')) (tail (pure a'))\n[PROOFSTEP]\nrw [h] at r \n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure a') (pure a')\nh : r' = a'\n\u22a2 R (tail (pure r')) (tail (pure a'))\n[PROOFSTEP]\nrw [tail_pure, tail_pure, h]\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' a' : \u03b1\nr : R (pure a') (pure a')\nh : r' = a'\n\u22a2 R (pure a') (pure a')\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' : \u03b1\na' : Computation \u03b1\nr : R (pure r') (think a')\nh : BisimO R (destruct (pure r')) (destruct (think a'))\n\u22a2 head (pure r') = head (think a') \u2227 R (tail (pure r')) (tail (think a'))\n[PROOFSTEP]\nrw [destruct_pure, destruct_think] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' : Computation \u03b1\nr' : \u03b1\na' : Computation \u03b1\nr : R (pure r') (think a')\nh : BisimO R (Sum.inl r') (Sum.inr a')\n\u22a2 head (pure r') = head (think a') \u2227 R (tail (pure r')) (tail (think a'))\n[PROOFSTEP]\nexact False.elim h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' : Computation \u03b1\na' : \u03b1\nr : R (think r') (pure a')\nh : BisimO R (destruct (think r')) (destruct (pure a'))\n\u22a2 head (think r') = head (pure a') \u2227 R (tail (think r')) (tail (pure a'))\n[PROOFSTEP]\nrw [destruct_pure, destruct_think] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' : Computation \u03b1\na' : \u03b1\nr : R (think r') (pure a')\nh : BisimO R (Sum.inr r') (Sum.inl a')\n\u22a2 head (think r') = head (pure a') \u2227 R (tail (think r')) (tail (pure a'))\n[PROOFSTEP]\nexact False.elim h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' a' : Computation \u03b1\nr : R (think r') (think a')\nh : BisimO R (destruct (think r')) (destruct (think a'))\n\u22a2 head (think r') = head (think a') \u2227 R (tail (think r')) (tail (think a'))\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr\u271d : R s\u2081 s\u2082\nt\u2081 t\u2082 : Stream' (Option \u03b1)\ne : \u2203 s s', \u2191s = t\u2081 \u2227 \u2191s' = t\u2082 \u2227 R s s'\ns s' r' a' : Computation \u03b1\nr : R (think r') (think a')\nh : R r' a'\n\u22a2 head (think r') = head (think a') \u2227 R (tail (think r')) (tail (think a'))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Computation \u03b1 \u2192 Computation \u03b1 \u2192 Prop\nbisim : IsBisimulation R\ns\u2081 s\u2082 : Computation \u03b1\nr : R s\u2081 s\u2082\n\u22a2 \u2203 s s', \u2191s = \u2191s\u2081 \u2227 \u2191s' = \u2191s\u2082 \u2227 R s s'\n[PROOFSTEP]\nexact \u27e8s\u2081, s\u2082, rfl, rfl, r\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u2191s m = some a \u2192 \u2191s n = some a\n[PROOFSTEP]\ncases' s with f al\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nm n : \u2115\nh : m \u2264 n\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 \u2191{ val := f, property := al } m = some a \u2192 \u2191{ val := f, property := al } n = some a\n[PROOFSTEP]\ninduction' h with n _ IH\n[GOAL]\ncase mk.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nm n : \u2115\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 \u2191{ val := f, property := al } m = some a \u2192 \u2191{ val := f, property := al } m = some a\ncase mk.step\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nm n\u271d : \u2115\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nn : \u2115\na\u271d : Nat.le m n\nIH : \u2191{ val := f, property := al } m = some a \u2192 \u2191{ val := f, property := al } n = some a\n\u22a2 \u2191{ val := f, property := al } m = some a \u2192 \u2191{ val := f, property := al } (Nat.succ n) = some a\n[PROOFSTEP]\nexacts [id, fun h2 => al (IH h2)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na b : \u03b1\nm : \u2115\nha : (fun b => some a = b) (Stream'.nth (\u2191s) m)\nn : \u2115\nhb : (fun b_1 => some b = b_1) (Stream'.nth (\u2191s) n)\n\u22a2 a = b\n[PROOFSTEP]\ninjection (le_stable s (le_max_left m n) ha.symm).symm.trans (le_stable s (le_max_right m n) hb.symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nx\u271d : Terminates s\na : \u03b1\nn : \u2115\nh : (fun b => some a = b) (Stream'.nth (\u2191s) n)\n\u22a2 Option.isSome (\u2191s n) = true\n[PROOFSTEP]\ndsimp [Stream'.nth] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nx\u271d : Terminates s\na : \u03b1\nn : \u2115\nh : some a = \u2191s n\n\u22a2 Option.isSome (\u2191s n) = true\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nx\u271d : Terminates s\na : \u03b1\nn : \u2115\nh : some a = \u2191s n\n\u22a2 Option.isSome (some a) = true\n[PROOFSTEP]\nexact rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : (fun b => some a = b) (Stream'.nth (\u2191(think s)) n)\n\u22a2 a \u2208 s\n[PROOFSTEP]\ncases' n with n'\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nh : some a = Stream'.nth (\u2191(think s)) Nat.zero\n\u22a2 a \u2208 s\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn' : \u2115\nh : some a = Stream'.nth (\u2191(think s)) (Nat.succ n')\n\u22a2 a \u2208 s\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn' : \u2115\nh : some a = Stream'.nth (\u2191(think s)) (Nat.succ n')\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact \u27e8n', h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nx\u271d : a \u2208 empty \u03b1\nn : \u2115\nh : (fun b => some a = b) (Stream'.nth (\u2191(empty \u03b1)) n)\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\n\u22a2 s = empty \u03b1\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\n\u22a2 \u2191s = \u2191(empty \u03b1)\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\nn : \u2115\n\u22a2 \u2191s n = \u2191(empty \u03b1) n\n[PROOFSTEP]\ninduction' h : s.val n with _\n[GOAL]\ncase a.h.none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\nn : \u2115\nx\u271d : Option \u03b1\nh\u271d : \u2191s n = x\u271d\nh : \u2191s n = none\n\u22a2 none = \u2191(empty \u03b1) n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\nn : \u2115\nx\u271d : Option \u03b1\nh\u271d : \u2191s n = x\u271d\nval\u271d : \u03b1\nh : \u2191s n = some val\u271d\n\u22a2 some val\u271d = \u2191(empty \u03b1) n\n[PROOFSTEP]\nrefine' absurd _ H\n[GOAL]\ncase a.h.some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nH : \u00acTerminates s\nn : \u2115\nx\u271d : Option \u03b1\nh\u271d : \u2191s n = x\u271d\nval\u271d : \u03b1\nh : \u2191s n = some val\u271d\n\u22a2 Terminates s\n[PROOFSTEP]\nexact \u27e8\u27e8_, _, h.symm\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\na : \u03b1\n\u22a2 get s = a \u2192 a \u2208 s\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh\u271d : Terminates s\na : \u03b1\nh : get s = a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh\u271d : Terminates s\na : \u03b1\nh : get s = a\n\u22a2 get s \u2208 s\n[PROOFSTEP]\napply get_mem\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\na : \u03b1\np : s ~> a\n\u22a2 a \u2208 s\n[PROOFSTEP]\ncases' h with h\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\np : s ~> a\nh : \u2203 a, a \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\ncases' h with a' h\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\np : s ~> a\na' : \u03b1\nh : a' \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [p h]\n[GOAL]\ncase mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\np : s ~> a\na' : \u03b1\nh : a' \u2208 s\n\u22a2 a' \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nT : Terminates s\na : \u03b1\nh : a \u2208 s\n\u22a2 Results s a (length s)\n[PROOFSTEP]\nrw [\u2190 get_eq_of_mem _ h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nT : Terminates s\na : \u03b1\nh : a \u2208 s\n\u22a2 Results s (get s) (length s)\n[PROOFSTEP]\napply results_of_terminates\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na b : \u03b1\nm n : \u2115\nh1 : Results s a m\nh2 : Results s b n\n\u22a2 m = n\n[PROOFSTEP]\nhaveI := h1.terminates\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na b : \u03b1\nm n : \u2115\nh1 : Results s a m\nh2 : Results s b n\nthis : Terminates s\n\u22a2 m = n\n[PROOFSTEP]\nhaveI := h2.terminates\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na b : \u03b1\nm n : \u2115\nh1 : Results s a m\nh2 : Results s b n\nthis\u271d this : Terminates s\n\u22a2 m = n\n[PROOFSTEP]\nrw [\u2190 h1.length, h2.length]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\n\u22a2 length (think s) = length s + 1\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\n\u22a2 length (think s) \u2264 length s + 1\n[PROOFSTEP]\nexact Nat.find_min' _ (Nat.find_spec ((terminates_def _).1 h))\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\n\u22a2 length s + 1 \u2264 length (think s)\n[PROOFSTEP]\nhave : (Option.isSome ((think s).val (length (think s))) : Prop) :=\n Nat.find_spec ((terminates_def _).1 s.think_terminates)\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nthis : Option.isSome (\u2191(think s) (length (think s))) = true\n\u22a2 length s + 1 \u2264 length (think s)\n[PROOFSTEP]\nrevert this\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\n\u22a2 Option.isSome (\u2191(think s) (length (think s))) = true \u2192 length s + 1 \u2264 length (think s)\n[PROOFSTEP]\ncases' length (think s) with n\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\n\u22a2 Option.isSome (\u2191(think s) Nat.zero) = true \u2192 length s + 1 \u2264 Nat.zero\n[PROOFSTEP]\nintro this\n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nn : \u2115\n\u22a2 Option.isSome (\u2191(think s) (Nat.succ n)) = true \u2192 length s + 1 \u2264 Nat.succ n\n[PROOFSTEP]\nintro this\n[GOAL]\ncase a.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nthis : Option.isSome (\u2191(think s) Nat.zero) = true\n\u22a2 length s + 1 \u2264 Nat.zero\n[PROOFSTEP]\nsimp [think, Stream'.cons] at this \n[GOAL]\ncase a.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nn : \u2115\nthis : Option.isSome (\u2191(think s) (Nat.succ n)) = true\n\u22a2 length s + 1 \u2264 Nat.succ n\n[PROOFSTEP]\napply Nat.succ_le_succ\n[GOAL]\ncase a.succ.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nn : \u2115\nthis : Option.isSome (\u2191(think s) (Nat.succ n)) = true\n\u22a2 length s \u2264 n\n[PROOFSTEP]\napply Nat.find_min'\n[GOAL]\ncase a.succ.a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nh : Terminates s\nn : \u2115\nthis : Option.isSome (\u2191(think s) (Nat.succ n)) = true\n\u22a2 Option.isSome (\u2191s n) = true\n[PROOFSTEP]\napply this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results s a n\nthis : Terminates s\n\u22a2 length (think s) = n + 1\n[PROOFSTEP]\nrw [length_think, h.length]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a n\n\u22a2 \u2203 m, Results s a m \u2227 n = m + 1\n[PROOFSTEP]\nhaveI := of_think_terminates h.terminates\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a n\nthis : Terminates s\n\u22a2 \u2203 m, Results s a m \u2227 n = m + 1\n[PROOFSTEP]\nhave := results_of_terminates' _ (of_think_mem h.mem)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a n\nthis\u271d : Terminates s\nthis : Results s a (length s)\n\u22a2 \u2203 m, Results s a m \u2227 n = m + 1\n[PROOFSTEP]\nexact \u27e8_, this, Results.len_unique h (results_think this)\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a (n + 1)\n\u22a2 Results s a n\n[PROOFSTEP]\nlet \u27e8n', r, e\u27e9 := of_results_think h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a (n + 1)\nn' : \u2115\nr : Results s a n'\ne : n + 1 = n' + 1\n\u22a2 Results s a n\n[PROOFSTEP]\ninjection e with h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a (n + 1)\nn' : \u2115\nr : Results s a n'\nh' : Nat.add n 0 = Nat.add n' 0\n\u22a2 Results s a n\n[PROOFSTEP]\nrw [Nat.add, Nat.add] at h' \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results (think s) a (n + 1)\nn' : \u2115\nr : Results s a n'\nh' : n = n'\n\u22a2 Results s a n\n[PROOFSTEP]\nrwa [h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\n\u22a2 Results (thinkN (pure a) n) a n\n[PROOFSTEP]\nhave := results_thinkN n (results_pure a)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\nthis : Results (thinkN (pure a) n) a (0 + n)\n\u22a2 Results (thinkN (pure a) n) a n\n[PROOFSTEP]\nrwa [Nat.zero_add] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\na : \u03b1\nn : \u2115\nh : Results s a n\n\u22a2 s = thinkN (pure a) n\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\n\u22a2 \u2200 {s : Computation \u03b1}, Results s a n \u2192 s = thinkN (pure a) n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\n\u22a2 \u2200 {s : Computation \u03b1}, Results s a Nat.zero \u2192 s = thinkN (pure a) Nat.zero\n[PROOFSTEP]\nintro s\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a n \u2192 s = thinkN (pure a) n\n\u22a2 \u2200 {s : Computation \u03b1}, Results s a (Nat.succ n) \u2192 s = thinkN (pure a) (Nat.succ n)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Computation \u03b1\n\u22a2 Results s a Nat.zero \u2192 s = thinkN (pure a) Nat.zero\n[PROOFSTEP]\napply recOn s (fun a' => _) fun s => _\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a n \u2192 s = thinkN (pure a) n\ns : Computation \u03b1\n\u22a2 Results s a (Nat.succ n) \u2192 s = thinkN (pure a) (Nat.succ n)\n[PROOFSTEP]\napply recOn s (fun a' => _) fun s => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Computation \u03b1\n\u22a2 \u2200 (a' : \u03b1), Results (pure a') a Nat.zero \u2192 pure a' = thinkN (pure a) Nat.zero\n[PROOFSTEP]\nintro a h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1), Results (think s) a Nat.zero \u2192 think s = thinkN (pure a) Nat.zero\n[PROOFSTEP]\nintro a h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a n \u2192 s = thinkN (pure a) n\ns : Computation \u03b1\n\u22a2 \u2200 (a' : \u03b1), Results (pure a') a (Nat.succ n) \u2192 pure a' = thinkN (pure a) (Nat.succ n)\n[PROOFSTEP]\nintro a h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a n \u2192 s = thinkN (pure a) n\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1), Results (think s) a (Nat.succ n) \u2192 think s = thinkN (pure a) (Nat.succ n)\n[PROOFSTEP]\nintro a h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns : Computation \u03b1\na : \u03b1\nh : Results (pure a) a\u271d Nat.zero\n\u22a2 pure a = thinkN (pure a\u271d) Nat.zero\n[PROOFSTEP]\nrw [\u2190 eq_of_pure_mem h.mem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns : Computation \u03b1\na : \u03b1\nh : Results (pure a) a\u271d Nat.zero\n\u22a2 pure a\u271d = thinkN (pure a\u271d) Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns a : Computation \u03b1\nh : Results (think a) a\u271d Nat.zero\n\u22a2 think a = thinkN (pure a\u271d) Nat.zero\n[PROOFSTEP]\ncases' of_results_think h with n h\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns a : Computation \u03b1\nh\u271d : Results (think a) a\u271d Nat.zero\nn : \u2115\nh : Results a a\u271d n \u2227 Nat.zero = n + 1\n\u22a2 think a = thinkN (pure a\u271d) Nat.zero\n[PROOFSTEP]\ncases h\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\ns a : Computation \u03b1\nh : Results (think a) a\u271d Nat.zero\nn : \u2115\nleft\u271d : Results a a\u271d n\nright\u271d : Nat.zero = n + 1\n\u22a2 think a = thinkN (pure a\u271d) Nat.zero\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a\u271d n \u2192 s = thinkN (pure a\u271d) n\ns : Computation \u03b1\na : \u03b1\nh : Results (pure a) a\u271d (Nat.succ n)\n\u22a2 pure a = thinkN (pure a\u271d) (Nat.succ n)\n[PROOFSTEP]\nhave := h.len_unique (results_pure _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a\u271d n \u2192 s = thinkN (pure a\u271d) n\ns : Computation \u03b1\na : \u03b1\nh : Results (pure a) a\u271d (Nat.succ n)\nthis : Nat.succ n = 0\n\u22a2 pure a = thinkN (pure a\u271d) (Nat.succ n)\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a\u271d n \u2192 s = thinkN (pure a\u271d) n\ns a : Computation \u03b1\nh : Results (think a) a\u271d (Nat.succ n)\n\u22a2 think a = thinkN (pure a\u271d) (Nat.succ n)\n[PROOFSTEP]\nrw [IH (results_think_iff.1 h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results s a\u271d n \u2192 s = thinkN (pure a\u271d) n\ns a : Computation \u03b1\nh : Results (think a) a\u271d (Nat.succ n)\n\u22a2 think (thinkN (pure a\u271d) n) = thinkN (pure a\u271d) (Nat.succ n)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\n\u22a2 C s\n[PROOFSTEP]\nhaveI T := terminates_of_mem M\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\nT : Terminates s\n\u22a2 C s\n[PROOFSTEP]\nrw [eq_thinkN' s, get_eq_of_mem s M]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\nT : Terminates s\n\u22a2 C (thinkN (pure a) (length s))\n[PROOFSTEP]\ngeneralize length s = n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\nT : Terminates s\nn : \u2115\n\u22a2 C (thinkN (pure a) n)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\nT : Terminates s\n\u22a2 C (thinkN (pure a) Nat.zero)\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nC : Computation \u03b1 \u2192 Sort v\na : \u03b1\ns : Computation \u03b1\nM : a \u2208 s\nh1 : C (pure a)\nh2 : (s : Computation \u03b1) \u2192 C s \u2192 C (think s)\nT : Terminates s\nn : \u2115\nIH : C (thinkN (pure a) n)\n\u22a2 C (thinkN (pure a) (Nat.succ n))\n[PROOFSTEP]\nexacts [h1, h2 _ IH]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\n\u22a2 Stream'.map (fun o => Option.casesOn o none (some \u2218 f)) s n = some b \u2192\n Stream'.map (fun o => Option.casesOn o none (some \u2218 f)) s (n + 1) = some b\n[PROOFSTEP]\ndsimp [Stream'.map, Stream'.nth]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\n\u22a2 Option.rec none (fun val => some (f val)) (s n) = some b \u2192\n Option.rec none (fun val => some (f val)) (s (n + 1)) = some b\n[PROOFSTEP]\ninduction' e : s n with a\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\ne : s n = none\n\u22a2 Option.rec none (fun val => some (f val)) none = some b \u2192\n Option.rec none (fun val => some (f val)) (s (n + 1)) = some b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\na : \u03b1\ne : s n = some a\n\u22a2 Option.rec none (fun val => some (f val)) (some a) = some b \u2192\n Option.rec none (fun val => some (f val)) (s (n + 1)) = some b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase none\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\ne : s n = none\nh : Option.rec none (fun val => some (f val)) none = some b\n\u22a2 Option.rec none (fun val => some (f val)) (s (n + 1)) = some b\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\na : \u03b1\ne : s n = some a\nh : Option.rec none (fun val => some (f val)) (some a) = some b\n\u22a2 Option.rec none (fun val => some (f val)) (s (n + 1)) = some b\n[PROOFSTEP]\nrw [al e]\n[GOAL]\ncase some\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nn : \u2115\nb : \u03b2\nx\u271d : Option \u03b1\ne\u271d : s n = x\u271d\na : \u03b1\ne : s n = some a\nh : Option.rec none (fun val => some (f val)) (some a) = some b\n\u22a2 Option.rec none (fun val => some (f val)) (some a) = some b\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 map f (think { val := s, property := al }) = think (map f { val := s, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 \u2191(map f (think { val := s, property := al })) = \u2191(think (map f { val := s, property := al }))\n[PROOFSTEP]\ndsimp [think, map]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) (Stream'.cons none s) =\n Stream'.cons none (Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) s)\n[PROOFSTEP]\nrw [Stream'.map_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 destruct (map f s) = lmap f (rmap (map f) (destruct s))\n[PROOFSTEP]\napply s.recOn\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1), destruct (map f (pure a)) = lmap f (rmap (map f) (destruct (pure a)))\n[PROOFSTEP]\nintro\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1), destruct (map f (think s)) = lmap f (rmap (map f) (destruct (think s)))\n[PROOFSTEP]\nintro\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\na\u271d : \u03b1\n\u22a2 destruct (map f (pure a\u271d)) = lmap f (rmap (map f) (destruct (pure a\u271d)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns s\u271d : Computation \u03b1\n\u22a2 destruct (map f (think s\u271d)) = lmap f (rmap (map f) (destruct (think s\u271d)))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 map id { val := f, property := al } = { val := f, property := al }\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 \u2191(map id { val := f, property := al }) = \u2191{ val := f, property := al }\n[PROOFSTEP]\nsimp [map, Function.comp]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some val) o) f = f\n[PROOFSTEP]\nhave e : @Option.rec \u03b1 (fun _ => Option \u03b1) none some = id := by ext \u27e8\u27e9 <;> rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\n\u22a2 Option.rec none some = id\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase h.none.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 Option.rec none some none \u2194 a\u271d \u2208 id none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.some.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\nval\u271d a\u271d : \u03b1\n\u22a2 a\u271d \u2208 Option.rec none some (some val\u271d) \u2194 a\u271d \u2208 id (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\ne : Option.rec none some = id\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some val) o) f = f\n[PROOFSTEP]\nhave h : ((fun x : Option \u03b1 => x) = id) := by rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\ne : Option.rec none some = id\n\u22a2 (fun x => x) = id\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, f n = some a \u2192 f (n + 1) = some a\ne : Option.rec none some = id\nh : (fun x => x) = id\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some val) o) f = f\n[PROOFSTEP]\nsimp [e, h, Stream'.map_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 map (g \u2218 f) { val := s, property := al } = map g (map f { val := s, property := al })\n[PROOFSTEP]\napply Subtype.eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 \u2191(map (g \u2218 f) { val := s, property := al }) = \u2191(map g (map f { val := s, property := al }))\n[PROOFSTEP]\ndsimp [map]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some (g (f val))) o) s =\n Stream'.map\n ((fun o => Option.rec none (fun val => some (g val)) o) \u2218 fun o => Option.rec none (fun val => some (f val)) o) s\n[PROOFSTEP]\napply congr_arg fun f : _ \u2192 Option \u03b3 => Stream'.map f s\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 (fun o => Option.rec none (fun val => some (g (f val))) o) =\n (fun o => Option.rec none (fun val => some (g val)) o) \u2218 fun o => Option.rec none (fun val => some (f val)) o\n[PROOFSTEP]\next \u27e8\u27e9\n[GOAL]\ncase a.h.none.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.rec none (fun val => some (g (f val))) none \u2194\n a\u271d \u2208\n ((fun o => Option.rec none (fun val => some (g val)) o) \u2218 fun o => Option.rec none (fun val => some (f val)) o)\n none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.some.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\nval\u271d : \u03b1\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 Option.rec none (fun val => some (g (f val))) (some val\u271d) \u2194\n a\u271d \u2208\n ((fun o => Option.rec none (fun val => some (g val)) o) \u2218 fun o => Option.rec none (fun val => some (f val)) o)\n (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\n\u22a2 bind (pure a) f = f a\n[PROOFSTEP]\napply eq_of_bisim fun c\u2081 c\u2082 => c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)) (destruct c\u2081) (destruct c\u2082)\n[PROOFSTEP]\nexact\n match c\u2081, c\u2082, h with\n | _, _, Or.inl \u27e8rfl, rfl\u27e9 => by\n simp [bind, Bind.f]\n cases' destruct (f a) with b cb <;> simp [Bind.g]\n | _, c, Or.inr rfl => by\n simp [Bind.f]\n cases' destruct c with b cb <;> simp [Bind.g]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082))\n (destruct (bind (pure a) f)) (destruct (f a))\n[PROOFSTEP]\nsimp [bind, Bind.f]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\n\u22a2 match\n match Bind.g (destruct (f a)) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n destruct (f a) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inl (pure a)) \u2227\n s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\ncases' destruct (f a) with b cb\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\nb : \u03b2\n\u22a2 match\n match Bind.g (Sum.inl b) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n Sum.inl b with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inl (pure a)) \u2227\n s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\nsimp [Bind.g]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\ncb : Computation \u03b2\n\u22a2 match\n match Bind.g (Sum.inr cb) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n Sum.inr cb with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inl (pure a)) \u2227\n s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\nsimp [Bind.g]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\nc : Computation \u03b2\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082))\n (destruct (corec (Bind.f f) (Sum.inr c))) (destruct c)\n[PROOFSTEP]\nsimp [Bind.f]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\nc : Computation \u03b2\n\u22a2 match\n match Bind.g (destruct c) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n destruct c with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s = bind (pure a) f \u2227 s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\ncases' destruct c with b cb\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\nc : Computation \u03b2\nb : \u03b2\n\u22a2 match\n match Bind.g (Sum.inl b) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n Sum.inl b with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s = bind (pure a) f \u2227 s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\nsimp [Bind.g]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = bind (pure a) f \u2227 c\u2082 = f a \u2228 c\u2081 = corec (Bind.f f) (Sum.inr c\u2082)\nc cb : Computation \u03b2\n\u22a2 match\n match Bind.g (Sum.inr cb) with\n | Sum.inl a => Sum.inl a\n | Sum.inr b =>\n Sum.inr\n (corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n b),\n Sum.inr cb with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' =>\n s = bind (pure a) f \u2227 s' = f a \u2228\n s =\n corec\n (fun x =>\n match x with\n | Sum.inl ca =>\n match destruct ca with\n | Sum.inl a => Bind.g (destruct (f a))\n | Sum.inr ca' => Sum.inr (Sum.inl ca')\n | Sum.inr cb => Bind.g (destruct cb))\n (Sum.inr s')\n | x, x_1 => False\n[PROOFSTEP]\nsimp [Bind.g]\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nf : \u03b1 \u2192 Computation \u03b2\n\u22a2 bind (pure a) f = bind (pure a) f \u2227 f a = f a \u2228 bind (pure a) f = corec (Bind.f f) (Sum.inr (f a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\n\u22a2 destruct (bind (think c) f) = Sum.inr (bind c f)\n[PROOFSTEP]\nsimp [bind, Bind.f]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 bind s (pure \u2218 f) = map f s\n[PROOFSTEP]\napply eq_of_bisim fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct c\u2081) (destruct c\u2082)\n[PROOFSTEP]\nexact\n match c\u2081, c\u2082, h with\n | _, c\u2082, Or.inl (Eq.refl _) => by cases' destruct c\u2082 with b cb <;> simp\n | _, _, Or.inr \u27e8s, rfl, rfl\u27e9 => by\n apply recOn s <;> intro s <;> simp\n exact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\nc\u2081 c\u2082\u271d : Computation \u03b2\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082\u271d = map f s\nc\u2082 : Computation \u03b2\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct c\u2082) (destruct c\u2082)\n[PROOFSTEP]\ncases' destruct c\u2082 with b cb\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\nc\u2081 c\u2082\u271d : Computation \u03b2\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082\u271d = map f s\nc\u2082 : Computation \u03b2\nb : \u03b2\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (Sum.inl b) (Sum.inl b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\nc\u2081 c\u2082\u271d : Computation \u03b2\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082\u271d = map f s\nc\u2082 cb : Computation \u03b2\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (Sum.inr cb) (Sum.inr cb)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct (bind s (pure \u2218 f)))\n (destruct (map f s))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct (bind (pure a) (pure \u2218 f)))\n (destruct (map f (pure a)))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct (bind (think s) (pure \u2218 f)))\n (destruct (map f (think s)))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d\u00b9 : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns\u271d : Computation \u03b1\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct (bind (pure s) (pure \u2218 f)))\n (destruct (map f (pure s)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d\u00b9 : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns\u271d s : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s) (destruct (bind (think s) (pure \u2218 f)))\n (destruct (map f (think s)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns\u271d\u00b9 : Computation \u03b1\nc\u2081 c\u2082 : Computation \u03b2\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure \u2218 f) \u2227 c\u2082 = map f s\ns\u271d s : Computation \u03b1\n\u22a2 bind s (pure \u2218 f) = map f s \u2228 \u2203 s_1, bind s (pure \u2218 f) = bind s_1 (pure \u2218 f) \u2227 map f s = map f s_1\n[PROOFSTEP]\nexact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\n\u22a2 bind s (pure \u2218 f) = map f s \u2228 \u2203 s_1, bind s (pure \u2218 f) = bind s_1 (pure \u2218 f) \u2227 map f s = map f s_1\n[PROOFSTEP]\nexact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\n\u22a2 bind s pure = s\n[PROOFSTEP]\napply eq_of_bisim fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s (pure) \u2227 c\u2082 = s\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct c\u2081) (destruct c\u2082)\n[PROOFSTEP]\nexact\n match c\u2081, c\u2082, h with\n | _, c\u2082, Or.inl (Eq.refl _) => by cases' destruct c\u2082 with b cb <;> simp\n | _, _, Or.inr \u27e8s, rfl, rfl\u27e9 => by apply recOn s <;> intro s <;> simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns c\u2081 c\u2082\u271d : Computation \u03b1\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082\u271d = s\nc\u2082 : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct c\u2082) (destruct c\u2082)\n[PROOFSTEP]\ncases' destruct c\u2082 with b cb\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns c\u2081 c\u2082\u271d : Computation \u03b1\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082\u271d = s\nc\u2082 : Computation \u03b1\nb : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (Sum.inl b) (Sum.inl b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns c\u2081 c\u2082\u271d : Computation \u03b1\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082\u271d = s\nc\u2082 cb : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (Sum.inr cb) (Sum.inr cb)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct (bind s pure)) (destruct s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct (bind (pure a) pure)) (destruct (pure a))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct (bind (think s) pure)) (destruct (think s))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\ns\u271d : Computation \u03b1\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct (bind (pure s) pure)) (destruct (pure s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 c\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s\ns\u271d s : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind s pure \u2227 c\u2082 = s) (destruct (bind (think s) pure)) (destruct (think s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\n\u22a2 bind s pure = s \u2228 \u2203 s_1, bind s pure = bind s_1 pure \u2227 s = s_1\n[PROOFSTEP]\nexact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\n\u22a2 bind (bind s f) g = bind s fun x => bind (f x) g\n[PROOFSTEP]\napply eq_of_bisim fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x : \u03b1 => bind (f x) g\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\n[PROOFSTEP]\nintro c\u2081 c\u2082 h\n[GOAL]\ncase bisim\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g) (destruct c\u2081)\n (destruct c\u2082)\n[PROOFSTEP]\nexact\n match c\u2081, c\u2082, h with\n | _, c\u2082, Or.inl (Eq.refl _) => by cases' destruct c\u2082 with b cb <;> simp\n | _, _, Or.inr \u27e8s, rfl, rfl\u27e9 => by\n apply recOn s <;> intro s <;> simp\n \u00b7 generalize f s = fs\n apply recOn fs <;> intro t <;> simp\n \u00b7 cases' destruct (g t) with b cb <;> simp\n \u00b7 exact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082\u271d : Computation \u03b3\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082\u271d = bind s fun x => bind (f x) g\nc\u2082 : Computation \u03b3\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g) (destruct c\u2082)\n (destruct c\u2082)\n[PROOFSTEP]\ncases' destruct c\u2082 with b cb\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082\u271d : Computation \u03b3\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082\u271d = bind s fun x => bind (f x) g\nc\u2082 : Computation \u03b3\nb : \u03b3\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g) (Sum.inl b)\n (Sum.inl b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082\u271d : Computation \u03b3\nh : c\u2081 = c\u2082\u271d \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082\u271d = bind s fun x => bind (f x) g\nc\u2082 cb : Computation \u03b3\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g) (Sum.inr cb)\n (Sum.inr cb)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g)\n (destruct (bind (bind s f) g)) (destruct (bind s fun x => bind (f x) g))\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g)\n (destruct (bind (bind (pure a) f) g)) (destruct (bind (pure a) fun x => bind (f x) g))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns : Computation \u03b1\n\u22a2 \u2200 (s : Computation \u03b1),\n BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g)\n (destruct (bind (bind (think s) f) g)) (destruct (bind (think s) fun x => bind (f x) g))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g)\n (destruct (bind (bind (pure s) f) g)) (destruct (bind (pure s) fun x => bind (f x) g))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d s : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g)\n (destruct (bind (bind (think s) f) g)) (destruct (bind (think s) fun x => bind (f x) g))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\n\u22a2 match destruct (bind (f s) g), destruct (bind (f s) g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\ngeneralize f s = fs\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\n\u22a2 match destruct (bind fs g), destruct (bind fs g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\napply recOn fs\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\n\u22a2 \u2200 (a : \u03b2),\n match destruct (bind (pure a) g), destruct (bind (pure a) g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nintro t\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\n\u22a2 \u2200 (s : Computation \u03b2),\n match destruct (bind (think s) g), destruct (bind (think s) g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nintro t\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\nt : \u03b2\n\u22a2 match destruct (bind (pure t) g), destruct (bind (pure t) g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs t : Computation \u03b2\n\u22a2 match destruct (bind (think t) g), destruct (bind (think t) g) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\nt : \u03b2\n\u22a2 match destruct (g t), destruct (g t) with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\ncases' destruct (g t) with b cb\n[GOAL]\ncase h1.h1.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\nt : \u03b2\nb : \u03b3\n\u22a2 match Sum.inl b, Sum.inl b with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1.h1.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d : Computation \u03b1\ns : \u03b1\nfs : Computation \u03b2\nt : \u03b2\ncb : Computation \u03b3\n\u22a2 match Sum.inr cb, Sum.inr cb with\n | Sum.inl a, Sum.inl a' => a = a'\n | Sum.inr s, Sum.inr s' => s = s' \u2228 \u2203 s_1, s = bind (bind s_1 f) g \u2227 s' = bind s_1 fun x => bind (f x) g\n | x, x_1 => False\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d\u00b9 : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\nc\u2081 c\u2082 : Computation \u03b3\nh : c\u2081 = c\u2082 \u2228 \u2203 s, c\u2081 = bind (bind s f) g \u2227 c\u2082 = bind s fun x => bind (f x) g\ns\u271d s : Computation \u03b1\n\u22a2 (bind (bind s f) g = bind s fun x => bind (f x) g) \u2228\n \u2203 s_1, bind (bind s f) g = bind (bind s_1 f) g \u2227 (bind s fun x => bind (f x) g) = bind s_1 fun x => bind (f x) g\n[PROOFSTEP]\nexact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\ncase r\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\ng : \u03b2 \u2192 Computation \u03b3\n\u22a2 (bind (bind s f) g = bind s fun x => bind (f x) g) \u2228\n \u2203 s_1, bind (bind s f) g = bind (bind s_1 f) g \u2227 (bind s fun x => bind (f x) g) = bind s_1 fun x => bind (f x) g\n[PROOFSTEP]\nexact Or.inr \u27e8s, rfl, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nh1 : Results s a m\nh2 : Results (f a) b n\n\u22a2 Results (bind s f) b (n + m)\n[PROOFSTEP]\nhave := h1.mem\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nh1 : Results s a m\nh2 : Results (f a) b n\nthis : a \u2208 s\n\u22a2 Results (bind s f) b (n + m)\n[PROOFSTEP]\nrevert m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\n\u22a2 \u2200 {m : \u2115}, Results s a m \u2192 Results (bind s f) b (n + m)\n[PROOFSTEP]\napply memRecOn this _ fun s IH => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\n\u22a2 \u2200 {m : \u2115}, Results (pure a) a m \u2192 Results (bind (pure a) f) b (n + m)\n[PROOFSTEP]\nintro _ h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\nm\u271d : \u2115\nh1 : Results (pure a) a m\u271d\n\u22a2 Results (bind (pure a) f) b (n + m\u271d)\n[PROOFSTEP]\nrw [ret_bind]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\nm\u271d : \u2115\nh1 : Results (pure a) a m\u271d\n\u22a2 Results (f a) b (n + m\u271d)\n[PROOFSTEP]\nrw [h1.len_unique (results_pure _)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\nm\u271d : \u2115\nh1 : Results (pure a) a m\u271d\n\u22a2 Results (f a) b (n + 0)\n[PROOFSTEP]\nexact h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\n\u22a2 \u2200 (s : Computation \u03b1),\n (\u2200 {m : \u2115}, Results s a m \u2192 Results (bind s f) b (n + m)) \u2192\n \u2200 {m : \u2115}, Results (think s) a m \u2192 Results (bind (think s) f) b (n + m)\n[PROOFSTEP]\nintro _ h3 _ h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\ns\u271d : Computation \u03b1\nh3 : \u2200 {m : \u2115}, Results s\u271d a m \u2192 Results (bind s\u271d f) b (n + m)\nm\u271d : \u2115\nh1 : Results (think s\u271d) a m\u271d\n\u22a2 Results (bind (think s\u271d) f) b (n + m\u271d)\n[PROOFSTEP]\nrw [think_bind]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\ns\u271d : Computation \u03b1\nh3 : \u2200 {m : \u2115}, Results s\u271d a m \u2192 Results (bind s\u271d f) b (n + m)\nm\u271d : \u2115\nh1 : Results (think s\u271d) a m\u271d\n\u22a2 Results (think (bind s\u271d f)) b (n + m\u271d)\n[PROOFSTEP]\ncases' of_results_think h1 with m' h\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\ns\u271d : Computation \u03b1\nh3 : \u2200 {m : \u2115}, Results s\u271d a m \u2192 Results (bind s\u271d f) b (n + m)\nm\u271d : \u2115\nh1 : Results (think s\u271d) a m\u271d\nm' : \u2115\nh : Results s\u271d a m' \u2227 m\u271d = m' + 1\n\u22a2 Results (think (bind s\u271d f)) b (n + m\u271d)\n[PROOFSTEP]\ncases' h with h1 e\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\ns\u271d : Computation \u03b1\nh3 : \u2200 {m : \u2115}, Results s\u271d a m \u2192 Results (bind s\u271d f) b (n + m)\nm\u271d : \u2115\nh1\u271d : Results (think s\u271d) a m\u271d\nm' : \u2115\nh1 : Results s\u271d a m'\ne : m\u271d = m' + 1\n\u22a2 Results (think (bind s\u271d f)) b (n + m\u271d)\n[PROOFSTEP]\nrw [e]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\nh2 : Results (f a) b n\nthis : a \u2208 s\ns\u271d : Computation \u03b1\nh3 : \u2200 {m : \u2115}, Results s\u271d a m \u2192 Results (bind s\u271d f) b (n + m)\nm\u271d : \u2115\nh1\u271d : Results (think s\u271d) a m\u271d\nm' : \u2115\nh1 : Results s\u271d a m'\ne : m\u271d = m' + 1\n\u22a2 Results (think (bind s\u271d f)) b (n + (m' + 1))\n[PROOFSTEP]\nexact results_think (h3 h1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nk : \u2115\n\u22a2 Results (bind s f) b k \u2192 \u2203 a m n, Results s a m \u2227 Results (f a) b n \u2227 k = n + m\n[PROOFSTEP]\ninduction' k with n IH generalizing s\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns : Computation \u03b1\n\u22a2 Results (bind s f) b Nat.zero \u2192 \u2203 a m n, Results s a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\napply recOn s (fun a => _) fun s' => _\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns : Computation \u03b1\n\u22a2 Results (bind s f) b (Nat.succ n) \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\napply recOn s (fun a => _) fun s' => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n Results (bind (pure a) f) b Nat.zero \u2192 \u2203 a_2 m n, Results (pure a) a_2 m \u2227 Results (f a_2) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\nintro e h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns : Computation \u03b1\n\u22a2 \u2200 (s' : Computation \u03b1),\n Results (bind (think s') f) b Nat.zero \u2192 \u2203 a m n, Results (think s') a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\nintro e h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns : Computation \u03b1\n\u22a2 \u2200 (a : \u03b1),\n Results (bind (pure a) f) b (Nat.succ n) \u2192\n \u2203 a_2 m n_1, Results (pure a) a_2 m \u2227 Results (f a_2) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nintro e h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns : Computation \u03b1\n\u22a2 \u2200 (s' : Computation \u03b1),\n Results (bind (think s') f) b (Nat.succ n) \u2192\n \u2203 a m n_1, Results (think s') a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nintro e h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns : Computation \u03b1\ne : \u03b1\nh : Results (bind (pure e) f) b Nat.zero\n\u22a2 \u2203 a m n, Results (pure e) a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\nsimp [thinkN] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns : Computation \u03b1\ne : \u03b1\nh : Results (f e) b 0\n\u22a2 \u2203 a m n, Results (pure e) a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\nrefine' \u27e8e, _, _, results_pure _, h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns e : Computation \u03b1\nh : Results (bind (think e) f) b Nat.zero\n\u22a2 \u2203 a m n, Results (think e) a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\nhave := congr_arg head (eq_thinkN h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\ns e : Computation \u03b1\nh : Results (bind (think e) f) b Nat.zero\nthis : head (bind (think e) f) = head (thinkN (pure b) Nat.zero)\n\u22a2 \u2203 a m n, Results (think e) a m \u2227 Results (f a) b n \u2227 Nat.zero = n + m\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns : Computation \u03b1\ne : \u03b1\nh : Results (bind (pure e) f) b (Nat.succ n)\n\u22a2 \u2203 a m n_1, Results (pure e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns : Computation \u03b1\ne : \u03b1\nh : Results (f e) b (Nat.succ n)\n\u22a2 \u2203 a m n_1, Results (pure e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nrefine' \u27e8e, _, n + 1, results_pure _, h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns e : Computation \u03b1\nh : Results (bind (think e) f) b (Nat.succ n)\n\u22a2 \u2203 a m n_1, Results (think e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns e : Computation \u03b1\nh : Results (bind e f) b n\n\u22a2 \u2203 a m n_1, Results (think e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nexact by\n let \u27e8a, m, n', h1, h2, e'\u27e9 := IH h\n rw [e']; exact \u27e8a, m.succ, n', results_think h1, h2, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns e : Computation \u03b1\nh : Results (bind e f) b n\n\u22a2 \u2203 a m n_1, Results (think e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nlet \u27e8a, m, n', h1, h2, e'\u27e9 := IH h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns e : Computation \u03b1\nh : Results (bind e f) b n\na : \u03b1\nm n' : \u2115\nh1 : Results e a m\nh2 : Results (f a) b n'\ne' : n = n' + m\n\u22a2 \u2203 a m n_1, Results (think e) a m \u2227 Results (f a) b n_1 \u2227 Nat.succ n = n_1 + m\n[PROOFSTEP]\nrw [e']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\nb : \u03b2\nn : \u2115\nIH : \u2200 {s : Computation \u03b1}, Results (bind s f) b n \u2192 \u2203 a m n_1, Results s a m \u2227 Results (f a) b n_1 \u2227 n = n_1 + m\ns e : Computation \u03b1\nh : Results (bind e f) b n\na : \u03b1\nm n' : \u2115\nh1 : Results e a m\nh2 : Results (f a) b n'\ne' : n = n' + m\n\u22a2 \u2203 a m_1 n, Results (think e) a m_1 \u2227 Results (f a) b n \u2227 Nat.succ (n' + m) = n + m_1\n[PROOFSTEP]\nexact \u27e8a, m.succ, n', results_think h1, h2, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nh1 : s ~> a\nh2 : f a ~> b\nb' : \u03b2\nbB : b' \u2208 bind s f\n\u22a2 b = b'\n[PROOFSTEP]\nrcases exists_of_mem_bind bB with \u27e8a', a's, ba'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nh1 : s ~> a\nh2 : f a ~> b\nb' : \u03b2\nbB : b' \u2208 bind s f\na' : \u03b1\na's : a' \u2208 s\nba' : b' \u2208 f a'\n\u22a2 b = b'\n[PROOFSTEP]\nrw [\u2190 h1 a's] at ba' \n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Computation \u03b1\nf : \u03b1 \u2192 Computation \u03b2\na : \u03b1\nb : \u03b2\nh1 : s ~> a\nh2 : f a ~> b\nb' : \u03b2\nbB : b' \u2208 bind s f\na' : \u03b1\na's : a' \u2208 s\nba' : b' \u2208 f a\n\u22a2 b = b'\n[PROOFSTEP]\nexact h2 ba'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Computation \u03b1\nm : a \u2208 s\n\u22a2 f a \u2208 map f s\n[PROOFSTEP]\nrw [\u2190 bind_pure]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Computation \u03b1\nm : a \u2208 s\n\u22a2 f a \u2208 bind s (pure \u2218 f)\n[PROOFSTEP]\napply mem_bind m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Computation \u03b1\nm : a \u2208 s\n\u22a2 f a \u2208 (pure \u2218 f) a\n[PROOFSTEP]\napply ret_mem\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Computation \u03b1\nh : b \u2208 map f s\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = b\n[PROOFSTEP]\nrw [\u2190 bind_pure] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Computation \u03b1\nh : b \u2208 bind s (pure \u2218 f)\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = b\n[PROOFSTEP]\nexact\n let \u27e8a, as, fb\u27e9 := exists_of_mem_bind h\n \u27e8a, as, mem_unique (ret_mem _) fb\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\ninst\u271d : Terminates s\n\u22a2 Terminates (map f s)\n[PROOFSTEP]\nrw [\u2190 bind_pure]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Computation \u03b1\ninst\u271d : Terminates s\n\u22a2 Terminates (bind s (pure \u2218 f))\n[PROOFSTEP]\nexact terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nc\u2082 : Computation \u03b1\n\u22a2 destruct (HOrElse.hOrElse (pure a) fun x => c\u2082) = Sum.inl a\n[PROOFSTEP]\nunfold HOrElse.hOrElse instHOrElse\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nc\u2082 : Computation \u03b1\n\u22a2 destruct ({ hOrElse := fun a b => OrElse.orElse a b }.1 (pure a) fun x => c\u2082) = Sum.inl a\n[PROOFSTEP]\nunfold OrElse.orElse instOrElse Alternative.orElse instAlternativeComputation\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nc\u2082 : Computation \u03b1\n\u22a2 destruct\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n (pure a) fun x => c\u2082) =\n Sum.inl a\n[PROOFSTEP]\nsimp [orElse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 : Computation \u03b1\na : \u03b1\n\u22a2 destruct (HOrElse.hOrElse (think c\u2081) fun x => pure a) = Sum.inl a\n[PROOFSTEP]\nunfold HOrElse.hOrElse instHOrElse\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 : Computation \u03b1\na : \u03b1\n\u22a2 destruct ({ hOrElse := fun a b => OrElse.orElse a b }.1 (think c\u2081) fun x => pure a) = Sum.inl a\n[PROOFSTEP]\nunfold OrElse.orElse instOrElse Alternative.orElse instAlternativeComputation\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 : Computation \u03b1\na : \u03b1\n\u22a2 destruct\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n (think c\u2081) fun x => pure a) =\n Sum.inl a\n[PROOFSTEP]\nsimp [orElse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\n\u22a2 destruct (HOrElse.hOrElse (think c\u2081) fun x => think c\u2082) = Sum.inr (HOrElse.hOrElse c\u2081 fun x => c\u2082)\n[PROOFSTEP]\nunfold HOrElse.hOrElse instHOrElse\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\n\u22a2 destruct ({ hOrElse := fun a b => OrElse.orElse a b }.1 (think c\u2081) fun x => think c\u2082) =\n Sum.inr ({ hOrElse := fun a b => OrElse.orElse a b }.1 c\u2081 fun x => c\u2082)\n[PROOFSTEP]\nunfold OrElse.orElse instOrElse Alternative.orElse instAlternativeComputation\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\n\u22a2 destruct\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n (think c\u2081) fun x => think c\u2082) =\n Sum.inr\n ({\n hOrElse := fun a b =>\n {\n orElse :=\n (let src := monad;\n Alternative.mk empty @orElse).3 }.1\n a b }.1\n c\u2081 fun x => c\u2082)\n[PROOFSTEP]\nsimp [orElse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\n\u22a2 (HOrElse.hOrElse (empty \u03b1) fun x => c) = c\n[PROOFSTEP]\napply eq_of_bisim (fun c\u2081 c\u2082 => (empty \u03b1 <|> c\u2082) = c\u2081) _ rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081\n[PROOFSTEP]\nintro s' s h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s) = s'\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081) (destruct s') (destruct s)\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s) = s'\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081) (destruct (HOrElse.hOrElse (empty \u03b1) fun x => s))\n (destruct s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s) = s'\n\u22a2 \u2200 (a : \u03b1),\n BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (empty \u03b1) fun x => pure a)) (destruct (pure a))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s) = s'\n\u22a2 \u2200 (s : Computation \u03b1),\n BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (empty \u03b1) fun x => think s)) (destruct (think s))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s\u271d) = s'\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (empty \u03b1) fun x => pure s)) (destruct (pure s))\n[PROOFSTEP]\nrw [think_empty]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s\u271d) = s'\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (empty \u03b1) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (empty \u03b1) fun x => think s)) (destruct (think s))\n[PROOFSTEP]\nrw [think_empty]\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s\u271d) = s'\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (think (empty \u03b1)) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (think (empty \u03b1)) fun x => pure s)) (destruct (pure s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s\u271d) = s'\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse (think (empty \u03b1)) fun x => c\u2082) = c\u2081)\n (destruct (HOrElse.hOrElse (think (empty \u03b1)) fun x => think s)) (destruct (think s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse (empty \u03b1) fun x => s\u271d) = s'\ns : Computation \u03b1\n\u22a2 (HOrElse.hOrElse (think (empty \u03b1)) fun x => s) = HOrElse.hOrElse (empty \u03b1) fun x => s\n[PROOFSTEP]\nrw [\u2190 think_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\n\u22a2 (HOrElse.hOrElse c fun x => empty \u03b1) = c\n[PROOFSTEP]\napply eq_of_bisim (fun c\u2081 c\u2082 => (c\u2082 <|> empty \u03b1) = c\u2081) _ rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc : Computation \u03b1\n\u22a2 IsBisimulation fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081\n[PROOFSTEP]\nintro s' s h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse s fun x => empty \u03b1) = s'\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081) (destruct s') (destruct s)\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse s fun x => empty \u03b1) = s'\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081) (destruct (HOrElse.hOrElse s fun x => empty \u03b1))\n (destruct s)\n[PROOFSTEP]\napply recOn s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse s fun x => empty \u03b1) = s'\n\u22a2 \u2200 (a : \u03b1),\n BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081)\n (destruct (HOrElse.hOrElse (pure a) fun x => empty \u03b1)) (destruct (pure a))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s : Computation \u03b1\nh : (HOrElse.hOrElse s fun x => empty \u03b1) = s'\n\u22a2 \u2200 (s : Computation \u03b1),\n BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081)\n (destruct (HOrElse.hOrElse (think s) fun x => empty \u03b1)) (destruct (think s))\n[PROOFSTEP]\nintro s\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s'\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081)\n (destruct (HOrElse.hOrElse (pure s) fun x => empty \u03b1)) (destruct (pure s))\n[PROOFSTEP]\nrw [think_empty]\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s'\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081)\n (destruct (HOrElse.hOrElse (think s) fun x => empty \u03b1)) (destruct (think s))\n[PROOFSTEP]\nrw [think_empty]\n[GOAL]\ncase h1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s'\ns : \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => think (empty \u03b1)) = c\u2081)\n (destruct (HOrElse.hOrElse (pure s) fun x => think (empty \u03b1))) (destruct (pure s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s'\ns : Computation \u03b1\n\u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => think (empty \u03b1)) = c\u2081)\n (destruct (HOrElse.hOrElse (think s) fun x => think (empty \u03b1))) (destruct (think s))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc s' s\u271d : Computation \u03b1\nh : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s'\ns : Computation \u03b1\n\u22a2 (HOrElse.hOrElse s fun x => think (empty \u03b1)) = HOrElse.hOrElse s fun x => empty \u03b1\n[PROOFSTEP]\nrw [\u2190 think_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : Computation \u03b1\na : \u03b1\nh1 : a \u2208 s\nh2 : a \u2208 t\na' : \u03b1\nma : a' \u2208 s\n\u22a2 a' \u2208 t\n[PROOFSTEP]\nrw [mem_unique ma h1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : Computation \u03b1\na : \u03b1\nh1 : a \u2208 s\nh2 : a \u2208 t\na' : \u03b1\nma : a' \u2208 s\n\u22a2 a \u2208 t\n[PROOFSTEP]\nexact h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : Computation \u03b1\na : \u03b1\nh1 : a \u2208 s\nh2 : a \u2208 t\na' : \u03b1\nma : a' \u2208 t\n\u22a2 a' \u2208 s\n[PROOFSTEP]\nrw [mem_unique ma h2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : Computation \u03b1\na : \u03b1\nh1 : a \u2208 s\nh2 : a \u2208 t\na' : \u03b1\nma : a' \u2208 t\n\u22a2 a \u2208 s\n[PROOFSTEP]\nexact h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\nh : c\u2081 ~ c\u2082\n\u22a2 Terminates c\u2081 \u2194 Terminates c\u2082\n[PROOFSTEP]\nsimp only [terminates_iff, exists_congr h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\nx\u271d : LiftRel (fun x x_1 => x = x_1) c\u2081 c\u2082\na : \u03b1\nh1 : \u2200 {a : \u03b1}, a \u2208 c\u2081 \u2192 \u2203 b, b \u2208 c\u2082 \u2227 (fun x x_1 => x = x_1) a b\nh2 : \u2200 {b : \u03b1}, b \u2208 c\u2082 \u2192 \u2203 a, a \u2208 c\u2081 \u2227 (fun x x_1 => x = x_1) a b\na1 : a \u2208 c\u2081\n\u22a2 a \u2208 c\u2082\n[PROOFSTEP]\nlet \u27e8b, b2, ab\u27e9 := h1 a1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\nx\u271d : LiftRel (fun x x_1 => x = x_1) c\u2081 c\u2082\na : \u03b1\nh1 : \u2200 {a : \u03b1}, a \u2208 c\u2081 \u2192 \u2203 b, b \u2208 c\u2082 \u2227 (fun x x_1 => x = x_1) a b\nh2 : \u2200 {b : \u03b1}, b \u2208 c\u2082 \u2192 \u2203 a, a \u2208 c\u2081 \u2227 (fun x x_1 => x = x_1) a b\na1 : a \u2208 c\u2081\nb : \u03b1\nb2 : b \u2208 c\u2082\nab : (fun x x_1 => x = x_1) a b\n\u22a2 a \u2208 c\u2082\n[PROOFSTEP]\nrwa [ab]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\nx\u271d : LiftRel (fun x x_1 => x = x_1) c\u2081 c\u2082\na : \u03b1\nh1 : \u2200 {a : \u03b1}, a \u2208 c\u2081 \u2192 \u2203 b, b \u2208 c\u2082 \u2227 (fun x x_1 => x = x_1) a b\nh2 : \u2200 {b : \u03b1}, b \u2208 c\u2082 \u2192 \u2203 a, a \u2208 c\u2081 \u2227 (fun x x_1 => x = x_1) a b\na2 : a \u2208 c\u2082\n\u22a2 a \u2208 c\u2081\n[PROOFSTEP]\nlet \u27e8b, b1, ab\u27e9 := h2 a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nc\u2081 c\u2082 : Computation \u03b1\nx\u271d : LiftRel (fun x x_1 => x = x_1) c\u2081 c\u2082\na : \u03b1\nh1 : \u2200 {a : \u03b1}, a \u2208 c\u2081 \u2192 \u2203 b, b \u2208 c\u2082 \u2227 (fun x x_1 => x = x_1) a b\nh2 : \u2200 {b : \u03b1}, b \u2208 c\u2082 \u2192 \u2203 a, a \u2208 c\u2081 \u2227 (fun x x_1 => x = x_1) a b\na2 : a \u2208 c\u2082\nb : \u03b1\nb1 : b \u2208 c\u2081\nab : (fun x x_1 => x = x_1) b a\n\u22a2 a \u2208 c\u2081\n[PROOFSTEP]\nrwa [\u2190 ab]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nrefl : \u2200 (x : \u03b1), R x x\nsymm : \u2200 {x y : \u03b1}, R x y \u2192 R y x\ntrans : \u2200 {x y z : \u03b1}, R x y \u2192 R y z \u2192 R x z\n\u22a2 \u2200 {x y : Computation \u03b1}, LiftRel R x y \u2192 LiftRel R y x\n[PROOFSTEP]\napply LiftRel.symm\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nrefl : \u2200 (x : \u03b1), R x x\nsymm : \u2200 {x y : \u03b1}, R x y \u2192 R y x\ntrans : \u2200 {x y z : \u03b1}, R x y \u2192 R y z \u2192 R x z\n\u22a2 Symmetric R\n[PROOFSTEP]\napply symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nrefl : \u2200 (x : \u03b1), R x x\nsymm : \u2200 {x y : \u03b1}, R x y \u2192 R y x\ntrans : \u2200 {x y z : \u03b1}, R x y \u2192 R y z \u2192 R x z\n\u22a2 \u2200 {x y z : Computation \u03b1}, LiftRel R x y \u2192 LiftRel R y z \u2192 LiftRel R x z\n[PROOFSTEP]\napply LiftRel.trans\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nrefl : \u2200 (x : \u03b1), R x x\nsymm : \u2200 {x y : \u03b1}, R x y \u2192 R y x\ntrans : \u2200 {x y z : \u03b1}, R x y \u2192 R y z \u2192 R x z\n\u22a2 Transitive R\n[PROOFSTEP]\napply trans\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\nl : \u2200 {a : \u03b1}, a \u2208 ca \u2192 \u2203 b, b \u2208 cb \u2227 R a b\nright\u271d : \u2200 {b : \u03b2}, b \u2208 cb \u2192 \u2203 a, a \u2208 ca \u2227 R a b\na : \u03b1\nb : \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\n\u22a2 R a b\n[PROOFSTEP]\nlet \u27e8b', mb', ab'\u27e9 := l ma\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\nl : \u2200 {a : \u03b1}, a \u2208 ca \u2192 \u2203 b, b \u2208 cb \u2227 R a b\nright\u271d : \u2200 {b : \u03b2}, b \u2208 cb \u2192 \u2203 a, a \u2208 ca \u2227 R a b\na : \u03b1\nb : \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nb' : \u03b2\nmb' : b' \u2208 cb\nab' : R a b'\n\u22a2 R a b\n[PROOFSTEP]\nrw [mem_unique mb mb']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\nl : \u2200 {a : \u03b1}, a \u2208 ca \u2192 \u2203 b, b \u2208 cb \u2227 R a b\nright\u271d : \u2200 {b : \u03b2}, b \u2208 cb \u2192 \u2203 a, a \u2208 ca \u2227 R a b\na : \u03b1\nb : \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nb' : \u03b2\nmb' : b' \u2208 cb\nab' : R a b'\n\u22a2 R a b'\n[PROOFSTEP]\nexact ab'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\nca : Computation \u03b1\ncb : Computation \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nab : R a b\na' : \u03b1\nma' : a' \u2208 ca\n\u22a2 \u2203 b, b \u2208 cb \u2227 R a' b\n[PROOFSTEP]\nrw [mem_unique ma' ma]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\nca : Computation \u03b1\ncb : Computation \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nab : R a b\na' : \u03b1\nma' : a' \u2208 ca\n\u22a2 \u2203 b, b \u2208 cb \u2227 R a b\n[PROOFSTEP]\nexact \u27e8b, mb, ab\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\nca : Computation \u03b1\ncb : Computation \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nab : R a b\nb' : \u03b2\nmb' : b' \u2208 cb\n\u22a2 \u2203 a, a \u2208 ca \u2227 R a b'\n[PROOFSTEP]\nrw [mem_unique mb' mb]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\nca : Computation \u03b1\ncb : Computation \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nab : R a b\nb' : \u03b2\nmb' : b' \u2208 cb\n\u22a2 \u2203 a, a \u2208 ca \u2227 R a b\n[PROOFSTEP]\nexact \u27e8a, ma, ab\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\nh : LiftRel R ca cb\na : \u03b1\nb : \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\n\u22a2 R a b\n[PROOFSTEP]\nlet \u27e8b', mb', ab\u27e9 := h.left ma\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\nh : LiftRel R ca cb\na : \u03b1\nb : \u03b2\nma : a \u2208 ca\nmb : b \u2208 cb\nb' : \u03b2\nmb' : b' \u2208 cb\nab : R a b'\n\u22a2 R a b\n[PROOFSTEP]\nrwa [mem_unique mb mb']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nx\u271d : \u2203 b, b \u2208 cb \u2227 R a b\nb : \u03b2\nmb : b \u2208 cb\nab : R a b\na' : \u03b1\nma' : a' \u2208 pure a\n\u22a2 \u2203 b, b \u2208 cb \u2227 R a' b\n[PROOFSTEP]\nrw [eq_of_pure_mem ma']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nx\u271d : \u2203 b, b \u2208 cb \u2227 R a b\nb : \u03b2\nmb : b \u2208 cb\nab : R a b\na' : \u03b1\nma' : a' \u2208 pure a\n\u22a2 \u2203 b, b \u2208 cb \u2227 R a b\n[PROOFSTEP]\nexact \u27e8b, mb, ab\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nx\u271d : \u2203 b, b \u2208 cb \u2227 R a b\nb : \u03b2\nmb : b \u2208 cb\nab : R a b\nb' : \u03b2\nmb' : b' \u2208 cb\n\u22a2 R a b'\n[PROOFSTEP]\nrw [mem_unique mb' mb]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nx\u271d : \u2203 b, b \u2208 cb \u2227 R a b\nb : \u03b2\nmb : b \u2208 cb\nab : R a b\nb' : \u03b2\nmb' : b' \u2208 cb\n\u22a2 R a b\n[PROOFSTEP]\nexact ab\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\nb : \u03b2\n\u22a2 LiftRel R ca (pure b) \u2194 \u2203 a, a \u2208 ca \u2227 R a b\n[PROOFSTEP]\nrw [LiftRel.swap, liftRel_pure_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\n\u22a2 LiftRel R (pure a) (pure b) \u2194 R a b\n[PROOFSTEP]\nrw [liftRel_pure_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\n\u22a2 (\u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1) \u2194 R a b\n[PROOFSTEP]\nexact \u27e8fun \u27e8b', mb', ab'\u27e9 => by rwa [eq_of_pure_mem mb'] at ab' , fun ab => \u27e8_, ret_mem _, ab\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\nx\u271d : \u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1\nb' : \u03b2\nmb' : b' \u2208 pure b\nab' : R a b'\n\u22a2 R a b\n[PROOFSTEP]\nrwa [eq_of_pure_mem mb'] at ab' \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\n\u22a2 LiftRel R ca (think cb) \u2194 LiftRel R ca cb\n[PROOFSTEP]\nrw [\u2190 LiftRel.swap R, \u2190 LiftRel.swap R]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\n\u22a2 LiftRel (swap R) (think cb) ca \u2194 LiftRel (swap R) cb ca\n[PROOFSTEP]\napply liftRel_think_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nS : \u03b3 \u2192 \u03b4 \u2192 Prop\ns1 : Computation \u03b1\ns2 : Computation \u03b2\nf1 : \u03b1 \u2192 \u03b3\nf2 : \u03b2 \u2192 \u03b4\nh1 : LiftRel R s1 s2\nh2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)\n\u22a2 LiftRel S (map f1 s1) (map f2 s2)\n[PROOFSTEP]\nrw [\u2190 bind_pure, \u2190 bind_pure]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nS : \u03b3 \u2192 \u03b4 \u2192 Prop\ns1 : Computation \u03b1\ns2 : Computation \u03b2\nf1 : \u03b1 \u2192 \u03b3\nf2 : \u03b2 \u2192 \u03b4\nh1 : LiftRel R s1 s2\nh2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)\n\u22a2 LiftRel S (bind s1 (pure \u2218 f1)) (bind s2 (pure \u2218 f2))\n[PROOFSTEP]\napply liftRel_bind _ _ h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nS : \u03b3 \u2192 \u03b4 \u2192 Prop\ns1 : Computation \u03b1\ns2 : Computation \u03b2\nf1 : \u03b1 \u2192 \u03b3\nf2 : \u03b2 \u2192 \u03b4\nh1 : LiftRel R s1 s2\nh2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)\n\u22a2 \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 LiftRel S ((pure \u2218 f1) a) ((pure \u2218 f2) b)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nS : \u03b3 \u2192 \u03b4 \u2192 Prop\ns1 : Computation \u03b1\ns2 : Computation \u03b2\nf1 : \u03b1 \u2192 \u03b3\nf2 : \u03b2 \u2192 \u03b4\nh1 : LiftRel R s1 s2\nh2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)\n\u22a2 \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 \u2203 a_2, a_2 \u2208 pure (f1 a) \u2227 S a_2 (f2 b)\n[PROOFSTEP]\nintros a b h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nS : \u03b3 \u2192 \u03b4 \u2192 Prop\ns1 : Computation \u03b1\ns2 : Computation \u03b2\nf1 : \u03b1 \u2192 \u03b3\nf2 : \u03b2 \u2192 \u03b4\nh1 : LiftRel R s1 s2\nh2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)\na : \u03b1\nb : \u03b2\nh : R a b\n\u22a2 \u2203 a_1, a_1 \u2208 pure (f1 a) \u2227 S a_1 (f2 b)\n[PROOFSTEP]\nexact \u27e8f1 a, \u27e8ret_mem _, @h2 a b h\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns1 s2 : Computation \u03b1\nf : \u03b1 \u2192 \u03b2\nh1 : s1 ~ s2\n\u22a2 map f s1 ~ map f s2\n[PROOFSTEP]\nrw [\u2190 lift_eq_iff_equiv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns1 s2 : Computation \u03b1\nf : \u03b1 \u2192 \u03b2\nh1 : s1 ~ s2\n\u22a2 LiftRel (fun x x_1 => x = x_1) (map f s1) (map f s2)\n[PROOFSTEP]\nexact liftRel_map Eq _ ((lift_eq_iff_equiv _ _).2 h1) fun {a} b => congr_arg _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\n\u22a2 LiftRelAux R C (Sum.inl a) (destruct cb) \u2194 \u2203 b, b \u2208 cb \u2227 R a b\n[PROOFSTEP]\napply cb.recOn (fun b => _) fun cb => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\n\u22a2 \u2200 (b : \u03b2), LiftRelAux R C (Sum.inl a) (destruct (pure b)) \u2194 \u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1\n[PROOFSTEP]\nintro b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nb : \u03b2\n\u22a2 LiftRelAux R C (Sum.inl a) (destruct (pure b)) \u2194 \u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1\n[PROOFSTEP]\nexact \u27e8fun h => \u27e8_, ret_mem _, h\u27e9, fun \u27e8b', mb, h\u27e9 => by rw [mem_unique (ret_mem _) mb]; exact h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nb : \u03b2\nx\u271d : \u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1\nb' : \u03b2\nmb : b' \u2208 pure b\nh : R a b'\n\u22a2 LiftRelAux R C (Sum.inl a) (destruct (pure b))\n[PROOFSTEP]\nrw [mem_unique (ret_mem _) mb]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\nb : \u03b2\nx\u271d : \u2203 b_1, b_1 \u2208 pure b \u2227 R a b_1\nb' : \u03b2\nmb : b' \u2208 pure b\nh : R a b'\n\u22a2 LiftRelAux R C (Sum.inl a) (destruct (pure b'))\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\n\u22a2 \u2200 (cb : Computation \u03b2), LiftRelAux R C (Sum.inl a) (destruct (think cb)) \u2194 \u2203 b, b \u2208 think cb \u2227 R a b\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb cb\u271d : Computation \u03b2\n\u22a2 LiftRelAux R C (Sum.inl a) (destruct (think cb\u271d)) \u2194 \u2203 b, b \u2208 think cb\u271d \u2227 R a b\n[PROOFSTEP]\nrw [destruct_think]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb cb\u271d : Computation \u03b2\n\u22a2 LiftRelAux R C (Sum.inl a) (Sum.inr cb\u271d) \u2194 \u2203 b, b \u2208 think cb\u271d \u2227 R a b\n[PROOFSTEP]\nexact \u27e8fun \u27e8b, h, r\u27e9 => \u27e8b, think_mem h, r\u27e9, fun \u27e8b, h, r\u27e9 => \u27e8b, of_think_mem h, r\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1 \u2295 Computation \u03b1\nb : \u03b2 \u2295 Computation \u03b2\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) b a = LiftRelAux R C a b\n[PROOFSTEP]\ncases' a with a ca\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nb : \u03b2 \u2295 Computation \u03b2\na : \u03b1\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) b (Sum.inl a) = LiftRelAux R C (Sum.inl a) b\n[PROOFSTEP]\ncases' b with b cb\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nb : \u03b2 \u2295 Computation \u03b2\nca : Computation \u03b1\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) b (Sum.inr ca) = LiftRelAux R C (Sum.inr ca) b\n[PROOFSTEP]\ncases' b with b cb\n[GOAL]\ncase inl.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\nb : \u03b2\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) (Sum.inl b) (Sum.inl a) = LiftRelAux R C (Sum.inl a) (Sum.inl b)\n[PROOFSTEP]\nsimp only [LiftRelAux]\n[GOAL]\ncase inl.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\na : \u03b1\ncb : Computation \u03b2\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) (Sum.inr cb) (Sum.inl a) = LiftRelAux R C (Sum.inl a) (Sum.inr cb)\n[PROOFSTEP]\nsimp only [LiftRelAux]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nca : Computation \u03b1\nb : \u03b2\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) (Sum.inl b) (Sum.inr ca) = LiftRelAux R C (Sum.inr ca) (Sum.inl b)\n[PROOFSTEP]\nsimp only [LiftRelAux]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nca : Computation \u03b1\ncb : Computation \u03b2\n\u22a2 LiftRelAux (Function.swap R) (Function.swap C) (Sum.inr cb) (Sum.inr ca) = LiftRelAux R C (Sum.inr ca) (Sum.inr cb)\n[PROOFSTEP]\nsimp only [LiftRelAux]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nb : \u03b2\nca : Computation \u03b1\n\u22a2 LiftRelAux R C (destruct ca) (Sum.inl b) \u2194 \u2203 a, a \u2208 ca \u2227 R a b\n[PROOFSTEP]\nrw [\u2190 LiftRelAux.swap, LiftRelAux.ret_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\ncb : Computation \u03b2\nHc : C ca cb\na : \u03b1\nha : a \u2208 ca\n\u22a2 LiftRel R ca cb\n[PROOFSTEP]\nrevert cb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\n\u22a2 \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb\n[PROOFSTEP]\nrefine' memRecOn (C := (\u03bb ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb)) ha _ (fun ca' IH => _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\n\u22a2 (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) (pure a)\n[PROOFSTEP]\nintro cb Hc\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\n\u22a2 (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) (think ca')\n[PROOFSTEP]\nintro cb Hc\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\ncb : Computation \u03b2\nHc : C (pure a) cb\n\u22a2 LiftRel R (pure a) cb\n[PROOFSTEP]\nhave h := H Hc\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\n\u22a2 LiftRel R (think ca') cb\n[PROOFSTEP]\nhave h := H Hc\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\ncb : Computation \u03b2\nHc : C (pure a) cb\nh : LiftRelAux R C (destruct (pure a)) (destruct cb)\n\u22a2 LiftRel R (pure a) cb\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\ncb : Computation \u03b2\nHc : C (pure a) cb\nh : \u2203 b, b \u2208 cb \u2227 R a b\n\u22a2 LiftRel R (pure a) cb\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n\u22a2 LiftRel R (think ca') cb\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\nh : LiftRelAux R C (destruct (think ca')) (destruct cb)\n\u22a2 LiftRel R ca' cb\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\n\u22a2 LiftRelAux R C (destruct (think ca')) (destruct cb) \u2192 LiftRel R ca' cb\n[PROOFSTEP]\napply cb.recOn (fun b => _) fun cb' => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\n\u22a2 \u2200 (b : \u03b2), LiftRelAux R C (destruct (think ca')) (destruct (pure b)) \u2192 LiftRel R ca' (pure b)\n[PROOFSTEP]\nintros _ h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\n\u22a2 \u2200 (cb' : Computation \u03b2), LiftRelAux R C (destruct (think ca')) (destruct (think cb')) \u2192 LiftRel R ca' (think cb')\n[PROOFSTEP]\nintros _ h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\nb\u271d : \u03b2\nh : LiftRelAux R C (destruct (think ca')) (destruct (pure b\u271d))\n\u22a2 LiftRel R ca' (pure b\u271d)\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\ncb'\u271d : Computation \u03b2\nh : LiftRelAux R C (destruct (think ca')) (destruct (think cb'\u271d))\n\u22a2 LiftRel R ca' (think cb'\u271d)\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\nb\u271d : \u03b2\nh : \u2203 a, a \u2208 ca' \u2227 R a b\u271d\n\u22a2 LiftRel R ca' (pure b\u271d)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\ncb'\u271d : Computation \u03b2\nh : C ca' cb'\u271d\n\u22a2 LiftRel R ca' (think cb'\u271d)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b2 \u2192 Prop\nC : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop\nH : \u2200 {ca : Computation \u03b1} {cb : Computation \u03b2}, C ca cb \u2192 LiftRelAux R C (destruct ca) (destruct cb)\nca : Computation \u03b1\na : \u03b1\nha : a \u2208 ca\nca' : Computation \u03b1\nIH : (fun ca => \u2200 (cb : Computation \u03b2), C ca cb \u2192 LiftRel R ca cb) ca'\ncb : Computation \u03b2\nHc : C (think ca') cb\ncb'\u271d : Computation \u03b2\nh : C ca' cb'\u271d\n\u22a2 LiftRel R ca' cb'\u271d\n[PROOFSTEP]\nexact IH _ h\n", "meta": {"mathlib_filename": "Mathlib.Data.Seq.Computation", "llama_tokens": 62856, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.4339814648038985, "lm_q1q2_score": 0.2522747604770126}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n\u22a2 (F \u2297\u22d9 H).\u03b5 \u226b NatTrans.app (NatTrans.mk src\u271d.app) (\ud835\udfd9_ C) = (G \u2297\u22d9 K).\u03b5\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n\u22a2 (H.\u03b5 \u226b H.map F.\u03b5) \u226b NatTrans.app \u03b2.toNatTrans (F.obj (\ud835\udfd9_ C)) \u226b K.map (NatTrans.app \u03b1.toNatTrans (\ud835\udfd9_ C)) =\n K.\u03b5 \u226b K.map G.\u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n\u22a2 K.\u03b5 \u226b K.map F.\u03b5 \u226b K.map (NatTrans.app \u03b1.toNatTrans (\ud835\udfd9_ C)) = K.\u03b5 \u226b K.map G.\u03b5\n[PROOFSTEP]\nconv_lhs => rw [\u2190 K.toFunctor.map_comp, \u03b1.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n| K.\u03b5 \u226b K.map F.\u03b5 \u226b K.map (NatTrans.app \u03b1.toNatTrans (\ud835\udfd9_ C))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n| K.\u03b5 \u226b K.map F.\u03b5 \u226b K.map (NatTrans.app \u03b1.toNatTrans (\ud835\udfd9_ C))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\n| K.\u03b5 \u226b K.map F.\u03b5 \u226b K.map (NatTrans.app \u03b1.toNatTrans (\ud835\udfd9_ C))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc (F \u2297\u22d9 H) X Y \u226b NatTrans.app (NatTrans.mk src\u271d.app) (X \u2297 Y) =\n (NatTrans.app (NatTrans.mk src\u271d.app) X \u2297 NatTrans.app (NatTrans.mk src\u271d.app) Y) \u226b LaxMonoidalFunctor.\u03bc (G \u2297\u22d9 K) X Y\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n\u22a2 (LaxMonoidalFunctor.\u03bc H (F.obj X) (F.obj Y) \u226b H.map (LaxMonoidalFunctor.\u03bc F X Y)) \u226b\n NatTrans.app \u03b2.toNatTrans (F.obj (X \u2297 Y)) \u226b K.map (NatTrans.app \u03b1.toNatTrans (X \u2297 Y)) =\n (NatTrans.app \u03b2.toNatTrans (F.obj X) \u226b K.map (NatTrans.app \u03b1.toNatTrans X) \u2297\n NatTrans.app \u03b2.toNatTrans (F.obj Y) \u226b K.map (NatTrans.app \u03b1.toNatTrans Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (G.obj X) (G.obj Y) \u226b K.map (LaxMonoidalFunctor.\u03bc G X Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n\u22a2 (NatTrans.app \u03b2.toNatTrans (F.obj X) \u2297 NatTrans.app \u03b2.toNatTrans (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (F.obj X) (F.obj Y) \u226b\n K.map (LaxMonoidalFunctor.\u03bc F X Y) \u226b K.map (NatTrans.app \u03b1.toNatTrans (X \u2297 Y)) =\n (NatTrans.app \u03b2.toNatTrans (F.obj X) \u2297 NatTrans.app \u03b2.toNatTrans (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (F.obj X) (F.obj Y) \u226b\n K.map (NatTrans.app \u03b1.toNatTrans X \u2297 NatTrans.app \u03b1.toNatTrans Y) \u226b K.map (LaxMonoidalFunctor.\u03bc G X Y)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 K.toFunctor.map_comp, \u03b1.tensor, K.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n| (NatTrans.app \u03b2.toNatTrans (F.obj X) \u2297 NatTrans.app \u03b2.toNatTrans (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (F.obj X) (F.obj Y) \u226b\n K.map (LaxMonoidalFunctor.\u03bc F X Y) \u226b K.map (NatTrans.app \u03b1.toNatTrans (X \u2297 Y))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.tensor, K.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n| (NatTrans.app \u03b2.toNatTrans (F.obj X) \u2297 NatTrans.app \u03b2.toNatTrans (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (F.obj X) (F.obj Y) \u226b\n K.map (LaxMonoidalFunctor.\u03bc F X Y) \u226b K.map (NatTrans.app \u03b1.toNatTrans (X \u2297 Y))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.tensor, K.toFunctor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\ninst\u271d : MonoidalCategory E\nF G : LaxMonoidalFunctor C D\nH K : LaxMonoidalFunctor D E\n\u03b1 : MonoidalNatTrans F G\n\u03b2 : MonoidalNatTrans H K\nsrc\u271d : F.toFunctor \u22d9 H.toFunctor \u27f6 G.toFunctor \u22d9 K.toFunctor := \u03b1.toNatTrans \u25eb \u03b2.toNatTrans\nX Y : C\n| (NatTrans.app \u03b2.toNatTrans (F.obj X) \u2297 NatTrans.app \u03b2.toNatTrans (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc K (F.obj X) (F.obj Y) \u226b\n K.map (LaxMonoidalFunctor.\u03bc F X Y) \u226b K.map (NatTrans.app \u03b1.toNatTrans (X \u2297 Y))\n[PROOFSTEP]\nrw [\u2190 K.toFunctor.map_comp, \u03b1.tensor, K.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 G : LaxMonoidalFunctor C D\napp : (X : C) \u2192 F.obj X \u2245 G.obj X\nnaturality' : \u2200 {X Y : C} (f : X \u27f6 Y), F.map f \u226b (app Y).hom = (app X).hom \u226b G.map f\nunit' : F.\u03b5 \u226b (app (\ud835\udfd9_ C)).hom = G.\u03b5\ntensor' :\n \u2200 (X Y : C), LaxMonoidalFunctor.\u03bc F X Y \u226b (app (X \u2297 Y)).hom = ((app X).hom \u2297 (app Y).hom) \u226b LaxMonoidalFunctor.\u03bc G X Y\nsrc\u271d : G.toFunctor \u27f6 F.toFunctor := (NatIso.ofComponents app).inv\n\u22a2 G.\u03b5 \u226b NatTrans.app (NatTrans.mk fun X => (app X).inv) (\ud835\udfd9_ C) = F.\u03b5\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 G : LaxMonoidalFunctor C D\napp : (X : C) \u2192 F.obj X \u2245 G.obj X\nnaturality' : \u2200 {X Y : C} (f : X \u27f6 Y), F.map f \u226b (app Y).hom = (app X).hom \u226b G.map f\nunit' : F.\u03b5 \u226b (app (\ud835\udfd9_ C)).hom = G.\u03b5\ntensor' :\n \u2200 (X Y : C), LaxMonoidalFunctor.\u03bc F X Y \u226b (app (X \u2297 Y)).hom = ((app X).hom \u2297 (app Y).hom) \u226b LaxMonoidalFunctor.\u03bc G X Y\nsrc\u271d : G.toFunctor \u27f6 F.toFunctor := (NatIso.ofComponents app).inv\n\u22a2 G.\u03b5 \u226b (app (\ud835\udfd9_ C)).inv = F.\u03b5\n[PROOFSTEP]\nrw [\u2190 unit', assoc, Iso.hom_inv_id, comp_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 G : LaxMonoidalFunctor C D\napp : (X : C) \u2192 F.obj X \u2245 G.obj X\nnaturality' : \u2200 {X Y : C} (f : X \u27f6 Y), F.map f \u226b (app Y).hom = (app X).hom \u226b G.map f\nunit' : F.\u03b5 \u226b (app (\ud835\udfd9_ C)).hom = G.\u03b5\ntensor' :\n \u2200 (X Y : C), LaxMonoidalFunctor.\u03bc F X Y \u226b (app (X \u2297 Y)).hom = ((app X).hom \u2297 (app Y).hom) \u226b LaxMonoidalFunctor.\u03bc G X Y\nsrc\u271d : G.toFunctor \u27f6 F.toFunctor := (NatIso.ofComponents app).inv\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc G X Y \u226b NatTrans.app (NatTrans.mk fun X => (app X).inv) (X \u2297 Y) =\n (NatTrans.app (NatTrans.mk fun X => (app X).inv) X \u2297 NatTrans.app (NatTrans.mk fun X => (app X).inv) Y) \u226b\n LaxMonoidalFunctor.\u03bc F X Y\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 G : LaxMonoidalFunctor C D\napp : (X : C) \u2192 F.obj X \u2245 G.obj X\nnaturality' : \u2200 {X Y : C} (f : X \u27f6 Y), F.map f \u226b (app Y).hom = (app X).hom \u226b G.map f\nunit' : F.\u03b5 \u226b (app (\ud835\udfd9_ C)).hom = G.\u03b5\ntensor' :\n \u2200 (X Y : C), LaxMonoidalFunctor.\u03bc F X Y \u226b (app (X \u2297 Y)).hom = ((app X).hom \u2297 (app Y).hom) \u226b LaxMonoidalFunctor.\u03bc G X Y\nsrc\u271d : G.toFunctor \u27f6 F.toFunctor := (NatIso.ofComponents app).inv\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc G X Y \u226b (app (X \u2297 Y)).inv = ((app X).inv \u2297 (app Y).inv) \u226b LaxMonoidalFunctor.\u03bc F X Y\n[PROOFSTEP]\nrw [Iso.comp_inv_eq, assoc, tensor', \u2190 tensor_comp_assoc, Iso.inv_hom_id, Iso.inv_hom_id, tensor_id, id_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 G : LaxMonoidalFunctor C D\napp : (X : C) \u2192 F.obj X \u2245 G.obj X\nnaturality : \u2200 {X Y : C} (f : X \u27f6 Y), F.map f \u226b (app Y).hom = (app X).hom \u226b G.map f\nunit : F.\u03b5 \u226b (app (\ud835\udfd9_ C)).hom = G.\u03b5\ntensor :\n \u2200 (X Y : C), LaxMonoidalFunctor.\u03bc F X Y \u226b (app (X \u2297 Y)).hom = ((app X).hom \u2297 (app Y).hom) \u226b LaxMonoidalFunctor.\u03bc G X Y\nX : C\n\u22a2 NatTrans.app (ofComponents app naturality unit tensor).inv.toNatTrans X = (app X).inv\n[PROOFSTEP]\nsimp [ofComponents]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc (LaxMonoidalFunctor.id C) X Y \u226b NatTrans.app (Equivalence.unit e) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit e) X \u2297 NatTrans.app (Equivalence.unit e) Y) \u226b\n LaxMonoidalFunctor.\u03bc (F.toLaxMonoidalFunctor \u2297\u22d9 (monoidalInverse F).toLaxMonoidalFunctor) X Y\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 \ud835\udfd9 (X \u2297 Y) \u226b NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n \u2191(Adjunction.homEquiv (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y))\n (F.obj X \u2297 F.obj Y))\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y))) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj X) \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj Y))) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)\n[PROOFSTEP]\nsimp only [Adjunction.homEquiv_unit, Adjunction.homEquiv_naturality_right, id_comp, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y))) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj X) \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj Y))) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)\n[PROOFSTEP]\nsimp only [\u2190 Functor.map_comp, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y))) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj X) \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit (F.obj Y)) \u226b\n LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)\n[PROOFSTEP]\nerw [e.counit_app_functor, e.counit_app_functor, F.toLaxMonoidalFunctor.\u03bc_natural, IsIso.inv_hom_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n (Functor.inv F.toLaxMonoidalFunctor.1).map\n (F.map (NatTrans.app (Equivalence.unitInv e) X \u2297 NatTrans.app (Equivalence.unitInv e) Y))\n[PROOFSTEP]\nsimp only [CategoryTheory.IsEquivalence.inv_fun_map]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toLaxMonoidalFunctor.1))\n (((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj X \u2297\n ((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj Y) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) ((\ud835\udfed C).obj X \u2297 (\ud835\udfed C).obj Y)\n[PROOFSTEP]\nslice_rhs 2 3 => erw [Iso.hom_inv_id_app]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toLaxMonoidalFunctor.1))\n (((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj X \u2297\n ((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y\ncase a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) ((\ud835\udfed C).obj X \u2297 (\ud835\udfed C).obj Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [Iso.hom_inv_id_app]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toLaxMonoidalFunctor.1))\n (((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj X \u2297\n ((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y\ncase a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) ((\ud835\udfed C).obj X \u2297 (\ud835\udfed C).obj Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [Iso.hom_inv_id_app]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toLaxMonoidalFunctor.1))\n (((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj X \u2297\n ((asEquivalence F.toFunctor).functor \u22d9 (asEquivalence F.toFunctor).inverse).obj Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y\ncase a.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) ((\ud835\udfed C).obj X \u2297 (\ud835\udfed C).obj Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [Iso.hom_inv_id_app]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (\ud835\udfd9\n ((\ud835\udfed C).obj\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297\n (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y))) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y)) \u226b\n NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) ((\ud835\udfed C).obj X \u2297 (\ud835\udfed C).obj Y)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (\ud835\udfd9 ((Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj X) \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj (F.obj Y)) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y)) \u226b\n NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\nsimp only [CategoryTheory.Category.id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y) \u226b\n NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\nslice_rhs 1 2 =>\n rw [\u2190 tensor_comp, Iso.hom_inv_id_app, Iso.hom_inv_id_app]\n dsimp\n rw [tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\n rw [\u2190 tensor_comp, Iso.hom_inv_id_app, Iso.hom_inv_id_app]\n dsimp\n rw [tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\n rw [\u2190 tensor_comp, Iso.hom_inv_id_app, Iso.hom_inv_id_app]\n dsimp\n rw [tensor_id]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) Y) \u226b\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor)) Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, Iso.hom_inv_id_app, Iso.hom_inv_id_app]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| \ud835\udfd9 ((\ud835\udfed C).obj X) \u2297 \ud835\udfd9 ((\ud835\udfed C).obj Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| \ud835\udfd9 X \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n| NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\nrw [tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : C\n\u22a2 NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n \ud835\udfd9 (X \u2297 Y) \u226b NatTrans.app (Equivalence.unit (asEquivalence F.toLaxMonoidalFunctor.1)) (X \u2297 Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 \u2200 (X : C), IsIso (NatTrans.app (monoidalUnit F).toNatTrans X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 \u2200 (X : C), IsIso (NatTrans.app (Equivalence.unit (asEquivalence F.toFunctor)) X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\n\u22a2 ((monoidalInverse F).toLaxMonoidalFunctor \u2297\u22d9 F.toLaxMonoidalFunctor).\u03b5 \u226b NatTrans.app (Equivalence.counit e) (\ud835\udfd9_ D) =\n (LaxMonoidalFunctor.id D).\u03b5\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\n\u22a2 (F.\u03b5 \u226b\n F.map\n (\u2191(Adjunction.homEquiv (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)) (\ud835\udfd9_ C) (\ud835\udfd9_ D))\n (inv F.\u03b5))) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D) =\n \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\nsimp only [comp_id, assoc, Functor.map_inv, Functor.map_comp, NatIso.inv_inv_app, IsIso.inv_comp,\n IsEquivalence.fun_inv_map, Adjunction.homEquiv_unit]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\n\u22a2 F.\u03b5 \u226b\n F.map (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit (\ud835\udfd9_ C)) \u226b\n NatTrans.app (asEquivalence F.toFunctor).counitIso.hom (F.obj (\ud835\udfd9_ C)) \u226b\n inv F.\u03b5 \u226b\n inv (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D) =\n \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\nerw [e.counit_app_functor, \u2190 e.functor.map_comp_assoc, Iso.hom_inv_id_app]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\n\u22a2 F.\u03b5 \u226b\n e.functor.map (\ud835\udfd9 ((\ud835\udfed C).obj (\ud835\udfd9_ C))) \u226b\n inv F.\u03b5 \u226b\n inv (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D) =\n \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\n\u22a2 F.\u03b5 \u226b\n F.map (\ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n inv F.\u03b5 \u226b\n inv (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (\ud835\udfd9_ D) =\n \ud835\udfd9 (\ud835\udfd9_ D)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc ((monoidalInverse F).toLaxMonoidalFunctor \u2297\u22d9 F.toLaxMonoidalFunctor) X Y \u226b\n NatTrans.app (Equivalence.counit e) (X \u2297 Y) =\n (NatTrans.app (Equivalence.counit e) X \u2297 NatTrans.app (Equivalence.counit e) Y) \u226b\n LaxMonoidalFunctor.\u03bc (LaxMonoidalFunctor.id D) X Y\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n F.map\n (\u2191(Adjunction.homEquiv (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y) (X \u2297 Y))\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y)))) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y) \u226b\n \ud835\udfd9 (X \u2297 Y)\n[PROOFSTEP]\nsimp only [Adjunction.homEquiv_unit, Adjunction.homEquiv_naturality_right, assoc, comp_id, Functor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n F.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n F.map\n ((Functor.inv F.toLaxMonoidalFunctor.1).map\n (inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)))) \u226b\n F.map\n ((Functor.inv F.toLaxMonoidalFunctor.1).map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nsimp only [IsEquivalence.fun_inv_map]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n F.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y))) \u226b\n (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor)) (X \u2297 Y)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [e.counit_app_functor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n F.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (e.functor.map\n (NatTrans.app (Equivalence.unitInv e)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y))) \u226b\n (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor)) (X \u2297 Y)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n F.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (asEquivalence F.toFunctor).functor.map\n (NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor)) (X \u2297 Y) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [\u2190 e.functor.map_comp_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n e.functor.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor))\n (F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj X) \u2297\n F.obj ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n NatTrans.app (Equivalence.counitInv (asEquivalence F.toFunctor)) (X \u2297 Y) \u226b\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) (X \u2297 Y) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nsimp only [CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.Iso.inv_hom_id_app_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n (asEquivalence F.toFunctor).functor.map\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).unit\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n NatTrans.app (Equivalence.unitInv (asEquivalence F.toFunctor))\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n \ud835\udfd9 ((\ud835\udfed D).obj (X \u2297 Y)) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [Iso.hom_inv_id_app, CategoryTheory.Functor.map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y) \u226b\n \ud835\udfd9\n ((asEquivalence F.toFunctor).functor.obj\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj X \u2297 (Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n inv\n (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor ((Functor.inv F.toLaxMonoidalFunctor.1).obj X)\n ((Functor.inv F.toLaxMonoidalFunctor.1).obj Y)) \u226b\n (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n \ud835\udfd9 ((\ud835\udfed D).obj (X \u2297 Y)) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nsimp only [id_comp, CategoryTheory.Iso.inv_hom_id_app, CategoryTheory.IsIso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 (NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y) \u226b\n \ud835\udfd9 ((\ud835\udfed D).obj (X \u2297 Y)) =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nerw [comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\ne : C \u224c D := asEquivalence F.toFunctor\nX Y : D\n\u22a2 NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit X \u2297\n NatTrans.app (Equivalence.toAdjunction (asEquivalence F.toLaxMonoidalFunctor.1)).counit Y =\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X \u2297\n NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) Y\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 \u2200 (X : D), IsIso (NatTrans.app (monoidalCounit F).toNatTrans X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\n\u22a2 \u2200 (X : D), IsIso (NatTrans.app (Equivalence.counit (asEquivalence F.toFunctor)) X)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.NaturalTransformation", "llama_tokens": 23044, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.252177968286533}} {"text": "[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hs)\n[GOAL]\ncase inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n \u2205 \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (\u2205 \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty \u2205 \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 \u2205 \u2227 A n = f' y)\n[PROOFSTEP]\nrefine' \u27e8fun _ => \u2205, fun _ => 0, _, _, _, _\u27e9\n[GOAL]\ncase inl.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\n\u22a2 \u2200 (n : \u2115), IsClosed ((fun x => \u2205) n)\n[PROOFSTEP]\nsimp\n -- we will use countably many linear maps. Select these from all the derivatives since the\n -- space of linear maps is second-countable\n[GOAL]\ncase inl.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\n\u22a2 \u2205 \u2286 \u22c3 (n : \u2115), (fun x => \u2205) n\n[PROOFSTEP]\nsimp\n -- we will use countably many linear maps. Select these from all the derivatives since the\n -- space of linear maps is second-countable\n[GOAL]\ncase inl.refine'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f ((fun x => 0) n) (\u2205 \u2229 (fun x => \u2205) n) (r ((fun x => 0) n))\n[PROOFSTEP]\nsimp\n -- we will use countably many linear maps. Select these from all the derivatives since the\n -- space of linear maps is second-countable\n[GOAL]\ncase inl.refine'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\n\u22a2 Set.Nonempty \u2205 \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 \u2205 \u2227 (fun x => 0) n = f' y\n[PROOFSTEP]\nsimp\n -- we will use countably many linear maps. Select these from all the derivatives since the\n -- space of linear maps is second-countable\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nobtain \u27e8T, T_count, hT\u27e9 :\n \u2203 T : Set s, T.Countable \u2227 \u22c3 x \u2208 T, ball (f' (x : E)) (r (f' x)) = \u22c3 x : s, ball (f' x) (r (f' x)) :=\n TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\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\n (0 : \u211d)\n -- `M n z` is the set of points `x` such that `f y - f x` is close to `f' z (y - x)` for `y`\n -- in the ball of radius `u n` around `x`.\n[GOAL]\ncase inr.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nlet M : \u2115 \u2192 T \u2192 Set E := fun n z =>\n {x | x \u2208 s \u2227 \u2200 y \u2208 s \u2229 ball x (u n), \u2016f y - f x - f' z (y - x)\u2016 \u2264 r (f' z) * \u2016y - x\u2016}\n -- As `f` is differentiable everywhere on `s`, the sets `M n z` cover `s` by design.\n[GOAL]\ncase inr.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nhave s_subset : \u2200 x \u2208 s, \u2203 (n : \u2115) (z : T), x \u2208 M n z :=\n by\n intro x xs\n obtain \u27e8z, zT, hz\u27e9 : \u2203 z \u2208 T, f' x \u2208 ball (f' (z : E)) (r (f' z)) :=\n by\n have : f' x \u2208 \u22c3 z \u2208 T, ball (f' (z : E)) (r (f' z)) :=\n by\n rw [hT]\n refine' mem_iUnion.2 \u27e8\u27e8x, xs\u27e9, _\u27e9\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n rwa [mem_iUnion\u2082, bex_def] at this \n obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 \u2016f' x - f' z\u2016 + \u03b5 \u2264 r (f' z) :=\n by\n refine' \u27e8r (f' z) - \u2016f' x - f' z\u2016, _, le_of_eq (by abel)\u27e9\n simpa only [sub_pos] using mem_ball_iff_norm.mp hz\n obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 (\u03b4 : \u211d), 0 < \u03b4 \u2227 ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) \u03b5pos)\n obtain \u27e8n, hn\u27e9 : \u2203 n, u n < \u03b4 := ((tendsto_order.1 u_lim).2 _ \u03b4pos).exists\n refine' \u27e8n, \u27e8z, zT\u27e9, \u27e8xs, _\u27e9\u27e9\n intro y hy\n calc\n \u2016f y - f x - (f' z) (y - x)\u2016 = \u2016f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)\u2016 :=\n by\n congr 1\n simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n abel\n _ \u2264 \u2016f y - f x - (f' x) (y - x)\u2016 + \u2016(f' x - f' z) (y - x)\u2016 := (norm_add_le _ _)\n _ \u2264 \u03b5 * \u2016y - x\u2016 + \u2016f' x - f' z\u2016 * \u2016y - x\u2016 :=\n by\n refine' add_le_add (h\u03b4 _) (ContinuousLinearMap.le_op_norm _ _)\n rw [inter_comm]\n exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy\n _ \u2264 r (f' z) * \u2016y - x\u2016 := by\n rw [\u2190 add_mul, add_comm]\n exact\n mul_le_mul_of_nonneg_right h\u03b5\n (norm_nonneg _)\n -- the sets `M n z` are relatively closed in `s`, as all the conditions defining it are clearly\n -- closed\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nintro x xs\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\n\u22a2 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nobtain \u27e8z, zT, hz\u27e9 : \u2203 z \u2208 T, f' x \u2208 ball (f' (z : E)) (r (f' z)) :=\n by\n have : f' x \u2208 \u22c3 z \u2208 T, ball (f' (z : E)) (r (f' z)) := by\n rw [hT]\n refine' mem_iUnion.2 \u27e8\u27e8x, xs\u27e9, _\u27e9\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n rwa [mem_iUnion\u2082, bex_def] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\n\u22a2 \u2203 z, z \u2208 T \u2227 f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n[PROOFSTEP]\nhave : f' x \u2208 \u22c3 z \u2208 T, ball (f' (z : E)) (r (f' z)) := by\n rw [hT]\n refine' mem_iUnion.2 \u27e8\u27e8x, xs\u27e9, _\u27e9\n simpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\n\u22a2 f' x \u2208 \u22c3 (z : \u2191s) (_ : z \u2208 T), ball (f' \u2191z) \u2191(r (f' \u2191z))\n[PROOFSTEP]\nrw [hT]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\n\u22a2 f' x \u2208 \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8\u27e8x, xs\u27e9, _\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\n\u22a2 f' x \u2208 ball (f' \u2191{ val := x, property := xs }) \u2191(r (f' \u2191{ val := x, property := xs }))\n[PROOFSTEP]\nsimpa only [mem_ball, Subtype.coe_mk, dist_self] using (rpos (f' x)).bot_lt\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nthis : f' x \u2208 \u22c3 (z : \u2191s) (_ : z \u2208 T), ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 \u2203 z, z \u2208 T \u2227 f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n[PROOFSTEP]\nrwa [mem_iUnion\u2082, bex_def] at this \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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 \u2016f' x - f' z\u2016 + \u03b5 \u2264 r (f' z) :=\n by\n refine' \u27e8r (f' z) - \u2016f' x - f' z\u2016, _, le_of_eq (by abel)\u27e9\n simpa only [sub_pos] using mem_ball_iff_norm.mp hz\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n[PROOFSTEP]\nrefine' \u27e8r (f' z) - \u2016f' x - f' z\u2016, _, le_of_eq (by abel)\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 \u2016f' x - f' \u2191z\u2016 + (\u2191(r (f' \u2191z)) - \u2016f' x - f' \u2191z\u2016) = \u2191(r (f' \u2191z))\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 \u2016f' x - f' \u2191z\u2016 + (\u2191(r (f' \u2191z)) - \u2016f' x - f' \u2191z\u2016) = \u2191(r (f' \u2191z))\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u22a2 0 < \u2191(r (f' \u2191z)) - \u2016f' x - f' \u2191z\u2016\n[PROOFSTEP]\nsimpa only [sub_pos] using mem_ball_iff_norm.mp hz\n[GOAL]\ncase intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u22a2 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 (\u03b4 : \u211d), 0 < \u03b4 \u2227 ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n Metric.mem_nhdsWithin_iff.1 (IsLittleO.def (hf' x xs) \u03b5pos)\n[GOAL]\ncase intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, u n < \u03b4 := ((tendsto_order.1 u_lim).2 _ \u03b4pos).exists\n[GOAL]\ncase intro.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\n\u22a2 \u2203 n z, x \u2208 M n z\n[PROOFSTEP]\nrefine' \u27e8n, \u27e8z, zT\u27e9, \u27e8xs, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\n\u22a2 \u2200 (y : E),\n y \u2208 s \u2229 ball x (u n) \u2192\n \u2016f y - f x - \u2191(f' \u2191\u2191{ val := z, property := zT }) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191{ val := z, property := zT })) * \u2016y - x\u2016\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase intro.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191{ val := z, property := zT }) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191{ val := z, property := zT })) * \u2016y - x\u2016\n[PROOFSTEP]\ncalc\n \u2016f y - f x - (f' z) (y - x)\u2016 = \u2016f y - f x - (f' x) (y - x) + (f' x - f' z) (y - x)\u2016 :=\n by\n congr 1\n simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n abel\n _ \u2264 \u2016f y - f x - (f' x) (y - x)\u2016 + \u2016(f' x - f' z) (y - x)\u2016 := (norm_add_le _ _)\n _ \u2264 \u03b5 * \u2016y - x\u2016 + \u2016f' x - f' z\u2016 * \u2016y - x\u2016 :=\n by\n refine' add_le_add (h\u03b4 _) (ContinuousLinearMap.le_op_norm _ _)\n rw [inter_comm]\n exact inter_subset_inter_right _ (ball_subset_ball hn.le) hy\n _ \u2264 r (f' z) * \u2016y - x\u2016 := by\n rw [\u2190 add_mul, add_comm]\n exact\n mul_le_mul_of_nonneg_right h\u03b5\n (norm_nonneg _)\n -- the sets `M n z` are relatively closed in `s`, as all the conditions defining it are clearly\n -- closed\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 \u2016f y - f x - \u2191(f' \u2191z) (y - x)\u2016 = \u2016f y - f x - \u2191(f' x) (y - x) + \u2191(f' x - f' \u2191z) (y - x)\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 f y - f x - \u2191(f' \u2191z) (y - x) = f y - f x - \u2191(f' x) (y - x) + \u2191(f' x - f' \u2191z) (y - x)\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 f y - f x - (\u2191(f' \u2191z) y - \u2191(f' \u2191z) x) =\n f y - f x - (\u2191(f' x) y - \u2191(f' x) x) + (\u2191(f' x) y - \u2191(f' \u2191z) y - (\u2191(f' x) x - \u2191(f' \u2191z) x))\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 f y - f x - (\u2191(f' \u2191z) y - \u2191(f' \u2191z) x) =\n f y - f x - (\u2191(f' x) y - \u2191(f' x) x) + (\u2191(f' x) y - \u2191(f' \u2191z) y - (\u2191(f' x) x - \u2191(f' \u2191z) x))\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 \u2016f y - f x - \u2191(f' x) (y - x)\u2016 + \u2016\u2191(f' x - f' \u2191z) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016 + \u2016f' x - f' \u2191z\u2016 * \u2016y - x\u2016\n[PROOFSTEP]\nrefine' add_le_add (h\u03b4 _) (ContinuousLinearMap.le_op_norm _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 y \u2208 ball x \u03b4 \u2229 s\n[PROOFSTEP]\nrw [inter_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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 y \u2208 s \u2229 ball x \u03b4\n[PROOFSTEP]\nexact inter_subset_inter_right _ (ball_subset_ball hn.le) hy\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 \u03b5 * \u2016y - x\u2016 + \u2016f' x - f' \u2191z\u2016 * \u2016y - x\u2016 \u2264 \u2191(r (f' \u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nrw [\u2190 add_mul, add_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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\nx : E\nxs : x \u2208 s\nz : \u2191s\nzT : z \u2208 T\nhz : f' x \u2208 ball (f' \u2191z) \u2191(r (f' \u2191z))\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : \u2016f' x - f' \u2191z\u2016 + \u03b5 \u2264 \u2191(r (f' \u2191z))\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball x \u03b4 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nn : \u2115\nhn : u n < \u03b4\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 (\u2016f' x - f' \u2191z\u2016 + \u03b5) * \u2016y - x\u2016 \u2264 \u2191(r (f' \u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nexact\n mul_le_mul_of_nonneg_right h\u03b5\n (norm_nonneg _)\n -- the sets `M n z` are relatively closed in `s`, as all the conditions defining it are clearly\n -- closed\n[GOAL]\ncase inr.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nhave closure_M_subset : \u2200 n z, s \u2229 closure (M n z) \u2286 M n z :=\n by\n rintro n z x \u27e8xs, hx\u27e9\n refine' \u27e8xs, fun y hy => _\u27e9\n obtain \u27e8a, aM, a_lim\u27e9 : \u2203 a : \u2115 \u2192 E, (\u2200 k, a k \u2208 M n z) \u2227 Tendsto a atTop (\ud835\udcdd x) := mem_closure_iff_seq_limit.1 hx\n have L1 : Tendsto (fun k : \u2115 => \u2016f y - f (a k) - (f' z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - (f' z) (y - x)\u2016) :=\n by\n apply Tendsto.norm\n have L : Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x)) :=\n by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).1\n apply Tendsto.sub (tendsto_const_nhds.sub L)\n exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)\n have L2 : Tendsto (fun k : \u2115 => (r (f' z) : \u211d) * \u2016y - a k\u2016) atTop (\ud835\udcdd (r (f' z) * \u2016y - x\u2016)) :=\n (tendsto_const_nhds.sub a_lim).norm.const_mul _\n have I : \u2200\u1da0 k in atTop, \u2016f y - f (a k) - (f' z) (y - a k)\u2016 \u2264 r (f' z) * \u2016y - a k\u2016 :=\n by\n have L : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x)) := tendsto_const_nhds.dist a_lim\n filter_upwards [(tendsto_order.1 L).2 _ hy.2]\n intro k hk\n exact (aM k).2 y \u27e8hy.1, hk\u27e9\n exact le_of_tendsto_of_tendsto L1 L2 I\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\n\u22a2 \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\n[PROOFSTEP]\nrintro n z x \u27e8xs, hx\u27e9\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\n\u22a2 x \u2208 M n z\n[PROOFSTEP]\nrefine' \u27e8xs, fun y hy => _\u27e9\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nobtain \u27e8a, aM, a_lim\u27e9 : \u2203 a : \u2115 \u2192 E, (\u2200 k, a k \u2208 M n z) \u2227 Tendsto a atTop (\ud835\udcdd x) := mem_closure_iff_seq_limit.1 hx\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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nhave L1 : Tendsto (fun k : \u2115 => \u2016f y - f (a k) - (f' z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - (f' z) (y - x)\u2016) :=\n by\n apply Tendsto.norm\n have L : Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x)) :=\n by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).1\n apply Tendsto.sub (tendsto_const_nhds.sub L)\n exact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\n[PROOFSTEP]\napply Tendsto.norm\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 Tendsto (fun x => f y - f (a x) - \u2191(f' \u2191\u2191z) (y - a x)) atTop (\ud835\udcdd (f y - f x - \u2191(f' \u2191\u2191z) (y - x)))\n[PROOFSTEP]\nhave L : Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x)) :=\n by\n apply (hf' x xs).continuousWithinAt.tendsto.comp\n apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n exact eventually_of_forall fun k => (aM k).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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x))\n[PROOFSTEP]\napply (hf' x xs).continuousWithinAt.tendsto.comp\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 Tendsto (fun k => a k) atTop (\ud835\udcdd[s] x)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ a_lim\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, a x \u2208 s\n[PROOFSTEP]\nexact eventually_of_forall fun k => (aM k).1\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL : Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun x => f y - f (a x) - \u2191(f' \u2191\u2191z) (y - a x)) atTop (\ud835\udcdd (f y - f x - \u2191(f' \u2191\u2191z) (y - x)))\n[PROOFSTEP]\napply Tendsto.sub (tendsto_const_nhds.sub L)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL : Tendsto (fun k => f (a k)) atTop (\ud835\udcdd (f x))\n\u22a2 Tendsto (fun x => \u2191(f' \u2191\u2191z) (y - a x)) atTop (\ud835\udcdd (\u2191(f' \u2191\u2191z) (y - x)))\n[PROOFSTEP]\nexact ((f' z).continuous.tendsto _).comp (tendsto_const_nhds.sub a_lim)\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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nhave L2 : Tendsto (fun k : \u2115 => (r (f' z) : \u211d) * \u2016y - a k\u2016) atTop (\ud835\udcdd (r (f' z) * \u2016y - x\u2016)) :=\n (tendsto_const_nhds.sub a_lim).norm.const_mul _\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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nhave I : \u2200\u1da0 k in atTop, \u2016f y - f (a k) - (f' z) (y - a k)\u2016 \u2264 r (f' z) * \u2016y - a k\u2016 :=\n by\n have L : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x)) := tendsto_const_nhds.dist a_lim\n filter_upwards [(tendsto_order.1 L).2 _ hy.2]\n intro k hk\n exact (aM k).2 y \u27e8hy.1, hk\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\n\u22a2 \u2200\u1da0 (k : \u2115) in atTop, \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016\n[PROOFSTEP]\nhave L : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x)) := tendsto_const_nhds.dist a_lim\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\nL : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x))\n\u22a2 \u2200\u1da0 (k : \u2115) in atTop, \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 L).2 _ hy.2]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\nL : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x))\n\u22a2 \u2200 (a_1 : \u2115), dist y (a a_1) < u n \u2192 \u2016f y - f (a a_1) - \u2191(f' \u2191\u2191z) (y - a a_1)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - a a_1\u2016\n[PROOFSTEP]\nintro k hk\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\nL : Tendsto (fun k => dist y (a k)) atTop (\ud835\udcdd (dist y x))\nk : \u2115\nhk : dist y (a k) < u n\n\u22a2 \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016\n[PROOFSTEP]\nexact (aM k).2 y \u27e8hy.1, hk\u27e9\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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nn : \u2115\nz : \u2191T\nx : E\nxs : x \u2208 s\nhx : x \u2208 closure (M n z)\ny : E\nhy : y \u2208 s \u2229 ball x (u n)\na : \u2115 \u2192 E\naM : \u2200 (k : \u2115), a k \u2208 M n z\na_lim : Tendsto a atTop (\ud835\udcdd x)\nL1 : Tendsto (fun k => \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016) atTop (\ud835\udcdd \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016)\nL2 : Tendsto (fun k => \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016) atTop (\ud835\udcdd (\u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016))\nI : \u2200\u1da0 (k : \u2115) in atTop, \u2016f y - f (a k) - \u2191(f' \u2191\u2191z) (y - a k)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - a k\u2016\n\u22a2 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016\n[PROOFSTEP]\nexact le_of_tendsto_of_tendsto L1 L2 I\n[GOAL]\ncase inr.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nrcases TopologicalSpace.exists_dense_seq E with\n \u27e8d, hd\u27e9\n -- split `M n z` into subsets `K n z p` of small diameters by intersecting with the ball\n -- `closedBall (d p) (u n / 3)`.\n[GOAL]\ncase inr.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nlet K : \u2115 \u2192 T \u2192 \u2115 \u2192 Set E := fun n z p =>\n closure (M n z) \u2229\n closedBall (d p)\n (u n / 3)\n -- on the sets `K n z p`, the map `f` is well approximated by `f' z` by design.\n[GOAL]\ncase inr.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nhave K_approx : \u2200 (n) (z : T) (p), ApproximatesLinearOn f (f' z) (s \u2229 K n z p) (r (f' z)) :=\n by\n intro n z p x hx y hy\n have yM : y \u2208 M n z := closure_M_subset _ _ \u27e8hy.1, hy.2.1\u27e9\n refine' yM.2 _ \u27e8hx.1, _\u27e9\n calc\n dist x y \u2264 dist x (d p) + dist y (d p) := dist_triangle_right _ _ _\n _ \u2264 u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)\n _ < u n := by\n linarith [u_pos n]\n -- the sets `K n z p` are also closed, again by design.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\n\u22a2 \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\n[PROOFSTEP]\nintro n z p x hx y hy\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nn : \u2115\nz : \u2191T\np : \u2115\nx : E\nhx : x \u2208 s \u2229 K n z p\ny : E\nhy : y \u2208 s \u2229 K n z p\n\u22a2 \u2016f x - f y - \u2191(f' \u2191\u2191z) (x - y)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016x - y\u2016\n[PROOFSTEP]\nhave yM : y \u2208 M n z := closure_M_subset _ _ \u27e8hy.1, hy.2.1\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nn : \u2115\nz : \u2191T\np : \u2115\nx : E\nhx : x \u2208 s \u2229 K n z p\ny : E\nhy : y \u2208 s \u2229 K n z p\nyM : y \u2208 M n z\n\u22a2 \u2016f x - f y - \u2191(f' \u2191\u2191z) (x - y)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016x - y\u2016\n[PROOFSTEP]\nrefine' yM.2 _ \u27e8hx.1, _\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nn : \u2115\nz : \u2191T\np : \u2115\nx : E\nhx : x \u2208 s \u2229 K n z p\ny : E\nhy : y \u2208 s \u2229 K n z p\nyM : y \u2208 M n z\n\u22a2 x \u2208 ball y (u n)\n[PROOFSTEP]\ncalc\n dist x y \u2264 dist x (d p) + dist y (d p) := dist_triangle_right _ _ _\n _ \u2264 u n / 3 + u n / 3 := (add_le_add hx.2.2 hy.2.2)\n _ < u n := by\n linarith [u_pos n]\n -- the sets `K n z p` are also closed, again by design.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nn : \u2115\nz : \u2191T\np : \u2115\nx : E\nhx : x \u2208 s \u2229 K n z p\ny : E\nhy : y \u2208 s \u2229 K n z p\nyM : y \u2208 M n z\n\u22a2 u n / 3 + u n / 3 < u n\n[PROOFSTEP]\nlinarith [u_pos n]\n -- the sets `K n z p` are also closed, again by design.\n[GOAL]\ncase inr.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nhave K_closed : \u2200 (n) (z : T) (p), IsClosed (K n z p) := fun n z p => isClosed_closure.inter isClosed_ball\n[GOAL]\ncase inr.intro.intro.intro.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nobtain \u27e8F, hF\u27e9 : \u2203 F : \u2115 \u2192 \u2115 \u00d7 T \u00d7 \u2115, Function.Surjective F :=\n by\n haveI : Encodable T := T_count.toEncodable\n have : Nonempty T := by\n rcases eq_empty_or_nonempty T with (rfl | hT)\n \u00b7 rcases hs with \u27e8x, xs\u27e9\n rcases s_subset x xs with \u27e8n, z, _\u27e9\n exact False.elim z.2\n \u00b7 exact hT.coe_sort\n inhabit \u21a5T\n exact\n \u27e8_, Encodable.surjective_decode_iget (\u2115 \u00d7 T \u00d7 \u2115)\u27e9\n -- these sets `t q = K n z p` will do\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\n\u22a2 \u2203 F, Function.Surjective F\n[PROOFSTEP]\nhaveI : Encodable T := T_count.toEncodable\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191T\n\u22a2 \u2203 F, Function.Surjective F\n[PROOFSTEP]\nhave : Nonempty T := by\n rcases eq_empty_or_nonempty T with (rfl | hT)\n \u00b7 rcases hs with \u27e8x, xs\u27e9\n rcases s_subset x xs with \u27e8n, z, _\u27e9\n exact False.elim z.2\n \u00b7 exact hT.coe_sort\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191T\n\u22a2 Nonempty \u2191T\n[PROOFSTEP]\nrcases eq_empty_or_nonempty T with (rfl | hT)\n[GOAL]\ncase inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nd : \u2115 \u2192 E\nhd : DenseRange d\nT_count : Set.Countable \u2205\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 \u2205), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nM : \u2115 \u2192 \u2191\u2205 \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191\u2205), s \u2229 closure (M n z) \u2286 M n z\nK : \u2115 \u2192 \u2191\u2205 \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191\u2205\n\u22a2 Nonempty \u2191\u2205\n[PROOFSTEP]\nrcases hs with \u27e8x, xs\u27e9\n[GOAL]\ncase inl.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nd : \u2115 \u2192 E\nhd : DenseRange d\nT_count : Set.Countable \u2205\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 \u2205), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nM : \u2115 \u2192 \u2191\u2205 \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191\u2205), s \u2229 closure (M n z) \u2286 M n z\nK : \u2115 \u2192 \u2191\u2205 \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191\u2205\nx : E\nxs : x \u2208 s\n\u22a2 Nonempty \u2191\u2205\n[PROOFSTEP]\nrcases s_subset x xs with \u27e8n, z, _\u27e9\n[GOAL]\ncase inl.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 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nd : \u2115 \u2192 E\nhd : DenseRange d\nT_count : Set.Countable \u2205\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 \u2205), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nM : \u2115 \u2192 \u2191\u2205 \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191\u2205), s \u2229 closure (M n z) \u2286 M n z\nK : \u2115 \u2192 \u2191\u2205 \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191\u2205) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191\u2205\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191\u2205\nh\u271d : x \u2208 M n z\n\u22a2 Nonempty \u2191\u2205\n[PROOFSTEP]\nexact False.elim z.2\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT\u271d : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nthis : Encodable \u2191T\nhT : Set.Nonempty T\n\u22a2 Nonempty \u2191T\n[PROOFSTEP]\nexact hT.coe_sort\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nthis\u271d : Encodable \u2191T\nthis : Nonempty \u2191T\n\u22a2 \u2203 F, Function.Surjective F\n[PROOFSTEP]\ninhabit \u21a5T\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nthis\u271d : Encodable \u2191T\nthis : Nonempty \u2191T\ninhabited_h : Inhabited \u2191T\n\u22a2 \u2203 F, Function.Surjective F\n[PROOFSTEP]\nexact\n \u27e8_, Encodable.surjective_decode_iget (\u2115 \u00d7 T \u00d7 \u2115)\u27e9\n -- these sets `t q = K n z p` will do\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\n\u22a2 \u2203 t A,\n (\u2200 (n : \u2115), IsClosed (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nrefine'\n \u27e8fun q => K (F q).1 (F q).2.1 (F q).2.2, fun q => f' (F q).2.1, fun n => K_closed _ _ _, fun x xs => _, fun q =>\n K_approx _ _ _, fun _ q => \u27e8(F q).2.1, (F q).2.1.1.2, rfl\u27e9\u27e9\n -- the only fact that needs further checking is that they cover `s`.\n -- we already know that any point `x \u2208 s` belongs to a set `M n z`.\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\n\u22a2 x \u2208 \u22c3 (n : \u2115), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n\n[PROOFSTEP]\nobtain \u27e8n, z, hnz\u27e9 : \u2203 (n : \u2115) (z : T), x \u2208 M n z := s_subset x xs\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\n\u22a2 x \u2208 \u22c3 (n : \u2115), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 : \u2203 p : \u2115, x \u2208 closedBall (d p) (u n / 3) :=\n by\n have : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]\n obtain \u27e8p, hp\u27e9 : \u2203 p : \u2115, d p \u2208 ball x (u n / 3) := hd.exists_mem_open isOpen_ball this\n exact\n \u27e8p, (mem_ball'.1 hp).le\u27e9\n -- choose `q` for which `t q = K n z p`.\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\n\u22a2 \u2203 p, x \u2208 closedBall (d p) (u n / 3)\n[PROOFSTEP]\nhave : Set.Nonempty (ball x (u n / 3)) := by simp only [nonempty_ball]; linarith [u_pos n]\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\n\u22a2 Set.Nonempty (ball x (u n / 3))\n[PROOFSTEP]\nsimp only [nonempty_ball]\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\n\u22a2 0 < u n / 3\n[PROOFSTEP]\nlinarith [u_pos n]\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\nthis : Set.Nonempty (ball x (u n / 3))\n\u22a2 \u2203 p, x \u2208 closedBall (d p) (u n / 3)\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 : \u2203 p : \u2115, d p \u2208 ball x (u n / 3) := hd.exists_mem_open isOpen_ball this\n[GOAL]\ncase intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\nthis : Set.Nonempty (ball x (u n / 3))\np : \u2115\nhp : d p \u2208 ball x (u n / 3)\n\u22a2 \u2203 p, x \u2208 closedBall (d p) (u n / 3)\n[PROOFSTEP]\nexact\n \u27e8p, (mem_ball'.1 hp).le\u27e9\n -- choose `q` for which `t q = K n z p`.\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\np : \u2115\nhp : x \u2208 closedBall (d p) (u n / 3)\n\u22a2 x \u2208 \u22c3 (n : \u2115), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 : \u2203 q, F q = (n, z, p) :=\n hF\n _\n -- then `x` belongs to `t q`.\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\np : \u2115\nhp : x \u2208 closedBall (d p) (u n / 3)\nq : \u2115\nhq : F q = (n, z, p)\n\u22a2 x \u2208 \u22c3 (n : \u2115), (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) n\n[PROOFSTEP]\napply mem_iUnion.2 \u27e8q, _\u27e9\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d : SecondCountableTopology F\u271d\nf : E \u2192 F\u271d\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\u271d\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F\u271d) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F\u271d), r A \u2260 0\nhs : Set.Nonempty s\nT : Set \u2191s\nT_count : Set.Countable T\nhT : \u22c3 (x : \u2191s) (_ : x \u2208 T), ball (f' \u2191x) \u2191(r (f' \u2191x)) = \u22c3 (x : \u2191s), ball (f' \u2191x) \u2191(r (f' \u2191x))\nu : \u2115 \u2192 \u211d\nleft\u271d : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nM : \u2115 \u2192 \u2191T \u2192 Set E :=\n fun n z => {x | x \u2208 s \u2227 \u2200 (y : E), y \u2208 s \u2229 ball x (u n) \u2192 \u2016f y - f x - \u2191(f' \u2191\u2191z) (y - x)\u2016 \u2264 \u2191(r (f' \u2191\u2191z)) * \u2016y - x\u2016}\ns_subset : \u2200 (x : E), x \u2208 s \u2192 \u2203 n z, x \u2208 M n z\nclosure_M_subset : \u2200 (n : \u2115) (z : \u2191T), s \u2229 closure (M n z) \u2286 M n z\nd : \u2115 \u2192 E\nhd : DenseRange d\nK : \u2115 \u2192 \u2191T \u2192 \u2115 \u2192 Set E := fun n z p => closure (M n z) \u2229 closedBall (d p) (u n / 3)\nK_approx : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), ApproximatesLinearOn f (f' \u2191\u2191z) (s \u2229 K n z p) (r (f' \u2191\u2191z))\nK_closed : \u2200 (n : \u2115) (z : \u2191T) (p : \u2115), IsClosed (K n z p)\nF : \u2115 \u2192 \u2115 \u00d7 \u2191T \u00d7 \u2115\nhF : Function.Surjective F\nx : E\nxs : x \u2208 s\nn : \u2115\nz : \u2191T\nhnz : x \u2208 M n z\np : \u2115\nhp : x \u2208 closedBall (d p) (u n / 3)\nq : \u2115\nhq : F q = (n, z, p)\n\u22a2 x \u2208 (fun q => K (F q).fst (F q).snd.fst (F q).snd.snd) q\n[PROOFSTEP]\nsimp (config := { zeta := false }) only [hq, subset_closure hnz, hp, mem_inter_iff, and_true, hnz]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\n\u22a2 \u2203 t A,\n Pairwise (Disjoint on t) \u2227\n (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nrcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with\n \u27e8t, A, t_closed, st, t_approx, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 (n : \u2115), t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 t A,\n Pairwise (Disjoint on t) \u2227\n (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n s \u2286 \u22c3 (n : \u2115), t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)\n[PROOFSTEP]\nrefine' \u27e8disjointed t, A, disjoint_disjointed _, MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 (n : \u2115), t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 (n : \u2115), disjointed t n\n[PROOFSTEP]\nrw [iUnion_disjointed]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 (n : \u2115), t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 (n : \u2115), t n\n[PROOFSTEP]\nexact st\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 (n : \u2115), t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 (n : \u2115), t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))\n[PROOFSTEP]\nexact (t_approx n).mono_set (inter_subset_inter_right _ (disjointed_subset _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\napply nhdsWithin_le_nhds\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nlet d :=\n ENNReal.ofReal\n |A.det|\n -- construct a small neighborhood of `A '' (closedBall 0 1)` with measure comparable to\n -- the determinant of `A`.\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 : \u2203 \u03b5 : \u211d, \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5 :=\n by\n have HC : IsCompact (A '' closedBall 0 1) := (ProperSpace.isCompact_closedBall _ _).image A.continuous\n have L0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n exact tendsto_measure_cthickening_of_isCompact HC\n have L1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) :=\n by\n apply L0.congr' _\n filter_upwards [self_mem_nhdsWithin] with r hr\n rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n have L2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1))) :=\n by\n convert L1\n exact (addHaar_image_continuousLinearMap _ _ _).symm\n have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne' measure_closedBall_lt_top.ne).2 hm\n have H : \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n (tendsto_order.1 L2).2 _ I\n exact (H.and self_mem_nhdsWithin).exists\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave HC : IsCompact (A '' closedBall 0 1) := (ProperSpace.isCompact_closedBall _ _).image A.continuous\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave L0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n exact tendsto_measure_cthickening_of_isCompact HC\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n[PROOFSTEP]\nexact tendsto_measure_cthickening_of_isCompact HC\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave L1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) :=\n by\n apply L0.congr' _\n filter_upwards [self_mem_nhdsWithin] with r hr\n rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n[PROOFSTEP]\napply L0.congr' _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 =>\n \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with r hr\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (cthickening r (\u2191A '' closedBall 0 1)) = \u2191\u2191\u03bc (closedBall 0 r + \u2191A '' closedBall 0 1)\n[PROOFSTEP]\nrw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave L2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1))) :=\n by\n convert L1\n exact (addHaar_image_continuousLinearMap _ _ _).symm\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\n[PROOFSTEP]\nconvert L1\n[GOAL]\ncase h.e'_5.h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\n\u22a2 d * \u2191\u2191\u03bc (closedBall 0 1) = \u2191\u2191\u03bc (\u2191A '' closedBall 0 1)\n[PROOFSTEP]\nexact (addHaar_image_continuousLinearMap _ _ _).symm\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne' measure_closedBall_lt_top.ne).2 hm\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nhave H : \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n (tendsto_order.1 L2).2 _ I\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\nHC : IsCompact (\u2191A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (cthickening \u03b5 (\u2191A '' closedBall 0 1))) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (\u2191A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (d * \u2191\u2191\u03bc (closedBall 0 1)))\nI : d * \u2191\u2191\u03bc (closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\nH : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (closedBall 0 b + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1) \u2227 0 < \u03b5\n[PROOFSTEP]\nexact (H.and self_mem_nhdsWithin).exists\n[GOAL]\ncase a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave : Iio (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply Iio_mem_nhds; exact \u03b5pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n[PROOFSTEP]\napply Iio_mem_nhds\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n[PROOFSTEP]\nexact \u03b5pos\n[GOAL]\ncase a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nfilter_upwards [this]\n -- fix a function `f` which is close enough to `A`.\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u22a2 \u2200 (a : \u211d\u22650),\n a \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2192\n \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nintro \u03b4 h\u03b4 s f hf\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave I : \u2200 x r, x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 m * \u03bc (closedBall x r) :=\n by\n intro x r xs r0\n have K : f '' (s \u2229 closedBall x r) \u2286 A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) :=\n by\n rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9\n apply Set.mem_add.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9\n \u00b7 apply mem_image_of_mem\n simpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr\n \u00b7 rw [mem_closedBall_iff_norm, add_sub_cancel]\n calc\n \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n _ \u2264 \u03b5 * r := mul_le_mul (le_of_lt h\u03b4) (mem_closedBall_iff_norm.1 zr) (norm_nonneg _) \u03b5pos.le\n \u00b7 simp only [map_sub, Pi.sub_apply]\n abel\n have : A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one, smul_zero,\n Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]\n rw [this] at K \n calc\n \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5)) := measure_mono K\n _ = ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add, measure_preimage_add]\n _ \u2264 ENNReal.ofReal (r ^ finrank \u211d E) * (m * \u03bc (closedBall 0 1)) := by rw [add_comm]; exact mul_le_mul_left' h\u03b5.le _\n _ = m * \u03bc (closedBall x r) := by simp only [addHaar_closedBall' \u03bc _ r0];\n ring\n -- covering `s` by closed balls with total measure very close to `\u03bc s`, one deduces that the\n -- measure of `f '' s` is at most `m * (\u03bc s + a)` for any positive `a`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nintro x r xs r0\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nhave K : f '' (s \u2229 closedBall x r) \u2286 A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) :=\n by\n rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9\n apply Set.mem_add.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9\n \u00b7 apply mem_image_of_mem\n simpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr\n \u00b7 rw [mem_closedBall_iff_norm, add_sub_cancel]\n calc\n \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n _ \u2264 \u03b5 * r := mul_le_mul (le_of_lt h\u03b4) (mem_closedBall_iff_norm.1 zr) (norm_nonneg _) \u03b5pos.le\n \u00b7 simp only [map_sub, Pi.sub_apply]\n abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n[PROOFSTEP]\nrintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z \u2208 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n[PROOFSTEP]\napply Set.mem_add.2 \u27e8A (z - x), f z - f x - A (z - x) + f x, _, _, _\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) \u2208 \u2191A '' closedBall 0 r\n[PROOFSTEP]\napply mem_image_of_mem\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 z - x \u2208 closedBall 0 r\n[PROOFSTEP]\nsimpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z - f x - \u2191A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r)\n[PROOFSTEP]\nrw [mem_closedBall_iff_norm, add_sub_cancel]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2016f z - f x - \u2191A (z - x)\u2016 \u2264 \u03b5 * r\n[PROOFSTEP]\ncalc\n \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n _ \u2264 \u03b5 * r := mul_le_mul (le_of_lt h\u03b4) (mem_closedBall_iff_norm.1 zr) (norm_nonneg _) \u03b5pos.le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A (z - x) + (f z - f x - \u2191A (z - x) + f x) = f z\n[PROOFSTEP]\nsimp only [map_sub, Pi.sub_apply]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A z - \u2191A x + (f z - f x - (\u2191A z - \u2191A x) + f x) = f z\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 \u2191A z - \u2191A x + (f z - f x - (\u2191A z - \u2191A x) + f x) = f z\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nhave : A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one, smul_zero,\n Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n[PROOFSTEP]\nrw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one, smul_zero, Real.norm_eq_abs,\n abs_of_nonneg r0, mul_one, mul_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nrw [this] at K \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\ncalc\n \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5)) := measure_mono K\n _ = ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add, measure_preimage_add]\n _ \u2264 ENNReal.ofReal (r ^ finrank \u211d E) * (m * \u03bc (closedBall 0 1)) := by rw [add_comm]; exact mul_le_mul_left' h\u03b5.le _\n _ = m * \u03bc (closedBall x r) := by simp only [addHaar_closedBall' \u03bc _ r0];\n ring\n -- covering `s` by closed balls with total measure very close to `\u03bc s`, one deduces that the\n -- measure of `f '' s` is at most `m * (\u03bc s + a)` for any positive `a`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u2191\u2191\u03bc ({f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)) =\n ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n[PROOFSTEP]\nsimp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add, measure_preimage_add]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5) \u2264\n ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1))\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) \u2264\n ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1))\n[PROOFSTEP]\nexact mul_le_mul_left' h\u03b5.le _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) = \u2191m * \u2191\u2191\u03bc (closedBall x r)\n[PROOFSTEP]\nsimp only [addHaar_closedBall' \u03bc _ r0]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u2191A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u2191A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u2191\u2191\u03bc (closedBall 0 1)) =\n \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (closedBall 0 1))\n[PROOFSTEP]\nring\n -- covering `s` by closed balls with total measure very close to `\u03bc s`, one deduces that the\n -- measure of `f '' s` is at most `m * (\u03bc s + a)` for any positive `a`.\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) :=\n by\n filter_upwards [self_mem_nhdsWithin] with a ha\n change 0 < a at ha \n obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n \u2203 (t : Set E) (r : E \u2192 \u211d),\n t.Countable \u2227\n t \u2286 s \u2227\n (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227 (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s fun x _ \u03b4 \u03b4pos =>\n \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9\n haveI : Encodable t := t_count.toEncodable\n calc\n \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) :=\n by\n rw [biUnion_eq_iUnion] at st \n apply measure_mono\n rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n exact image_subset _ (subset_inter (Subset.refl _) st)\n _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) := (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left];\n exact\n mul_le_mul_left' \u03bct\n _\n -- taking the limit in `a`, one obtains the conclusion\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with a ha\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : a \u2208 Ioi 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nchange 0 < a at ha \n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nobtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n \u2203 (t : Set E) (r : E \u2192 \u211d),\n t.Countable \u2227\n t \u2286 s \u2227\n (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227 (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s fun x _ \u03b4 \u03b4pos =>\n \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4\u271d : \u211d\u22650\nh\u03b4 : \u03b4\u271d \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\u271d\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nx : E\nx\u271d : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 \u03b4 / 2 \u2208 (fun x => Ioi 0) x \u2229 Ioo 0 \u03b4\n[PROOFSTEP]\nsimp [half_pos \u03b4pos, \u03b4pos]\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : E) (_ : x \u2208 t), closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nhaveI : Encodable t := t_count.toEncodable\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : E) (_ : x \u2208 t), closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\ncalc\n \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) :=\n by\n rw [biUnion_eq_iUnion] at st \n apply measure_mono\n rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n exact image_subset _ (subset_inter (Subset.refl _) st)\n _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := (measure_iUnion_le _)\n _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) := (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left];\n exact\n mul_le_mul_left' \u03bct\n _\n -- taking the limit in `a`, one obtains the conclusion\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : E) (_ : x \u2208 t), closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 (x : \u2191t), f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))\n[PROOFSTEP]\nrw [biUnion_eq_iUnion] at st \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : \u2191t), closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 (x : \u2191t), f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : \u2191t), closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 \u22c3 (x : \u2191t), f '' (s \u2229 closedBall (\u2191x) (r \u2191x))\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 inter_iUnion]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : \u2191t), closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 (i : \u2191t), closedBall (\u2191i) (r \u2191i))\n[PROOFSTEP]\nexact image_subset _ (subset_inter (Subset.refl _) st)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : E) (_ : x \u2208 t), closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2211' (x : \u2191t), \u2191m * \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nrw [ENNReal.tsum_mul_left]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : Set.Countable t\nts : t \u2286 s\nrpos : \u2200 (x : E), x \u2208 t \u2192 0 < r x\nst : s \u2286 \u22c3 (x : E) (_ : x \u2208 t), closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191\u2191\u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191m * \u2211' (i : \u2191t), \u2191\u2191\u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n[PROOFSTEP]\nexact\n mul_le_mul_left' \u03bct\n _\n -- taking the limit in `a`, one obtains the conclusion\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave L : Tendsto (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)\n simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))\n[PROOFSTEP]\napply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\n\u22a2 \u2191\u2191\u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * (\u2191\u2191\u03bc s + 0)))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [add_zero] at L \n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b5 : \u211d\nh\u03b5 : \u2191\u2191\u03bc (closedBall 0 \u03b5 + \u2191A '' closedBall 0 1) < \u2191m * \u2191\u2191\u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) } \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u2191\u2191\u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * (\u2191\u2191\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u2191\u2191\u03bc s + a)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191m * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact ge_of_tendsto L J\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\napply nhdsWithin_le_nhds\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrcases eq_or_lt_of_le (zero_le m) with (rfl | mpos)\n[GOAL]\ncase a.inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase a.inl.hp\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nhm : \u21910 < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2200 (x : \u211d\u22650) (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s x \u2192 \u21910 * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nsimp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero]\n[GOAL]\ncase a.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave hA : A.det \u2260 0 := by intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\n\u22a2 ContinuousLinearMap.det A \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nh : ContinuousLinearMap.det A = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm \n[GOAL]\ncase a.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nlet B := A.toContinuousLinearEquivOfDetNeZero hA\n[GOAL]\ncase a.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave I : ENNReal.ofReal |(B.symm : E \u2192L[\u211d] E).det| < (m\u207b\u00b9 : \u211d\u22650) :=\n by\n simp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2\n exact NNReal.inv_lt_inv mpos.ne' hm\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, abs_inv, Real.toNNReal_inv, ContinuousLinearEquiv.det_coe_symm,\n ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero, ENNReal.coe_lt_coe] at hm \u22a2\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nhm : m < Real.toNNReal |ContinuousLinearMap.det A|\n\u22a2 (Real.toNNReal |ContinuousLinearMap.det A|)\u207b\u00b9 < m\u207b\u00b9\n[PROOFSTEP]\nexact NNReal.inv_lt_inv mpos.ne' hm\n[GOAL]\ncase a.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nobtain \u27e8\u03b4\u2080, \u03b4\u2080pos, h\u03b4\u2080\u27e9 :\n \u2203 \u03b4 : \u211d\u22650,\n 0 < \u03b4 \u2227 \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n by\n have :\n \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n addHaar_image_le_mul_of_det_lt \u03bc B.symm I\n rcases(this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, h, h'\u27e9\n exact\n \u27e8\u03b4\u2080, h', h\u27e9\n -- record smallness conditions for `\u03b4` that will be needed to apply `h\u03b4\u2080` below.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave :\n \u2200\u1da0 \u03b4 : \u211d\u22650 in \ud835\udcdd[>] 0,\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (B.symm : E \u2192L[\u211d] E) t \u03b4 \u2192 \u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u03bc t :=\n addHaar_image_le_mul_of_det_lt \u03bc B.symm I\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases(this.and self_mem_nhdsWithin).exists with \u27e8\u03b4\u2080, h, h'\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\nthis :\n \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u03b4\u2080 : \u211d\u22650\nh :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh' : 0 < \u03b4\u2080\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact\n \u27e8\u03b4\u2080, h', h\u27e9\n -- record smallness conditions for `\u03b4` that will be needed to apply `h\u03b4\u2080` below.\n[GOAL]\ncase a.inr.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave L1 : \u2200\u1da0 \u03b4 in \ud835\udcdd (0 : \u211d\u22650), Subsingleton E \u2228 \u03b4 < \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 :=\n by\n by_cases Subsingleton E\n \u00b7 simp only [h, true_or_iff, eventually_const]\n simp only [h, false_or_iff]\n apply Iio_mem_nhds\n simpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\nby_cases Subsingleton E\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\nby_cases Subsingleton E\n[GOAL]\ncase pos\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : Subsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\nsimp only [h, true_or_iff, eventually_const]\n[GOAL]\ncase neg\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\nsimp only [h, false_or_iff]\n[GOAL]\ncase neg\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0,\n \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\napply Iio_mem_nhds\n[GOAL]\ncase neg.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nh : \u00acSubsingleton E\n\u22a2 0 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\n[PROOFSTEP]\nsimpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos\n[GOAL]\ncase a.inr.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nhave L2 : \u2200\u1da0 \u03b4 in \ud835\udcdd (0 : \u211d\u22650), \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a * (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080 :=\n by\n have :\n Tendsto (fun \u03b4 => \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a * (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a * (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) :=\n by\n rcases eq_or_ne \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H)\n \u00b7 simpa only [H, zero_mul] using tendsto_const_nhds\n refine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id\n refine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv\u2080 _\n simpa only [tsub_zero, inv_eq_zero, Ne.def] using H\n simp only [mul_zero] at this \n exact (tendsto_order.1 this).2 \u03b4\u2080 \u03b4\u2080pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n[PROOFSTEP]\nhave :\n Tendsto (fun \u03b4 => \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a * (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a * (\u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0)) :=\n by\n rcases eq_or_ne \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H)\n \u00b7 simpa only [H, zero_mul] using tendsto_const_nhds\n refine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id\n refine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv\u2080 _\n simpa only [tsub_zero, inv_eq_zero, Ne.def] using H\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\n[PROOFSTEP]\nrcases eq_or_ne \u2016(B.symm : E \u2192L[\u211d] E)\u2016\u208a 0 with (H | H)\n[GOAL]\ncase inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a = 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\n[PROOFSTEP]\nsimpa only [H, zero_mul] using tendsto_const_nhds\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\n[PROOFSTEP]\nrefine' Tendsto.mul (tendsto_const_nhds.mul _) tendsto_id\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 Tendsto (fun \u03b4 => (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9) (\ud835\udcdd 0) (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9)\n[PROOFSTEP]\nrefine' (Tendsto.sub tendsto_const_nhds tendsto_id).inv\u2080 _\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nH : \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a \u2260 0\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0 \u2260 0\n[PROOFSTEP]\nsimpa only [tsub_zero, inv_eq_zero, Ne.def] using H\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nthis :\n Tendsto (fun \u03b4 => \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4) (\ud835\udcdd 0)\n (\ud835\udcdd (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - 0)\u207b\u00b9 * 0))\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n[PROOFSTEP]\nsimp only [mul_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nthis :\n Tendsto\n (fun \u03b4 =>\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4)\n (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n[PROOFSTEP]\nexact (tendsto_order.1 this).2 \u03b4\u2080 \u03b4\u2080pos\n[GOAL]\ncase a.inr.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)) x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nfilter_upwards [L1, L2]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u22a2 \u2200 (a : \u211d\u22650),\n Subsingleton E \u2228\n a < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 \u2192\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - a)\u207b\u00b9 *\n a <\n \u03b4\u2080 \u2192\n \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nintro \u03b4 h1\u03b4 h2\u03b4 s f hf\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nhave hf' : ApproximatesLinearOn f (B : E \u2192L[\u211d] E) s \u03b4 := by convert hf\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 ApproximatesLinearOn f (\u2191B) s \u03b4\n[PROOFSTEP]\nconvert hf\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nlet F := hf'.toLocalEquiv h1\u03b4\n[GOAL]\ncase h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nsuffices H : \u03bc (F.symm '' F.target) \u2264 (m\u207b\u00b9 : \u211d\u22650) * \u03bc F.target\n[GOAL]\ncase h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nchange (m : \u211d\u22650\u221e) * \u03bc F.source \u2264 \u03bc F.target\n[GOAL]\ncase h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m * \u2191\u2191\u03bc F.source \u2264 \u2191\u2191\u03bc F.target\n[PROOFSTEP]\nrwa [\u2190 F.symm_image_target_eq_source, mul_comm, \u2190 ENNReal.le_div_iff_mul_le, div_eq_mul_inv, mul_comm, \u2190\n ENNReal.coe_inv mpos.ne']\n[GOAL]\ncase h.h0\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0 \u2228 \u2191\u2191\u03bc F.target \u2260 0\n[PROOFSTEP]\napply Or.inl\n[GOAL]\ncase h.h0.h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 0\n[PROOFSTEP]\nsimpa only [ENNReal.coe_eq_zero, Ne.def] using mpos.ne'\n[GOAL]\ncase h.ht\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\nH : \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n\u22a2 \u2191m \u2260 \u22a4 \u2228 \u2191\u2191\u03bc F.target \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.coe_ne_top, true_or_iff, Ne.def, not_false_iff]\n -- as `f\u207b\u00b9` is well approximated by `B\u207b\u00b9`, the conclusion follows from `h\u03b4\u2080`\n -- and our choice of `\u03b4`.\n[GOAL]\ncase H\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\nmpos : 0 < m\nhA : ContinuousLinearMap.det A \u2260 0\nB : E \u2243L[\u211d] E := ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA\nI : ENNReal.ofReal |ContinuousLinearMap.det \u2191(ContinuousLinearEquiv.symm B)| < \u2191m\u207b\u00b9\n\u03b4\u2080 : \u211d\u22650\n\u03b4\u2080pos : 0 < \u03b4\u2080\nh\u03b4\u2080 :\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g (\u2191(ContinuousLinearEquiv.symm B)) t \u03b4\u2080 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc t\nL1 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9\nL2 : \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd 0, \u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a * (\u2016\u2191(ContinuousLinearEquiv.symm B)\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 * \u03b4 < \u03b4\u2080\n\u03b4 : \u211d\u22650\nh1\u03b4 :\n Subsingleton E \u2228 \u03b4 < \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9\nh2\u03b4 :\n \u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a *\n (\u2016\u2191(ContinuousLinearEquiv.symm (ContinuousLinearMap.toContinuousLinearEquivOfDetNeZero A hA))\u2016\u208a\u207b\u00b9 - \u03b4)\u207b\u00b9 *\n \u03b4 <\n \u03b4\u2080\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nhf' : ApproximatesLinearOn f (\u2191B) s \u03b4\nF : LocalEquiv E E := ApproximatesLinearOn.toLocalEquiv hf' h1\u03b4\n\u22a2 \u2191\u2191\u03bc (\u2191(LocalEquiv.symm F) '' F.target) \u2264 \u2191m\u207b\u00b9 * \u2191\u2191\u03bc F.target\n[PROOFSTEP]\nexact h\u03b4\u2080 _ _ ((hf'.to_inv h1\u03b4).mono_num h2\u03b4.le)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, \u2016f' x - A\u2016\u208a \u2264 \u03b4\n[PROOFSTEP]\nfilter_upwards [Besicovitch.ae_tendsto_measure_inter_div \u03bc s, ae_restrict_mem hs]\n -- start from a Lebesgue density point `x`, belonging to `s`.\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2200 (a : E),\n Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1) \u2192 a \u2208 s \u2192 \u2016f' a - A\u2016\u208a \u2264 \u03b4\n[PROOFSTEP]\nintro x hx xs\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\n\u22a2 \u2016f' x - A\u2016\u208a \u2264 \u03b4\n[PROOFSTEP]\napply\n ContinuousLinearMap.op_norm_le_bound _ \u03b4.2 fun z =>\n ?_\n -- to show that `\u2016(f' x - A) z\u2016 \u2264 \u03b4 \u2016z\u2016`, it suffices to do it up to some error that vanishes\n -- asymptotically in terms of `\u03b5 > 0`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016\n[PROOFSTEP]\nsuffices H : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2016(f' x - A) z\u2016 \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016\n[PROOFSTEP]\nhave :\n Tendsto (fun \u03b5 : \u211d => ((\u03b4 : \u211d) + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[>] 0)\n (\ud835\udcdd ((\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0)) :=\n Tendsto.mono_left (Continuous.tendsto (by continuity) 0) nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n\u22a2 Continuous fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\ncontinuity\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd ((\u2191\u03b4 + 0) * (\u2016z\u2016 + 0) + \u2016f' x - A\u2016 * 0))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016\n[PROOFSTEP]\nsimp only [add_zero, mul_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 \u2191\u03b4 * \u2016z\u2016\n[PROOFSTEP]\napply le_of_tendsto_of_tendsto tendsto_const_nhds this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 (fun x_1 => \u2016\u2191(f' x - A) z\u2016) \u2264\u1da0[\ud835\udcdd[Ioi 0] 0] fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\nH : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\nthis : Tendsto (fun \u03b5 => (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u03b4 * \u2016z\u2016))\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + a) * (\u2016z\u2016 + a) + \u2016f' x - A\u2016 * a\n[PROOFSTEP]\nexact H\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nhave B\u2081 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), (s \u2229 ({ x } + r \u2022 closedBall z \u03b5)).Nonempty :=\n eventually_nonempty_inter_smul_of_density_one \u03bc s x hx _ measurableSet_closedBall\n (measure_closedBall_pos \u03bc z \u03b5pos).ne'\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03c1, \u03c1pos, h\u03c1\u27e9 : \u2203 \u03c1 > 0, ball x \u03c1 \u2229 s \u2286 {y : E | \u2016f y - f x - (f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016} :=\n mem_nhdsWithin_iff.1\n (IsLittleO.def (hf' x xs) \u03b5pos)\n -- for small enough `r`, the rescaled ball `r \u2022 closedBall z \u03b5` is included in the set where\n -- `f y - f x` is well approximated by `f' x (y - x)`.\n[GOAL]\ncase H.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nhave B\u2082 : \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), { x } + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 :=\n by\n apply nhdsWithin_le_nhds\n exact\n eventually_singleton_add_smul_subset bounded_closedBall\n (ball_mem_nhds x \u03c1pos)\n -- fix a small positive `r` satisfying the above properties, as well as a corresponding `y`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\n[PROOFSTEP]\napply nhdsWithin_le_nhds\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\n\u22a2 {x_1 | (fun r => {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1) x_1} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nexact\n eventually_singleton_add_smul_subset bounded_closedBall\n (ball_mem_nhds x \u03c1pos)\n -- fix a small positive `r` satisfying the above properties, as well as a corresponding `y`.\n[GOAL]\ncase H.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nobtain \u27e8r, \u27e8y, \u27e8ys, hy\u27e9\u27e9, r\u03c1, rpos\u27e9 :\n \u2203 r : \u211d, (s \u2229 ({ x } + r \u2022 closedBall z \u03b5)).Nonempty \u2227 { x } + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1 \u2227 0 < r :=\n (B\u2081.and (B\u2082.and self_mem_nhdsWithin)).exists\n[GOAL]\ncase H.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nobtain \u27e8a, az, ya\u27e9 : \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a :=\n by\n simp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy \n rcases hy with \u27e8a, az, ha\u27e9\n exact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a\n[PROOFSTEP]\nsimp only [mem_smul_set, image_add_left, mem_preimage, singleton_add] at hy \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\nhy : \u2203 y_1, y_1 \u2208 closedBall z \u03b5 \u2227 r \u2022 y_1 = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a\n[PROOFSTEP]\nrcases hy with \u27e8a, az, ha\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nha : r \u2022 a = -x + y\n\u22a2 \u2203 a, a \u2208 closedBall z \u03b5 \u2227 y = x + r \u2022 a\n[PROOFSTEP]\nexact \u27e8a, az, by simp only [ha, add_neg_cancel_left]\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nha : r \u2022 a = -x + y\n\u22a2 y = x + r \u2022 a\n[PROOFSTEP]\nsimp only [ha, add_neg_cancel_left]\n[GOAL]\ncase H.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nhave norm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5 :=\n calc\n \u2016a\u2016 = \u2016z + (a - z)\u2016 := by simp only [add_sub_cancel'_right]\n _ \u2264 \u2016z\u2016 + \u2016a - z\u2016 := (norm_add_le _ _)\n _ \u2264 \u2016z\u2016 + \u03b5 :=\n add_le_add_left (mem_closedBall_iff_norm.1 az)\n _\n -- use the approximation properties to control `(f' x - A) a`, and then `(f' x - A) z` as `z` is\n -- close to `a`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\n\u22a2 \u2016a\u2016 = \u2016z + (a - z)\u2016\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right]\n[GOAL]\ncase H.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nhave I : r * \u2016(f' x - A) a\u2016 \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) :=\n calc\n r * \u2016(f' x - A) a\u2016 = \u2016(f' x - A) (r \u2022 a)\u2016 := by\n simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n _ = \u2016f y - f x - A (y - x) - (f y - f x - (f' x) (y - x))\u2016 :=\n by\n congr 1\n simp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub', eq_self_iff_true,\n sub_left_inj, Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub]\n _ \u2264 \u2016f y - f x - A (y - x)\u2016 + \u2016f y - f x - (f' x) (y - x)\u2016 := (norm_sub_le _ _)\n _ \u2264 \u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 := (add_le_add (hf _ ys _ xs) (h\u03c1 \u27e8r\u03c1 hy, ys\u27e9))\n _ = r * (\u03b4 + \u03b5) * \u2016a\u2016 :=\n by\n simp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n ring\n _ \u2264 r * (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) := mul_le_mul_of_nonneg_left norm_a (mul_nonneg rpos.le (add_nonneg \u03b4.2 \u03b5pos.le))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 r * \u2016\u2191(f' x - A) a\u2016 = \u2016\u2191(f' x - A) (r \u2022 a)\u2016\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2016\u2191(f' x - A) (r \u2022 a)\u2016 = \u2016f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191(f' x - A) (r \u2022 a) = f y - f x - \u2191A (y - x) - (f y - f x - \u2191(f' x) (y - x))\n[PROOFSTEP]\nsimp only [ya, add_sub_cancel', sub_sub_sub_cancel_left, ContinuousLinearMap.coe_sub', eq_self_iff_true, sub_left_inj,\n Pi.sub_apply, ContinuousLinearMap.map_smul, smul_sub]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191\u03b4 * \u2016y - x\u2016 + \u03b5 * \u2016y - x\u2016 = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016\n[PROOFSTEP]\nsimp only [ya, add_sub_cancel', norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\n\u22a2 \u2191\u03b4 * (r * \u2016a\u2016) + \u03b5 * (r * \u2016a\u2016) = r * (\u2191\u03b4 + \u03b5) * \u2016a\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase H.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nshow \u2016(f' x - A) z\u2016 \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[GOAL]\ncase H.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) z\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5\n[PROOFSTEP]\nexact\n calc\n \u2016(f' x - A) z\u2016 = \u2016(f' x - A) a + (f' x - A) (z - a)\u2016 :=\n by\n congr 1\n simp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n abel\n _ \u2264 \u2016(f' x - A) a\u2016 + \u2016(f' x - A) (z - a)\u2016 := (norm_add_le _ _)\n _ \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u2016z - a\u2016 :=\n by\n apply add_le_add\n \u00b7 rw [mul_assoc] at I ; exact (mul_le_mul_left rpos).1 I\n \u00b7 apply ContinuousLinearMap.le_op_norm\n _ \u2264 (\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u03b5 :=\n add_le_add le_rfl (mul_le_mul_of_nonneg_left (mem_closedBall_iff_norm'.1 az) (norm_nonneg _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) z\u2016 = \u2016\u2191(f' x - A) a + \u2191(f' x - A) (z - a)\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x - A) z = \u2191(f' x - A) a + \u2191(f' x - A) (z - a)\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_sub', map_sub, Pi.sub_apply]\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x) z - \u2191A z = \u2191(f' x) a - \u2191A a + (\u2191(f' x) z - \u2191A z - (\u2191(f' x) a - \u2191A a))\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2191(f' x) z - \u2191A z = \u2191(f' x) a - \u2191A a + (\u2191(f' x) z - \u2191A z - (\u2191(f' x) a - \u2191A a))\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) a\u2016 + \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5) + \u2016f' x - A\u2016 * \u2016z - a\u2016\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n[PROOFSTEP]\nrw [mul_assoc] at I \n[GOAL]\ncase h\u2081\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * ((\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5))\n\u22a2 \u2016\u2191(f' x - A) a\u2016 \u2264 (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n[PROOFSTEP]\nexact (mul_le_mul_left rpos).1 I\n[GOAL]\ncase h\u2082\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nA : E \u2192L[\u211d] E\n\u03b4 : \u211d\u22650\nhf : ApproximatesLinearOn f A s \u03b4\nhs : MeasurableSet s\nf' : E \u2192 E \u2192L[\u211d] E\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nx : E\nhx : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 1)\nxs : x \u2208 s\nz : E\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nB\u2081 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, Set.Nonempty (s \u2229 ({x} + r \u2022 closedBall z \u03b5))\n\u03c1 : \u211d\n\u03c1pos : \u03c1 > 0\nh\u03c1 : ball x \u03c1 \u2229 s \u2286 {y | \u2016f y - f x - \u2191(f' x) (y - x)\u2016 \u2264 \u03b5 * \u2016y - x\u2016}\nB\u2082 : \u2200\u1da0 (r : \u211d) in \ud835\udcdd[Ioi 0] 0, {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nr : \u211d\ny : E\nys : y \u2208 s\nhy : y \u2208 {x} + r \u2022 closedBall z \u03b5\nr\u03c1 : {x} + r \u2022 closedBall z \u03b5 \u2286 ball x \u03c1\nrpos : 0 < r\na : E\naz : a \u2208 closedBall z \u03b5\nya : y = x + r \u2022 a\nnorm_a : \u2016a\u2016 \u2264 \u2016z\u2016 + \u03b5\nI : r * \u2016\u2191(f' x - A) a\u2016 \u2264 r * (\u2191\u03b4 + \u03b5) * (\u2016z\u2016 + \u03b5)\n\u22a2 \u2016\u2191(f' x - A) (z - a)\u2016 \u2264 \u2016f' x - A\u2016 * \u2016z - a\u2016\n[PROOFSTEP]\napply ContinuousLinearMap.le_op_norm\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0\n[PROOFSTEP]\nrefine' le_antisymm _ (zero_le _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0\n[PROOFSTEP]\nhave :\n \u2200 A : E \u2192L[\u211d] E,\n \u2203 \u03b4 : \u211d\u22650,\n 0 < \u03b4 \u2227 \u2200 (t : Set E), ApproximatesLinearOn f A t \u03b4 \u2192 \u03bc (f '' t) \u2264 (Real.toNNReal |A.det| + 1 : \u211d\u22650) * \u03bc t :=\n by\n intro A\n let m : \u211d\u22650 := Real.toNNReal |A.det| + 1\n have I : ENNReal.ofReal |A.det| < m := by\n simp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe]\n rcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet m : \u211d\u22650 := Real.toNNReal |A.det| + 1\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + 1\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I : ENNReal.ofReal |A.det| < m := by\n simp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + 1\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, lt_add_iff_pos_right, zero_lt_one, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + 1\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + 1\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\nh' : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\nthis :\n \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0\n[PROOFSTEP]\nchoose \u03b4 h\u03b4 using this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0\n[PROOFSTEP]\nobtain \u27e8t, A, _, _, t_cover, ht, -\u27e9 :\n \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n Pairwise (Disjoint on t) \u2227\n (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n (s \u2286 \u22c3 n : \u2115, t n) \u2227\n (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227\n (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = fderivWithin \u211d f s y) :=\n exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s (fderivWithin \u211d f s)\n (fun x xs => (hf x xs).hasFDerivWithinAt) \u03b4 fun A => (h\u03b4 A).1.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0\n[PROOFSTEP]\ncalc\n \u03bc (f '' s) \u2264 \u03bc (\u22c3 n, f '' (s \u2229 t n)) := by\n apply measure_mono\n rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n exact image_subset f (subset_inter Subset.rfl t_cover)\n _ \u2264 \u2211' n, \u03bc (f '' (s \u2229 t n)) := (measure_iUnion_le _)\n _ \u2264 \u2211' n, (Real.toNNReal |(A n).det| + 1 : \u211d\u22650) * \u03bc (s \u2229 t n) :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply (h\u03b4 (A n)).2\n exact ht n\n _ \u2264 \u2211' n, ((Real.toNNReal |(A n).det| + 1 : \u211d\u22650) : \u211d\u22650\u221e) * 0 :=\n by\n refine' ENNReal.tsum_le_tsum fun n => mul_le_mul_left' _ _\n exact le_trans (measure_mono (inter_subset_left _ _)) (le_of_eq hs)\n _ = 0 := by simp only [tsum_zero, mul_zero]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 (n : \u2115), f '' (s \u2229 t n))\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 f '' s \u2286 \u22c3 (n : \u2115), f '' (s \u2229 t n)\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 inter_iUnion]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 (i : \u2115), t i)\n[PROOFSTEP]\nexact image_subset f (subset_inter Subset.rfl t_cover)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply (h\u03b4 (A n)).2\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n[PROOFSTEP]\nexact ht n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * \u2191\u2191\u03bc (s \u2229 t n) \u2264\n \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * 0\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => mul_le_mul_left' _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 t n) \u2264 0\n[PROOFSTEP]\nexact le_trans (measure_mono (inter_subset_left _ _)) (le_of_eq hs)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf : DifferentiableOn \u211d f s\nhs : \u2191\u2191\u03bc s = 0\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + 1) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nleft\u271d\u00b9 : Pairwise (Disjoint on t)\nleft\u271d : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + 1) * 0 = 0\n[PROOFSTEP]\nsimp only [tsum_zero, mul_zero]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | h's)\n[GOAL]\ncase inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nR : \u211d\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhf' : \u2200 (x : E), x \u2208 \u2205 \u2192 HasFDerivWithinAt f (f' x) \u2205 x\nhs : \u2205 \u2286 closedBall 0 R\nh'f' : \u2200 (x : E), x \u2208 \u2205 \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u2205) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nsimp only [measure_empty, zero_le, image_empty]\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nhave :\n \u2200 A : E \u2192L[\u211d] E,\n \u2203 \u03b4 : \u211d\u22650,\n 0 < \u03b4 \u2227 \u2200 (t : Set E), ApproximatesLinearOn f A t \u03b4 \u2192 \u03bc (f '' t) \u2264 (Real.toNNReal |A.det| + \u03b5 : \u211d\u22650) * \u03bc t :=\n by\n intro A\n let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n have I : ENNReal.ofReal |A.det| < m := by simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n rcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n exact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I : ENNReal.ofReal |A.det| < m := by simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, h'\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\nh' : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact \u27e8\u03b4, h', fun t ht => h t f ht\u27e9\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\nthis :\n \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t \u03b4 \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nchoose \u03b4 h\u03b4 using this\n[GOAL]\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nobtain \u27e8t, A, t_disj, t_meas, t_cover, ht, Af'\u27e9 :\n \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n Pairwise (Disjoint on t) \u2227\n (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n (s \u2286 \u22c3 n : \u2115, t n) \u2227\n (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227 (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' \u03b4 fun A => (h\u03b4 A).1.ne'\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\ncalc\n \u03bc (f '' s) \u2264 \u03bc (\u22c3 n, f '' (s \u2229 t n)) := by\n apply measure_mono\n rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n exact image_subset f (subset_inter Subset.rfl t_cover)\n _ \u2264 \u2211' n, \u03bc (f '' (s \u2229 t n)) := (measure_iUnion_le _)\n _ \u2264 \u2211' n, (Real.toNNReal |(A n).det| + \u03b5 : \u211d\u22650) * \u03bc (s \u2229 t n) :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply (h\u03b4 (A n)).2\n exact ht n\n _ = \u2211' n, \u03b5 * \u03bc (s \u2229 t n) := by\n congr with n\n rcases Af' h's n with \u27e8y, ys, hy\u27e9\n simp only [hy, h'f' y ys, Real.toNNReal_zero, abs_zero, zero_add]\n _ \u2264 \u03b5 * \u2211' n, \u03bc (closedBall 0 R \u2229 t n) := by\n rw [ENNReal.tsum_mul_left]\n refine' mul_le_mul_left' (ENNReal.tsum_le_tsum fun n => measure_mono _) _\n exact inter_subset_inter_left _ hs\n _ = \u03b5 * \u03bc (\u22c3 n, closedBall 0 R \u2229 t n) := by\n rw [measure_iUnion]\n \u00b7 exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n \u00b7 intro n\n exact measurableSet_closedBall.inter (t_meas n)\n _ \u2264 \u03b5 * \u03bc (closedBall 0 R) := by\n rw [\u2190 inter_iUnion]\n exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 (n : \u2115), f '' (s \u2229 t n))\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 \u22c3 (n : \u2115), f '' (s \u2229 t n)\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 inter_iUnion]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 (i : \u2115), t i)\n[PROOFSTEP]\nexact image_subset f (subset_inter Subset.rfl t_cover)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply (h\u03b4 (A n)).2\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n[PROOFSTEP]\nexact ht n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\ncongr with n\n[GOAL]\ncase e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\nrcases Af' h's n with \u27e8y, ys, hy\u27e9\n[GOAL]\ncase e_f.h.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\ny : E\nys : y \u2208 s\nhy : A n = f' y\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (A n)| + \u03b5) * \u2191\u2191\u03bc (s \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\nsimp only [hy, h'f' y ys, Real.toNNReal_zero, abs_zero, zero_add]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2211' (n : \u2115), \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n)\n[PROOFSTEP]\nrw [ENNReal.tsum_mul_left]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i) \u2264 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n)\n[PROOFSTEP]\nrefine' mul_le_mul_left' (ENNReal.tsum_le_tsum fun n => measure_mono _) _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 s \u2229 t n \u2286 closedBall 0 R \u2229 t n\n[PROOFSTEP]\nexact inter_subset_inter_left _ hs\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2211' (n : \u2115), \u2191\u2191\u03bc (closedBall 0 R \u2229 t n) = \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 (n : \u2115), closedBall 0 R \u2229 t n)\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 Pairwise (Disjoint on fun n => closedBall 0 R \u2229 t n)\n[PROOFSTEP]\nexact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (i : \u2115), MeasurableSet (closedBall 0 R \u2229 t i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 MeasurableSet (closedBall 0 R \u2229 t n)\n[PROOFSTEP]\nexact measurableSet_closedBall.inter (t_meas n)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 (n : \u2115), closedBall 0 R \u2229 t n) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nrw [\u2190 inter_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nR : \u211d\nhs : s \u2286 closedBall 0 R\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nh's : Set.Nonempty s\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n \u2200 (t : Set E),\n ApproximatesLinearOn f A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (f '' t) \u2264 \u2191(Real.toNNReal |ContinuousLinearMap.det A| + \u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nAf' : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R \u2229 \u22c3 (i : \u2115), t i) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nexact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0\n[PROOFSTEP]\nsuffices H : \u2200 R, \u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) = 0\n[PROOFSTEP]\napply le_antisymm _ (zero_le _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 0\n[PROOFSTEP]\nrw [\u2190 iUnion_inter_closedBall_nat s 0]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u22c3 (n : \u2115), s \u2229 closedBall 0 \u2191n) \u2264 0\n[PROOFSTEP]\ncalc\n \u03bc (f '' \u22c3 n : \u2115, s \u2229 closedBall 0 n) \u2264 \u2211' n : \u2115, \u03bc (f '' (s \u2229 closedBall 0 n)) := by rw [image_iUnion];\n exact measure_iUnion_le _\n _ \u2264 0 := by simp only [H, tsum_zero, nonpos_iff_eq_zero]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (f '' \u22c3 (n : \u2115), s \u2229 closedBall 0 \u2191n) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n))\n[PROOFSTEP]\nrw [image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f '' (s \u2229 closedBall 0 \u2191i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n))\n[PROOFSTEP]\nexact measure_iUnion_le _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nH : \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 \u2191n)) \u2264 0\n[PROOFSTEP]\nsimp only [H, tsum_zero, nonpos_iff_eq_zero]\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\n\u22a2 \u2200 (R : \u211d), \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n[PROOFSTEP]\nintro R\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n[PROOFSTEP]\nhave A : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u03b5 * \u03bc (closedBall 0 R) := fun \u03b5 \u03b5pos =>\n addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux \u03bc (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) R\n (inter_subset_right _ _) \u03b5 \u03b5pos fun x hx => h'f' x hx.1\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n[PROOFSTEP]\nhave B : Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) :=\n by\n have : Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr measure_closedBall_lt_top.ne)\n simp only [zero_mul, ENNReal.coe_zero] at this \n exact Tendsto.mono_left this nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun \u03b5 : \u211d\u22650 => (\u03b5 : \u211d\u22650\u221e) * \u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (((0 : \u211d\u22650) : \u211d\u22650\u221e) * \u03bc (closedBall 0 R))) :=\n ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr measure_closedBall_lt_top.ne)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd (\u21910 * \u2191\u2191\u03bc (closedBall 0 R)))\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [zero_mul, ENNReal.coe_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nthis : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nexact Tendsto.mono_left this nhdsWithin_le_nhds\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) = 0\n[PROOFSTEP]\napply le_antisymm _ (zero_le _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 0\n[PROOFSTEP]\napply ge_of_tendsto B\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191c * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nh'f' : \u2200 (x : E), x \u2208 s \u2192 ContinuousLinearMap.det (f' x) = 0\nR : \u211d\nA : \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)\nB : Tendsto (fun \u03b5 => \u2191\u03b5 * \u2191\u2191\u03bc (closedBall 0 R)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0)\n\u22a2 \u2200 (a : \u211d\u22650), a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' (s \u2229 closedBall 0 R)) \u2264 \u2191a * \u2191\u2191\u03bc (closedBall 0 R)\n[PROOFSTEP]\nexact A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable f'\n[PROOFSTEP]\nrefine' aemeasurable_of_unif_approx fun \u03b5 \u03b5pos => _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nlet \u03b4 : \u211d\u22650 := \u27e8\u03b5, le_of_lt \u03b5pos\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nhave \u03b4pos : 0 < \u03b4 := \u03b5pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8t, A, t_disj, t_meas, t_cover, ht, _\u27e9 :\n \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n Pairwise (Disjoint on t) \u2227\n (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n (s \u2286 \u22c3 n : \u2115, t n) \u2227\n (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4) \u2227 (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' (fun _ => \u03b4) fun _ => \u03b4pos.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8g, g_meas, hg\u27e9 : \u2203 g : E \u2192 E \u2192L[\u211d] E, Measurable g \u2227 \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n :=\n exists_measurable_piecewise t t_meas (fun n _ => A n) (fun n => measurable_const) <|\n t_disj.mono fun i j h => by simp only [h.inter_eq, eqOn_empty]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ni j : \u2115\nh : (Disjoint on t) i j\n\u22a2 EqOn ((fun n x => A n) i) ((fun n x => A n) j) (t i \u2229 t j)\n[PROOFSTEP]\nsimp only [h.inter_eq, eqOn_empty]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2203 f, AEMeasurable f \u2227 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (f x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nrefine'\n \u27e8g, g_meas.aemeasurable, _\u27e9\n -- reduce to checking that `f'` and `g` are close on almost all of `s \u2229 t n`, for all `n`.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nsuffices H : \u2200\u1d50 x : E \u2202sum fun n => \u03bc.restrict (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nhave : \u03bc.restrict s \u2264 sum fun n => \u03bc.restrict (s \u2229 t n) :=\n by\n have : s = \u22c3 n, s \u2229 t n := by\n rw [\u2190 inter_iUnion]\n exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n conv_lhs => rw [this]\n exact restrict_iUnion_le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n[PROOFSTEP]\nhave : s = \u22c3 n, s \u2229 t n := by\n rw [\u2190 inter_iUnion]\n exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 t n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n\u22a2 s = s \u2229 \u22c3 (i : \u2115), t i\n[PROOFSTEP]\nexact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n[PROOFSTEP]\nconv_lhs => rw [this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n| Measure.restrict \u03bc s\n[PROOFSTEP]\nrw [this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n| Measure.restrict \u03bc s\n[PROOFSTEP]\nrw [this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n| Measure.restrict \u03bc s\n[PROOFSTEP]\nrw [this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Measure.restrict \u03bc (\u22c3 (n : \u2115), s \u2229 t n) \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n[PROOFSTEP]\nexact restrict_iUnion_le\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nH : \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\nthis : Measure.restrict \u03bc s \u2264 sum fun n => Measure.restrict \u03bc (s \u2229 t n)\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nexact ae_mono this H\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202sum fun n => Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nrefine'\n ae_sum_iff.2 fun n =>\n _\n -- on almost all `s \u2229 t n`, `f' x` is close to `A n` thanks to\n -- `ApproximatesLinearOn.norm_fderiv_sub_le`.\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nhave E\u2081 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4 :=\n (ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n (hf' x hx.1).mono\n (inter_subset_left _ _)\n -- moreover, `g x` is equal to `A n` there.\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nhave E\u2082 : \u2200\u1d50 x : E \u2202\u03bc.restrict (s \u2229 t n), g x = A n :=\n by\n suffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n\n exact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H\n filter_upwards [ae_restrict_mem (t_meas n)]\n exact hg n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\n[PROOFSTEP]\nsuffices H : \u2200\u1d50 x : E \u2202\u03bc.restrict (t n), g x = A n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nH : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n\n[PROOFSTEP]\nexact ae_mono (restrict_mono (inter_subset_right _ _) le_rfl) H\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (t n), g x = A n\n[PROOFSTEP]\nfilter_upwards [ae_restrict_mem (t_meas n)]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\n\u22a2 \u2200 (a : E), a \u2208 t n \u2192 g a = A n\n[PROOFSTEP]\nexact hg n\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nfilter_upwards [E\u2081, E\u2082] with x hx1 hx2\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : \u2016f' x - A n\u2016\u208a \u2264 { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nrw [\u2190 nndist_eq_nnnorm] at hx1 \n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\nhx2 : g x = A n\n\u22a2 dist (g x) (f' x) \u2264 \u03b5\n[PROOFSTEP]\nrw [hx2, dist_comm]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u03b4 : \u211d\u22650 := { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\n\u03b4pos : 0 < \u03b4\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) \u03b4\nright\u271d : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\ng : E \u2192 E \u2192L[\u211d] E\ng_meas : Measurable g\nhg : \u2200 (n : \u2115) (x : E), x \u2208 t n \u2192 g x = A n\nn : \u2115\nE\u2081 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), \u2016f' x - A n\u2016\u208a \u2264 \u03b4\nE\u2082 : \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc (s \u2229 t n), g x = A n\nx : E\nhx1 : nndist (f' x) (A n) \u2264 { val := \u03b5, property := (_ : 0 \u2264 \u03b5) }\nhx2 : g x = A n\n\u22a2 dist (f' x) (A n) \u2264 \u03b5\n[PROOFSTEP]\nexact hx1\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\napply ENNReal.measurable_ofReal.comp_aemeasurable\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\nrefine' continuous_abs.measurable.comp_aemeasurable _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => ContinuousLinearMap.det (f' x)\n[PROOFSTEP]\nrefine' ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => f' x\n[PROOFSTEP]\nexact aemeasurable_fderivWithin \u03bc hs hf'\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => Real.toNNReal |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\napply measurable_real_toNNReal.comp_aemeasurable\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\nrefine' continuous_abs.measurable.comp_aemeasurable _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => ContinuousLinearMap.det (f' x)\n[PROOFSTEP]\nrefine' ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 AEMeasurable fun x => f' x\n[PROOFSTEP]\nexact aemeasurable_fderivWithin \u03bc hs hf'\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave :\n \u2200 A : E \u2192L[\u211d] E,\n \u2203 \u03b4 : \u211d\u22650,\n 0 < \u03b4 \u2227\n (\u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u03b4 \u2192 |B.det - A.det| \u2264 \u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E), ApproximatesLinearOn g A t \u03b4 \u2192 \u03bc (g '' t) \u2264 (ENNReal.ofReal |A.det| + \u03b5) * \u03bc t :=\n by\n intro A\n let m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n have I : ENNReal.ofReal |A.det| < m := by simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n rcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9\n obtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos\n let \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\u27e9\n refine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\u27e9\n \u00b7 intro B hB\n rw [\u2190 Real.dist_eq]\n apply (h\u03b4' B _).le\n rw [dist_eq_norm]\n calc\n \u2016B - A\u2016 \u2264 (min \u03b4 \u03b4'' : \u211d\u22650) := hB\n _ \u2264 \u03b4'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]\n _ < \u03b4' := half_lt_self \u03b4'pos\n \u00b7 intro t g htg\n exact h t g (htg.mono_num (min_le_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet m : \u211d\u22650 := Real.toNNReal |A.det| + \u03b5\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I : ENNReal.ofReal |A.det| < m := by simp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, lt_add_iff_pos_right, \u03b5pos, ENNReal.coe_lt_coe]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases((addHaar_image_le_mul_of_det_lt \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nobtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\nintro B hB\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\nrw [\u2190 Real.dist_eq]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) \u2264 \u2191\u03b5\n[PROOFSTEP]\napply (h\u03b4' B _).le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 dist B A < \u03b4'\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 \u2016B - A\u2016 < \u03b4'\n[PROOFSTEP]\ncalc\n \u2016B - A\u2016 \u2264 (min \u03b4 \u03b4'' : \u211d\u22650) := hB\n _ \u2264 \u03b4'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]\n _ < \u03b4' := half_lt_self \u03b4'pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 \u2191(min \u03b4 \u03b4'') \u2264 \u2191\u03b4''\n[PROOFSTEP]\nsimp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro t g htg\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| + \u03b5\nI : ENNReal.ofReal |ContinuousLinearMap.det A| < \u2191m\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u2191m * \u2191\u2191\u03bc s\n\u03b4pos : 0 < \u03b4\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\n\u22a2 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n[PROOFSTEP]\nexact h t g (htg.mono_num (min_le_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nthis :\n \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nchoose \u03b4 h\u03b4 using this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nobtain \u27e8t, A, t_disj, t_meas, t_cover, ht, -\u27e9 :\n \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n Pairwise (Disjoint on t) \u2227\n (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n (s \u2286 \u22c3 n : \u2115, t n) \u2227\n (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227 (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' \u03b4 fun A => (h\u03b4 A).1.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\ncalc\n \u03bc (f '' s) \u2264 \u03bc (\u22c3 n, f '' (s \u2229 t n)) := by\n apply measure_mono\n rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n exact image_subset f (subset_inter Subset.rfl t_cover)\n _ \u2264 \u2211' n, \u03bc (f '' (s \u2229 t n)) := (measure_iUnion_le _)\n _ \u2264 \u2211' n, (ENNReal.ofReal |(A n).det| + \u03b5) * \u03bc (s \u2229 t n) :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply (h\u03b4 (A n)).2.2\n exact ht n\n _ = \u2211' n, \u222b\u207b _ in s \u2229 t n, ENNReal.ofReal |(A n).det| + \u03b5 \u2202\u03bc := by\n simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n _ \u2264 \u2211' n, \u222b\u207b x in s \u2229 t n, ENNReal.ofReal |(f' x).det| + 2 * \u03b5 \u2202\u03bc :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply lintegral_mono_ae\n filter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)]\n intro x hx\n have I : |(A n).det| \u2264 |(f' x).det| + \u03b5 :=\n calc\n |(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| := by congr 1; abel\n _ \u2264 |(f' x).det| + |(f' x).det - (A n).det| := (abs_sub _ _)\n _ \u2264 |(f' x).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx)\n calc\n ENNReal.ofReal |(A n).det| + \u03b5 \u2264 ENNReal.ofReal (|(f' x).det| + \u03b5) + \u03b5 :=\n add_le_add (ENNReal.ofReal_le_ofReal I) le_rfl\n _ = ENNReal.ofReal |(f' x).det| + 2 * \u03b5 := by\n simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n _ = \u222b\u207b x in \u22c3 n, s \u2229 t n, ENNReal.ofReal |(f' x).det| + 2 * \u03b5 \u2202\u03bc :=\n by\n have M : \u2200 n : \u2115, MeasurableSet (s \u2229 t n) := fun n => hs.inter (t_meas n)\n rw [lintegral_iUnion M]\n exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n _ = \u222b\u207b x in s, ENNReal.ofReal |(f' x).det| + 2 * \u03b5 \u2202\u03bc :=\n by\n have : s = \u22c3 n, s \u2229 t n := by\n rw [\u2190 inter_iUnion]\n exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n rw [\u2190 this]\n _ = (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * \u03b5 * \u03bc s := by\n simp only [lintegral_add_right' _ aemeasurable_const, set_lintegral_const]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2191\u2191\u03bc (\u22c3 (n : \u2115), f '' (s \u2229 t n))\n[PROOFSTEP]\napply measure_mono\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 f '' s \u2286 \u22c3 (n : \u2115), f '' (s \u2229 t n)\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 inter_iUnion]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 (i : \u2115), t i)\n[PROOFSTEP]\nexact image_subset f (subset_inter Subset.rfl t_cover)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t n)) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5) * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\napply (h\u03b4 (A n)).2.2\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n[PROOFSTEP]\nexact ht n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5) * \u2191\u2191\u03bc (s \u2229 t n) =\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nsimp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc \u2264\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc \u2264\n \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\napply lintegral_mono_ae\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u2200\u1d50 (a : E) \u2202Measure.restrict \u03bc (s \u2229 t n),\n ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2264 ENNReal.ofReal |ContinuousLinearMap.det (f' a)| + 2 * \u2191\u03b5\n[PROOFSTEP]\nfilter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\n\u22a2 \u2200 (a : E),\n \u2016f' a - A n\u2016\u208a \u2264 \u03b4 (A n) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2264 ENNReal.ofReal |ContinuousLinearMap.det (f' a)| + 2 * \u2191\u03b5\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2264 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5\n[PROOFSTEP]\nhave I : |(A n).det| \u2264 |(f' x).det| + \u03b5 :=\n calc\n |(A n).det| = |(f' x).det - ((f' x).det - (A n).det)| := by congr 1; abel\n _ \u2264 |(f' x).det| + |(f' x).det - (A n).det| := (abs_sub _ _)\n _ \u2264 |(f' x).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 |ContinuousLinearMap.det (A n)| =\n |ContinuousLinearMap.det (f' x) - (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))|\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (A n) =\n ContinuousLinearMap.det (f' x) - (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (A n) =\n ContinuousLinearMap.det (f' x) - (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (A n)| \u2264 |ContinuousLinearMap.det (f' x)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2264 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5\n[PROOFSTEP]\ncalc\n ENNReal.ofReal |(A n).det| + \u03b5 \u2264 ENNReal.ofReal (|(f' x).det| + \u03b5) + \u03b5 :=\n add_le_add (ENNReal.ofReal_le_ofReal I) le_rfl\n _ = ENNReal.ofReal |(f' x).det| + 2 * \u03b5 := by\n simp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (A n)| \u2264 |ContinuousLinearMap.det (f' x)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal (|ContinuousLinearMap.det (f' x)| + \u2191\u03b5) + \u2191\u03b5 = ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_add, abs_nonneg, two_mul, add_assoc, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc =\n \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nhave M : \u2200 n : \u2115, MeasurableSet (s \u2229 t n) := fun n => hs.inter (t_meas n)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nM : \u2200 (n : \u2115), MeasurableSet (s \u2229 t n)\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc =\n \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iUnion M]\n[GOAL]\ncase hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nM : \u2200 (n : \u2115), MeasurableSet (s \u2229 t n)\n\u22a2 Pairwise (Disjoint on fun i => s \u2229 t i)\n[PROOFSTEP]\nexact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nhave : s = \u22c3 n, s \u2229 t n := by\n rw [\u2190 inter_iUnion]\n exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 t n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 s = s \u2229 \u22c3 (i : \u2115), t i\n[PROOFSTEP]\nexact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\nthis : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192 \u2191\u2191\u03bc (g '' t) \u2264 (ENNReal.ofReal |ContinuousLinearMap.det A| + \u2191\u03b5) * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| + 2 * \u2191\u03b5 \u2202\u03bc =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nsimp only [lintegral_add_right' _ aemeasurable_const, set_lintegral_const]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nhave :\n Tendsto (fun \u03b5 : \u211d\u22650 => (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0)\n (\ud835\udcdd ((\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) + 2 * (0 : \u211d\u22650) * \u03bc s)) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n refine' tendsto_const_nhds.add _\n refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.add _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\nrefine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))\n[PROOFSTEP]\nexact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nsimp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\napply ge_of_tendsto this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u22a2 \u2200 (a : \u211d\u22650),\n a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191a * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrintro \u03b5 (\u03b5pos : 0 < \u03b5)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun \u03b5 => \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0)\n (\ud835\udcdd (\u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact addHaar_image_le_lintegral_abs_det_fderiv_aux1 \u03bc hs hf' \u03b5pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nlet u n := disjointed (spanningSets \u03bc) n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nhave u_meas : \u2200 n, MeasurableSet (u n) := by\n intro n\n apply MeasurableSet.disjointed fun i => ?_\n exact measurable_spanningSets \u03bc i\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2200 (n : \u2115), MeasurableSet (u n)\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn : \u2115\n\u22a2 MeasurableSet (u n)\n[PROOFSTEP]\napply MeasurableSet.disjointed fun i => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn i : \u2115\n\u22a2 MeasurableSet (spanningSets \u03bc i)\n[PROOFSTEP]\nexact measurable_spanningSets \u03bc i\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nhave A : s = \u22c3 n, s \u2229 u n := by rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 u n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\ncalc\n \u03bc (f '' s) \u2264 \u2211' n, \u03bc (f '' (s \u2229 u n)) := by\n conv_lhs => rw [A, image_iUnion]\n exact measure_iUnion_le _\n _ \u2264 \u2211' n, \u222b\u207b x in s \u2229 u n, ENNReal.ofReal |(f' x).det| \u2202\u03bc :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply\n addHaar_image_le_lintegral_abs_det_fderiv_aux2 \u03bc (hs.inter (u_meas n)) _ fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)\n have : \u03bc (u n) < \u221e := lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n)\n exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)\n _ = \u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc := by\n conv_rhs => rw [A]\n rw [lintegral_iUnion]\n \u00b7 intro n; exact hs.inter (u_meas n)\n \u00b7 exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (f '' s) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))\n[PROOFSTEP]\nconv_lhs => rw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u2115), f '' (s \u2229 u i)) \u2264 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))\n[PROOFSTEP]\nexact measure_iUnion_le _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 u n)) \u2264 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\napply\n addHaar_image_le_lintegral_abs_det_fderiv_aux2 \u03bc (hs.inter (u_meas n)) _ fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4\n[PROOFSTEP]\nhave : \u03bc (u n) < \u221e := lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\nthis : \u2191\u2191\u03bc (u n) < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4\n[PROOFSTEP]\nexact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nconv_rhs => rw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iUnion]\n[GOAL]\ncase hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 u i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 MeasurableSet (s \u2229 u n)\n[PROOFSTEP]\nexact hs.inter (u_meas n)\n[GOAL]\ncase hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 u n)\n[PROOFSTEP]\nexact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave :\n \u2200 A : E \u2192L[\u211d] E,\n \u2203 \u03b4 : \u211d\u22650,\n 0 < \u03b4 \u2227\n (\u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u03b4 \u2192 |B.det - A.det| \u2264 \u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |A.det| * \u03bc t \u2264 \u03bc (g '' t) + \u03b5 * \u03bc t :=\n by\n intro A\n obtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos\n let \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\u27e9\n have I'' : \u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |B.det - A.det| \u2264 \u2191\u03b5 :=\n by\n intro B hB\n rw [\u2190 Real.dist_eq]\n apply (h\u03b4' B _).le\n rw [dist_eq_norm]\n exact hB.trans_lt (half_lt_self \u03b4'pos)\n rcases eq_or_ne A.det 0 with (hA | hA)\n \u00b7 refine' \u27e8\u03b4'', half_pos \u03b4'pos, I'', _\u27e9\n simp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff, zero_le, abs_zero]\n let m : \u211d\u22650 := Real.toNNReal |A.det| - \u03b5\n have I : (m : \u211d\u22650\u221e) < ENNReal.ofReal |A.det| :=\n by\n simp only [ENNReal.ofReal, ENNReal.coe_sub]\n apply ENNReal.sub_lt_self ENNReal.coe_ne_top\n \u00b7 simpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA\n \u00b7 simp only [\u03b5pos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff]\n rcases((mul_le_addHaar_image_of_lt_det \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9\n refine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\u27e9\n \u00b7 intro B hB\n apply I'' _ (hB.trans _)\n simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]\n \u00b7 intro t g htg\n rcases eq_or_ne (\u03bc t) \u221e with (ht | ht)\n \u00b7 simp only [ht, \u03b5pos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne.def, not_false_iff, _root_.add_top]\n have := h t g (htg.mono_num (min_le_left _ _))\n rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this \n simp only [ht, imp_true_iff, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nobtain \u27e8\u03b4', \u03b4'pos, h\u03b4'\u27e9 : \u2203 (\u03b4' : \u211d), 0 < \u03b4' \u2227 \u2200 B, dist B A < \u03b4' \u2192 dist B.det A.det < \u2191\u03b5 :=\n continuousAt_iff.1 ContinuousLinearMap.continuous_det.continuousAt \u03b5 \u03b5pos\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet \u03b4'' : \u211d\u22650 := \u27e8\u03b4' / 2, (half_pos \u03b4'pos).le\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I'' : \u2200 B : E \u2192L[\u211d] E, \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |B.det - A.det| \u2264 \u2191\u03b5 :=\n by\n intro B hB\n rw [\u2190 Real.dist_eq]\n apply (h\u03b4' B _).le\n rw [dist_eq_norm]\n exact hB.trans_lt (half_lt_self \u03b4'pos)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\nintro B hB\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\nrw [\u2190 Real.dist_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) \u2264 \u2191\u03b5\n[PROOFSTEP]\napply (h\u03b4' B _).le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 dist B A < \u03b4'\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191\u03b4''\n\u22a2 \u2016B - A\u2016 < \u03b4'\n[PROOFSTEP]\nexact hB.trans_lt (half_lt_self \u03b4'pos)\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases eq_or_ne A.det 0 with (hA | hA)\n[GOAL]\ncase intro.intro.inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' \u27e8\u03b4'', half_pos \u03b4'pos, I'', _\u27e9\n[GOAL]\ncase intro.intro.inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A = 0\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4'' \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimp only [hA, forall_const, zero_mul, ENNReal.ofReal_zero, imp_true_iff, zero_le, abs_zero]\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nlet m : \u211d\u22650 := Real.toNNReal |A.det| - \u03b5\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave I : (m : \u211d\u22650\u221e) < ENNReal.ofReal |A.det| :=\n by\n simp only [ENNReal.ofReal, ENNReal.coe_sub]\n apply ENNReal.sub_lt_self ENNReal.coe_ne_top\n \u00b7 simpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA\n \u00b7 simp only [\u03b5pos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n[PROOFSTEP]\nsimp only [ENNReal.ofReal, ENNReal.coe_sub]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|)\n[PROOFSTEP]\napply ENNReal.sub_lt_self ENNReal.coe_ne_top\n[GOAL]\ncase ha\u2080\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2260 0\n[PROOFSTEP]\nsimpa only [abs_nonpos_iff, Real.toNNReal_eq_zero, ENNReal.coe_eq_zero, Ne.def] using hA\n[GOAL]\ncase hb\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\n\u22a2 \u2191\u03b5 \u2260 0\n[PROOFSTEP]\nsimp only [\u03b5pos.ne', ENNReal.coe_eq_zero, Ne.def, not_false_iff]\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases((mul_le_addHaar_image_of_lt_det \u03bc A I).and self_mem_nhdsWithin).exists with \u27e8\u03b4, h, \u03b4pos\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrefine' \u27e8min \u03b4 \u03b4'', lt_min \u03b4pos (half_pos \u03b4'pos), _, _\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'') \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\nintro B hB\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\n[PROOFSTEP]\napply I'' _ (hB.trans _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nB : E \u2192L[\u211d] E\nhB : \u2016B - A\u2016 \u2264 \u2191(min \u03b4 \u03b4'')\n\u22a2 \u2191(min \u03b4 \u03b4'') \u2264 \u2191\u03b4''\n[PROOFSTEP]\nsimp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (min \u03b4 \u03b4'') \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nintro t g htg\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrcases eq_or_ne (\u03bc t) \u221e with (ht | ht)\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2.inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t = \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nsimp only [ht, \u03b5pos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne.def, not_false_iff, _root_.add_top]\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nhave := h t g (htg.mono_num (min_le_left _ _))\n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : \u2191m * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n[PROOFSTEP]\nrwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this \n[GOAL]\ncase intro.intro.inr.intro.intro.refine'_2.inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nA : E \u2192L[\u211d] E\n\u03b4' : \u211d\n\u03b4'pos : 0 < \u03b4'\nh\u03b4' : \u2200 (B : E \u2192L[\u211d] E), dist B A < \u03b4' \u2192 dist (ContinuousLinearMap.det B) (ContinuousLinearMap.det A) < \u2191\u03b5\n\u03b4'' : \u211d\u22650 := { val := \u03b4' / 2, property := (_ : 0 \u2264 \u03b4' / 2) }\nI'' : \u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4'' \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5\nhA : ContinuousLinearMap.det A \u2260 0\nm : \u211d\u22650 := Real.toNNReal |ContinuousLinearMap.det A| - \u03b5\nI : \u2191m < ENNReal.ofReal |ContinuousLinearMap.det A|\n\u03b4 : \u211d\u22650\nh : \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u2191m * \u2191\u2191\u03bc s \u2264 \u2191\u2191\u03bc (f '' s)\n\u03b4pos : 0 < \u03b4\nt : Set E\ng : E \u2192 E\nhtg : ApproximatesLinearOn g A t (min \u03b4 \u03b4'')\nht : \u2191\u2191\u03bc t \u2260 \u22a4\nthis : (\u2191(Real.toNNReal |ContinuousLinearMap.det A|) - \u2191\u03b5) * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t)\n\u22a2 0 < \u2191\u03b5 \u2192 \u2191\u03b5 < \u2191(Real.toNNReal |ContinuousLinearMap.det A|) \u2192 \u2191\u2191\u03bc t \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ht, imp_true_iff, Ne.def, not_false_iff]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nthis :\n \u2200 (A : E \u2192L[\u211d] E),\n \u2203 \u03b4,\n 0 < \u03b4 \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191\u03b4 \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t \u03b4 \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nchoose \u03b4 h\u03b4 using this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nobtain \u27e8t, A, t_disj, t_meas, t_cover, ht, -\u27e9 :\n \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] E),\n Pairwise (Disjoint on t) \u2227\n (\u2200 n : \u2115, MeasurableSet (t n)) \u2227\n (s \u2286 \u22c3 n : \u2115, t n) \u2227\n (\u2200 n : \u2115, ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))) \u2227 (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y) :=\n exists_partition_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' \u03b4 fun A => (h\u03b4 A).1.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nhave s_eq : s = \u22c3 n, s \u2229 t n := by\n rw [\u2190 inter_iUnion]\n exact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 t n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\n\u22a2 s = s \u2229 \u22c3 (i : \u2115), t i\n[PROOFSTEP]\nexact Subset.antisymm (subset_inter Subset.rfl t_cover) (inter_subset_left _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\ncalc\n (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) = \u2211' n, \u222b\u207b x in s \u2229 t n, ENNReal.ofReal |(f' x).det| \u2202\u03bc :=\n by\n conv_lhs => rw [s_eq]\n rw [lintegral_iUnion]\n \u00b7 exact fun n => hs.inter (t_meas n)\n \u00b7 exact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n _ \u2264 \u2211' n, \u222b\u207b _ in s \u2229 t n, ENNReal.ofReal |(A n).det| + \u03b5 \u2202\u03bc :=\n by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply lintegral_mono_ae\n filter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)]\n intro x hx\n have I : |(f' x).det| \u2264 |(A n).det| + \u03b5 :=\n calc\n |(f' x).det| = |(A n).det + ((f' x).det - (A n).det)| := by congr 1; abel\n _ \u2264 |(A n).det| + |(f' x).det - (A n).det| := (abs_add _ _)\n _ \u2264 |(A n).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx)\n calc\n ENNReal.ofReal |(f' x).det| \u2264 ENNReal.ofReal (|(A n).det| + \u03b5) := ENNReal.ofReal_le_ofReal I\n _ = ENNReal.ofReal |(A n).det| + \u03b5 := by\n simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n _ = \u2211' n, (ENNReal.ofReal |(A n).det| * \u03bc (s \u2229 t n) + \u03b5 * \u03bc (s \u2229 t n)) := by\n simp only [set_lintegral_const, lintegral_add_right _ measurable_const]\n _ \u2264 \u2211' n, (\u03bc (f '' (s \u2229 t n)) + \u03b5 * \u03bc (s \u2229 t n) + \u03b5 * \u03bc (s \u2229 t n)) :=\n by\n refine' ENNReal.tsum_le_tsum fun n => add_le_add_right _ _\n exact (h\u03b4 (A n)).2.2 _ _ (ht n)\n _ = \u03bc (f '' s) + 2 * \u03b5 * \u03bc s := by\n conv_rhs => rw [s_eq]\n rw [image_iUnion, measure_iUnion]; rotate_left\n \u00b7 intro i j hij\n apply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)\n exact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (t_disj hij)\n \u00b7 intro i\n exact\n measurable_image_of_fderivWithin (hs.inter (t_meas i)) (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n (hf.mono (inter_subset_left _ _))\n rw [measure_iUnion]; rotate_left\n \u00b7 exact pairwise_disjoint_mono t_disj fun i => inter_subset_right _ _\n \u00b7 exact fun i => hs.inter (t_meas i)\n rw [\u2190 ENNReal.tsum_mul_left, \u2190 ENNReal.tsum_add]\n congr 1\n ext1 i\n rw [mul_assoc, two_mul, add_assoc]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nconv_lhs => rw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iUnion]\n[GOAL]\ncase hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)\n[PROOFSTEP]\nexact fun n => hs.inter (t_meas n)\n[GOAL]\ncase hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t n)\n[PROOFSTEP]\nexact pairwise_disjoint_mono t_disj fun n => inter_subset_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\n\u22a2 \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc\n[PROOFSTEP]\napply lintegral_mono_ae\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\n\u22a2 \u2200\u1d50 (a : E) \u2202Measure.restrict \u03bc (s \u2229 t n),\n ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n[PROOFSTEP]\nfilter_upwards [(ht n).norm_fderiv_sub_le \u03bc (hs.inter (t_meas n)) f' fun x hx =>\n (hf' x hx.1).mono (inter_subset_left _ _)]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\n\u22a2 \u2200 (a : E),\n \u2016f' a - A n\u2016\u208a \u2264 \u03b4 (A n) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det (f' a)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n[PROOFSTEP]\nhave I : |(f' x).det| \u2264 |(A n).det| + \u03b5 :=\n calc\n |(f' x).det| = |(A n).det + ((f' x).det - (A n).det)| := by congr 1; abel\n _ \u2264 |(A n).det| + |(f' x).det - (A n).det| := (abs_add _ _)\n _ \u2264 |(A n).det| + \u03b5 := add_le_add le_rfl ((h\u03b4 (A n)).2.1 _ hx)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 |ContinuousLinearMap.det (f' x)| =\n |ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))|\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (f' x) =\n ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\n\u22a2 ContinuousLinearMap.det (f' x) =\n ContinuousLinearMap.det (A n) + (ContinuousLinearMap.det (f' x) - ContinuousLinearMap.det (A n))\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2264 ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n[PROOFSTEP]\ncalc\n ENNReal.ofReal |(f' x).det| \u2264 ENNReal.ofReal (|(A n).det| + \u03b5) := ENNReal.ofReal_le_ofReal I\n _ = ENNReal.ofReal |(A n).det| + \u03b5 := by\n simp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\nx : E\nhx : \u2016f' x - A n\u2016\u208a \u2264 \u03b4 (A n)\nI : |ContinuousLinearMap.det (f' x)| \u2264 |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n\u22a2 ENNReal.ofReal (|ContinuousLinearMap.det (A n)| + \u2191\u03b5) = ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_add, abs_nonneg, NNReal.zero_le_coe, ENNReal.ofReal_coe_nnreal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 t n, ENNReal.ofReal |ContinuousLinearMap.det (A n)| + \u2191\u03b5 \u2202\u03bc =\n \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n))\n[PROOFSTEP]\nsimp only [set_lintegral_const, lintegral_add_right _ measurable_const]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) \u2264\n \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n))\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => add_le_add_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\nn : \u2115\n\u22a2 ENNReal.ofReal |ContinuousLinearMap.det (A n)| * \u2191\u2191\u03bc (s \u2229 t n) \u2264 \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)\n[PROOFSTEP]\nexact (h\u03b4 (A n)).2.2 _ _ (ht n)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nconv_rhs => rw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n| \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrw [s_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2191\u2191\u03bc (f '' \u22c3 (n : \u2115), s \u2229 t n) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 (n : \u2115), s \u2229 t n)\n[PROOFSTEP]\nrw [image_iUnion, measure_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 (n : \u2115), s \u2229 t n)\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i))\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 t i))\n[PROOFSTEP]\nintro i j hij\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 t i)) i j\n[PROOFSTEP]\napply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 t i) (s \u2229 t j)\n[PROOFSTEP]\nexact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (t_disj hij)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 t i))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 t i))\n[PROOFSTEP]\nexact\n measurable_image_of_fderivWithin (hs.inter (t_meas i)) (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n (hf.mono (inter_subset_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (\u22c3 (n : \u2115), s \u2229 t n)\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t n)\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 t n)\n[PROOFSTEP]\nexact pairwise_disjoint_mono t_disj fun i => inter_subset_right _ _\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 t i)\n[PROOFSTEP]\nexact fun i => hs.inter (t_meas i)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2211' (i : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2211' (i : \u2115), \u2191\u2191\u03bc (s \u2229 t i)\n[PROOFSTEP]\nrw [\u2190 ENNReal.tsum_mul_left, \u2190 ENNReal.tsum_add]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 \u2211' (n : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) =\n \u2211' (a : \u2115), (\u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\n\u22a2 (fun n => \u2191\u2191\u03bc (f '' (s \u2229 t n)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t n)) = fun a =>\n \u2191\u2191\u03bc (f '' (s \u2229 t a)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t a)\n[PROOFSTEP]\next1 i\n[GOAL]\ncase e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u03b4 : (E \u2192L[\u211d] E) \u2192 \u211d\u22650\nh\u03b4 :\n \u2200 (A : E \u2192L[\u211d] E),\n 0 < \u03b4 A \u2227\n (\u2200 (B : E \u2192L[\u211d] E), \u2016B - A\u2016 \u2264 \u2191(\u03b4 A) \u2192 |ContinuousLinearMap.det B - ContinuousLinearMap.det A| \u2264 \u2191\u03b5) \u2227\n \u2200 (t : Set E) (g : E \u2192 E),\n ApproximatesLinearOn g A t (\u03b4 A) \u2192\n ENNReal.ofReal |ContinuousLinearMap.det A| * \u2191\u2191\u03bc t \u2264 \u2191\u2191\u03bc (g '' t) + \u2191\u03b5 * \u2191\u2191\u03bc t\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] E\nt_disj : Pairwise (Disjoint on t)\nt_meas : \u2200 (n : \u2115), MeasurableSet (t n)\nt_cover : s \u2286 \u22c3 (n : \u2115), t n\nht : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (\u03b4 (A n))\ns_eq : s = \u22c3 (n : \u2115), s \u2229 t n\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (f '' (s \u2229 t i)) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) + \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i) = \u2191\u2191\u03bc (f '' (s \u2229 t i)) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc (s \u2229 t i)\n[PROOFSTEP]\nrw [mul_assoc, two_mul, add_assoc]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nhave : Tendsto (fun \u03b5 : \u211d\u22650 => \u03bc (f '' s) + 2 * \u03b5 * \u03bc s) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (f '' s) + 2 * (0 : \u211d\u22650) * \u03bc s)) :=\n by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n refine' tendsto_const_nhds.add _\n refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.add _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910 * \u2191\u2191\u03bc s))\n[PROOFSTEP]\nrefine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Tendsto (fun \u03b5 => 2 * \u2191\u03b5) (\ud835\udcdd 0) (\ud835\udcdd (2 * \u21910))\n[PROOFSTEP]\nexact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s) + 2 * \u21910 * \u2191\u2191\u03bc s))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nsimp only [add_zero, zero_mul, mul_zero, ENNReal.coe_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\napply ge_of_tendsto this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200\u1da0 (c : \u211d\u22650) in \ud835\udcdd[Ioi 0] 0,\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191c * \u2191\u2191\u03bc s\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u22a2 \u2200 (a : \u211d\u22650),\n a \u2208 Ioi 0 \u2192 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191a * \u2191\u2191\u03bc s\n[PROOFSTEP]\nrintro \u03b5 (\u03b5pos : 0 < \u03b5)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nh's : \u2191\u2191\u03bc s \u2260 \u22a4\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : Tendsto (fun \u03b5 => \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191\u2191\u03bc (f '' s)))\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s) + 2 * \u2191\u03b5 * \u2191\u2191\u03bc s\n[PROOFSTEP]\nexact lintegral_abs_det_fderiv_le_addHaar_image_aux1 \u03bc hs hf' hf \u03b5pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nlet u n := disjointed (spanningSets \u03bc) n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nhave u_meas : \u2200 n, MeasurableSet (u n) := by\n intro n\n apply MeasurableSet.disjointed fun i => ?_\n exact measurable_spanningSets \u03bc i\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\n\u22a2 \u2200 (n : \u2115), MeasurableSet (u n)\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn : \u2115\n\u22a2 MeasurableSet (u n)\n[PROOFSTEP]\napply MeasurableSet.disjointed fun i => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nn i : \u2115\n\u22a2 MeasurableSet (spanningSets \u03bc i)\n[PROOFSTEP]\nexact measurable_spanningSets \u03bc i\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nhave A : s = \u22c3 n, s \u2229 u n := by rw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\n\u22a2 s = \u22c3 (n : \u2115), s \u2229 u n\n[PROOFSTEP]\nrw [\u2190 inter_iUnion, iUnion_disjointed, iUnion_spanningSets, inter_univ]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\ncalc\n (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) = \u2211' n, \u222b\u207b x in s \u2229 u n, ENNReal.ofReal |(f' x).det| \u2202\u03bc :=\n by\n conv_lhs => rw [A]\n rw [lintegral_iUnion]\n \u00b7 intro n; exact hs.inter (u_meas n)\n \u00b7 exact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _\n _ \u2264 \u2211' n, \u03bc (f '' (s \u2229 u n)) := by\n apply ENNReal.tsum_le_tsum fun n => ?_\n apply\n lintegral_abs_det_fderiv_le_addHaar_image_aux2 \u03bc (hs.inter (u_meas n)) _\n (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) (hf.mono (inter_subset_left _ _))\n have : \u03bc (u n) < \u221e := lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n)\n exact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)\n _ = \u03bc (f '' s) := by\n conv_rhs => rw [A, image_iUnion]\n rw [measure_iUnion]\n \u00b7 intro i j hij\n apply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)\n exact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (disjoint_disjointed _ hij)\n \u00b7 intro i\n exact\n measurable_image_of_fderivWithin (hs.inter (u_meas i)) (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n (hf.mono (inter_subset_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nconv_lhs => rw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [A]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u222b\u207b (x : E) in \u22c3 (n : \u2115), s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc =\n \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_iUnion]\n[GOAL]\ncase hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (s \u2229 u i)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase hm\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 MeasurableSet (s \u2229 u n)\n[PROOFSTEP]\nexact hs.inter (u_meas n)\n[GOAL]\ncase hd\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 Pairwise (Disjoint on fun n => s \u2229 u n)\n[PROOFSTEP]\nexact pairwise_disjoint_mono (disjoint_disjointed _) fun n => inter_subset_right _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264\n \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n))\n[PROOFSTEP]\napply ENNReal.tsum_le_tsum fun n => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 \u222b\u207b (x : E) in s \u2229 u n, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| \u2202\u03bc \u2264 \u2191\u2191\u03bc (f '' (s \u2229 u n))\n[PROOFSTEP]\napply\n lintegral_abs_det_fderiv_le_addHaar_image_aux2 \u03bc (hs.inter (u_meas n)) _\n (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _)) (hf.mono (inter_subset_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4\n[PROOFSTEP]\nhave : \u03bc (u n) < \u221e := lt_of_le_of_lt (measure_mono (disjointed_subset _ _)) (measure_spanningSets_lt_top \u03bc n)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\nn : \u2115\nthis : \u2191\u2191\u03bc (u n) < \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 u n) \u2260 \u22a4\n[PROOFSTEP]\nexact ne_of_lt (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) this)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) = \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nconv_rhs => rw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n| \u2191\u2191\u03bc (f '' s)\n[PROOFSTEP]\nrw [A, image_iUnion]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2211' (n : \u2115), \u2191\u2191\u03bc (f '' (s \u2229 u n)) = \u2191\u2191\u03bc (\u22c3 (i : \u2115), f '' (s \u2229 u i))\n[PROOFSTEP]\nrw [measure_iUnion]\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 Pairwise (Disjoint on fun i => f '' (s \u2229 u i))\n[PROOFSTEP]\nintro i j hij\n[GOAL]\ncase hn\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 (Disjoint on fun i => f '' (s \u2229 u i)) i j\n[PROOFSTEP]\napply Disjoint.image _ hf (inter_subset_left _ _) (inter_subset_left _ _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\ni j : \u2115\nhij : i \u2260 j\n\u22a2 Disjoint (s \u2229 u i) (s \u2229 u j)\n[PROOFSTEP]\nexact Disjoint.mono (inter_subset_right _ _) (inter_subset_right _ _) (disjoint_disjointed _ hij)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\n\u22a2 \u2200 (i : \u2115), MeasurableSet (f '' (s \u2229 u i))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : \u2115 \u2192 Set E := fun n => disjointed (spanningSets \u03bc) n\nu_meas : \u2200 (n : \u2115), MeasurableSet (u n)\nA : s = \u22c3 (n : \u2115), s \u2229 u n\ni : \u2115\n\u22a2 MeasurableSet (f '' (s \u2229 u i))\n[PROOFSTEP]\nexact\n measurable_image_of_fderivWithin (hs.inter (u_meas i)) (fun x hx => (hf' x hx.1).mono (inter_subset_left _ _))\n (hf.mono (inter_subset_left _ _))\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\n\u22a2 Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\napply Measure.ext fun t ht => ?_\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\nt : Set E\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =\n \u2191\u2191(Measure.restrict \u03bc (f '' s)) t\n[PROOFSTEP]\nrw [map_apply h'f ht, withDensity_apply _ (h'f ht), Measure.restrict_apply ht, restrict_restrict (h'f ht),\n lintegral_abs_det_fderiv_eq_addHaar_image \u03bc ((h'f ht).inter hs)\n (fun x hx => (hf' x hx.2).mono (inter_subset_right _ _)) (hf.mono (inter_subset_right _ _)),\n image_preimage_inter]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nobtain \u27e8u, u_meas, uf\u27e9 : \u2203 u, Measurable u \u2227 EqOn u f s := by\n classical\n refine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\n refine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\n have : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\n exact this.continuousOn\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u2203 u, Measurable u \u2227 EqOn u f s\n[PROOFSTEP]\nclassical\nrefine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\nrefine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\nhave : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\nexact this.continuousOn\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 \u2203 u, Measurable u \u2227 EqOn u f s\n[PROOFSTEP]\nrefine' \u27e8piecewise s f 0, _, piecewise_eqOn _ _ _\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 Measurable (piecewise s f 0)\n[PROOFSTEP]\nrefine' ContinuousOn.measurable_piecewise _ continuous_zero.continuousOn hs\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nhave : DifferentiableOn \u211d f s := fun x hx => (hf' x hx).differentiableWithinAt\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nthis : DifferentiableOn \u211d f s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nexact this.continuousOn\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nhave u' : \u2200 x \u2208 s, HasFDerivWithinAt u (f' x) s x := fun x hx => (hf' x hx).congr (fun y hy => uf hy) (uf hx)\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nset F : s \u2192 E := u \u2218 (\u2191) with hF\n[GOAL]\ncase intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nhave A : Measure.map F (comap (\u2191) (\u03bc.withDensity fun x => ENNReal.ofReal |(f' x).det|)) = \u03bc.restrict (u '' s) :=\n by\n rw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs, restrict_withDensity hs]\n exact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (u '' s)\n[PROOFSTEP]\nrw [hF, \u2190 Measure.map_map u_meas measurable_subtype_coe, map_comap_subtype_coe hs, restrict_withDensity hs]\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\n\u22a2 Measure.map u (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n Measure.restrict \u03bc (u '' s)\n[PROOFSTEP]\nexact map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs u' (hf.congr uf.symm) u_meas\n[GOAL]\ncase intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (u '' s)\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nrw [uf.image_eq] at A \n[GOAL]\ncase intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nhave : F = s.restrict f := by\n ext x\n exact uf x.2\n[GOAL]\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n\u22a2 F = Set.restrict s f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\nx : \u2191s\n\u22a2 F x = Set.restrict s f x\n[PROOFSTEP]\nexact uf x.2\n[GOAL]\ncase intro.intro\nE : Type u_1\nF\u271d : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\u271d\ninst\u271d\u00b3 : NormedSpace \u211d F\u271d\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nu : E \u2192 E\nu_meas : Measurable u\nuf : EqOn u f s\nu' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt u (f' x) s x\nF : \u2191s \u2192 E := u \u2218 Subtype.val\nhF : F = u \u2218 Subtype.val\nA :\n Measure.map F (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\nthis : F = Set.restrict s f\n\u22a2 Measure.map (Set.restrict s f)\n (Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) =\n Measure.restrict \u03bc (f '' s)\n[PROOFSTEP]\nrwa [this] at A \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (x : E) in f '' s, g x \u2202\u03bc = \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n (measurableEmbedding_of_fderivWithin hs hf' hf).lintegral_map]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u2191s),\n g\n (Set.restrict s f\n a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n g\n (Set.restrict s f\n a) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : \u2191s),\n (g \u2218 f) \u2191a \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 (MeasurableEmbedding.subtype_coe hs).lintegral_map, map_comap_subtype_coe hs,\n set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080 _ _ _ hs]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b\u207b (a : E) in s, ((fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) * fun a => (g \u2218 f) a) a \u2202\u03bc =\n \u222b\u207b (x : E) in s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| * g (f x) \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u2200\u1d50 (x : E) \u2202Measure.restrict \u03bc s, ENNReal.ofReal |ContinuousLinearMap.det (f' x)| < \u22a4\n[PROOFSTEP]\nsimp only [eventually_true, ENNReal.ofReal_lt_top]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 \u211d\u22650\u221e\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 AEMeasurable fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\nexact aemeasurable_ofReal_abs_det_fderivWithin \u03bc hs hf'\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 IntegrableOn g (f '' s) \u2194 IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)) s\n[PROOFSTEP]\nrw [IntegrableOn, \u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n (measurableEmbedding_of_fderivWithin hs hf' hf).integrable_map_iff]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 Integrable (g \u2218 Set.restrict s f) \u2194 IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)) s\n[PROOFSTEP]\nchange Integrable ((g \u2218 f) \u2218 ((\u2191) : s \u2192 E)) _ \u2194 _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 Integrable ((g \u2218 f) \u2218 Subtype.val) \u2194 IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)) s\n[PROOFSTEP]\nrw [\u2190 (MeasurableEmbedding.subtype_coe hs).integrable_map_iff, map_comap_subtype_coe hs]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 Integrable (g \u2218 f) \u2194 IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)) s\n[PROOFSTEP]\nsimp only [ENNReal.ofReal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 Integrable (g \u2218 f) \u2194 IntegrableOn (fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)) s\n[PROOFSTEP]\nrw [restrict_withDensity hs, integrable_withDensity_iff_integrable_coe_smul\u2080, IntegrableOn]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 (Integrable fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|) \u2022 (g \u2218 f) x) \u2194\n Integrable fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrw [iff_iff_eq]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 (Integrable fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|) \u2022 (g \u2218 f) x) =\n Integrable fun x => |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\ncongr 2 with x\n[GOAL]\ncase e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nx : E\n\u22a2 \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|) \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrw [Real.coe_toNNReal _ (abs_nonneg _)]\n[GOAL]\ncase e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nx : E\n\u22a2 |ContinuousLinearMap.det (f' x)| \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hf\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 AEMeasurable fun x => Real.toNNReal |ContinuousLinearMap.det (f' x)|\n[PROOFSTEP]\nexact aemeasurable_toNNReal_abs_det_fderivWithin \u03bc hs hf'\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 \u222b (x : E) in f '' s, g x \u2202\u03bc = \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 restrict_map_withDensity_abs_det_fderiv_eq_addHaar \u03bc hs hf' hf,\n (measurableEmbedding_of_fderivWithin hs hf' hf).integral_map]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\n\u22a2 \u222b (x : \u2191s),\n g\n (Set.restrict s f\n x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc\n[PROOFSTEP]\nhave : \u2200 x : s, g (s.restrict f x) = (g \u2218 f) x := fun x => rfl\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n g\n (Set.restrict s f\n x) \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc\n[PROOFSTEP]\nsimp only [this, ENNReal.ofReal]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (x : \u2191s),\n (g \u2218 f) \u2191x \u2202Measure.comap Subtype.val (withDensity \u03bc fun x => \u2191(Real.toNNReal |ContinuousLinearMap.det (f' x)|)) =\n \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 (MeasurableEmbedding.subtype_coe hs).integral_map, map_comap_subtype_coe hs,\n set_integral_withDensity_eq_set_integral_smul\u2080 (aemeasurable_toNNReal_abs_det_fderivWithin \u03bc hs hf') _ hs]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\n\u22a2 \u222b (a : E) in s, Real.toNNReal |ContinuousLinearMap.det (f' a)| \u2022 (g \u2218 f) a \u2202\u03bc =\n \u222b (x : E) in s, |ContinuousLinearMap.det (f' x)| \u2022 g (f x) \u2202\u03bc\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_f.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n\u22a2 Real.toNNReal |ContinuousLinearMap.det (f' x)| \u2022 (g \u2218 f) x = |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n| |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n| |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E \u2192 F\nthis : \u2200 (x : \u2191s), g (Set.restrict s f x) = (g \u2218 f) \u2191x\nx : E\n| |ContinuousLinearMap.det (f' x)| \u2022 g (f x)\n[PROOFSTEP]\nrw [\u2190 Real.coe_toNNReal _ (abs_nonneg (f' x).det)]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\n\u22a2 ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v\n[PROOFSTEP]\nhave : (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c).smulRight v = v \u2022 (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) :=\n by\n ext1\n simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, Algebra.id.smul_eq_mul, one_mul,\n ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\n\u22a2 ContinuousLinearMap.smulRight 1 v = v \u2022 1\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\n\u22a2 \u2191(ContinuousLinearMap.smulRight 1 v) 1 = \u2191(v \u2022 1) 1\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, Algebra.id.smul_eq_mul, one_mul,\n ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\nthis : ContinuousLinearMap.smulRight 1 v = v \u2022 1\n\u22a2 ContinuousLinearMap.det (ContinuousLinearMap.smulRight 1 v) = v\n[PROOFSTEP]\nrw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\nthis : ContinuousLinearMap.smulRight 1 v = v \u2022 1\n\u22a2 \u2191LinearMap.det (v \u2022 \u21911) = v\n[PROOFSTEP]\nrw [show ((1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) : \ud835\udd5c \u2192\u2097[\ud835\udd5c] \ud835\udd5c) = LinearMap.id from rfl]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\n\ud835\udd5c : Type u_3\ninst\u271d : NormedField \ud835\udd5c\nv : \ud835\udd5c\nthis : ContinuousLinearMap.smulRight 1 v = v \u2022 1\n\u22a2 \u2191LinearMap.det (v \u2022 LinearMap.id) = v\n[PROOFSTEP]\nrw [LinearMap.det_smul, FiniteDimensional.finrank_self, LinearMap.det_id, pow_one, mul_one]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\ns : Set \u211d\nf f' : \u211d \u2192 \u211d\nhs : MeasurableSet s\nhf' : \u2200 (x : \u211d), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : \u211d \u2192 F\n\u22a2 IntegrableOn g (f '' s) \u2194 IntegrableOn (fun x => |f' x| \u2022 g (f x)) s\n[PROOFSTEP]\nsimpa only [det_one_smulRight] using\n integrableOn_image_iff_integrableOn_abs_det_fderiv_smul volume hs (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ns : Set \u211d\nf f' : \u211d \u2192 \u211d\ninst\u271d : CompleteSpace F\nhs : MeasurableSet s\nhf' : \u2200 (x : \u211d), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : \u211d \u2192 F\n\u22a2 \u222b (x : \u211d) in f '' s, g x = \u222b (x : \u211d) in s, |f' x| \u2022 g (f x)\n[PROOFSTEP]\nsimpa only [det_one_smulRight] using\n integral_image_eq_integral_abs_det_fderiv_smul volume hs (fun x hx => (hf' x hx).hasFDerivWithinAt) hf g\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivAt (\u2191f) (f' x) x\ng : E \u2192 F\n\u22a2 \u222b (x : E) in f.target, g x \u2202\u03bc = \u222b (x : E) in f.source, |ContinuousLinearMap.det (f' x)| \u2022 g (\u2191f x) \u2202\u03bc\n[PROOFSTEP]\nhave : f '' f.source = f.target := LocalEquiv.image_source_eq_target f.toLocalEquiv\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivAt (\u2191f) (f' x) x\ng : E \u2192 F\nthis : \u2191f '' f.source = f.target\n\u22a2 \u222b (x : E) in f.target, g x \u2202\u03bc = \u222b (x : E) in f.source, |ContinuousLinearMap.det (f' x)| \u2022 g (\u2191f x) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivAt (\u2191f) (f' x) x\ng : E \u2192 F\nthis : \u2191f '' f.source = f.target\n\u22a2 \u222b (x : E) in \u2191f '' f.source, g x \u2202\u03bc = \u222b (x : E) in f.source, |ContinuousLinearMap.det (f' x)| \u2022 g (\u2191f x) \u2202\u03bc\n[PROOFSTEP]\napply integral_image_eq_integral_abs_det_fderiv_smul \u03bc f.open_source.measurableSet _ f.injOn\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivAt (\u2191f) (f' x) x\ng : E \u2192 F\nthis : \u2191f '' f.source = f.target\n\u22a2 \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivWithinAt (\u2191f) (f' x) f.source x\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : \u2200 (x : E), x \u2208 f.source \u2192 HasFDerivAt (\u2191f) (f' x) x\ng : E \u2192 F\nthis : \u2191f '' f.source = f.target\nx : E\nhx : x \u2208 f.source\n\u22a2 HasFDerivWithinAt (\u2191f) (f' x) f.source x\n[PROOFSTEP]\nexact (hf' x hx).hasFDerivWithinAt\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.Jacobian", "llama_tokens": 315556, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.4073334000459302, "lm_q1q2_score": 0.2520499379587779}} {"text": "[GOAL]\nC : Type u\nX Y Z : F C\n\u22a2 Discrete.functor (normalizeObj (tensor (tensor X Y) Z)) \u27f6 Discrete.functor (normalizeObj (tensor X (tensor Y Z)))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nX Y Z : F C\n\u22a2 (Discrete.functor fun x => normalizeObj Z (normalizeObj Y (normalizeObj X x).as).as) \u27f6\n Discrete.functor fun x => normalizeObj Z (normalizeObj Y (normalizeObj X x).as).as\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\nX\u271d Y\u271d Z\u271d : F C\n\u22a2 Discrete.functor (normalizeObj (tensor X\u271d (tensor Y\u271d Z\u271d))) \u27f6\n Discrete.functor (normalizeObj (tensor (tensor X\u271d Y\u271d) Z\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nX\u271d Y\u271d Z\u271d : F C\n\u22a2 (Discrete.functor fun x => normalizeObj Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x).as).as) \u27f6\n Discrete.functor fun x => normalizeObj Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x).as).as\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj (tensor Unit a\u271d)) \u27f6 Discrete.functor (normalizeObj a\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 (Discrete.functor fun x => normalizeObj a\u271d x) \u27f6 Discrete.functor (normalizeObj a\u271d)\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj a\u271d) \u27f6 Discrete.functor (normalizeObj (tensor Unit a\u271d))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj a\u271d) \u27f6 Discrete.functor fun x => normalizeObj a\u271d x\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj (tensor a\u271d Unit)) \u27f6 Discrete.functor (normalizeObj a\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 (Discrete.functor fun x => { as := (normalizeObj a\u271d x).as }) \u27f6 Discrete.functor (normalizeObj a\u271d)\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj a\u271d) \u27f6 Discrete.functor (normalizeObj (tensor a\u271d Unit))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\na\u271d : F C\n\u22a2 Discrete.functor (normalizeObj a\u271d) \u27f6 Discrete.functor fun x => { as := (normalizeObj a\u271d x).as }\n[PROOFSTEP]\nexact Discrete.natTrans (fun _ => \ud835\udfd9 _)\n[GOAL]\nC : Type u\nT X\u271d Y\u271d W : F C\nf : T \u27f6\u1d50 Y\u271d\ng : X\u271d \u27f6\u1d50 W\n\u22a2 Discrete.functor (normalizeObj (tensor T X\u271d)) \u27f6 Discrete.functor (normalizeObj (tensor Y\u271d W))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nT X\u271d Y\u271d W : F C\nf : T \u27f6\u1d50 Y\u271d\ng : X\u271d \u27f6\u1d50 W\n\u22a2 (Discrete.functor fun x => normalizeObj X\u271d (normalizeObj T x).as) \u27f6\n Discrete.functor fun x => normalizeObj W (normalizeObj Y\u271d x).as\n[PROOFSTEP]\nexact\n Discrete.natTrans\n (fun \u27e8X\u27e9 =>\n (normalizeMapAux g).app (normalizeObj T X) \u226b\n (Discrete.functor (normalizeObj W) : _ \u2964 N C).map ((normalizeMapAux f).app \u27e8X\u27e9))\n[GOAL]\nC : Type u\nX Y : F C\n\u22a2 \u2200 (a b : X \u27f6\u1d50 Y), a \u2248 b \u2192 normalizeMapAux a = normalizeMapAux b\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\nZ : F C\nn n' : (Discrete \u2218 NormalMonoidalObject) C\nf : n \u27f6 n'\n\u22a2 ((tensorFunc C).obj Z).map f = inclusion.map f \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\ncases n\n[GOAL]\ncase mk\nC : Type u\nZ : F C\nn' : (Discrete \u2218 NormalMonoidalObject) C\nas\u271d : NormalMonoidalObject C\nf : { as := as\u271d } \u27f6 n'\n\u22a2 ((tensorFunc C).obj Z).map f = inclusion.map f \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\ncases n'\n[GOAL]\ncase mk.mk\nC : Type u\nZ : F C\nas\u271d\u00b9 as\u271d : NormalMonoidalObject C\nf : { as := as\u271d\u00b9 } \u27f6 { as := as\u271d }\n\u22a2 ((tensorFunc C).obj Z).map f = inclusion.map f \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrcases f with \u27e8\u27e8h\u27e9\u27e9\n[GOAL]\ncase mk.mk.up.up\nC : Type u\nZ : F C\nas\u271d\u00b9 as\u271d : NormalMonoidalObject C\nh : { as := as\u271d\u00b9 }.as = { as := as\u271d }.as\n\u22a2 ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase mk.mk.up.up\nC : Type u\nZ : F C\nas\u271d\u00b9 as\u271d : NormalMonoidalObject C\nh : as\u271d\u00b9 = as\u271d\n\u22a2 ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mk.mk.up.up\nC : Type u\nZ : F C\nas\u271d : NormalMonoidalObject C\n\u22a2 ((tensorFunc C).obj Z).map { down := { down := (_ : as\u271d = as\u271d) } } =\n inclusion.map { down := { down := (_ : as\u271d = as\u271d) } } \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\nX : F C\n\u22a2 \u2200 {X_1 Y : (Discrete \u2218 NormalMonoidalObject) C} (f : X_1 \u27f6 Y),\n ((tensorFunc C).obj X).map f \u226b (normalizeIsoApp C X Y).hom =\n (normalizeIsoApp C X X_1).hom \u226b ((normalize' C).obj X).map f\n[PROOFSTEP]\nrintro \u27e8X\u27e9 \u27e8Y\u27e9 \u27e8\u27e8f\u27e9\u27e9\n[GOAL]\ncase mk.mk.up.up\nC : Type u\nX\u271d : F C\nX Y : NormalMonoidalObject C\nf : { as := X }.as = { as := Y }.as\n\u22a2 ((tensorFunc C).obj X\u271d).map { down := { down := f } } \u226b (normalizeIsoApp C X\u271d { as := Y }).hom =\n (normalizeIsoApp C X\u271d { as := X }).hom \u226b ((normalize' C).obj X\u271d).map { down := { down := f } }\n[PROOFSTEP]\ndsimp at f \n[GOAL]\ncase mk.mk.up.up\nC : Type u\nX\u271d : F C\nX Y : NormalMonoidalObject C\nf : X = Y\n\u22a2 ((tensorFunc C).obj X\u271d).map { down := { down := f } } \u226b (normalizeIsoApp C X\u271d { as := Y }).hom =\n (normalizeIsoApp C X\u271d { as := X }).hom \u226b ((normalize' C).obj X\u271d).map { down := { down := f } }\n[PROOFSTEP]\nsubst f\n[GOAL]\ncase mk.mk.up.up\nC : Type u\nX\u271d : F C\nX : NormalMonoidalObject C\n\u22a2 ((tensorFunc C).obj X\u271d).map { down := { down := (_ : X = X) } } \u226b (normalizeIsoApp C X\u271d { as := X }).hom =\n (normalizeIsoApp C X\u271d { as := X }).hom \u226b ((normalize' C).obj X\u271d).map { down := { down := (_ : X = X) } }\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.up.up\nC : Type u\nX\u271d : F C\nX : NormalMonoidalObject C\n\u22a2 (Discrete.functor fun n => inclusionObj n \u2297 X\u271d).map { down := { down := (_ : X = X) } } \u226b\n (normalizeIsoApp C X\u271d { as := X }).hom =\n (normalizeIsoApp C X\u271d { as := X }).hom \u226b\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).obj\n (Discrete.functor (normalizeObj X\u271d))).map\n { down := { down := (_ : X = X) } }\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\n\u22a2 \u2200 {X Y : F C} (f : X \u27f6 Y),\n (tensorFunc C).map f \u226b (normalizeIsoAux C Y).hom = (normalizeIsoAux C X).hom \u226b (normalize' C).map f\n[PROOFSTEP]\nrintro X Y f\n[GOAL]\nC : Type u\nX Y : F C\nf : X \u27f6 Y\n\u22a2 (tensorFunc C).map f \u226b (normalizeIsoAux C Y).hom = (normalizeIsoAux C X).hom \u226b (normalize' C).map f\n[PROOFSTEP]\ninduction' f using Quotient.recOn with f\n[GOAL]\ncase f\nC : Type u\nX Y : F C\nf : X \u27f6\u1d50 Y\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X Y) f) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) f)\ncase h\nC : Type u\nX Y : F C\nf : X \u27f6 Y\na\u271d b\u271d : X \u27f6\u1d50 Y\np\u271d : a\u271d \u2248 b\u271d\n\u22a2 (_ :\n (tensorFunc C).map (Quotient.mk (setoidHom X Y) b\u271d) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) b\u271d)) =\n (_ :\n (tensorFunc C).map (Quotient.mk (setoidHom X Y) b\u271d) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) b\u271d))\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nC : Type u\nX Y : F C\nf : X \u27f6 Y\na\u271d b\u271d : X \u27f6\u1d50 Y\np\u271d : a\u271d \u2248 b\u271d\n\u22a2 (_ :\n (tensorFunc C).map (Quotient.mk (setoidHom X Y) b\u271d) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) b\u271d)) =\n (_ :\n (tensorFunc C).map (Quotient.mk (setoidHom X Y) b\u271d) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) b\u271d))\ncase f\nC : Type u\nX Y : F C\nf : X \u27f6\u1d50 Y\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X Y) f) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) f)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f\nC : Type u\nX Y : F C\nf : X \u27f6\u1d50 Y\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X Y) f) \u226b (normalizeIsoAux C Y).hom =\n (normalizeIsoAux C X).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X Y) f)\n[PROOFSTEP]\ninduction' f with _ X\u2081 X\u2082 X\u2083 _ _ _ _ _ _ _ _ _ _ _ _ h\u2081 h\u2082 X\u2081 X\u2082 Y\u2081 Y\u2082 f g h\u2081 h\u2082\n[GOAL]\ncase f.id\nC : Type u\nX Y X\u271d : F C\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X\u271d X\u271d) (Hom.id X\u271d)) \u226b (normalizeIsoAux C X\u271d).hom =\n (normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d X\u271d) (Hom.id X\u271d))\n[PROOFSTEP]\nsimp only [mk_id, Functor.map_id, Category.comp_id, Category.id_comp]\n[GOAL]\ncase f.\u03b1_hom\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\n\u22a2 (tensorFunc C).map\n (Quotient.mk (setoidHom (tensor (tensor X\u2081 X\u2082) X\u2083) (tensor X\u2081 (tensor X\u2082 X\u2083))) (Hom.\u03b1_hom X\u2081 X\u2082 X\u2083)) \u226b\n (normalizeIsoAux C (tensor X\u2081 (tensor X\u2082 X\u2083))).hom =\n (normalizeIsoAux C (tensor (tensor X\u2081 X\u2082) X\u2083)).hom \u226b\n (normalize' C).map\n (Quotient.mk (setoidHom (tensor (tensor X\u2081 X\u2082) X\u2083) (tensor X\u2081 (tensor X\u2082 X\u2083))) (Hom.\u03b1_hom X\u2081 X\u2082 X\u2083))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map\n (Quotient.mk (setoidHom (tensor (tensor X\u2081 X\u2082) X\u2083) (tensor X\u2081 (tensor X\u2082 X\u2083))) (Hom.\u03b1_hom X\u2081 X\u2082 X\u2083)) \u226b\n (normalizeIsoAux C (tensor X\u2081 (tensor X\u2082 X\u2083))).hom)\n n =\n NatTrans.app\n ((normalizeIsoAux C (tensor (tensor X\u2081 X\u2082) X\u2083)).hom \u226b\n (normalize' C).map\n (Quotient.mk (setoidHom (tensor (tensor X\u2081 X\u2082) X\u2083) (tensor X\u2081 (tensor X\u2082 X\u2083))) (Hom.\u03b1_hom X\u2081 X\u2082 X\u2083)))\n n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n =>\n \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom ((X\u2081 \u2297 X\u2082) \u2297 X\u2083) (X\u2081 \u2297 X\u2082 \u2297 X\u2083)) (Hom.\u03b1_hom X\u2081 X\u2082 X\u2083)) \u226b\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u2081 (X\u2082 \u2297 X\u2083)).symm \u226a\u226b\n (normalizeIsoApp C X\u2081 x \u2297 Iso.refl (X\u2082 \u2297 X\u2083)) \u226a\u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u2081 x.as).as) X\u2082 X\u2083).symm \u226a\u226b\n (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 x.as) \u2297 Iso.refl X\u2083) \u226a\u226b\n normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x.as).as)).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (X\u2081 \u2297 X\u2082) X\u2083).symm \u226a\u226b\n ((\u03b1_ (inclusionObj x.as) X\u2081 X\u2082).symm \u226a\u226b\n (normalizeIsoApp C X\u2081 x \u2297 Iso.refl X\u2082) \u226a\u226b normalizeIsoApp C X\u2082 (normalizeObj X\u2081 x.as) \u2297\n Iso.refl X\u2083) \u226a\u226b\n normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x.as).as)).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x =>\n \ud835\udfd9 ((Discrete.functor fun x => normalizeObj X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x).as).as).obj x)))\n n\n[PROOFSTEP]\nrw [mk_\u03b1_hom, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).hom) n \u226b\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u2081 (X\u2082 \u2297 X\u2083)).symm \u226a\u226b\n (normalizeIsoApp C X\u2081 x \u2297 Iso.refl (X\u2082 \u2297 X\u2083)) \u226a\u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u2081 x.as).as) X\u2082 X\u2083).symm \u226a\u226b\n (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 x.as) \u2297 Iso.refl X\u2083) \u226a\u226b\n normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x.as).as)).hom\n n =\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (X\u2081 \u2297 X\u2082) X\u2083).symm \u226a\u226b\n ((\u03b1_ (inclusionObj x.as) X\u2081 X\u2082).symm \u226a\u226b\n (normalizeIsoApp C X\u2081 x \u2297 Iso.refl X\u2082) \u226a\u226b normalizeIsoApp C X\u2082 (normalizeObj X\u2081 x.as) \u2297\n Iso.refl X\u2083) \u226a\u226b\n normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x.as).as)).hom\n n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x =>\n \ud835\udfd9 ((Discrete.functor fun x => normalizeObj X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x).as).as).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).hom) \u226b\n (\u03b1_ (inclusionObj n.as) X\u2081 (X\u2082 \u2297 X\u2083)).inv \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 (X\u2082 \u2297 X\u2083)) \u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u2081 n.as).as) X\u2082 X\u2083).inv \u226b\n ((normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297 \ud835\udfd9 X\u2083) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom =\n ((\u03b1_ (inclusionObj n.as) (X\u2081 \u2297 X\u2082) X\u2083).inv \u226b\n ((\u03b1_ (inclusionObj n.as) X\u2081 X\u2082).inv \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 X\u2082) \u226b (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297\n \ud835\udfd9 X\u2083) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom) \u226b\n \ud835\udfd9\n (inclusionObj\n ((Discrete.functor fun x => normalizeObj X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 x).as).as).obj n).as)\n[PROOFSTEP]\nsimp only [comp_tensor_id, associator_conjugation, tensor_id, Category.comp_id]\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).hom) \u226b\n (\u03b1_ (inclusionObj n.as) X\u2081 (X\u2082 \u2297 X\u2083)).inv \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 (X\u2082 \u2297 X\u2083)) \u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u2081 n.as).as) X\u2082 X\u2083).inv \u226b\n ((normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297 \ud835\udfd9 X\u2083) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom =\n (\u03b1_ (inclusionObj n.as) (X\u2081 \u2297 X\u2082) X\u2083).inv \u226b\n (((\u03b1_ (inclusionObj n.as) X\u2081 X\u2082).inv \u2297 \ud835\udfd9 X\u2083) \u226b\n ((\u03b1_ (inclusionObj n.as \u2297 X\u2081) X\u2082 X\u2083).hom \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 (X\u2082 \u2297 X\u2083)) \u226b\n (\u03b1_ (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj n).as) X\u2082 X\u2083).inv) \u226b\n ((normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297 \ud835\udfd9 X\u2083)) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase f.\u03b1_hom.w.h\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (((((\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).hom) \u226b (\u03b1_ (inclusionObj n.as) X\u2081 (X\u2082 \u2297 X\u2083)).inv) \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 (X\u2082 \u2297 X\u2083))) \u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u2081 n.as).as) X\u2082 X\u2083).inv) \u226b\n ((normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297 \ud835\udfd9 X\u2083)) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom =\n ((((((\u03b1_ (inclusionObj n.as) (X\u2081 \u2297 X\u2082) X\u2083).inv \u226b ((\u03b1_ (inclusionObj n.as) X\u2081 X\u2082).inv \u2297 \ud835\udfd9 X\u2083)) \u226b\n (\u03b1_ (inclusionObj n.as \u2297 X\u2081) X\u2082 X\u2083).hom) \u226b\n ((normalizeIsoApp C X\u2081 n).hom \u2297 \ud835\udfd9 (X\u2082 \u2297 X\u2083))) \u226b\n (\u03b1_ (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj n).as) X\u2082 X\u2083).inv) \u226b\n ((normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n.as)).hom \u2297 \ud835\udfd9 X\u2083)) \u226b\n (normalizeIsoApp C X\u2083 (normalizeObj X\u2082 (normalizeObj X\u2081 n.as).as)).hom\n[PROOFSTEP]\ncongr 4\n[GOAL]\ncase f.\u03b1_hom.w.h.e_a.e_a.e_a.e_a\nC : Type u\nX Y X\u2081 X\u2082 X\u2083 : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).hom) \u226b (\u03b1_ (inclusionObj n.as) X\u2081 (X\u2082 \u2297 X\u2083)).inv =\n ((\u03b1_ (inclusionObj n.as) (X\u2081 \u2297 X\u2082) X\u2083).inv \u226b ((\u03b1_ (inclusionObj n.as) X\u2081 X\u2082).inv \u2297 \ud835\udfd9 X\u2083)) \u226b\n (\u03b1_ (inclusionObj n.as \u2297 X\u2081) X\u2082 X\u2083).hom\n[PROOFSTEP]\nsimp only [Category.assoc, \u2190 cancel_epi (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u2081 X\u2082 X\u2083).inv),\n pentagon_inv_assoc (inclusionObj n.as) X\u2081 X\u2082 X\u2083, tensor_inv_hom_id_assoc, tensor_id, Category.id_comp, Iso.inv_hom_id,\n Category.comp_id]\n[GOAL]\ncase f.\u03b1_inv\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\n\u22a2 (tensorFunc C).map\n (Quotient.mk (setoidHom (tensor X\u271d (tensor Y\u271d Z\u271d)) (tensor (tensor X\u271d Y\u271d) Z\u271d)) (Hom.\u03b1_inv X\u271d Y\u271d Z\u271d)) \u226b\n (normalizeIsoAux C (tensor (tensor X\u271d Y\u271d) Z\u271d)).hom =\n (normalizeIsoAux C (tensor X\u271d (tensor Y\u271d Z\u271d))).hom \u226b\n (normalize' C).map\n (Quotient.mk (setoidHom (tensor X\u271d (tensor Y\u271d Z\u271d)) (tensor (tensor X\u271d Y\u271d) Z\u271d)) (Hom.\u03b1_inv X\u271d Y\u271d Z\u271d))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.\u03b1_inv.w.h\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map\n (Quotient.mk (setoidHom (tensor X\u271d (tensor Y\u271d Z\u271d)) (tensor (tensor X\u271d Y\u271d) Z\u271d)) (Hom.\u03b1_inv X\u271d Y\u271d Z\u271d)) \u226b\n (normalizeIsoAux C (tensor (tensor X\u271d Y\u271d) Z\u271d)).hom)\n n =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u271d (tensor Y\u271d Z\u271d))).hom \u226b\n (normalize' C).map\n (Quotient.mk (setoidHom (tensor X\u271d (tensor Y\u271d Z\u271d)) (tensor (tensor X\u271d Y\u271d) Z\u271d)) (Hom.\u03b1_inv X\u271d Y\u271d Z\u271d)))\n n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase f.\u03b1_inv.w.h\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n =>\n \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom (X\u271d \u2297 Y\u271d \u2297 Z\u271d) ((X\u271d \u2297 Y\u271d) \u2297 Z\u271d)) (Hom.\u03b1_inv X\u271d Y\u271d Z\u271d)) \u226b\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (X\u271d \u2297 Y\u271d) Z\u271d).symm \u226a\u226b\n ((\u03b1_ (inclusionObj x.as) X\u271d Y\u271d).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl Y\u271d) \u226a\u226b normalizeIsoApp C Y\u271d (normalizeObj X\u271d x.as) \u2297\n Iso.refl Z\u271d) \u226a\u226b\n normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x.as).as)).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (Y\u271d \u2297 Z\u271d)).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (Y\u271d \u2297 Z\u271d)) \u226a\u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u271d x.as).as) Y\u271d Z\u271d).symm \u226a\u226b\n (normalizeIsoApp C Y\u271d (normalizeObj X\u271d x.as) \u2297 Iso.refl Z\u271d) \u226a\u226b\n normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x.as).as)).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x =>\n \ud835\udfd9 ((Discrete.functor fun x => normalizeObj Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x).as).as).obj x)))\n n\n[PROOFSTEP]\nrw [mk_\u03b1_inv, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.\u03b1_inv.w.h\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u271d Y\u271d Z\u271d).inv) n \u226b\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (X\u271d \u2297 Y\u271d) Z\u271d).symm \u226a\u226b\n ((\u03b1_ (inclusionObj x.as) X\u271d Y\u271d).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl Y\u271d) \u226a\u226b normalizeIsoApp C Y\u271d (normalizeObj X\u271d x.as) \u2297\n Iso.refl Z\u271d) \u226a\u226b\n normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x.as).as)).hom\n n =\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (Y\u271d \u2297 Z\u271d)).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (Y\u271d \u2297 Z\u271d)) \u226a\u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u271d x.as).as) Y\u271d Z\u271d).symm \u226a\u226b\n (normalizeIsoApp C Y\u271d (normalizeObj X\u271d x.as) \u2297 Iso.refl Z\u271d) \u226a\u226b\n normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x.as).as)).hom\n n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x =>\n \ud835\udfd9 ((Discrete.functor fun x => normalizeObj Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x).as).as).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.\u03b1_inv.w.h\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03b1_ X\u271d Y\u271d Z\u271d).inv) \u226b\n (\u03b1_ (inclusionObj n.as) (X\u271d \u2297 Y\u271d) Z\u271d).inv \u226b\n ((\u03b1_ (inclusionObj n.as) X\u271d Y\u271d).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 Y\u271d) \u226b (normalizeIsoApp C Y\u271d (normalizeObj X\u271d n.as)).hom \u2297\n \ud835\udfd9 Z\u271d) \u226b\n (normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d n.as).as)).hom =\n ((\u03b1_ (inclusionObj n.as) X\u271d (Y\u271d \u2297 Z\u271d)).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (Y\u271d \u2297 Z\u271d)) \u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u271d n.as).as) Y\u271d Z\u271d).inv \u226b\n ((normalizeIsoApp C Y\u271d (normalizeObj X\u271d n.as)).hom \u2297 \ud835\udfd9 Z\u271d) \u226b\n (normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d n.as).as)).hom) \u226b\n \ud835\udfd9\n (inclusionObj\n ((Discrete.functor fun x => normalizeObj Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d x).as).as).obj n).as)\n[PROOFSTEP]\nsimp only [Category.assoc, comp_tensor_id, tensor_id, Category.comp_id, pentagon_inv_assoc, \u2190\n associator_inv_naturality_assoc]\n[GOAL]\ncase f.\u03b1_inv.w.h\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\u03b1_ (inclusionObj n.as) X\u271d (Y\u271d \u2297 Z\u271d)).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (Y\u271d \u2297 Z\u271d)) \u226b\n (\u03b1_ (inclusionObj ((Discrete.functor (normalizeObj X\u271d)).obj n).as) Y\u271d Z\u271d).inv \u226b\n ((normalizeIsoApp C Y\u271d (normalizeObj X\u271d n.as)).hom \u2297 \ud835\udfd9 Z\u271d) \u226b\n (normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d n.as).as)).hom =\n (\u03b1_ (inclusionObj n.as) X\u271d (Y\u271d \u2297 Z\u271d)).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (Y\u271d \u2297 Z\u271d)) \u226b\n (\u03b1_ (inclusionObj (normalizeObj X\u271d n.as).as) Y\u271d Z\u271d).inv \u226b\n ((normalizeIsoApp C Y\u271d (normalizeObj X\u271d n.as)).hom \u2297 \ud835\udfd9 Z\u271d) \u226b\n (normalizeIsoApp C Z\u271d (normalizeObj Y\u271d (normalizeObj X\u271d n.as).as)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f.l_hom\nC : Type u\nX Y X\u271d : F C\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom (tensor Unit X\u271d) X\u271d) (Hom.l_hom X\u271d)) \u226b (normalizeIsoAux C X\u271d).hom =\n (normalizeIsoAux C (tensor Unit X\u271d)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor Unit X\u271d) X\u271d) (Hom.l_hom X\u271d))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.l_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor Unit X\u271d) X\u271d) (Hom.l_hom X\u271d)) \u226b (normalizeIsoAux C X\u271d).hom) n =\n NatTrans.app\n ((normalizeIsoAux C (tensor Unit X\u271d)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor Unit X\u271d) X\u271d) (Hom.l_hom X\u271d)))\n n\n[PROOFSTEP]\ndsimp [Functor.comp]\n[GOAL]\ncase f.l_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom (\ud835\udfd9_ (F C) \u2297 X\u271d) X\u271d) (Hom.l_hom X\u271d)) \u226b\n (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (\ud835\udfd9_ (F C)) X\u271d).symm \u226a\u226b\n (\u03c1_ (inclusionObj x.as) \u2297 Iso.refl X\u271d) \u226a\u226b normalizeIsoApp C X\u271d { as := x.as }).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor fun x => normalizeObj X\u271d x).obj x)))\n n\n[PROOFSTEP]\nrw [mk_l_hom, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.l_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).hom) n \u226b\n NatTrans.app (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom n =\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (\ud835\udfd9_ (F C)) X\u271d).symm \u226a\u226b\n (\u03c1_ (inclusionObj x.as) \u2297 Iso.refl X\u271d) \u226a\u226b normalizeIsoApp C X\u271d { as := x.as }).hom\n n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor fun x => normalizeObj X\u271d x).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.l_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).hom) \u226b (normalizeIsoApp C X\u271d n).hom =\n ((\u03b1_ (inclusionObj n.as) (\ud835\udfd9_ (F C)) X\u271d).inv \u226b\n ((\u03c1_ (inclusionObj n.as)).hom \u2297 \ud835\udfd9 X\u271d) \u226b (normalizeIsoApp C X\u271d { as := n.as }).hom) \u226b\n \ud835\udfd9 (inclusionObj ((Discrete.functor fun x => normalizeObj X\u271d x).obj n).as)\n[PROOFSTEP]\nsimp only [triangle_assoc_comp_right_assoc, Category.assoc, Category.comp_id]\n[GOAL]\ncase f.l_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).hom) \u226b (normalizeIsoApp C X\u271d n).hom =\n (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).hom) \u226b (normalizeIsoApp C X\u271d { as := n.as }).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f.l_inv\nC : Type u\nX Y X\u271d : F C\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X\u271d (tensor Unit X\u271d)) (Hom.l_inv X\u271d)) \u226b\n (normalizeIsoAux C (tensor Unit X\u271d)).hom =\n (normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d (tensor Unit X\u271d)) (Hom.l_inv X\u271d))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.l_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom X\u271d (tensor Unit X\u271d)) (Hom.l_inv X\u271d)) \u226b\n (normalizeIsoAux C (tensor Unit X\u271d)).hom)\n n =\n NatTrans.app\n ((normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d (tensor Unit X\u271d)) (Hom.l_inv X\u271d))) n\n[PROOFSTEP]\ndsimp [Functor.comp]\n[GOAL]\ncase f.l_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom X\u271d (\ud835\udfd9_ (F C) \u2297 X\u271d)) (Hom.l_inv X\u271d)) \u226b\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (\ud835\udfd9_ (F C)) X\u271d).symm \u226a\u226b\n (\u03c1_ (inclusionObj x.as) \u2297 Iso.refl X\u271d) \u226a\u226b normalizeIsoApp C X\u271d { as := x.as }).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor (normalizeObj X\u271d)).obj x)))\n n\n[PROOFSTEP]\nrw [mk_l_inv, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.l_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).inv) n \u226b\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) (\ud835\udfd9_ (F C)) X\u271d).symm \u226a\u226b\n (\u03c1_ (inclusionObj x.as) \u2297 Iso.refl X\u271d) \u226a\u226b normalizeIsoApp C X\u271d { as := x.as }).hom\n n =\n NatTrans.app (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor (normalizeObj X\u271d)).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.l_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03bb_ X\u271d).inv) \u226b\n (\u03b1_ (inclusionObj n.as) (\ud835\udfd9_ (F C)) X\u271d).inv \u226b\n ((\u03c1_ (inclusionObj n.as)).hom \u2297 \ud835\udfd9 X\u271d) \u226b (normalizeIsoApp C X\u271d { as := n.as }).hom =\n (normalizeIsoApp C X\u271d n).hom \u226b \ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u271d)).obj n).as)\n[PROOFSTEP]\nsimp only [triangle_assoc_comp_left_inv_assoc, inv_hom_id_tensor_assoc, tensor_id, Category.id_comp, Category.comp_id]\n[GOAL]\ncase f.l_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (normalizeIsoApp C X\u271d { as := n.as }).hom = (normalizeIsoApp C X\u271d n).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f.\u03c1_hom\nC : Type u\nX Y X\u271d : F C\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u271d Unit) X\u271d) (Hom.\u03c1_hom X\u271d)) \u226b (normalizeIsoAux C X\u271d).hom =\n (normalizeIsoAux C (tensor X\u271d Unit)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u271d Unit) X\u271d) (Hom.\u03c1_hom X\u271d))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.\u03c1_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u271d Unit) X\u271d) (Hom.\u03c1_hom X\u271d)) \u226b (normalizeIsoAux C X\u271d).hom) n =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u271d Unit)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u271d Unit) X\u271d) (Hom.\u03c1_hom X\u271d)))\n n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase f.\u03c1_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom (X\u271d \u2297 \ud835\udfd9_ (F C)) X\u271d) (Hom.\u03c1_hom X\u271d)) \u226b\n (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (\ud835\udfd9_ (F C))).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (\ud835\udfd9_ (F C))) \u226a\u226b \u03c1_ (inclusionObj (normalizeObj X\u271d x.as).as)).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor fun x => { as := (normalizeObj X\u271d x).as }).obj x)))\n n\n[PROOFSTEP]\nrw [mk_\u03c1_hom, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.\u03c1_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03c1_ X\u271d).hom) n \u226b\n NatTrans.app (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom n =\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (\ud835\udfd9_ (F C))).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (\ud835\udfd9_ (F C))) \u226a\u226b \u03c1_ (inclusionObj (normalizeObj X\u271d x.as).as)).hom\n n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor fun x => { as := (normalizeObj X\u271d x).as }).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.\u03c1_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03c1_ X\u271d).hom) \u226b (normalizeIsoApp C X\u271d n).hom =\n ((\u03b1_ (inclusionObj n.as) X\u271d (\ud835\udfd9_ (F C))).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ (F C))) \u226b (\u03c1_ (inclusionObj (normalizeObj X\u271d n.as).as)).hom) \u226b\n \ud835\udfd9 (inclusionObj ((Discrete.functor fun x => { as := (normalizeObj X\u271d x).as }).obj n).as)\n[PROOFSTEP]\nsimp only [\u2190 (Iso.inv_comp_eq _).2 (rightUnitor_tensor _ _), Category.assoc, \u2190 rightUnitor_naturality, Category.comp_id]\n[GOAL]\ncase f.\u03c1_hom.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\u03b1_ (inclusionObj n.as) X\u271d (\ud835\udfd9_ (F C))).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 tensorUnit') \u226b\n (\u03c1_ (inclusionObj ((Discrete.functor (normalizeObj X\u271d)).obj n).as)).hom =\n (\u03b1_ (inclusionObj n.as) X\u271d (\ud835\udfd9_ (F C))).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ (F C))) \u226b (\u03c1_ (inclusionObj (normalizeObj X\u271d n.as).as)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f.\u03c1_inv\nC : Type u\nX Y X\u271d : F C\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X\u271d (tensor X\u271d Unit)) (Hom.\u03c1_inv X\u271d)) \u226b\n (normalizeIsoAux C (tensor X\u271d Unit)).hom =\n (normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d (tensor X\u271d Unit)) (Hom.\u03c1_inv X\u271d))\n[PROOFSTEP]\next n\n[GOAL]\ncase f.\u03c1_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom X\u271d (tensor X\u271d Unit)) (Hom.\u03c1_inv X\u271d)) \u226b\n (normalizeIsoAux C (tensor X\u271d Unit)).hom)\n n =\n NatTrans.app\n ((normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d (tensor X\u271d Unit)) (Hom.\u03c1_inv X\u271d))) n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase f.\u03c1_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app\n ((Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 Quotient.mk (setoidHom X\u271d (X\u271d \u2297 \ud835\udfd9_ (F C))) (Hom.\u03c1_inv X\u271d)) \u226b\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (\ud835\udfd9_ (F C))).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (\ud835\udfd9_ (F C))) \u226a\u226b \u03c1_ (inclusionObj (normalizeObj X\u271d x.as).as)).hom)\n n =\n NatTrans.app\n ((NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom \u226b\n ((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor (normalizeObj X\u271d)).obj x)))\n n\n[PROOFSTEP]\nrw [mk_\u03c1_inv, NatTrans.comp_app, NatTrans.comp_app]\n[GOAL]\ncase f.\u03c1_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 NatTrans.app (Discrete.natTrans fun n => \ud835\udfd9 (inclusionObj n.as) \u2297 (\u03c1_ X\u271d).inv) n \u226b\n NatTrans.app\n (NatIso.ofComponents fun x =>\n (\u03b1_ (inclusionObj x.as) X\u271d (\ud835\udfd9_ (F C))).symm \u226a\u226b\n (normalizeIsoApp C X\u271d x \u2297 Iso.refl (\ud835\udfd9_ (F C))) \u226a\u226b \u03c1_ (inclusionObj (normalizeObj X\u271d x.as).as)).hom\n n =\n NatTrans.app (NatIso.ofComponents (normalizeIsoApp C X\u271d)).hom n \u226b\n NatTrans.app\n (((whiskeringRight (Discrete (NormalMonoidalObject C)) (Discrete (NormalMonoidalObject C)) (F C)).obj\n (Discrete.functor inclusionObj)).map\n (Discrete.natTrans fun x => \ud835\udfd9 ((Discrete.functor (normalizeObj X\u271d)).obj x)))\n n\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp]\n[GOAL]\ncase f.\u03c1_inv.w.h\nC : Type u\nX Y X\u271d : F C\nn : (Discrete \u2218 NormalMonoidalObject) C\n\u22a2 (\ud835\udfd9 (inclusionObj n.as) \u2297 (\u03c1_ X\u271d).inv) \u226b\n (\u03b1_ (inclusionObj n.as) X\u271d (\ud835\udfd9_ (F C))).inv \u226b\n ((normalizeIsoApp C X\u271d n).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ (F C))) \u226b (\u03c1_ (inclusionObj (normalizeObj X\u271d n.as).as)).hom =\n (normalizeIsoApp C X\u271d n).hom \u226b \ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u271d)).obj n).as)\n[PROOFSTEP]\nsimp only [\u2190 (Iso.eq_comp_inv _).1 (rightUnitor_tensor_inv _ _), rightUnitor_conjugation, Category.assoc,\n Iso.hom_inv_id_assoc, Iso.inv_hom_id_assoc, Iso.inv_hom_id, Discrete.functor, Category.comp_id, Function.comp]\n[GOAL]\ncase f.comp\nC : Type u\nX Y X\u271d Y\u271d Z\u271d : F C\nf\u271d : X\u271d \u27f6\u1d50 Y\u271d\ng\u271d : Y\u271d \u27f6\u1d50 Z\u271d\nh\u2081 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u271d Y\u271d) f\u271d) \u226b (normalizeIsoAux C Y\u271d).hom =\n (normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d Y\u271d) f\u271d)\nh\u2082 :\n (tensorFunc C).map (Quotient.mk (setoidHom Y\u271d Z\u271d) g\u271d) \u226b (normalizeIsoAux C Z\u271d).hom =\n (normalizeIsoAux C Y\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom Y\u271d Z\u271d) g\u271d)\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom X\u271d Z\u271d) (Hom.comp f\u271d g\u271d)) \u226b (normalizeIsoAux C Z\u271d).hom =\n (normalizeIsoAux C X\u271d).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u271d Z\u271d) (Hom.comp f\u271d g\u271d))\n[PROOFSTEP]\nrw [mk_comp, Functor.map_comp, Functor.map_comp, Category.assoc, h\u2082, reassoc_of% h\u2081]\n[GOAL]\ncase f.tensor\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nh\u2081 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom =\n (normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)\nh\u2082 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom =\n (normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g)\n\u22a2 (tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom =\n (normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g))\n[PROOFSTEP]\next \u27e8n\u27e9\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nh\u2081 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom =\n (normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)\nh\u2082 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom =\n (normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g)\nn : NormalMonoidalObject C\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom)\n { as := n } =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\nreplace h\u2081 := NatTrans.congr_app h\u2081 \u27e8n\u27e9\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nh\u2082 :\n (tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom =\n (normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g)\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom)\n { as := n } =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\nreplace h\u2082 := NatTrans.congr_app h\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj \u27e8n\u27e9)\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom)\n { as := n } =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\nhave h\u2083 := (normalizeIsoAux _ Y\u2082).hom.naturality ((normalizeMapAux f).app \u27e8n\u27e9)\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom)\n { as := n } =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\nhave h\u2084 :\n \u2200 (X\u2083 Y\u2083 : N C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n \u21a6 inclusionObj n \u2297 Y\u2082).map \u03c6 :=\n by\n rintro \u27e8X\u2083\u27e9 \u27e8Y\u2083\u27e9 \u03c6\n obtain rfl : X\u2083 = Y\u2083 := \u03c6.1.1\n simp only [discrete_functor_map_eq_id, tensor_id]\n rfl\n[GOAL]\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\n\u22a2 \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n[PROOFSTEP]\nrintro \u27e8X\u2083\u27e9 \u27e8Y\u2083\u27e9 \u03c6\n[GOAL]\ncase mk.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\nX\u2083 Y\u2083 : NormalMonoidalObject C\n\u03c6 : { as := X\u2083 } \u27f6 { as := Y\u2083 }\n\u22a2 (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n[PROOFSTEP]\nobtain rfl : X\u2083 = Y\u2083 := \u03c6.1.1\n[GOAL]\ncase mk.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\nX\u2083 : NormalMonoidalObject C\n\u03c6 : { as := X\u2083 } \u27f6 { as := X\u2083 }\n\u22a2 (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n[PROOFSTEP]\nsimp only [discrete_functor_map_eq_id, tensor_id]\n[GOAL]\ncase mk.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\nX\u2083 : NormalMonoidalObject C\n\u03c6 : { as := X\u2083 } \u27f6 { as := X\u2083 }\n\u22a2 \ud835\udfd9 ((Discrete.functor inclusionObj).obj { as := X\u2083 } \u2297 Y\u2082) =\n \ud835\udfd9 ((Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).obj { as := X\u2083 })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoAux C Y\u2081).hom) { as := n } =\n NatTrans.app ((normalizeIsoAux C X\u2081).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b (normalizeIsoAux C Y\u2082).hom)\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app ((normalizeIsoAux C X\u2082).hom \u226b (normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 NatTrans.app\n ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)) \u226b\n (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom)\n { as := n } =\n NatTrans.app\n ((normalizeIsoAux C (tensor X\u2081 X\u2082)).hom \u226b\n (normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\nrw [NatTrans.comp_app, NatTrans.comp_app] at h\u2081 h\u2082 \u22a2\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n } \u226b\n NatTrans.app (normalizeIsoAux C Y\u2081).hom { as := n } =\n NatTrans.app (normalizeIsoAux C X\u2081).hom { as := n } \u226b\n NatTrans.app ((normalize' C).map (Quotient.mk (setoidHom X\u2081 Y\u2081) f)) { as := n }\nh\u2082 :\n NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C X\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n NatTrans.app ((normalize' C).map (Quotient.mk (setoidHom X\u2082 Y\u2082) g))\n ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })\nh\u2083 :\n ((tensorFunc C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n }) =\n NatTrans.app (normalizeIsoAux C Y\u2082).hom ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }) \u226b\n ((normalize' C).obj Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 NatTrans.app ((tensorFunc C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n } \u226b\n NatTrans.app (normalizeIsoAux C (tensor Y\u2081 Y\u2082)).hom { as := n } =\n NatTrans.app (normalizeIsoAux C (tensor X\u2081 X\u2082)).hom { as := n } \u226b\n NatTrans.app ((normalize' C).map (Quotient.mk (setoidHom (tensor X\u2081 X\u2082) (tensor Y\u2081 Y\u2082)) (Hom.tensor f g)))\n { as := n }\n[PROOFSTEP]\ndsimp [NatIso.ofComponents, normalizeMapAux, whiskeringRight, whiskerRight, Functor.comp, Discrete.natTrans] at h\u2081 h\u2082 h\u2083\n \u22a2\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoApp C Y\u2081 { as := n }).hom =\n (normalizeIsoApp C X\u2081 { as := n }).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2082 :\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C X\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }))\nh\u2083 :\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082)) (Hom.tensor f g)) \u226b\n (\u03b1_ (inclusionObj n) Y\u2081 Y\u2082).inv \u226b\n ((normalizeIsoApp C Y\u2081 { as := n }).hom \u2297 \ud835\udfd9 Y\u2082) \u226b (normalizeIsoApp C Y\u2082 (normalizeObj Y\u2081 n)).hom =\n ((\u03b1_ (inclusionObj n) X\u2081 X\u2082).inv \u226b\n ((normalizeIsoApp C X\u2081 { as := n }).hom \u2297 \ud835\udfd9 X\u2082) \u226b (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n)).hom) \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) (normalizeObj X\u2081 n) \u226b\n (Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\n[PROOFSTEP]\nrw [mk_tensor, associator_inv_naturality_assoc, \u2190 tensor_comp_assoc, h\u2081, Category.assoc, Category.comp_id, \u2190\n @Category.id_comp (F C) _ _ _ (@Quotient.mk _ _ g), tensor_comp, Category.assoc, Category.assoc, Functor.map_comp]\n[GOAL]\ncase f.tensor.w.h.mk\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoApp C Y\u2081 { as := n }).hom =\n (normalizeIsoApp C X\u2081 { as := n }).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2082 :\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C X\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }))\nh\u2083 :\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 (\u03b1_ (inclusionObj n) X\u2081 X\u2082).inv \u226b\n ((normalizeIsoApp C X\u2081 { as := n }).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n ((Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n }) \u2297\n Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 (normalizeObj Y\u2081 n)).hom =\n (\u03b1_ (inclusionObj n) X\u2081 X\u2082).inv \u226b\n ((normalizeIsoApp C X\u2081 { as := n }).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n)).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux g) (normalizeObj X\u2081 n)) \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase f.tensor.w.h.mk.e_a.e_a\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoApp C Y\u2081 { as := n }).hom =\n (normalizeIsoApp C X\u2081 { as := n }).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2082 :\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C X\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }))\nh\u2083 :\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 ((Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n }) \u2297\n Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 (normalizeObj Y\u2081 n)).hom =\n (normalizeIsoApp C X\u2082 (normalizeObj X\u2081 n)).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux g) (normalizeObj X\u2081 n)) \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\n[PROOFSTEP]\nerw [\u2190 reassoc_of% h\u2082]\n[GOAL]\ncase f.tensor.w.h.mk.e_a.e_a\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoApp C Y\u2081 { as := n }).hom =\n (normalizeIsoApp C X\u2081 { as := n }).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2082 :\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C X\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }))\nh\u2083 :\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 ((Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n }) \u2297\n Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 (normalizeObj Y\u2081 n)).hom =\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\n[PROOFSTEP]\nrw [\u2190 h\u2083, \u2190 Category.assoc, \u2190 id_tensor_comp_tensor_id, h\u2084]\n[GOAL]\ncase f.tensor.w.h.mk.e_a.e_a\nC : Type u\nX Y X\u2081 X\u2082 Y\u2081 Y\u2082 : F C\nf : X\u2081 \u27f6\u1d50 Y\u2081\ng : X\u2082 \u27f6\u1d50 Y\u2082\nn : NormalMonoidalObject C\nh\u2081 :\n (\ud835\udfd9 (inclusionObj n) \u2297 Quotient.mk (setoidHom X\u2081 Y\u2081) f) \u226b (normalizeIsoApp C Y\u2081 { as := n }).hom =\n (normalizeIsoApp C X\u2081 { as := n }).hom \u226b\n (Discrete.functor inclusionObj).map (NatTrans.app (normalizeMapAux f) { as := n })\nh\u2082 :\n (\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C X\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n (NatTrans.app (normalizeMapAux g) ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }))\nh\u2083 :\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n }) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom =\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj X\u2081)).obj { as := n })).hom \u226b\n (Discrete.functor inclusionObj).map\n ((Discrete.functor (normalizeObj Y\u2082)).map (NatTrans.app (normalizeMapAux f) { as := n }))\nh\u2084 :\n \u2200 (X\u2083 Y\u2083 : (Discrete \u2218 NormalMonoidalObject) C) (\u03c6 : X\u2083 \u27f6 Y\u2083),\n (Discrete.functor inclusionObj).map \u03c6 \u2297 \ud835\udfd9 Y\u2082 = (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map \u03c6\n\u22a2 ((\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })) \u226b\n (normalizeIsoApp C Y\u2082 (normalizeObj Y\u2081 n)).hom =\n ((\ud835\udfd9 (inclusionObj ((Discrete.functor (normalizeObj X\u2081)).obj { as := n }).as) \u2297 Quotient.mk (setoidHom X\u2082 Y\u2082) g) \u226b\n (Discrete.functor fun n => inclusionObj n \u2297 Y\u2082).map (NatTrans.app (normalizeMapAux f) { as := n })) \u226b\n (normalizeIsoApp C Y\u2082 ((Discrete.functor (normalizeObj Y\u2081)).obj { as := n })).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\n\u22a2 \u2200 {X Y : F C} (f : X \u27f6 Y),\n (\ud835\udfed (F C)).map f \u226b\n ((fun X => (\u03bb_ X).symm \u226a\u226b ((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }) Y).hom =\n ((fun X => (\u03bb_ X).symm \u226a\u226b ((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }) X).hom \u226b\n (fullNormalize C \u22d9 inclusion).map f\n[PROOFSTEP]\nintro X Y f\n[GOAL]\nC : Type u\nX Y : F C\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed (F C)).map f \u226b ((fun X => (\u03bb_ X).symm \u226a\u226b ((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }) Y).hom =\n ((fun X => (\u03bb_ X).symm \u226a\u226b ((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }) X).hom \u226b\n (fullNormalize C \u22d9 inclusion).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nX Y : F C\nf : X \u27f6 Y\n\u22a2 f \u226b (\u03bb_ Y).inv \u226b (((normalizeIso C).app Y).app { as := NormalMonoidalObject.Unit }).hom =\n ((\u03bb_ X).inv \u226b (((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }).hom) \u226b\n (fullNormalize C \u22d9 Discrete.functor inclusionObj).map f\n[PROOFSTEP]\nrw [leftUnitor_inv_naturality_assoc, Category.assoc, Iso.cancel_iso_inv_left]\n[GOAL]\nC : Type u\nX Y : F C\nf : X \u27f6 Y\n\u22a2 (\ud835\udfd9 tensorUnit' \u2297 f) \u226b (((normalizeIso C).app Y).app { as := NormalMonoidalObject.Unit }).hom =\n (((normalizeIso C).app X).app { as := NormalMonoidalObject.Unit }).hom \u226b\n (fullNormalize C \u22d9 Discrete.functor inclusionObj).map f\n[PROOFSTEP]\nexact\n congr_arg (fun f => NatTrans.app f (Discrete.mk NormalMonoidalObject.Unit)) ((normalizeIso.{u} C).hom.naturality f)\n[GOAL]\nC : Type u\nX Y : F C\nf g : X \u27f6 Y\n\u22a2 f = g\n[PROOFSTEP]\nhave hfg : (fullNormalize C).map f = (fullNormalize C).map g := Subsingleton.elim _ _\n[GOAL]\nC : Type u\nX Y : F C\nf g : X \u27f6 Y\nhfg : (fullNormalize C).map f = (fullNormalize C).map g\n\u22a2 f = g\n[PROOFSTEP]\nhave hf := NatIso.naturality_2 (fullNormalizeIso.{u} C) f\n[GOAL]\nC : Type u\nX Y : F C\nf g : X \u27f6 Y\nhfg : (fullNormalize C).map f = (fullNormalize C).map g\nhf :\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map f \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n (\ud835\udfed (F C)).map f\n\u22a2 f = g\n[PROOFSTEP]\nhave hg := NatIso.naturality_2 (fullNormalizeIso.{u} C) g\n[GOAL]\nC : Type u\nX Y : F C\nf g : X \u27f6 Y\nhfg : (fullNormalize C).map f = (fullNormalize C).map g\nhf :\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map f \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n (\ud835\udfed (F C)).map f\nhg :\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map g \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n (\ud835\udfed (F C)).map g\n\u22a2 f = g\n[PROOFSTEP]\nexact hf.symm.trans (Eq.trans (by simp only [Functor.comp_map, hfg]) hg)\n[GOAL]\nC : Type u\nX Y : F C\nf g : X \u27f6 Y\nhfg : (fullNormalize C).map f = (fullNormalize C).map g\nhf :\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map f \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n (\ud835\udfed (F C)).map f\nhg :\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map g \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n (\ud835\udfed (F C)).map g\n\u22a2 NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map f \u226b NatTrans.app (fullNormalizeIso C).inv Y =\n NatTrans.app (fullNormalizeIso C).hom X \u226b\n (fullNormalize C \u22d9 inclusion).map g \u226b NatTrans.app (fullNormalizeIso C).inv Y\n[PROOFSTEP]\nsimp only [Functor.comp_map, hfg]\n[GOAL]\nC : Type u\nsrc\u271d : Category.{u, u} (F C) := inferInstance\nX\u271d Y\u271d : F C\n\u22a2 \u2200 (a b : X\u271d \u27f6\u1d50 Y\u271d),\n a \u2248 b \u2192\n (fun f => Quotient.mk (setoidHom Y\u271d X\u271d) (inverseAux f)) a =\n (fun f => Quotient.mk (setoidHom Y\u271d X\u271d) (inverseAux f)) b\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Free.Coherence", "llama_tokens": 28460, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.2514923173202992}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\n\u22a2 foldl (argAux r) o [] = none \u2194 [] = [] \u2227 o = none\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd : \u03b1\n\u22a2 (foldl (argAux r) o tl = none \u2194 tl = [] \u2227 o = none) \u2192\n (foldl (argAux r) o (tl ++ [hd]) = none \u2194 tl ++ [hd] = [] \u2227 o = none)\n[PROOFSTEP]\nsimp [argAux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd : \u03b1\n\u22a2 (foldl (fun a b => Option.rec (some b) (fun val => if r b val then some b else some val) a) o tl = none \u2194\n tl = [] \u2227 o = none) \u2192\n \u00acOption.rec (some hd) (fun val => if r hd val then some hd else some val)\n (foldl (fun a b => Option.rec (some b) (fun val => if r b val then some b else some val) a) o tl) =\n none\n[PROOFSTEP]\ncases foldl (argAux r) o tl\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd : \u03b1\n\u22a2 (none = none \u2194 tl = [] \u2227 o = none) \u2192\n \u00acOption.rec (some hd) (fun val => if r hd val then some hd else some val) none = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd val\u271d : \u03b1\n\u22a2 (some val\u271d = none \u2194 tl = [] \u2227 o = none) \u2192\n \u00acOption.rec (some hd) (fun val => if r hd val then some hd else some val) (some val\u271d) = none\n[PROOFSTEP]\nsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd val\u271d : \u03b1\n\u22a2 (tl = [] \u2192 \u00aco = none) \u2192 \u00ac(if r hd val\u271d then some hd else some val\u271d) = none\n[PROOFSTEP]\ntry split_ifs <;> simp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd val\u271d : \u03b1\n\u22a2 (tl = [] \u2192 \u00aco = none) \u2192 \u00ac(if r hd val\u271d then some hd else some val\u271d) = none\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd val\u271d : \u03b1\nh\u271d : r hd val\u271d\n\u22a2 (tl = [] \u2192 \u00aco = none) \u2192 \u00acFalse\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ntl : List \u03b1\nhd val\u271d : \u03b1\nh\u271d : \u00acr hd val\u271d\n\u22a2 (tl = [] \u2192 \u00aco = none) \u2192 \u00acFalse\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na m : \u03b1\nl : List \u03b1\n\u22a2 \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) [] \u2192 m \u2208 [a]\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na m : \u03b1\nl : List \u03b1\n\u22a2 \u2200 (l : List \u03b1) (a : \u03b1),\n (\u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) l \u2192 m \u2208 a :: l) \u2192\n \u2200 (a_2 m : \u03b1), m \u2208 foldl (argAux r) (some a_2) (l ++ [a]) \u2192 m \u2208 a_2 :: (l ++ [a])\n[PROOFSTEP]\nintro tl hd ih a m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m : \u03b1\n\u22a2 m \u2208 foldl (argAux r) (some a) (tl ++ [hd]) \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsimp only [foldl_append, foldl_cons, foldl_nil, argAux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m : \u03b1\n\u22a2 m \u2208\n Option.rec (some hd) (fun val => if r hd val then some hd else some val)\n (foldl (fun a b => Option.rec (some b) (fun val => if r b val then some b else some val) a) (some a) tl) \u2192\n m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\ncases hf : foldl (argAux r) (some a) tl\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m : \u03b1\nhf : foldl (argAux r) (some a) tl = none\n\u22a2 m \u2208 Option.rec (some hd) (fun val => if r hd val then some hd else some val) none \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\n\u22a2 m \u2208 Option.rec (some hd) (fun val => if r hd val then some hd else some val) (some val\u271d) \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\n\u22a2 (m \u2208 if r hd val\u271d then some hd else some val\u271d) \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\nh\u271d : r hd val\u271d\n\u22a2 m \u2208 some hd \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m \u2208 a :: tl\na m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\nh\u271d : \u00acr hd val\u271d\n\u22a2 m \u2208 some val\u271d \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsimp only [List.mem_cons] at ih \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd a m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\nh\u271d : \u00acr hd val\u271d\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m = a \u2228 m \u2208 tl\n\u22a2 m \u2208 some val\u271d \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nrcases ih _ _ hf with rfl | H\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\nl tl : List \u03b1\nhd m val\u271d : \u03b1\nh\u271d : \u00acr hd val\u271d\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m = a \u2228 m \u2208 tl\nhf : foldl (argAux r) (some val\u271d) tl = some val\u271d\n\u22a2 m \u2208 some val\u271d \u2192 m \u2208 val\u271d :: (tl ++ [hd])\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [Option.mem_def, Option.some.injEq, find?, eq_comm, mem_cons, mem_append,\n mem_singleton, true_or, implies_true]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nl tl : List \u03b1\nhd a m val\u271d : \u03b1\nhf : foldl (argAux r) (some a) tl = some val\u271d\nh\u271d : \u00acr hd val\u271d\nih : \u2200 (a m : \u03b1), m \u2208 foldl (argAux r) (some a) tl \u2192 m = a \u2228 m \u2208 tl\nH : val\u271d \u2208 tl\n\u22a2 m \u2208 some val\u271d \u2192 m \u2208 a :: (tl ++ [hd])\n[PROOFSTEP]\nsimp (config := { contextual := true }) [@eq_comm _ _ m, H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\n\u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 l \u2192 m \u2208 foldl (argAux r) o l \u2192 \u00acr a m\n[PROOFSTEP]\ninduction' l using List.reverseRecOn with tl a ih\n[GOAL]\ncase H0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\n\u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 [] \u2192 m \u2208 foldl (argAux r) o [] \u2192 \u00acr a m\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\n\u22a2 \u2200 {a_1 m : \u03b1} {o : Option \u03b1}, a_1 \u2208 tl ++ [a] \u2192 m \u2208 foldl (argAux r) o (tl ++ [a]) \u2192 \u00acr a_1 m\n[PROOFSTEP]\nintro b m o hb ho\n[GOAL]\ncase H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 foldl (argAux r) o (tl ++ [a])\n\u22a2 \u00acr b m\n[PROOFSTEP]\nrw [foldl_append, foldl_cons, foldl_nil, argAux] at ho \n[GOAL]\ncase H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\n\u22a2 \u00acr b m\n[PROOFSTEP]\ncases' hf : foldl (argAux r) o tl with c\n[GOAL]\ncase H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m\n[PROOFSTEP]\nrw [hf] at ho \n[GOAL]\ncase H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m\n[PROOFSTEP]\nrw [foldl_argAux_eq_none] at hf \n[GOAL]\ncase H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : tl = [] \u2227 o = none\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsimp_all [hf.1, hf.2, hr\u2080 _]\n[GOAL]\ncase H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m\n[PROOFSTEP]\nrw [hf, Option.mem_def] at ho \n[GOAL]\ncase H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (Option.casesOn (some c) (some a) fun c => if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m\n[PROOFSTEP]\ndsimp only at ho \n[GOAL]\ncase H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsplit_ifs at ho with hac\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nho : some a = some m\n\u22a2 \u00acr b m\n[PROOFSTEP]\ncases' mem_append.1 hb with h h\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nho : some c = some m\n\u22a2 \u00acr b m\n[PROOFSTEP]\ncases' mem_append.1 hb with h h\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nho : some a = some m\nh : b \u2208 tl\n\u22a2 \u00acr b m\n[PROOFSTEP]\ninjection ho with ho\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nho : some a = some m\nh : b \u2208 [a]\n\u22a2 \u00acr b m\n[PROOFSTEP]\ninjection ho with ho\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nho : some c = some m\nh : b \u2208 tl\n\u22a2 \u00acr b m\n[PROOFSTEP]\ninjection ho with ho\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nho : some c = some m\nh : b \u2208 [a]\n\u22a2 \u00acr b m\n[PROOFSTEP]\ninjection ho with ho\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 tl\nho : a = m\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsubst ho\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 [a]\nho : a = m\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsubst ho\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 tl\nho : c = m\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsubst ho\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 [a]\nho : c = m\n\u22a2 \u00acr b m\n[PROOFSTEP]\nsubst ho\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 tl\n\u22a2 \u00acr b a\n[PROOFSTEP]\nexact fun hba => ih h hf (hr\u2081 hba hac)\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 [a]\n\u22a2 \u00acr b a\n[PROOFSTEP]\nsimp_all [hr\u2080 _]\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 tl\n\u22a2 \u00acr b c\n[PROOFSTEP]\nexact ih h hf\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 [a]\n\u22a2 \u00acr b c\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 argmax f (l ++ [a]) = Option.casesOn (argmax f l) (some a) fun c => if f c < f a then some a else some c\n[PROOFSTEP]\nrw [argmax, argmax]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 foldl (argAux fun b c => f c < f b) none (l ++ [a]) =\n Option.casesOn (foldl (argAux fun b c => f c < f b) none l) (some a) fun c => if f c < f a then some a else some c\n[PROOFSTEP]\nsimp [argAux]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d m : \u03b1\n\u22a2 m \u2208 argmax f [] \u2192 m \u2208 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d hd : \u03b1\ntl : List \u03b1\nm : \u03b1\n\u22a2 m \u2208 argmax f (hd :: tl) \u2192 m \u2208 hd :: tl\n[PROOFSTEP]\nsimpa [argmax, argAux] using foldl_argAux_mem _ tl hd m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\n\u22a2 argmax f l = none \u2194 l = []\n[PROOFSTEP]\nsimp [argmax]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\n\u22a2 argmax f (a :: (hd ++ [tl])) =\n Option.casesOn (argmax f (hd ++ [tl])) (some a) fun c => if f a < f c then some c else some a\n[PROOFSTEP]\nrw [\u2190 cons_append, argmax_concat, ih, argmax_concat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\n\u22a2 (Option.casesOn (Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a) (some tl)\n fun c => if f c < f tl then some tl else some c) =\n Option.casesOn (Option.casesOn (argmax f hd) (some tl) fun c => if f c < f tl then some tl else some c) (some a)\n fun c => if f a < f c then some c else some a\n[PROOFSTEP]\ncases' h : argmax f hd with m\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nh : argmax f hd = none\n\u22a2 (Option.casesOn (Option.casesOn none (some a) fun c => if f a < f c then some c else some a) (some tl) fun c =>\n if f c < f tl then some tl else some c) =\n Option.casesOn (Option.casesOn none (some tl) fun c => if f c < f tl then some tl else some c) (some a) fun c =>\n if f a < f c then some c else some a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\n\u22a2 (Option.casesOn (Option.casesOn (some m) (some a) fun c => if f a < f c then some c else some a) (some tl) fun c =>\n if f c < f tl then some tl else some c) =\n Option.casesOn (Option.casesOn (some m) (some tl) fun c => if f c < f tl then some tl else some c) (some a) fun c =>\n if f a < f c then some c else some a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\n\u22a2 Option.rec (some tl) (fun val => if f val < f tl then some tl else some val) (if f a < f m then some m else some a) =\n Option.rec (some a) (fun val => if f a < f val then some val else some a) (if f m < f tl then some tl else some m)\n[PROOFSTEP]\nrw [\u2190 apply_ite, \u2190 apply_ite]\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\n\u22a2 Option.rec (some tl) (fun val => if f val < f tl then some tl else some val) (some (if f a < f m then m else a)) =\n Option.rec (some a) (fun val => if f a < f val then some val else some a) (some (if f m < f tl then tl else m))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\n\u22a2 (if f (if f a < f m then m else a) < f tl then some tl else some (if f a < f m then m else a)) =\n if f a < f (if f m < f tl then tl else m) then some (if f m < f tl then tl else m) else some a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : f a < f m\nh\u271d\u00b9 : f m < f tl\nh\u271d : f a < f tl\n\u22a2 some tl = some tl\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : f a < f m\nh\u271d\u00b9 : f m < f tl\nh\u271d : f a < f tl\n\u22a2 some tl = some tl\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : f a < f m\nh\u271d\u00b9 : f m < f tl\nh\u271d : \u00acf a < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : f a < f m\nh\u271d\u00b9 : f m < f tl\nh\u271d : \u00acf a < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b9 : f a < f m\nh\u271d : \u00acf m < f tl\n\u22a2 some m = some m\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b9 : f a < f m\nh\u271d : \u00acf m < f tl\n\u22a2 some m = some m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : f a < f tl\nh\u271d : f m < f tl\n\u22a2 some tl = some tl\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : f a < f tl\nh\u271d : f m < f tl\n\u22a2 some tl = some tl\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : f a < f tl\nh\u271d : \u00acf m < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : f a < f tl\nh\u271d : \u00acf m < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : \u00acf a < f tl\nh\u271d : f m < f tl\n\u22a2 some a = some a\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : \u00acf a < f tl\nh\u271d : f m < f tl\n\u22a2 some a = some a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : \u00acf a < f tl\nh\u271d : \u00acf m < f tl\n\u22a2 some a = some a\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : \u00acf a < f tl\nh\u271d : \u00acf m < f tl\n\u22a2 some a = some a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : f a < f m\nh\u271d\u00b9 : f m < f tl\nh\u271d : \u00acf a < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\nexact absurd (lt_trans \u2039f a < f m\u203a \u2039_\u203a) \u2039_\u203a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b2\nf\u271d : \u03b1 \u2192 \u03b2\nl\u271d : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl hd : List \u03b1\ntl : \u03b1\nih : argmax f (a :: hd) = Option.casesOn (argmax f hd) (some a) fun c => if f a < f c then some c else some a\nm : \u03b1\nh : argmax f hd = some m\nh\u271d\u00b2 : \u00acf a < f m\nh\u271d\u00b9 : f a < f tl\nh\u271d : \u00acf m < f tl\n\u22a2 some tl = some a\n[PROOFSTEP]\ncases (\u2039f a < f tl\u203a.lt_or_lt _).elim \u2039_\u203a \u2039_\u203a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nm : \u03b1\nx\u271d\u00b3 : m \u2208 argmax f []\nx\u271d\u00b2 : \u03b1\nx\u271d\u00b9 : x\u271d\u00b2 \u2208 []\nx\u271d : f m \u2264 f x\u271d\u00b2\n\u22a2 indexOf m [] \u2264 indexOf x\u271d\u00b2 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm : \u03b1\nhm : m \u2208 argmax f (hd :: tl)\na : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\n\u22a2 indexOf m (hd :: tl) \u2264 indexOf a (hd :: tl)\n[PROOFSTEP]\nsimp only [indexOf_cons, argmax_cons, Option.mem_def] at hm \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nhm : Option.rec (some hd) (fun val => if f hd < f val then some val else some hd) (argmax f tl) = some m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\ncases h : argmax f tl\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nhm : Option.rec (some hd) (fun val => if f hd < f val then some val else some hd) (argmax f tl) = some m\nh : argmax f tl = none\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nrw [h] at hm \n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nhm : Option.rec (some hd) (fun val => if f hd < f val then some val else some hd) none = some m\nh : argmax f tl = none\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nhm : Option.rec (some hd) (fun val => if f hd < f val then some val else some hd) (argmax f tl) = some m\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nrw [h] at hm \n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nval\u271d : \u03b1\nhm : Option.rec (some hd) (fun val => if f hd < f val then some val else some hd) (some val\u271d) = some m\nh : argmax f tl = some val\u271d\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\ndsimp only at hm \n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nha : a \u2208 hd :: tl\nham : f m \u2264 f a\nval\u271d : \u03b1\nhm : (if f hd < f val\u271d then some val\u271d else some hd) = some m\nh : argmax f tl = some val\u271d\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nobtain ha | ha := ha\n[GOAL]\ncase some.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm val\u271d : \u03b1\nhm : (if f hd < f val\u271d then some val\u271d else some hd) = some m\nh : argmax f tl = some val\u271d\nham : f m \u2264 f hd\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\nsplit_ifs at hm \n[GOAL]\ncase some.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nham : f m \u2264 f a\nval\u271d : \u03b1\nhm : (if f hd < f val\u271d then some val\u271d else some hd) = some m\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nsplit_ifs at hm \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm val\u271d : \u03b1\nh : argmax f tl = some val\u271d\nham : f m \u2264 f hd\nh\u271d : f hd < f val\u271d\nhm : some val\u271d = some m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\ninjection hm with hm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm val\u271d : \u03b1\nh : argmax f tl = some val\u271d\nham : f m \u2264 f hd\nh\u271d : \u00acf hd < f val\u271d\nhm : some hd = some m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\ninjection hm with hm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nham : f m \u2264 f a\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nhm : some val\u271d = some m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\ninjection hm with hm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nham : f m \u2264 f a\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : \u00acf hd < f val\u271d\nhm : some hd = some m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\ninjection hm with hm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm val\u271d : \u03b1\nh : argmax f tl = some val\u271d\nham : f m \u2264 f hd\nh\u271d : f hd < f val\u271d\nhm : val\u271d = m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\nsubst hm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm val\u271d : \u03b1\nh : argmax f tl = some val\u271d\nham : f m \u2264 f hd\nh\u271d : \u00acf hd < f val\u271d\nhm : hd = m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\nsubst hm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nham : f m \u2264 f a\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nhm : val\u271d = m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nsubst hm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nm a : \u03b1\nham : f m \u2264 f a\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : \u00acf hd < f val\u271d\nhm : hd = m\n\u22a2 (if m = hd then 0 else Nat.succ (indexOf m tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nsubst hm\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f hd\n\u22a2 (if val\u271d = hd then 0 else Nat.succ (indexOf val\u271d tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\ncases not_le_of_lt \u2039_\u203a \u2039_\u203a\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\nval\u271d : \u03b1\nh : argmax f tl = some val\u271d\nh\u271d : \u00acf hd < f val\u271d\nham : f hd \u2264 f hd\n\u22a2 (if hd = hd then 0 else Nat.succ (indexOf hd tl)) \u2264 if hd = hd then 0 else Nat.succ (indexOf hd tl)\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 (if val\u271d = hd then 0 else Nat.succ (indexOf val\u271d tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nrw [if_neg, if_neg]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 Nat.succ (indexOf val\u271d tl) \u2264 Nat.succ (indexOf a tl)\ncase pos.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 \u00aca = hd\ncase pos.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 \u00acval\u271d = hd\n[PROOFSTEP]\nexact Nat.succ_le_succ (index_of_argmax h (by assumption) ham)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 a \u2208 tl\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pos.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 \u00aca = hd\n[PROOFSTEP]\nexact ne_of_apply_ne f (lt_of_lt_of_le \u2039_\u203a \u2039_\u203a).ne'\n[GOAL]\ncase pos.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : f hd < f val\u271d\nham : f val\u271d \u2264 f a\n\u22a2 \u00acval\u271d = hd\n[PROOFSTEP]\nexact ne_of_apply_ne _ \u2039f hd < f _\u203a.ne'\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : \u00acf hd < f val\u271d\nham : f hd \u2264 f a\n\u22a2 (if hd = hd then 0 else Nat.succ (indexOf hd tl)) \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nrw [if_pos rfl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na\u271d\u00b9 m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhd : \u03b1\ntl : List \u03b1\na val\u271d : \u03b1\nh : argmax f tl = some val\u271d\na\u271d : Mem a tl\nh\u271d : \u00acf hd < f val\u271d\nham : f hd \u2264 f a\n\u22a2 0 \u2264 if a = hd then 0 else Nat.succ (indexOf a tl)\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (m \u2208 l \u2227 (\u2200 (a : \u03b1), a \u2208 l \u2192 f a \u2264 f m) \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 f m \u2264 f a \u2192 indexOf m l \u2264 indexOf a l) \u2192 m \u2208 argmax f l\n[PROOFSTEP]\nrintro \u27e8hml, ham, hma\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhml : m \u2208 l\nham : \u2200 (a : \u03b1), a \u2208 l \u2192 f a \u2264 f m\nhma : \u2200 (a : \u03b1), a \u2208 l \u2192 f m \u2264 f a \u2192 indexOf m l \u2264 indexOf a l\n\u22a2 m \u2208 argmax f l\n[PROOFSTEP]\ncases' harg : argmax f l with n\n[GOAL]\ncase intro.intro.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhml : m \u2208 l\nham : \u2200 (a : \u03b1), a \u2208 l \u2192 f a \u2264 f m\nhma : \u2200 (a : \u03b1), a \u2208 l \u2192 f m \u2264 f a \u2192 indexOf m l \u2264 indexOf a l\nharg : argmax f l = none\n\u22a2 m \u2208 none\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase intro.intro.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhml : m \u2208 l\nham : \u2200 (a : \u03b1), a \u2208 l \u2192 f a \u2264 f m\nhma : \u2200 (a : \u03b1), a \u2208 l \u2192 f m \u2264 f a \u2192 indexOf m l \u2264 indexOf a l\nn : \u03b1\nharg : argmax f l = some n\n\u22a2 m \u2208 some n\n[PROOFSTEP]\nhave :=\n _root_.le_antisymm (hma n (argmax_mem harg) (le_of_mem_argmax hml harg))\n (index_of_argmax harg hml (ham _ (argmax_mem harg)))\n[GOAL]\ncase intro.intro.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\ninst\u271d : DecidableEq \u03b1\nhml : m \u2208 l\nham : \u2200 (a : \u03b1), a \u2208 l \u2192 f a \u2264 f m\nhma : \u2200 (a : \u03b1), a \u2208 l \u2192 f m \u2264 f a \u2192 indexOf m l \u2264 indexOf a l\nn : \u03b1\nharg : argmax f l = some n\nthis : indexOf m l = indexOf n l\n\u22a2 m \u2208 some n\n[PROOFSTEP]\nrw [(indexOf_inj hml (argmax_mem harg)).1 this, Option.mem_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nl : List \u03b1\na m : \u03b1\nha : a \u2208 l\n\u22a2 \u00acmaximum l < \u2191a\n[PROOFSTEP]\ncases h : l.maximum\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nl : List \u03b1\na m : \u03b1\nha : a \u2208 l\nh : maximum l = none\n\u22a2 \u00acnone < \u2191a\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : DecidableRel fun x x_1 => x < x_1\nl : List \u03b1\na m : \u03b1\nha : a \u2208 l\nval\u271d : \u03b1\nh : maximum l = some val\u271d\n\u22a2 \u00acsome val\u271d < \u2191a\n[PROOFSTEP]\nsimp [WithBot.some_eq_coe, WithBot.coe_lt_coe, not_lt_maximum_of_mem ha h, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl : List \u03b1\n\u22a2 maximum (l ++ [a]) = max (maximum l) \u2191a\n[PROOFSTEP]\nsimp only [maximum, argmax_concat, id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl : List \u03b1\n\u22a2 Option.rec (some a) (fun val => if val < a then some a else some val) (argmax (fun a => a) l) =\n max (argmax (fun a => a) l) \u2191a\n[PROOFSTEP]\ncases h : argmax id l\n[GOAL]\ncase none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl : List \u03b1\nh : argmax id l = none\n\u22a2 Option.rec (some a) (fun val => if val < a then some a else some val) none = max none \u2191a\n[PROOFSTEP]\nexact (max_eq_right bot_le).symm\n[GOAL]\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl : List \u03b1\nval\u271d : \u03b1\nh : argmax id l = some val\u271d\n\u22a2 Option.rec (some a) (fun val => if val < a then some a else some val) (some val\u271d) = max (some val\u271d) \u2191a\n[PROOFSTEP]\nsimp [WithBot.some_eq_coe, max_def_lt, WithBot.coe_lt_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl : List \u03b1\n\u22a2 maximum [a] = max (\u2191a) (maximum [])\n[PROOFSTEP]\nsimp [@max_eq_left (WithBot \u03b1) _ _ _ bot_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl\u271d : List \u03b1\na\u271d m a : \u03b1\nl tl : List \u03b1\nhd : \u03b1\nih : maximum (a :: tl) = max (\u2191a) (maximum tl)\n\u22a2 maximum (a :: (tl ++ [hd])) = max (\u2191a) (maximum (tl ++ [hd]))\n[PROOFSTEP]\nrw [\u2190 cons_append, maximum_concat, ih, maximum_concat, max_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 maximum l = \u2191m \u2194 m \u2208 l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 a \u2264 m\n[PROOFSTEP]\nrw [maximum, \u2190 WithBot.some_eq_coe, argmax_eq_some_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 (m \u2208 l \u2227 (\u2200 (a : \u03b1), a \u2208 l \u2192 id a \u2264 id m) \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 id m \u2264 id a \u2192 indexOf m l \u2264 indexOf a l) \u2194\n m \u2208 l \u2227 \u2200 (a : \u03b1), a \u2208 l \u2192 a \u2264 m\n[PROOFSTEP]\nsimp only [id_eq, and_congr_right_iff, and_iff_left_iff_imp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 m \u2208 l \u2192 (\u2200 (a : \u03b1), a \u2208 l \u2192 a \u2264 m) \u2192 \u2200 (a : \u03b1), a \u2208 l \u2192 m \u2264 a \u2192 indexOf m l \u2264 indexOf a l\n[PROOFSTEP]\nintro _ h a hal hma\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na\u271d\u00b9 m : \u03b1\na\u271d : m \u2208 l\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 a \u2264 m\na : \u03b1\nhal : a \u2208 l\nhma : m \u2264 a\n\u22a2 indexOf m l \u2264 indexOf a l\n[PROOFSTEP]\nrw [_root_.le_antisymm hma (h a hal)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 \u2191a \u2264 maximum l \u2194 \u2203 b, b \u2208 l \u2227 a \u2264 b\n[PROOFSTEP]\ninduction l with\n| nil => simp\n| cons h t ih => simp [maximum_cons, ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 \u2191a \u2264 maximum l \u2194 \u2203 b, b \u2208 l \u2227 a \u2264 b\n[PROOFSTEP]\ninduction l with\n| nil => simp\n| cons h t ih => simp [maximum_cons, ih]\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 \u2191a \u2264 maximum [] \u2194 \u2203 b, b \u2208 [] \u2227 a \u2264 b\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\n\u22a2 \u2191a \u2264 maximum [] \u2194 \u2203 b, b \u2208 [] \u2227 a \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m h : \u03b1\nt : List \u03b1\nih : \u2191a \u2264 maximum t \u2194 \u2203 b, b \u2208 t \u2227 a \u2264 b\n\u22a2 \u2191a \u2264 maximum (h :: t) \u2194 \u2203 b, b \u2208 h :: t \u2227 a \u2264 b\n[PROOFSTEP]\n\n| cons h t ih => simp [maximum_cons, ih]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m h : \u03b1\nt : List \u03b1\nih : \u2191a \u2264 maximum t \u2194 \u2203 b, b \u2208 t \u2227 a \u2264 b\n\u22a2 \u2191a \u2264 maximum (h :: t) \u2194 \u2203 b, b \u2208 h :: t \u2227 a \u2264 b\n[PROOFSTEP]\nsimp [maximum_cons, ih]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\nh : l \u2260 []\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nx\u271d : head\u271d :: tail\u271d \u2260 []\n\u22a2 maximum (head\u271d :: tail\u271d) \u2260 \u22a5\n[PROOFSTEP]\nsimp [maximum_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\nh : 0 < length l\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nx\u271d : 0 < length (head\u271d :: tail\u271d)\n\u22a2 maximum (head\u271d :: tail\u271d) \u2260 \u22a5\n[PROOFSTEP]\nsimp [maximum_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\nh : a \u2208 l\nw : 0 < length l\n\u22a2 a \u2264 maximum_of_length_pos w\n[PROOFSTEP]\nsimp [le_maximum_of_length_pos_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\nh : a \u2208 l\nw : 0 < length l\n\u22a2 \u2191a \u2264 maximum l\n[PROOFSTEP]\nexact le_maximum_of_mem' h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\ni : \u2115\nw : i < length l\nh : optParam (0 < length l) (_ : 0 < length l)\n\u22a2 l[i] \u2264 maximum_of_length_pos h\n[PROOFSTEP]\napply le_maximum_of_length_pos_of_mem\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrder \u03b1\nl : List \u03b1\na m : \u03b1\ni : \u2115\nw : i < length l\nh : optParam (0 < length l) (_ : 0 < length l)\n\u22a2 l[i] \u2208 l\n[PROOFSTEP]\nexact get_mem l i w\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh : l \u2260 []\n\u22a2 \u2191(foldr max \u22a5 l) = maximum l\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh\u271d : l \u2260 []\nh : [] \u2260 []\n\u22a2 \u2191(foldr max \u22a5 []) = maximum []\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh\u271d : l \u2260 []\nhd : \u03b1\ntl : List \u03b1\nIH : tl \u2260 [] \u2192 \u2191(foldr max \u22a5 tl) = maximum tl\nh : hd :: tl \u2260 []\n\u22a2 \u2191(foldr max \u22a5 (hd :: tl)) = maximum (hd :: tl)\n[PROOFSTEP]\nrw [maximum_cons, foldr, WithBot.coe_max]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh\u271d : l \u2260 []\nhd : \u03b1\ntl : List \u03b1\nIH : tl \u2260 [] \u2192 \u2191(foldr max \u22a5 tl) = maximum tl\nh : hd :: tl \u2260 []\n\u22a2 max \u2191hd \u2191(foldr max \u22a5 tl) = max (\u2191hd) (maximum tl)\n[PROOFSTEP]\nby_cases h : tl = []\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh\u271d\u00b9 : l \u2260 []\nhd : \u03b1\ntl : List \u03b1\nIH : tl \u2260 [] \u2192 \u2191(foldr max \u22a5 tl) = maximum tl\nh\u271d : hd :: tl \u2260 []\nh : tl = []\n\u22a2 max \u2191hd \u2191(foldr max \u22a5 tl) = max (\u2191hd) (maximum tl)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl : List \u03b1\nh\u271d\u00b9 : l \u2260 []\nhd : \u03b1\ntl : List \u03b1\nIH : tl \u2260 [] \u2192 \u2191(foldr max \u22a5 tl) = maximum tl\nh\u271d : hd :: tl \u2260 []\nh : \u00actl = []\n\u22a2 max \u2191hd \u2191(foldr max \u22a5 tl) = max (\u2191hd) (maximum tl)\n[PROOFSTEP]\nsimp [IH h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d l : List \u03b1\na : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2264 a\n\u22a2 foldr max \u22a5 l \u2264 a\n[PROOFSTEP]\ninduction' l with y l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d l : List \u03b1\na : \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2264 a\nh : \u2200 (x : \u03b1), x \u2208 [] \u2192 x \u2264 a\n\u22a2 foldr max \u22a5 [] \u2264 a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na : \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 x \u2264 a\ny : \u03b1\nl : List \u03b1\nIH : (\u2200 (x : \u03b1), x \u2208 l \u2192 x \u2264 a) \u2192 foldr max \u22a5 l \u2264 a\nh : \u2200 (x : \u03b1), x \u2208 y :: l \u2192 x \u2264 a\n\u22a2 foldr max \u22a5 (y :: l) \u2264 a\n[PROOFSTEP]\nsimpa [h y (mem_cons_self _ _)] using IH fun x hx => h x <| mem_cons_of_mem _ hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d l : List \u03b1\na x : \u03b1\nhx : x \u2208 l\nh : a \u2264 x\n\u22a2 a \u2264 foldr max \u22a5 l\n[PROOFSTEP]\ninduction' l with y l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d l : List \u03b1\na x : \u03b1\nhx\u271d : x \u2208 l\nh : a \u2264 x\nhx : x \u2208 []\n\u22a2 a \u2264 foldr max \u22a5 []\n[PROOFSTEP]\nexact absurd hx (not_mem_nil _)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx\u271d : x \u2208 l\u271d\nh : a \u2264 x\ny : \u03b1\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\nhx : x \u2208 y :: l\n\u22a2 a \u2264 foldr max \u22a5 (y :: l)\n[PROOFSTEP]\nobtain hl | hl := hx\n[GOAL]\ncase cons.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx : x \u2208 l\u271d\nh : a \u2264 x\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\n\u22a2 a \u2264 foldr max \u22a5 (x :: l)\ncase cons.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx : x \u2208 l\u271d\nh : a \u2264 x\ny : \u03b1\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\na\u271d : Mem x l\n\u22a2 a \u2264 foldr max \u22a5 (y :: l)\n[PROOFSTEP]\nsimp only [foldr, foldr_cons]\n[GOAL]\ncase cons.head\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx : x \u2208 l\u271d\nh : a \u2264 x\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\n\u22a2 a \u2264 max x (foldr max \u22a5 l)\n[PROOFSTEP]\nexact le_max_of_le_left h\n[GOAL]\ncase cons.tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx : x \u2208 l\u271d\nh : a \u2264 x\ny : \u03b1\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\na\u271d : Mem x l\n\u22a2 a \u2264 foldr max \u22a5 (y :: l)\n[PROOFSTEP]\nexact le_max_of_le_right (IH (by assumption))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderBot \u03b1\nl\u271d\u00b9 l\u271d : List \u03b1\na x : \u03b1\nhx : x \u2208 l\u271d\nh : a \u2264 x\ny : \u03b1\nl : List \u03b1\nIH : x \u2208 l \u2192 a \u2264 foldr max \u22a5 l\na\u271d : Mem x l\n\u22a2 x \u2208 l\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Data.List.MinMax", "llama_tokens": 26135, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.2510938934196687}}